Handbook of Applied Superconductivity

Handbook of Applied Superconductivity Volume 1: Fundamental theory, basic hardware and low-temperature science and technology

Edited by

Bernd Seeber University of Geneva

Institute of Physics Publishing Bristol and Philadelphia

Copyright © 1998 IOP Publishing Ltd

© IOP Publishing Ltd 1998 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with the Committee of Vice-Chancellors and Principals. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0377 8 Library of Congress Cataloging-in-Publication Data are available

Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in TEX using the IOP Bookmaker Macros Printed in the UK by J W Arrowsmith Ltd, Bristol

Copyright © 1998 IOP Publishing Ltd

For Alexandra, David and Ricardo

Copyright © 1998 IOP Publishing Ltd

In memorium

vii

In memoriam: Anthony Derek Appleton In the early stages of the Handbook of Applied Superconductivity project, in February 1990, I wrote a letter to Dr A D Appleton who was at that time a technical director of NEI (Northern Engineering Industries, Newcastle upon Tyne, UK). In this letter I explained the handbook project and asked him if he would be interested in participating. It should be mentioned that I did know Tony from the literature and from conferences, but not personally. Within a few days he replied:

‘I would be very happy to participate in the programme which is outlined in the Working Paper and I am pleased to note the emphasis on applications. My contributions will be on the Power Engineering Applications, and you will be interested to know that I will shortly be spending almost all my time on these applications.’ A proposal was sent to Brussels and at the end of August I received the good news that the project was accepted. Soon we set up a Power Applications working group and several meetings were organized. Tony gave precious advice and was very helpful in finding authors. In 1992 he retired from NEI and founded his own company Appleton Associates International. There was good progress with the Power Applications part, and it was characteristic of Tony to help others though this meant he was delayed with his own work. For instance he did a considerable amount of editorial work for most of the contributions in this part before he had finished his own contributions. The character of the man can be illustrated by another anecdote. After a distressing medical treatment he wrote in February 1993: ’During all of this time I had to continue with a heavy work load and my life was extremely difficult. It was during this period that I lost a lot of my previous work on your project when trying to transfer from a laptop computer onto a disk; I must admit that I was not thinking too clearly.’ When Tony left us for ever in December 1994 his introductory chapter to the part Power Applications of Superconductivity, as well as his contribution Direct Current Machines were almost finished. Dr D H Prothero, who worked for many years with Tony at NEI, took over the commitment to complete his contributions. Professor J Watson from the University of Southampton also helped to complete the sections by providing photographic material at Tony’s bequest. The following extracts are from Tony’s obituary which Professor Watson contributed to Superconductor Science and Technology (1995 8 119–20): ‘With the death of Tony Appleton on 19 December 1994, after a short illness, the world lost one of the most vigorous pioneers who have been working towards the application of superconductivity to the development of large-scale electrical machinery. … Tony Appleton contributed significantly to the development of superconducting motors and generators, superconducting magnetic energy storage, methods of limiting faults in electrical networks such as those that blacked-out the east coast of the United States in the late 1960s and early 1970s, and the generation of electrical power using magnetohydrodynamics. … As a design engineer he developed the theory of superconducting homopolar DC machines and … a model superconducting 50 hp homopolar motor was built and commissioned in 1966. This machine is now exhibited in the Science Museum, South Kensington, London. … From 1976, he started many new projects, such as applying magnetic separation to a wide range of areas from water treatment to medicine and mining. In this period, work started on a superconducting fault-current limiter, superconducting AC generators,

Copyright © 1998 IOP Publishing Ltd

viii

In memorian

and electrical power generation using magnetohydrodynamics with superconducting magnets to supply the background field. … His technical competence, his clear thinking, his pleasant and cooperative manner, his desire to serve the profession of which he was proud, and the fact that he was never self-seeking made his services greatly in demand as an advisor to various government departments, professional organizations and as an invited speaker at international conferences. … Tony Appleton was a very modest man with great enthusiasm, energy and determination. He was extremely well-liked and greatly respected by his colleagues and had many friends throughout the world. He was also a devoted family man who found great pleasure and pride in his children.’ It was always a lot of fun working with Tony and I deeply regret that he did not get the opportunity to see the finished Handbook of Applied Superconductivity. Bernd Seeber Geneva, July 1997

Dr A D Appleton Reproduced with kind permission of Professor J Watson

Copyright © 1998 IOP Publishing Ltd

ix

Contents List of contributors Foreward Preface

xiv xxi xxiii

VOLUME 1: FUNDAMENTAL THEORY, BASIC HARDWARE AND LOW-TEMPERATURE SCIENCE AND TECHNOLOGY PART A

INTRODUCTION

1

A1

3

A2 A3

PART B

The evolution of superconducting theories A A Golubov Type II superconductivity A A Golubov High-temperature superconductivity A A Golubov

37 53

SUPERCONDUCTING WIRES AND CABLES

63

B1

65

B2 B3 B3.1 B3.2 B3.3 B3.4 B4 B4.1 B4.2 B4.3

Field distributions in superconductors A M Campbell Current distribution in superconductors S Takacs Stability of superconducting wires and cables Normal zone in composites R G Mints Flux-jump instability R G Mints Practical stability design L Bottura Cable-in-conduits L Bottura Losses in superconducting wires Introduction to a.c. losses A M Campbell Hysteresis losses in superconductors A M Campbell Coupling-current losses in composites and cables: analytical calculations J L Duchateau, B Turck and D Ciazynski

Copyright © 1998 IOP Publishing Ltd

79 99 99 120 139 151 173 173 186 205

x B4.4 B5 B6 B7 B7.1 B7.2 B7.3 B7.4

B7.5 B8 B8.1 B8.2 B9 B9.1 B9.2 B9.3

Numerical calculation of a.c. losses E M J Niessen and A J M Roovers Rutherford-type cables: interstand coupling currents A Verweij Cable-in-conduit superconductors J-L Duchateau Measurement techniques for the characterization of superconducting wires and cables Critical temperature J R Cave Critical fields J R Cave Critical current of wires B Seeber Critical current measurements of superconducting cables by the transformer method P Fabbricatore and R Musenich A.c. losses in superconducting wires and cables I Hlasnik, M Majorors and L Jansak Commercially available superconducting wires Conductors for d.c. applications H Krauth Low-Tc superconductos for 50-20 Hz applications T Verhaege, Y Laumond and A Lacaze New superconducting wires Chevrel phases B Seeber General aspects of high-temperature superconductor wires and tapes J Tenbrink The case of Bi(223) tapes R Flükiger and G Grasso

PART C SUPERCONDUCTING MAGNETS C1 C2 C3 C4

C5

Basics of superconducting magnet design F Zerobin and B Seeber Practical aspects of superconducting magnet manufacturing F Zerobin and B Seeber Quench propagation and magnet protection K H Meβ Quench propagation and protection of cable-in-conduit superconductors L Bottura Radiation effects on superconducting fusion magnet components H W Weber

Copyright © 1998 IOP Publishing Ltd

232 249 265 281 281 295 307

325 344 397 397 415 429 429 446 466

489 491 513 527

557 573

Contents PART D

xi COOLING TECHNOLOGY FOR SUPERCONDUCTORS

601

D1

603

D2 D3 D4 D5 D6 D7 D8 D9 D10 D11 D11.1 D11.2 D12

PART E

INSTRUMENTATION E1 E2 E3 E4 E5

PART F

The properties of cryogens R N Richardson and V arp Dielectric properties of cryogens J Gerhold The thermodynamics of cryocycles J Weber Refrigerators H Quack Examples of cryogenic plants M Wanner and B Ziegler Small cryocoolers A Ravex Magnetic refrigeration P Seyfert Cryostats N H Balshaw Bath cryostats for superfluid helium cooling G Claudet Current leads P F Herrmann Forced cooling Forced-flow cooling of superconducting magnets I L Horvath Some aspects of cryogenic magnets A Hofmann Safety with cryogens R N Richardson

Cryogenic fluid-level indicators J Casas Gas flow-rate meters J Casas and L Serio Pressure measurements of cryogenic gases and liquids J A Zichy Thermometry F Pavese Methods and instrumentation for magnetic field measurement K N Henrichsen, C Reymond and M Tkatchenko

MATERIALS AND LOW TEMPERATURES F1 Mechanical properties of engineering metals and alloys F1.1 Basic aspects of tensile properties B Obst

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639 657 657 695 721 747 763 795 801 845 845 862 875

899 891 897 909 919 951 967 969 969

xii F1.2 F2 F3 F4 F5 F6 F7 F8

Structural stainless steel materials A Nyilas Properties of fibre composites G Hartwig Electrical resistivity B Seeber Thermal conductivity B Seeber and G K White Specific heat G K White Thermal expansion G K White Dielectric properties J Gerhold Thermoelectric effects of superconductors A B Kaiser and C Uher

994 1007 1067 1083 1095 1107 1121 1139

VOLUME 2: APPLICATIONS PART G

PRESENTS APPLICATIONS OF SUPERCONDUCTIVITY 1165 G1 Ultra-high-field magnets for research applications 1167 N Kerley G2 Medical, biological and chemical applications 1191 G2.1 Nuclear magnetic resonance spectroscopy for chemical applications 1191 W H Tschopp and D D Laukien G2.2 Magnetic resonance imaging and spectroscopy (medical applications) 1213 W H-G Müller and D Höpfel G2.3 SQUID sensors for medical applications 1249 O Dössel, B David, M Fuchs and H-A Wischmann G3 Superconducting magnets for thermonuclear fusion 1261 J-L Duchateau G4 Superconducting magnets for particle acclerators (dipoles, multipoles) 1289 R Perin and D Leroy G5 Superconducting magnetic seperation 1319 M N Wilson G6 Superconducting magnetic separation 1345 J H P Watson G7 High-frequency cavities 1371 W Weingarten G8 A superconducting transportation system 1407 E Suzuki, S Fujiwara, K Sawada and Y Nakamichi G9 Superconducting magnetic bearings 1441 T A Coombs G10 Magnetic shielding 1461 F Pavese

Copyright © 1998 IOP Publishing Ltd

Contents PART H

xiii POWER APPLICATIONS OF SUPERCONDUCTIVITY

1485

H1

1487

H2 H2.1 H2.2 H3 H4 H5 H6 H7 H7.1 H7.2

PART I

An introduction to the power applications of superconductivity A D Appleton and D H Prothero Synchoronous machines H Köfler Generators with superconducting field windings H Köfler Fully superconducting generators P Tixador Direct current machines A D Appleton and D H Prothero Transformers Y Laumond Power transmission J Gerhold Fault current limiters T Verhaege and Y Laumond Energy storage Small and fast-acting SMES Systems H W Lorenzen, U Brammer, M Harke and F Rosenbauer The impact of SC magnet energy storage on power system opertion E Handschin and Th Stephanblome

1497 1497 1553 1579 1613 1627 1691 1703 1703 1735

SUPERCONDUCTING ELECTRONICS

1757

I1

1759

I2 I3 I4 I5 I5.1 I5.2 I6 I6.2

Josephson junctions H Rogalla SQUID sensors J Flokstra Single-flux quantum electronics K Nakajima Josephson voltage standards J Niemeyer Signal processing applications Analogue processing by passive devices P Hartemann Analogue-to-digital converters G J Gerritsma Thermal detection and antennas P Hartemann Superconducting heterodyne receivers T Noguchi and S-C Shi

Glossary Index

Copyright © 1998 IOP Publishing Ltd

1777 1795 1813 1835 1835 1860 1875 1899

G1 I1

xiv

List of contributors

List of contributors A D Appleton (H1, H3) Deceased

V Arp (D1)

Cryodata Inc., PO Box 558, Niwot, CO 80544, USA

N H Balshaw (D8)

Oxford Instruments, Scientific Research Division, Tubney Woods, Abingdon, Oxon 0X13 5QX, UK

L Bottura (B3.3, B3.4, C4) CERN, Division LHC, CH-1211 Genève 23, Switzerland

U Brammer (H7.1)

Lehrstuhl für Elektrische Maschinen und Geräte, Technische Universität München, D-80290 München, Germany

A M Campbell (B1, B4.1, B4.2) IRC in Superconductivity, University of Cambridge, Madingley Road, Cambridge CB3 OHE, UK

J Casas (E1, E2)

CERN, Division LHC, CH-1211 Genève 23, Switzerland

J R Cave (B7.1, B7.2)

Institut de Recherche d’Hydro Québec, 1800 boulevard Lionel-Boulet, Varennes (Quebec), Canada J3X ISI

D Cianzynski (B4.3)

Commissariat à l’Energie Atomique de Cadarache, Département de Recherche sur la Fusion Contrôlée, F-13108 St Paul Lez Durances Cedex, France

Copyright © 1998 IOP Publishing Ltd

G Claudet (D9)

Commissariat à l’Energie Atomique de Grenoble, Département de Recherche Fondamentale sur la Matière Condensée, Service des Basses Températures, F-38054 Grenoble Cedex 9, France

T A Coombs (G9)

IRC in Superconductivity, University of Cambridge, Madingley Road, Cambridge CBS OHE, UK

B David (G2.3)

Philips GmbH Forschungslaboratorien, Forschungsableilung Technische Systeme Hamburg, Rontgenstraβe 24–26, D-22335 Hamburg, Germany

O Dössel (G2.3)

Philips GmbH Forschungslaboratorien, Forschungsabteilung Technische Systeme Hamburg, Röntgenstrabe 24–26, D-22335 Hamburg, Germany Present address:: Institut für Biomedizinische Technik, Universität Karlsruhe, Kaiserstrabe 12, D-76128 Karlsruhe, Germany

J-L Duchateau (B4.3, B6, G3)

Commissariat à l’Energie Atomique de Cadarache, Département de Recherche sur la Fusion Contrôlée, F-13108 St Paul Lez. Durances Cedex, France

P Fabbricatore (B7.4)

Istituto Nazionale di Fisica Nucleare, Via Dodecaneso 33, 1-16146 Genova, Italy

List of contributors J Flokstra (I2)

Applied Physics, Low Temperature Division, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands

R Flükiger (B9.3)

Department de Physique de la Matière Condensèe, Universitè de Genève 4, CH-1211 Genève 4, Switzerland

M Fuchs (G2.3)

Philips GmbH Forschungslaboratorien, Forschungsabteilung Technische Systeme Hamburg, Röntgenstrabe 24–26, D-22335 Hamburg, Germany

S Fujiwara (G8)

Railway Technical Research Institute, Fujiwara Laboratory (Maglev System Technology), Kokubunji City, Tokyo 185, Japan

J Gerhold (D2, F7, H5)

Technische Universität Graz, Institut für Elektrische Maschinen und Antriebstechnik, Kopernikusgasse 24, A-8010 Graz, Austria

G J Gerritsma (15.2) University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands

A A Golubov (A1, A2, A3)

Institute of Solid State Physics, Russian Academy of Sciences, 142 432 Chernogolovka, Moscow District, Russia Present address:: Department of Applied Physics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands

G Grasso (B9.3)

Départment de Physique de la Matière Condensée, Université de Genève, CH-1211 Genève 4, Switzerland

Copyright © 1998 IOP Publishing Ltd

xv E Handschin (H7.2)

Universität Dortmund, Lehrstuhl für Electrische Energieversorgung, D-44221 Dortmund, Germany

M Harke (H7.1) Lehrstuhl für Elektrische Maschinen und Geräte, Technische Universität München, D-80290 München, Germany P Hartemann (I5.1, I6.1)

Thomson-CSF, Laboratoire Central de Recherches, F-91404 Orsay Cedex, France

G Hartwig (F2) Forschungszentrum Karlsruhe, Institut für Materialforschung II, Postfach 3640, D-76021 Karlsruhe, Germany K N Henrichsen (E5) CERN, Division LHC, CH-1211 Genève 23, Switzerland P F Herrmann (D10)

Alcatel-Alslhom-Recherche, Route de Nozay, F-91460 Marcoussis, France

I Hlasnik (B7.5)

Institute of Electrical Engineering SAS, Dübravská cesta 9, 842 39 Bratislava, Slovak Republic

A Hofmann (D11.2) Forschungszentrum Karlsruhe, Institut für Technische Physik, Postfach 3640, D-76021 Karlsruhe, Germany D Höpfel (G2.2) Fachhochschule Karlsruhe, FB Naturwissenschaften, Postfach 2440, D-76012 Karlsruhe, Germany

xvi

List of contributors

I L Horvath (D11.1)

Swiss Institute of Technology Zürich, Laboratory for High Energy Physics, CH-8093 Zürich, Switzerland

L Jansak (B7.5)

Institute of Electrical Engineering, Slovak Academy of Sciences, Dúbravská cesta 9, 842 39 Bratislava. Slovak Republic

A B Kaiser (F8)

Victoria University of Wellington PO Box 600, Wellington, New Zealand

N Kerley (G1)

Oxford Instruments, Scientific Research Division, Tubney Woods, Abingdon, Oxon OX13 5QX, UK

H Köfler (H2.1, H2.2)

Technische Universität Graz, Institut für Elektrische Maschinen und Antriebstechnik, Kopernikusgasse 24, A-8010 Graz, Austria

H Krauth (B8.1)

Vacuumschmelze GmbH, Grüner Weg 37, D-63450 Hanau, Germany

A Lacaze (B8.2)

GEC Alsthom, F-90018 Belfort, France

D Laukien (G2.1)

Bruker Instruments Inc., 19 Fortune Drive, Manning Park, Billerica, MA 01821-3991, USA

Y Laumond (B8.2, H4, H6) GEC Alsthom, F-90018 Belfort, France

Copyright © 1998 IOP Publishing Ltd

D Leroy (G4)

CERN, Division LHC, CH-1211 Genève 23, Switzerland

H W Lorenzen (H7.1)

Lehrstuhl für Elektrische Maschinen und Geräte, Technische Universität München, D-80290 München, Germany

M Majoros (B7.5)

Institute of Electrical Engineering SAS, Dúbravská cesta 9, 842 39 Bratislava, Slovak Republic

K H Meβ (C3)

Deutsches Elektronen Synchrotron DESY, D-22607 Hamburg, Germany

RG Mints (B3.1, B3.2)

School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

W H-G Müller (G2.2)

FH Hildesheim-Holzminden, Fachbereich PMF in Göttingen, Von-Ossietzky Strabe 99, D-37085 Göttingen, Germany

R Musenich (B7.4)

Istituto Nazionale di Fisica Nucleare, Via Dodecaneso 33, 1-16146 Genova, Italy

K Nakajima (I3)

Research Institute of Electrical Communication, Tohoku University, 2-1-1 Katahira, Aobaku Sendai 980-77, Japan

Y Nakamichi (G8)

Railway Technical Research Institute, Nakamichi Laboratory (Maglev Power Supply), Kokubunji City, Tokyo 185, Japan

List of contributors J Niemeyer (I4)

Physikalisch-Technische Bundesanstalt, Postfach 3345, D-38023 Braunschweig, Germany

E M J Niessen (B4.4) University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands and Stadhouderslaan 235, NL-6171 KK Stein, The Netherlands

T Noguchi (I6.2)

Nobeyama Radio Observatory, National Astronomical Observatory of Japan, 411 Nobeyama Minamimaki-mura, Nagano 384-13, Japan

A Nyilas (F1.2)

Forschungszentrum Karlsruhe GmbH, Technik und Umwelt, Institut für Technische Physik, Postfach 3640, D-76021 Karlsruhe, Germany

B Obst (F1.1)

Forschungszentrum Karlsruhe GmbH, Technik und Umwelt, Institut für Technische Physik, Postfach 3640, D-76021 Karlsruhe, Germany

F Pavese (E4, G10)

Istituto di Metrologia ‘G Colonnetti’, Strada Delle Cacce 73, 1-10135 Torino, Italy

R Perin (G4)

CERN, Divison SPL, CH-1211 Genève 23, Switzerland

D H Prothero (H1, H3)

Rolls Royce—Industrial Power Group, International Research & Development Ltd, Shields Road, Newcastle upon Tyne NE6 2YD, UK

Copyright © 1998 IOP Publishing Ltd

xvii H Quack (D4)

Technische Universität Dresden, Lehrshuhl für Kälte und Kryotechnik, D-01062 Dresden, Germany

A Ravex (D6)

Commissariat à l’Energie Atomique de Grenoble, Département de Recherche Fondamentale sur la Matière Condensée, Service des Basses Températures, F-38054 Grenoble Cedex 9, France

C Reymond (E5)

Metrolab Instruments SA, 110 Chemin du Pont-de-Centenaire, CH-1228 Genève, Switzerland

R N Richardson (D1, D12) Institute of Cryogenics, University of Southampton, Highfield, Southampton SO17 IBJ, UK

H Rogalla (I1)

Low Temperature Division, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands

A J M Roovers (B4.4) University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands Present address:: PTT Telecom BV, PO Box 30150, 2500 GD The Hague, The Netherlands

F Rosenbauer (H7.1)

Lehrstuhl für Elektrische Maschinen und Geräte, Technische Universität München, D-80290 München. Germany

K Sawada (G8)

Railway Technical Research Institute, Sawada Laboratory (Guideway Engineering), Kokubunji City, Tokyo 185, Japan

xviii B Seeber (B7.3, B9.1, Cl, C2, F3, F4) Groupe de Physique Appliquée, Université de Genève, 20 rue de l’Ecole-de-Médecine, CH-1211 Genève 4, Switzerland

L Serio (E2)

CERN. Division LHC, CH-1211 Genève 23, Switzerland

P Seyfert (D7)

Commissariat à l’Energie Atomique de Grenoble, Département de Recherche Fondamentale sur la Matière Condensée, Service des Basses Températures, F-38054 Grenoble Cede 9, France

S-C Shi (I6.2)

Nobeyama Radio Observatory, National Astronomical Observatory of Japan, Nobeyama, Nagano 384-13, Japan Present address:: Purple Mountain Observatory, 2 West Beijing Road, Nanjing, Jiangsu 210008, People’s Republic of China

Th Stephanblome (H7.2)

Gesellschaft für Innovative Energieumwandlung und -speicherung mbH, Schwarzmühlenstrabe 104, D-45884 Gelsenkirchen, Germany

E Suzuki (G8)

Railway Technical Research Institute, Suzuki Laboratory (Cryogenic Technology}, Kokubunji City, Tokyo 185, Japan

S Takács (B2)

Institute of Electrical Engineering, Slovak Academy of Sciences, Dúbravská cesta 9, 842 39 Bratislava, Slovak Republic

Copyright © 1998 IOP Publishing Ltd

List of contributors J Tenbrink (B9.2)

Vacuumschmelze GmbH, Grüner Weg 37, D-63450 Hanau, Germany

P Tixador (H2.3)

Centre National de la Recherche Scientifique, CRTBT, F-38042 Grenoble Cedex 9, France

M Tkatchenko (E5)

Commissariat à l’Energie Atomique de Saclay, DSM-LNS-SAP, F-91191 Gif-sur-Yvette, France

W H Tschopp (G2.1) Spectrospin AG, Industriestrabe 26, CH-8117 Fällanden, Switzerland

B Turck (B4.3)

Commissariat à l’Energie Atomique de Cadarache, Dépaneinent de Recherche sur la Fusion Contrôlée, F-13108 St Paul Lez Durances Cedex. France

C Uher (F8)

Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120, USA

T Verhaege (B8.2, H6)

Alcatel-Alsthom-Recherche, Route de Nozay, F-91460 Marcoussis, France

A Verweij (B5)

CERN, Division LHC-MMS, CH-1211 Genève 23, Switzerland

M Wanner (D5)

Max-Planck Institut für Plasmaphysik, Boltzmannstrabe 2, D-85748 Garching. Germany

J H P Watson (G6)

Institute of Cryogenics, University of Southampton, Highfield, Southampton SO17 IBJ, UK

List of contributors H W Weber (C5)

Alorninstitut der Österreichischen Universitäten, Schüttelstrabe 115, A-1020 Wien, Austria

J Weber (D3)

Linde AG, D-82049 Höllriegelskreuth, Germany

W Weingarten (G7)

CERN, Division TIS, CH-1211 Genève 23, Switzerland

G K White (F4, F5, F6)

CSIRO, National Measurement Laboratory, PO Box 218, Lindfield, NSW 2070, Australia

M N Wilson (G5)

Oxford Instruments, Research Instruments Group, Tubney Woods, Abingdon, Oxon OX13 5QX, UK

Copyright © 1998 IOP Publishing Ltd

xix H-A Wischmann (G2.3)

Philips GmbH Forschungslaboratorien, Forschungsabteilung Technische Systeme Hamburg, Röntgenstrabe 24–26, D-22335 Hamburg, Germany

F Zerobin (C1, C2)

Elin Energieanwendung GmbH, Magnet Technology, A-8160 Weiz., Austria Present address:: Tridonic Bauelemente GmbH, Jahnstrabe 11, A-8280 Fürstenfeld, Austria

JA Zichy (E3)

Paul Scherrer Insitut (PSI), CH-5232 Villigen. Switzerland

B Ziegler (D5)

Linde Kryotechnik AG, Dättlikonerstrabe 5, CH-8422 Pfungen, Switzerland

Foreword

xxi

Foreword While writing this foreword for the Handbook of Applied Superconductivity, we recalled that, in exactly this month in 1986, we were making the final revisions to our manuscript on high-Tc superconductivity in La— Ba—Cu-oxide. As we continued our research on the magnetic properties of the cuprate superconductors, we were almost certain that our results would meet with substantial scepticism within the scientific community. Our surprise was all the greater then when, by the end of 1986, we learned that numerous groups throughout the world had started to follow our approach to the search for high-Tc superconductors. Chemical modification of the original compound led to the discovery of new cuprate phases, and the record transition temperatures rose at a rapid pace. This progress, however, also raised expectations concerning the timescale for realizing practical applications of these discoveries. As scientists and engineers faced numerous technical problems arising from the specific properties of cuprates, it was at first not possible for practical applications to keep pace with the speed at which new compounds were being discovered. Nevertheless the list of achievements made in the past decade is impressive and the continuing progress reflects a concentrated worldwide effort. We recalled such obstacles as anisotropy, short coherence lengths and grain-boundary effects, which appeared in the beginning to rule out any meaningful application of the ceramic layered cuprates. With the fabrication of epitaxial films, it became possible to demonstrate critical current densities at 77 K up to several million amperes per square centimetre, compared with a few hundreds of amperes per square centimetre in bulk ceramics. Subsequently, thin films assumed an important role as model systems for the study of anisotropy effects in flux pinning, interlayer coupling, dimensionality crossover and grainboundary effects. These key experiments had a significant impact on the methods of processing bulk superconductors, in which grain alignment and enlargement of the grain-boundary area, together with the introduction of suitable defects, proved to enhance the critical current densities up to some tens of thousands of amperes per square centimetre. Today, eleven years after the discovery of high-Tc superconductors, there are no longer merely prospects of applications but several of them have become reality. Bulk superconductors in the form of massive components, wires or tapes have been used for a wide variety of prototypes ranging from magnetic bearings to magnets, motors, generators and flexible cables for power transmission. Current leads, moreover, are already employed in combination with low-Tc magnets. Further examples of the substantial progress made in the past decade are a current limiter installed in a hydroelectric power facility and a transformer connecting an industrial plant to the local electrical utility. Closer to the market, however, are applications based on thin epitaxial films. Although the fabrication of reasonable Josephson junctions was originally thought to be out of the question because of the short coherence length of the cuprates, superconducting quantum interference devices (SQUIDs) based on single grain-boundary junctions have reached a performance with a low noise level comparable to those of the low-Tc versions. The majority of applications in this area, however, are probably microwave components such as high-Q resonators, stable oscillators, antennas, filters and delay lines. Until now, information on the scientific and technological breakthroughs that led to such tremendous progress has been published in thousands of articles in various journals, some of them highly specialized. Thus it is a daunting task to keep abreast of the many relevant disciplines in science, technology and engineering.

Copyright © 1998 IOP Publishing Ltd

xxii

Foreword

The time has come to collect a discussion of the fundamental properties of high-Tc superconductors in a handbook, and to set them in context with current technological requirements. This Handbook of Applied Superconductivity links science with engineering aspects in a concise form. It not only initiates the newcomer to the field, it also synchronizes various relevant disciplines in physics, chemistry, materials science and engineering by establishing terminology standards. From this point of view, we are convinced that this handbook will be highly appreciated. J Georg Bednorz and K Alex Müller Rüschlikon, April 1997

Copyright © 1998 IOP Publishing Ltd

Preface

xxiii

Preface This handbook has grown out of the European Community COMETT initiative on the cooperation between universities and industry for training in the field of technology. The project started from the premise that superconductivity is likely to be a key technology in the immediate future, and that it is important that scientists and engineers become familiar with current applications of superconductivity as well as its potential for future applications. The lack of selected and easily accessible information is considered as a serious obstacle to advances in superconducting technology; the material currently available is mainly addressed to scientists working directly on superconductivity and its applications, rather than to engineers. This handbook attempts to summarize the essentials of applied superconductivity as well as its supporting technologies and to demonstrate what can be achieved by the use of superconducting technologies. It is hoped that this approach will encourage scientists and engineers in the implementation of these technologies in new areas of both academic and industrial research and development. The emphasis in general is on well established techniques, but areas of the subject which remain controversial are included and, where appropriate, the impact of high-Tc superconductivity is discussed. Volume 1 begins with an introduction to the theoretical background of both low-Tc, and high-Tc superconductivity, followed by detailed discussions of superconducting wires, cables and magnets. Subsequent chapters deal with the necessary supporting technology, with sections devoted to cryogenics, instrumentation and the properties of materials at low temperatures. Volume 2 covers present and future applications of superconductivity. While it is not claimed that this coverage is complete, it does offer a representative selection of practical examples of the applications of superconducting technologies. Particularly in the field of the impact of high-Tc superconductivity on electronics, the subject is changing very rapidly; the handbook devotes major sections to modern applications such as high-field magnets, medical applications, including magnetic resonance imaging and spectroscopy, magnetic separation, and transport systems. This is followed by a discussion of power applications, dealing with power transmission, rotating machines and energy storage, as well as electronic applications, including Josephson junctions and superconducting quantum interference devices, singleflux quantum electronics, Josephson voltage standards, digital and analogue signal processing and electromagnetic wave receivers. Contributors to the handbook have been drawn from industrial and academic research, and have all made significant contributions to their chosen fields of research. In addition, a distinguished team of consultants has given valuable advice on the content of the handbook, paying attention to the future needs of industry. To keep cross-references to a minimum, and to make individual contributions easier to read, some duplication of material has been tolerated, as have occasional departures from standard nomenclature, but where such departures occur, they are clearly indicated as such. Furthermore, it has to be said that, in some cases, considerations of industrial secrecy have prevented the disclosure of technical details. Although this has, on occasion, limited the amount of quantitative data in some contributions, it has not restricted the ability of contributors to address important practical issues in superconductivity and its applications. It is my pleasant duty to thank all the contributors. In a multi-author work such as this, there are inevitable delays in assembling all the contributions, and the patience of all those who were affected is gratefully acknowledged. Publication of the handbook would not have been possible without financial

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Preface

support from the European Community and the endorsement of the proposal from Dr A D Appleton (formerly of Northern Engineering Industries-International Research & Development Ltd, UK), H Fillunger (Elin GmbH, Austria), Dr D Lambrecht and Dr L Intichar (Siemens-KWU, Germany) and J L Sabrie and Y Laumond (GEC-Alsthom, France). Equally important was the constant interest and encouragement shown by Professor ∅ Fischer and Professor R Flükiger of my own Institute at the University of Geneva. Finally, I would like to thank V Schröter and L Erbüke for technical assistance with illustrations and electronic file management, as well as secretarial help from C Bayala, C Chappuis, C Dotti and A M Guarnero-Ruffieux. Bernd Seeber Geneva, December 1997

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A1 The evolution of superconducting theories

A A Golubov

A1.0.1 Basic properties of a superconducting state In 1908 Kammerlingh Onnes liquefied helium in his laboratory in Leiden. Three years later, in 1911 he found that the resistance of mercury, Hg, dropped to zero at temperatures below 4.19 K (Kammerlingh Onnes 1911). Importantly, the resistance drop was discontinous, so it was clear that a phase transition to a qualitatively new state with zero resistance took place. This new state of a metal was called ‘superconducting’. The temperature of this phase transition was called the transition temperature Tc. Later, Kammerlingh Onnes found similar transitions in lead and tin. Among the elements, Nb has the highest Tc of about 9.3 K. Many other nonmagnetic metals and alloys are superconducting. Before the discovery of a new class of so-called high-temperature superconductors in 1986, a highest Tc of about 23.2 K was achieved in Nb3Ge. From the practical point of view, zero resistance is advantageous for applications such as in high-field electromagnets, because the power dissipation should be negligible. Experimentally, the characteristic decay time of persistent current (supercurrent) in a superconducting ring was estimated to be at least 105 years, which corresponds to upper limits of 10–24 Ω cm for the resistivity of a superconductor. In many practical situations, due to various physical mechanisms, losses in superconductors may still be nonzero. The study of the mechanisms of losses in different types of superconducting material is of great importance for applications. Soon after the discovery of superconductivity it was found that superconductivity is destroyed not only by heating of a sample, but also by magnetic field. Since the 1960s, after the discovery of type II superconductivity, the issue of the behaviour of a superconductor in a magnetic field has become the subject of systematic study. It was found in particular that losses in such superconductors may become quite large, especially at finite frequencies, unless special treatment is made. In order to discuss the properties of real practical superconductors one should start with the behaviour of an ideal superconductor as addressed in various theoretical approaches. Prior to giving a short comprehensive description of superconducting theories let us first introduce the basic characteristics of a superconducting state.

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The evolution of SC theories

A1.0.1.1 The basic quantities Tc, Hc and Ic The important magnetic characteristic of a bulk superconductor is the thermodynamic critical field Hc. It has to be distinguished from critical fields of thin films, thin filaments, small spheres etc. Meissner and Ochsenfeld (1933) found that when a superconductor is cooled below Tc in a weak magnetic field H < Hc, the field is expelled from the sample. This perfect diamagnetism is a fundamental property of a superconductor and is called the Meissner effect. The physical picture is that screening supercurrents flow in a thin surface layer of a sample, exactly cancelling the external field. As a result, the magnetic field inside a superconductor is zero. At some field H > Hc the spatially homogeneous superconducting state is unstable and a transition to a normal state with finite resistance occurs. The dependence of Hc on temperature is well described by the empirical relation

with Hc(0) values for elements typically being less than 103 Oe. The critical field vanishes as T gets close to Tc. Schematically the temperature dependence of Hc is shown in figure A1.0.1.

Figure A1.0.1. The temperature dependence of the critical field Hc.

This figure is in fact the H—T phase diagram of an ideal superconductor. At H < Hc penetration of a homogeneous magnetic field in superconductor is thermodynamically unfavourable. Thus, a superconductor can be characterized by perfect conductivity and perfect diamagnetism. An important characteristic of a superconductor is the maximum possible transport current which can flow without dissipation, i.e. the critical current Ic. The value of the critical current depends on the sample geometry and sample quality. According to Silsbee’s criterion, a superconductor loses its zero resistance when at any point on the surface the total magnetic field strength, due to the transport current and applied magnetic field, exceeds the critical field strength Hc. This quantity Ic is called the thermodynamic critical current or the depairing current and depends on the external magnetic field. Its typical values are of the order of Jc = 107–108 A cm–2. In most practical superconductors Ic is much smaller than the thermodynamic critical current due to the penetration of magnetic flux into a superconductor at magnetic fields lower than Hc. In this respect, according to Abrikosov (1952), superconductors are classified into two kinds: type I and type II superconductors. Silsbee’s criterion of depairing current holds only for type I superconductors, whereas for type II superconductors the complete flux expulsion at H < Hc does not take place.

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Basic properties of a superconducting state

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A1.0.1.2 Two kinds of superconductor: type I and type II superconductors In type I superconductors the magnetic field H < Hc is completely screened due to the Meissner effect and zero resistance is preserved in the fields up to Hc . Most type I superconductors are pure elements like Al, Hg, Sn, etc. In many real situations geometrical (sample-shape-dependent) effects may cause magnetic fields exceeding Hc in some parts of the sample. As a result, fields smaller than Hc can, in principle, penetrate into a type I superconductor due to a large demagnetization factor. The simplest example is a thin superconducting film in a perpendicular magnetic field. Then in some volume fraction of a sample a transition from the superconducting to a normal state takes place, i.e. a sample is in the socalled intermediate state. Type II superconductors are characterized by incomplete flux expulsion, even in a small magnetic field, which is a fundamental property of these materials, regardless of shape-dependent effects. Magnetic field penetrates type II superconductors in the form of superconducting vortices. Each vortex carries a magnetic flux equal to a superconducting flux quantum Φ0

where h is Planck’s constant 6.6262 × 10–34 J s, and e is the charge of an electron 1.60219 × 10–19 C. If magnetic vortices are present in a sample, they start to move under external current and, as a result, electric field is generated. Therefore, a truly zero-resistance state does not generally occur in a sample due to the motion of the magnetic vortices. Most practical superconducting metals and alloys are type II superconductors. It is important that the zero-resistance state (or a state with extremely small resistance) is still possible in these materials, provided the magnetic flux pattern (vortex lattice) interacts with the crystal lattice and therefore cannot move. This effect is called vortex pinning and is very important for practical applications. Crossover from type I to type II behaviour can take place in a material with an increase in the number of defects, i.e. with a decrease of electron mean free path. The physical properties of type II superconductors will be discussed separately in chapter A2. A1.0.1.3 Flux quantization and the Josephson effect Superconductivity is by its nature a quantum effect. It originates from quantum coherence in a macroscopically large sample. That means that all electrons carrying the current in a sample of macroscopic size can be described by a wavefunction with a single phase. This leads to a number of observable macroscopic quantum effects, some of which are of great practical importance. Historically, the first phenomenon discovered experimentally was flux quantization. If a superconducting ring carries a supercurrent, magnetic flux inside the ring can have only values which are integer multiples of a superconducting flux quantum Φ0. Thus Φ0 is the unit of magnetic flux distributing within a superconductor. This fact was established experimentally by Deaver and Fairbank (1961) and by Doll and Näbauer (1961). Another manifestation of the quantum nature of superconductivity is the Josephson effect. This phenomenon has a large number of applications in microelectronics, the best known examples being SQUIDs (superconducting quantum interference devices). This effect was predicted theoretically by Josephson (1962) and first realized experimentally a few years later. If two superconductors are brought into weak electrical contact then nondissipative superconducting current can flow through such a contact with zero voltage drop (the d.c. Josephson effect). The maximum possible supercurrent is called the critical current Ic. Typical values of the critical current density are of the order of 1-104 A cm–2, i.e. much smaller than the depairing current density in type I superconductors. This effect is called the stationary Josephson effect.

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The evolution of SC theories

When a current through a Josephson junction exceeds Ic, a voltage difference V across the junction is generated. The most important effect is that this voltage difference has a component oscillating with frequency determined by the relation hω = 2eV (the a.c. Josephson effect). Therefore a Josephson junction is a tunable generator of electromagnetic field with the frequency determined by an applied voltage. Moreover, such a junction can be used for the detection and mixing of electromagnetic signals. The properties of Josephson junctions will be discussed in more detail in section A1.0.5.2. A1.0.1.4 Magnetic properties of type I superconductors Even without going into details of the microscopic explanation of the phenomenon of superconductivity, it is reasonable to assume that the vanishing of the magnetic induction at the interior of a superconductor is due to induced surface currents. In the presence of an external magnetic field, the magnitude and distribution of this current is just such as to create an opposing interior field cancelling out the applied one. Let us introduce the magnetic induction B in a sample, the external field H and the magnetization per unit volume M. The formal description is the following: in the interior of the sample B = 0, H i = 0, M = 0, at the surface J s ≠ 0 (where J s is the surface current density, H i is the interior field), and outside B = H + H s (where H s is due to surface currents). It is more convenient to use the equivalent description which treats the superconductor as a magnetic body with B = 0, Hi ≠ 0, M ≠ 0 in the interior of the sample, Js= 0 at the surface and B = H + H s outside. Here H s is the field due to the magnetization of the sample. For a cylindrical superconducting sample placed in a homogeneous external field, the relation Hi = H holds. Considering the projections of the vectors on the axis of the cylinder, one can write down the relation between the magnetic induction in a sample B, the external field H and the magnetization M

The dependence M(H) is usually called a magnetization curve. The Meissner effect means complete flux expulsion, i.e. B = 0 and therefore H = - 4πM. Magnetization curves are most convenient to plot in coordinates - 4πM. versus H. An ideal magnetization curve for a type I superconductor is shown schematically in figure A 1.0.2 and has a simple physical meaning: at sufficiently small fields, H < Hc, the magnetic field in a sample is zero because of the Meissner effect, therefore a superconductor possesses a magnetic moment due to the screening currents which flow near the sample surface. At larger fields, H > Hc, flux expulsion no longer takes place, the magnetic field penetrates a sample and the magnetic moment M becomes zero. It is important to note that there is a jump-like drop of the magnetic moment at H = Hc, i.e. a phase transition of the first order from a superconducting to a normal state takes place in a magnetic field. This fact follows from simple thermodynamic considerations. A thermodynamic equilibrium in an external magnetic field H under fixed temperature T and magnetic induction B corresponds to a minimum of the Helmholtz free energy F. Since H 2/8π is the magnetic energy density, the difference between the Helmholtz free energy of a superconductor in a magnetic field FsH and in zero field Fs0 is equal to H2/8π. Transition from the superconducting to the normal state takes place when the free energies become equal: FsH = Fn. Therefore the free energy difference between the normal and superconducting states per unit volume is

which means that the superconducting state has a lower energy and is preserved up to Hc, where the firstorder phase transition to the normal state takes place. The phase diagram of a type I superconductor in a magnetic field is shown in figure A1.0.1.

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Basic properties of a superconducting state

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Figure A1.0.2. The B(H) dependence (a) and ideal magnetization curve (b) of a type I superconductor.

From equation (A1.0.4), using the relation between the entropy S, the specific heat C and free energy F, S = -(∂ F / ∂ T), and C = -T (∂ 2F / ∂ T 2), one can express the entropy difference and specific heat difference between the superconducting and normal states in terms of the thermodynamic critical field Hc

As is illustrated in figure A1.0.1, in the vicinity of a critical temperature Tc the critical field becomes zero, Hc = 0, but the derivative ∂Ηc(Τc) / ∂Τ ≠ 0. Then at T = Tc it follows from equation (A1.0.5) that Ss = Sn, and therefore the phase transition at T = Tc is a second-order phase transition. On the other hand, as follows from equation (A1.0.6), Cs ≠ Cn at T = Tc , i.e. there is a specific heat jump at the superconducting transition. At temperatures below Tc the transition from the superconducting to normal state is due to the application of a magnetic field. According to equation (A1.0.5) the entropy difference SS - Sn ≠ 0 at T < Tc, so the transition from the superconducting to normal state in an external magnetic field is a first-order phase transition. Therefore, quite independently of the detailed shape of the magnetic critical field curve, its negative slope indicates that the superconducting phase has a lower entropy than the normal one. The thermodynamic treatment developed thus far links the magnetic and thermal properties of a superconductor, but has ignored any changes in the volume at the transition, as well as any dependence of Hc on pressure. In reality many mechanical properties of the superconducting and normal states are thermodynamically related to the free energies of these states. Taking these into account one could consider first the magnetostriction effect. Differentiating equation (A1.0.4) with respect to the pressure p and using V = (∂G / ∂p)T .H one obtains the actual volume change at the transition as

The derivatives of (A1.0.7) with respect to T and p yield expressions for the changes at the transition of the coefficient of thermal expansion α = (1/V)(∂V/∂Τ) and of the bulk modulus k = -V( ∂p / ∂V). At

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The evolution of SC theories

T = Tc, Hc = 0, this yields and

The magnitudes of the above mechanical effects are exceedingly small, but measurable. Typical values for ∂Hc /∂p are very small, thus yielding the fractional length change of a long rod of a few parts in 10–8. Using the above thermodynamic relations this yields a difference in thermal expansion coefficient of about 10–7 per degree, and a fractional change in compressibility of one part in 105. Therefore there is an extremely small change in volume when a normal material becomes superconducting and the thermal expansion coefficient and the bulk modulus of elasticity are only slightly different in the superconducting and normal states. A1.0.1.5 The intermediate state of type I superconductors The diamagnetic mode of description used above can be generalized to take into account the effect of a more complicated noncylindrical superconductor. The simplest case is an ellipsoidal superconducting specimen in an external field parallel to the major axis. Then

where D is the demagnetization factor for the specimen. Combining equations (A1.0.5) and (A1.0.11) yields

The demagnetization factor for an ellipsoid of revolution is given by

where a and b are the semi-major and semi-minor axes, respectively, and e = (1 - b2 / a2)1/2. For an infinite cylinder with its axis parallel to H, D = 0; for an infinite cylinder transverse to the field D = -12, for a sphere, D = -13 and for a thin film perpendicular to the field, D = 1. The transition from the superconducting to normal state for a specimen with D ≠ 0 takes place in a broad field interval as illustrated schematically by the broken line in figure A1.0.2. In the field range Hc(1 – D) < H < Hc the entire specimen is subdivided into a small-scale arrangement of alternating normal and superconducting regions. In the normal regions B = Hc and in the others B = 0. This state is called the intermediate state, reflecting the fact that in this state the specimen is neither entirely normal nor entirely superconducting. The distribution of these regions varies in such a way that the total magnetization per unit volume changes linearly from M = -H /4π (1-D) = -Hc/4π at H = Hc (1-D) to M = 0 at H = Hc. Magnetization curves at the field range (1-D) Hc ≤ H ≤ Hc are given by

The detailed structure of the intermediate state can be rather complicated depending on the specific shape. For any geometry, except a quasi-infinite cylindrical sample parallel to the external field, the intermediate state exists in some field interval sufficiently close to Hc. Two examples are shown in figure A1.0.3: (a) a schematic representation of the domain structure in a thin slab in a perpendicular field and (b) the domain structure in a superconducting wire of radius a carrying a current larger than the critical current I > Ic. The first case is the simplest example of the

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Basic properties of a superconducting state

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Figure A1.0.3. (a) The domain structure in a thin slab of type I superconductor in a perpendicular magnetic field. (b) The domain structure in a current-carrying wire of type I superconductor.

intermediate state. The fraction ρ = ds / (dn + ds ) of superconducting (S) regions is fixed directly by the condition of conservation of magnetic flux and is given by ρ = 1 - H / Hc. However, even in this case the microscopic structure, i.e. the spatial distribution of normal (N) and S domains and their size, is quite complicated and is controlled by the surface energy of the NS interface (Landau 1937). The existence of such lamina structure was first demonstrated by Meshkovskii and Shal’nikov (1947) in a tin sphere. For a complex shape the domain patterns of the intermediate state in the cross-section perpendicular to the magnetic field can be very irregular. Figure A1.0.3(b) shows a sketch of the distribution of normal and superconducting regions in a current-carrying wire. The field at the surface of the wire is H (a) = 2 I /ca. According to the Silsbee’s criterion, the wire can be completely superconducting when H (a) < Hc. This yields a critical current

When the current I becomes larger than Ic, the resulting domain structure becomes like that shown in figure A1.0.3(b). There is an exterior normal region, R < r < a, where B = H and an intermediate

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The evolution of SC theories

interior region, 0 < r < R. At the boundary between these two regions H (R) = Hc. The thickness of the surface normal layer, R - a, will grow in proportion to the excess (over the critical value) current I - Ic. The sensitivity of the critical current of a superconducting wire to external magnetic field is the basis for the operation of the so-called cryotron: the current passing through the superconducting wire which is placed in a solenoid is controlled by a current flowing through the solenoid. When the field in the coil reaches Hc the wire becomes normal. The gain coefficient of the cryotron is 2π Na / L, where N is the number of turns, L is the length of the coil and a is the radius of the wire. The intermediate state discussed above for type I superconductors can only exist when the surface energy between superconducting and nonsuperconducting regions is positive, because the system will be stabilized at a certain period of the domain structure, corresponding to the balance between magnetic and surface energies. In the opposite case of negative surface energy the system decreases its energy by creating new domain walls between N and S domains. Therefore it is energetically favourable to create the largest possible number of walls, and it is evident that the magnetic properties should be considerably different from those considered previously. In type II superconductors the magnetic flux penetrates in the form of Abrikosov vortices (Abrikosov 1957). The issue of the surface energy is considered below in section A1.0.3.3. A1.0.2 The London theory In order to treat the electromagnetic properties of a superconductor, the general approach demonstrated above, based on thermodynamical relations for a superconductor, should be completed by electrodynamical relations describing the magnetic field distribution inside a superconductor. Such relations are quite different from those in normal metals. Prior to their derivation from microscopic theory by Bardeen, Cooper and Schrieffer (1957) (the BCS theory) the electrodynamics of superconductors was quite successively described phenomenologically, first in the framework of the London theory (London and London 1935) and then on the basis of the more general Ginzburg-Landau theory (Ginzburg and Landau 1950). Both theories operate with phenomenological parameters which can be estimated from experiment, even without specifying a microscopic mechanism of superconductivity. This gives a rather satisfactory description of many practical situations. Importantly, the relation between these phenomenological theories and the BCS microscopic theory has recently been established, which allows one to express the parameters of the phenomenological theories through material constants of real superconductors. The aim below is to give a brief description of these theories in order to introduce the main quantities, notations, basic predictions and their relation to various practical situations starting with the London theory. A1.0.2.1 Equations of two-fluid electrodynamics The basic assumption first made by Gorter and Casimir (1934) is that a system exhibiting superconductivity possesses an ordered state, the total energy of which is characterized by an order parameter. This parameter varies from zero at T = Tc to unity at T = 0 K, and thus indicates the fraction w of the total system which is in the ordered state. Another part of the system is in a disordered, or noncondensed, state and its behaviour is taken to be similar to that of the equivalent nonsuperconducting system. The fact that the superconducting state is more ordered than the normal state follows in fact from simple thermodynamic considerations: according to equation (A1.0.5) the entropy difference between S and N states is negative. This description is called the two-fluid model: all electrons are divided into two subsystems, into the superconducting electrons of density ns (the superfluid), and into the normal electrons of density nn (the normal fluid). The ordered fraction is w = ns + nn.

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The London theory

11

Within the London model, the set of Maxwell’s equations

should be completed by the equation of motion of superconducting electrons

and by the equation describing the relation between the magnetic field and current density in a superconductor

Here curl A = H and φ are vector and scalar potentials respectively. The parameter λL introduced above has the dimensions of length and is called the London penetration depth

At T <
which simply expresses a change of the energy of a superconductor in a magnetic field as the sum of two terms: the magnetic energy and the kinetic energy of the superfluid component ns. In the most general case the total current density J is the sum of a normal current density J n and a supercurrent density J s. This relation should be added to the above set of equations

where σn is a conductivity associated with normal electrons. From the above equations (A1.0.17)–(A1.0.21) one can work out the distributions of fields and currents in a superconducting specimen under various conditions. The static distribution of a magnetic field and current within a superconductor of an arbitrary shape can be found, as well as the response of a superconductor to an external high-frequency electromagnetic field. A1.0.2.2 London penetration depth The application of Maxwell’s equations to the second London equation leads to the equation describing the penetration of a magnetic field into a superconductor

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The evolution of SC theories

This equation can be used in principle to find the distributions of flux density within any superconducting body. One needs to apply to the solutions of this equation the boundary conditions which follow from the shape of the body. Let us use this equation to determine the distribution of the magnetic field inside a superconductor when it is placed in a uniform magnetic field. A superconductor occupies the semi-space x > 0 and a magnetic field is applied parallel to the plane x = 0. Taking into account the symmetry of the problem, one can write equation (A1.0.22) in the one-dimensional form

with the boundary conditions H (0) = H0, H (∞) = 0. The solution of equation (A1.0.23) is

Equation (A1.0.24) shows that the magnetic field H decays exponentially upon penetrating into a superconducting specimen in accordance with the Meissner effect. Thus the London equations do not yield the complete expulsion of a magnetic field from the interior of a superconductor, but predict, in fact, that it penetrates to a depth λL inside the sample. This result, established here for a semi-infinite slab, is easily generalized to a macroscopic specimen of an arbitrary shape. A1.0.23 Current and field distributions for simple geometries (examples) The penetration of a magnetic field into a superconducting specimen is most easily illustrated by reference to the case of a parallel-sided plate with a magnetic field H0 applied parallel to the surfaces of the plate. The length of the plate in the direction of the field and its width are both assumed to be much larger than its thickness d. If the direction normal to the surfaces of the plate is chosen as the x direction, the variation of the flux density with x is easily found from the London equation (A1.0.22) with the boundary conditions H (±d / 2) = H0

The supercurrent density is found by applying Maxwell’s equations and is given by

This example also demonstrates the effect of the field penetration on the critical magnetic field Hc of small specimens. The magnetic moment per unit area of the plate is given by

i.e. magnetization is reduced in comparison with the case of λL = 0 due to flux penetration into the specimen. Adding this magnetic contribution to the Gibbs free energy density, G = F - HH0/4π, yields the critical field H′c of a thin plate

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The London theory

13

It is seen from equation (A1.0.28) that H’c > Hc for any relation between d and λL. Therefore the critical magnetic field is increased as a result of the penetration of the magnetic flux. The effect is the largest for a very thin film d << λL when H’c = 2p3HcλL/d Another example is the distribution of the magnetic field and the current in a parallel-sided currentcarrying plate. The plate has a width w and the linear current density equals J = I/w where I is the total current. The direction normal to the surfaces of the plate is chosen as the x direction and the thickness equals d. The solution of the problem is found from the London equation (A1.0.22) with the boundary conditions H (±d/2) = HJ at the surfaces of the plate with the magnetic field HJ = 2π J/c parallel to the surfaces

The supercurrent density is found by applying Maxwell’s equations

In the particular case of a thin film with d << λL the current flows homogeneously over the film and the solutions are given by

A practically important example is a superconducting film above a superconducting screen (‘ground plane’), since such a configuration represents a superconducting transmission line. To model this situation let us consider a structure consisting of two parallel superconducting films with a current I flowing through one of the films. The linear current density J = I / w. The second film then plays the role of a superconducting screen. The distance between the two films is d. The magnetic field is concentrated in a spacing between the films and is given by

The magnetic field distribution inside each of the films is given by the solution (A1.0.29). Let us calculate the inductance per unit length of the structure. In real situations both films have finite thickness, therefore one should take into account the variation of the field (A1.0.29). Generally, the inductance L is defined as the sum of the magnetic part LM and of the kinetic part LK

where

and

In normal conductors LK is typically negligibly small in comparison with LM; in a superconductor both contributions should be taken into account. Substitution of the solutions (A1.0.29) and (A1.0.30) into equations (A1.0.33)-(A1.0.35) yields the result for the inductance per unit length

where d1.2 and λL1.2 are the thicknesses and London penetration depths of both films respectively.

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The evolution of SC theories

For the particular case where both films are thin in comparison with the corresponding penetration depths, d1.2 << λL1.2, the inductance is given by L = (4π/W )(λ2L1/d1 + λ2L2/d2), i.e. it increases with the decrease of d1, d2. For another case where both films are thick, d1.2 >> λL1.2,equation (A1.0.36) can be simplified as

The expressions (A1.0.36) and (A1.0.37) are good approximations even for a high-frequency electromagnetic field, provided the frequency is not too high, ν < 1 GHz, and the temperature is not too close to the critical temperature, T/Tc < 0.95. The expression (A1.0.37) has to be compared with the inductance of the superconducting semi-space (per square), which can be calculated analogously

The comparison of equations (A1.0.37) and (A1.0.38) shows that to reduce the total inductance of the system consisting of the superconducting film and the superconducting screen (‘ground plane’) the spacing d should be kept small. In the limit of very small spacing d << λ the inductance is simply twice the inductance of the superconducting semi-space. In the latter case it is seen explicitly from (A1.0.38) that LM = LK . It follows from the fact that L is mainly proportional to λL that the penetration depth characterizes the inertial properties of charge carriers in a superconductor. Taking typical values of λL ~ 50 nm yields typical magnitudes of the inductance per square of L ~ µ0λL ~ 6 × 10−14 H. Such a superconducting waveguide allows the propagation of transverse electromagnetic (TEM) pulses with signal decay and dispersion typically much less than in an ohmic waveguide. This property is advantageous for practical applications. A practical interesting example is a thin-film loop placed near the top of the superconducting screen. When the thicknesses of the loop and of the screen are large in comparison with the penetration depths, d1.2 >> λL1.2, and the mean radius of the loop r is much larger than its width w, application of equation (A1.0.37) yields the total inductance of the loop

This inductance has to be compared with the inductance of a normally conducting loop of the same geometry

Substituting as typical values r = 0.1 mm, d = 0.3 µm, w = 100 µm, λL1 = λL2 = 50 nm, one obtains L1 / L0= 0.074. Therefore there is a strong reduction of the inductance due to the use of a superconducting screen. A1.0.2.4 Complex conductivity and surface impedance So far static distributions of fields and currents have been considered, but the London equations (A1.0.17)(A1.0.21) allow one to consider a much broader class of problems concerning the high-frequency properties of superconductors. In fact the London theory gives a good phenomenological basis for predicting the properties of superconducting waveguides, cavity resonators etc. A more complicated approach was developed by Pippard (1953) by phenomenological account of the so-called nonlocal effects and then by Mattis and Bardeen (1958) and Abrikosov et al (1958) in the framework of the microscopic BCS theory. Here we discuss first two-fluid electrodynamics in the framework of the London theory. In section A1.0.4 the relation to the Mattis-Bardeen theory is given.

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The London theory

15

The response of a superconductor to an external electromagnetic field is fully determined by the equations (A1.0.17)-(A1.0.21) in which a conductivity associated with normal electrons, σn, is introduced phenomenologically in order to account for the behaviour of the normally conducting electron fluid. The response of a superconductor to a weak alternating electromagnetic field of frequency ω, Eeiωt, is then determined by the London equations (A1.0.17)-(A1.0.21)

where the complex conductivity of a superconductor is given by

where τ is the momentum relaxation time in the normal component. The complex conductivity is a basic property of a superconductor and its explanation is a great success of the London model. The real part of conductivity σ1 is determined purely by the normal component (Drude conductance) whereas both the components, normal and superconducting, contribute to the imaginary part σ2. For sufficiently low frequencies (ωτ)2 << 1 (typically for f = ω/2π < 1011 Hz) and not too close to Tc the expressions for the complex conductivity (A1.0.42) and (A1.0.43) can be simplified and written in the following form

where n = nn + ns is the total density of conducting electrons and σn = nne2τ/m is the static conductance of a metal in a normal state. The penetration of alternating field into a superconductor is then given by the frequency-dependent skin depth. Based on the result for the effective complex conductivity, one can obtain the skin depth δ by the generalization of the corresponding expression for a normal conductor

With the increase of frequency w the skin depth decreases, therefore the London penetration depth of a static field, λL, gives the upper bound for the penetration of electromagnetic field into a superconductor. For frequencies below 1011 Hz the difference is, however, rather small. Nondissipative current can flow through a superconductor only in the static case, while at finite frequencies energy dissipation is present. The losses in a metallic specimen are generally characterized by surface impedance per square

For the considered case of a superconductor, the generalization is made by substitution in (A1.0.46) of the expression for σ (A1.0.44), which for temperatures not too close to Tc yields

The real part Rs determines the surface losses of a superconductor due to normal electrons, while the imaginary part gives the inductive resistance of a superconductor. In this low-frequency limit, ωτ << 1, the inductance per square given by equation (A1.0.47) coincides with the static expression (A1.0.38).

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The evolution of SC theories

The temperature dependences of various physical quantities below Tc are introduced in the London model empirically, by postulating that nn(t) = nt 4, ns = n (1 - t 4), where t = T / Tc is the reduced temperature. As a result, the London penetration depth is given by

and equation (A1.0.47) yields

These expressions give a good qualitative description of the temperature dependence of the surface impedance in a temperature range not too close to Tc where the condition (ns/nn)(ω τ)2 << 1 holds. Namely, the surface resistance decreases strongly with the decrease of temperature and frequency, while the penetration depth diverges near Tc when transition to a normal state takes place. Qualitatively, the London theory provides a quite satisfactory agreement with the experimental data for many real superconductors. For not too low temperatures and frequencies below 10 GHz it can be used for practical estimates of losses in superconducting resonators and transmission lines. However, in quantitative comparison of temperature and frequency dependences of the surface impedance predicted by the London theory with experimental data some differences occur. Experimentally the behaviour of the surface resistance is described well by the following relation

where A is some material-dependent constant, kB is the Boltzmann constant and ∆(T) is the so-called energy gap which depends on temperature and at T = 0 is of the order of 1 mV for most low-Tc superconductors. The physical meaning of ∆(T) is that it is the gap in the excitation spectrum of quasiparticles in a superconductor. The microscopic origin of this energy gap is explained in the framework of the microscopic BCS theory, as discussed below in section A1.0.4, and is therefore beyond the framework of the phenomenological London theory. The existence of the gap leads to the absence of lowenergy excitations in a superconductor and results in strong modification of microwave losses at low temperatures when the gap becomes large. Indeed, as seen from the comparison of equations (A1.0.47) and (A1.0.50), the experimental surface resistance decreases exponentially at low temperatures whereas the London theory predicts a slower power-law decrease. A1.0.2.5 Advantages and limitations of the London theory A great advantage of the London theory is that it qualitatively correctly describes the linear electrodynamical response of a superconductor, properly taking into account the basic properties of zero resistance and perfect diamagnetism. This allows one to design superconducting circuits and estimate their parameters in a very simple way, using the London equations, which is extremely important for engineering applications. On the other hand, the London theory is limited in several ways. One limitation is that the London theory is essentially a classical theory which treats the electrons as classical particles whereas superconductivity is a quantum phenomenon. The modification of the excitation spectrum of quasiparticles in a superconductor is not taken into account in the London theory. It is, in particular, the reason for the large difference between the predictions of the London theory for the lowtemperature behaviour of various quantities and experimental data. Another source of deviation between the London theory and experiment is that this theory assumes the electrodynamical response of a superconductor to be local in space, whereas in clean materials with

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The Ginzburg-Landau theory

17

large mean free paths the response is nonlocal. Phenomenological extension of the London theory to account for nonlocal effects was first carried out by Pippard (1953). Other important assumptions of the London theory are that the penetration depth λL is independent of the strength of the applied magnetic field and of the dimensions of the specimen, i.e. the superfluid density ns(T) is spatially homogeneous in the interior of the superconducting body. Neither of these assumptions is fulfilled at sufficiently high magnetic fields or in spatially inhomogeneous superconducting structures consisting of different materials or different superconducting phases (composite materials). The London theory is essentially a weak-field theory. As a result, it can only predict linear magnetization curves whereas in practice magnetization curves become strongly nonlinear at high magnetic fields. It is the last limitation which is relaxed in the Ginzburg-Landau theory. A1.0.3 The Ginzburg-Landau theory A1.0.3.1 The Ginzburg-Landau equations The Ginzburg-Landau (GL) theory (1950) is an alternative to the London theory. It is also a phenomenological theory because it is based on Landau’s theory of second-order phase transitions rather than on consideration of microscopic interactions in a material. On the other hand, the GL approach uses quantum mechanics to predict the effect of a magnetic field and makes it possible to discuss a general case of nonlinear response. The GL theory introduces a complex wavefunction

as an order parameter. The absolute value of the order parameter is related to the density of superconducting electrons ns as

whereas φ is the macroscopic phase of the order parameter. The description of all superconducting electrons by a single phase suggests a macroscopic quantum coherence. This fact has its explanation in the microscopic BCS theory. Unlike the London theory, the GL order parameter is spatially dependent, where r denotes a spatial coordinate. In the spirit of the theory of second-order phase transitions it is then assumed that the free energy of the superconducting state can be written as a power series in |ψ(r )|2. Near the critical temperature it is sufficient to retain only the first two terms in this expansion. Furthermore, if a wavefunction ψ(r ) is not constant in space, this gives rise to kinetic energy, and to take account of this an additional term is added to the free energy, which is proportional to the square of the gradient of |ψ(r )|2. As a result the difference between the free energy densities of superconducting and normal phases is written in the following form

where H = curlA is the exact microscopic magnetic field at a given point of a superconductor, m* = 2m, where m is the electron mass. The GL expansion (A1.0.53) is valid only near Tc where the order parameter ψ(r ) is small enough. The GL coefficients α and β are phenomenological parameters and all measurable physical quantities are expressed through them. Alternatively, α and β can be determined from a microscopic approach. Both these relations will be given below. The central problem of the GL approach is now to find functions Ψ(r ) and A(r ) which make the total free energy of the specimen a minimum, subject to appropriate boundary conditions.

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The evolution of SC theories

In the absence of the magnetic field and in the absence of spatial variations the order parameter is obtained from ∂(δFsn )/δns. For α > 0, the minimum of the free energy is at |Ψ|2 = 0, the normal state. For α > 0 then

In equation (A1.0.54), |Ψ∞| is defined as the Ψ value in the interior of the sample, far from any gradients in this parameter (bulk value of the order parameter). At Tc there is a sign change of α, namely α = α’ (t – 1) with temperature-independent α′ > 0, whereas b has no singularities at Tc. This temperature dependence demonstrates explicitly the second-order phase transition at Tc: the order parameter grows continuously from zero at T > Tc to

at T < Tc. Thus GL theory gives the result Hc(T) ∝ (1 – t) which is different from the behaviour approximately found in experiment (1 – t2 ), equation (A1.0.1), as well as from the two-fluid result (1 – t 4 ). This demonstrates that the GL theory is a good approximation only in the vicinity of Tc . The condition of the minimum of a the GL functional yields the first GL equation

This equation should be completed by the equation for the electrical current

which together with Maxwell’s equation (A1.0.15) forms the second GL equation. Bearing in mind the relation (A1.0.52) between the GL order parameter and the density ns in the London theory, one may note that the second London equation (A1.0.18) is a particular case of the second GL equation (A1.0.58) when there are no spatial gradients of the order parameter (the first term in the right-hand side of (A1.0.58) becomes zero). The evident extension of the London theory by GL is the introduction of the spatial gradients of the modulus and of the phase of Ψ. From the equation for the current (A1.0.58) one can demonstrate the effect of quantization of a magnetic flux. Let us rewrite this equation in the form

Consider a superconducting disc with a hole in it and integrate Js around the hole along a path deep in the superconductor, where the screening currents Js can be neglected. Since the order parameter Ψ(r) = |Ψ(r)|exp[φ (r)] must be single valued, the phase φ (r) changes 2πn in going around the hole, n being an integer number. Using Stokes’ theorem, the total flux is obtained as being an integer multiple of the flux quantum Φ0

The flux quantum Φ0 introduced above in equation (A1.0.2) gives a fundamental magnetic flux scale for a superconductor.

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The Ginzburg-Landau theory

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A1.0.3.2 Two length scales: ξ(T) and λ(T) The GL theory introduces two length scales, the coherence length ξ(T) and the penetration depth λ(T), which are of fundamental importance for the classification of superconductors according to their behaviour in a magnetic field. Consider first the GL equation (A1.0.57) in the absence of a magnetic field. For one-dimensional geometry it can be written in the following dimensionless form

where the order parameter is normalized to the absolute value |Ψ∞| far from the interfaces and gradients (bulk value): ψ = Ψ/|Ψ∞| and the coherence length is defined as

This length describes the variation of ψ in space due to a small disturbance, as is evident from equation (A1.0.61). According to (A1.0.56) and (A1.0.62), the GL coherence length is temperature dependent ξ (T) ~ (1 – T/Tc )-1/2 and diverges in the vicinity of Tc . This is the general property of secondorder phase transitions. To introduce the other characteristic length, λ(T), consider the second GL equation (A1.0.58) in the following form

where λ(T) describes the screening of the external magnetic field, i.e. it is the characteristic length of the variation of A inside a superconductor and is given by

Note the equivalence of this definition to the definition of the London penetration depth equation (A1.0.19). Similarly to the GL coherence length, λ(T) depends on temperature as (1 – T/Tc )-1/2. The important dimensionless GL parameter is

In the framework of the GL theory this parameter is temperature independent and therefore is a characteristic of a given material. The microscopic expression for the GL parameter through material characteristics is given below in section A1.0.4.2. A1.0.3.3 The boundary between superconducting and normal phases. Two kinds of superconductor When a superconductor is placed in a magnetic field, a state of coexistence between superconducting (S) and normal-conducting (N) regions is realized in the bulk of a material. A boundary between N and S regions is called the NS interface. The surface energy of the NS interface is the most important parameter in the classification of superconductors into types I or II. Consider the NS interface in a magnetic field. A sketch of the behaviour of the order parameter and the magnetic field near the boundary is shown in figure A1.0.4. The surface energy σns per square of the interface is given by the difference of Gibbs free energies

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The evolution of SC theories

Figure A1.0.4. A sketch of the behaviour of the order parameter and the magnetic field near the interface between a normal phase and a superconducting phase for two cases of k < 1 and k > 1.

where GsH = δ Fsn – HHc /4π and δ Fsn is given by equation (A1.0.53). The result is

The expression (A1.0.67) shows that the contribution to the surface energy from the field penetration, namely the term H( H – Hc )/2Hc2, is always negative. Such a contribution is present in the London theory and leads to the increase of the critical field as discussed above (see equation (A1.0.28)). In the London theory the surface energy is always negative because of the condition of constant order parameter ns such that spatial gradients of the order parameter are absent. The GL theory predicts that an additional positive contribution to the surface energy exists due to the spatial gradient of ψ (the first term in parenthesis in equation (A1.0.67)). As a result the sign of the surface energy σns can be not only negative but positive as well, depending on the relative contribution to the energy from the spatial gradients (positive) and field penetration into a superconductor (negative). Exact calculation shows that

while

In the first limit, ξ << λ, the surface energy is positive and therefore it is thermodynamically unfavourable to create new NS interfaces in a magnetic field. In contrast, in the second limit ξ >> λ, the surface energy is negative. Therefore in a material with k >> 1 the total energy in a magnetic field is reduced by the creation of new interface boundaries. As is shown in the original GL paper, the crossover

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The Ginzburg-Landau theory

21

from positive to negative surface energy occurs for κ 1/p2. The physical consequences of this fact were later pointed out by Abrikosov (1952). Superconductors with κ <1/p2 (i.e. with ξ < λp2 ) are called type I superconductors. As considered above, a specimen of a type I superconductor with a demagnetization coefficient D is in the intermediate state when (1 – D) Hc < H < Hc. Superconductors with κ >1/p 2 (i.e. with ξ >λ p2) are called type II superconductors. The magnetic field penetrates a type II superconductor in the form of magnetic vortices. The basic properties of the vortex state will be discussed below in chapter A2. A1.0.3.4 The proximity effect In the GL approach the effect of the interface can be simply taken into account by means of a boundary condition at the interface. This boundary condition for the superconductor-normal-metal contact (NS junction) has the following form (de Gennes 1964, Deutscher and de Gennes 1969)

where n designates the unit vector normal to the NS interface. Here b is the penetration length of the order parameter for the normal metal. Physically it means that a thin layer of normal region adjacent to a superconductor becomes superconducting itself. This effect is called the proximity effect. The length b can be expressed through microscopic parameters of N and S materials. Specific expressions for b depend on the transparency of the NS interface (i.e. on the strength of a potential barrier for electrons) and the electron mean free paths of N and S materials and can be found on the basis of a microscopic theory. For a clean N metal (large electron mean free path ln) the length b is given by the order of magnitude as b ~ ξn , where ξn is the coherence length in a normal metal

where υFn is the Fermi velocity in N. At low temperatures T << Tc the order parameter can penetrate into normal metal for quite large distances (~1 µ m). The possibility of proximity-induced superconductivity in a macroscopic region of a normal metal is important for the fabrication of SNS Josephson junctions (two superconducting electrodes connected by nonsuperconducting metal), as well as for the design of composite current-carrying superconducting cables. Boundary conditions and their applications for SNS Josephson junctions are discussed in detail by Likharev (1979) and Kupriyanov and Likharev (1990). The latter review includes the discussion of SNS devices based on high-temperature superconductors. Similar phenomena play an important role in Josephson junctions with an interlayer consisting of a degenerate semiconductor Sm. These structures are the subject of current active research and appear to be very promising for practical applications (Klapwijk 1994). Another consequence of the proximity effect is a reduction of a superconducting order parameter from the S side of the NS interface in a region of the order of the coherence length ξ(T). As a result, superconductivity in a thin superconducting film can be strongly suppressed by bringing it into electrical contact with a normal metal. Such a suppression can manifest itself in particular in the reduction of the critical temperature of the superconducting film. The case of contact with a dielectric or vacuum can be formally treated in the same manner with the parameter b–1 being extremely small (de Gennes 1964) ξ(0)/b ~ a0 /ξ(0) << 1 (where a0 is of the order of the interatomic distance) and thus one can set 1/b = 0 in the right-hand side of equation (A1.0.70). Therefore there is no suppression of the order parameter near the superconductor-insulator interface. The exception can be the case of extremely short coherence length ξ ∼ a0 which is characteristic for the high-Tc superconductors. According to Deutscher and Müller (1987), this provides an intrinsic mechanism of the suppression of the order parameter near an interface in an high-Tc superconductor.

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The evolution of SC theories

A1.0.3.5 The critical field and critical current of a thin film The GL theory provides the framework to predict qualitatively a large number of physical phenomena in a superconductor. To apply it to some specific examples, consider first the particularly simple cases where the magnitude of the order parameter |ψ| is constant in space. However, even in these simple cases |ψ| is not necessarily equal to the bulk equilibrium value |ψ ∞|. In section A2.0.2 the more complicated situation of an Abrikosov vortex is considered when |ψ| varies in space. As first applications consider the critical field of a thin film of thickness d placed in a magnetic field H0 applied parallel to the surface of the film along the y direction, where the direction normal to the surfaces of the film is chosen as the x direction. In a gauge with the vector potential Ay directed along y the solution of the GL equations has the form

where the field-dependent order parameter ψ (Ηο) in small fields Ho << Hc is given by

With further increase of the magnetic field H0 the order parameter decreases to zero at Hc. The order of the phase transition at Hc depends on the thickness of the thin film. For d < p5λ the phase transition is of second order, whereas for d > p5λ it is of first order. In the former case of a thin superconducting film the critical field is given by

This result should be compared with the one found from the London theory equation (A1.0.28). Both expressions coincide up to the difference in numerical factor which is due to the neglect of the order parameter variation in the London theory. Consider a film which carries a current of density J flowing along the y axis. The critical current is found by the same method using the boundary condition H(±d/2) = + HI, where HI is the magnetic field produced by the current. For a thin film d << λL the critical current density Jc is given by

Therefore for a thin superconducting film d << λL, the critical field (A1.0.75) increases strongly with the decrease of d in comparison with the thermodynamic field Hc, whereas the critical current density Jc does not depend on d in this limit. The relation (A1.0.76) is referred to as GL depairing critical current density Jc . In conclusion, the GL theory is general enough to solve most practical problems in superconductors including nonlinear response to a static magnetic field. Its later extension to a high-frequency response is called the time-dependent GL theory. The GL theory deals successfully with complicated problems of vortex nucleation in type II superconductors (see chapter A2). One disadvantage is that the GL theory is valid quantitatively only in the vicinity of Tc when the order parameter ψ is small enough. Therefore the expressions for critical fields and currents (e.g. (A1.0.75) and (A1.0.76)) cannot be directly extrapolated to absolute zero temperature (see also discussion in section A1.0.4.2). However, for most practical purposes, in order to obtain qualitative estimates the GL theory may be used at lower temperatures as well.

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The BCS theory

23

A1.0.4 The BCS theory The microscopic theory of superconductivity was proposed by Bardeen, Cooper and Schrieffer (1957) almost half a century after the discovery of this phenomenon. This theory is called the BCS theory and it gives an excellent account of the equilibrium properties of superconductors. The BCS theory was able to explain the following basic properties of superconductors: (a) superconductivity is essentially connected with some change in the behaviour of the conduction electrons which shows up in the appearance of long-range order (at the scale of coherence length) and an energy gap in their excitation spectrum of the order of 10–3 eV; zero resistance of an ideal superconductor and the Meissner effect are shown to be the consequences of this coherence; (b) the crystal lattice shows an extremely small change of properties, but nevertheless plays a very important role in establishing superconductivity because the critical temperature depends on the atomic mass (the isotope effect); (c) the transition from a superconducting to a normal state in zero magnetic field is a second-order phase transition. The BCS theory involves many-body quantum mechanical equations and the details are given in many textbooks (see the further reading list in chapter A3). Here we shall give only qualitative aspects and conclusions of this theory, in particular its relation to the earlier phenomenological London and GL theories. Importantly, the GL theory is fully justified on the basis of the microscopic approach when the temperature is sufficiently close to Tc, and the parameters of the GL theory introduced above are related to material constants (Gor’kov 1959). Therefore many practical problems can still be treated in the framework of the GL theory. A1.0.4.1 Energy spectrum, energy gap and density of states The basis of the BCS theory is Cooper pairs. It is shown that a net attractive interaction can exist between two electrons as a result of exchange by a phonon, provided their energies are within hωc of the Fermi energy EF. Here hωc is the so-called cut-off energy related to a maximum phonon energy hωD (Debye energy) in a metal. This net interaction is generally a result of interplay between the attractive phonon-mediated interaction and the repulsive screened Coulomb interaction. Therefore the state of the lowest energy (the ground state) of a superconductor occurs when all electrons with momenta in the range δp = hωc /υF about the Fermi momentum pF are coupled together in Cooper pairs having opposite momentum and spin. This state is often referred to as a condensed state to stress that the paired electrons are described by a single quantum mechanical wavefunction and therefore must be regarded as all belonging to the same quantum state. This wavefunction is closely related to the GL order parameter ψ and the paired electrons can be qualitatively regarded as forming the superfluid fraction. The BCS interaction is often called phonon-mediated pairing to distinguish it from other possible nonphonon mechanisms of superconductivity. All classical low-Tc superconductors are known to be BCS superconductors with an electron-phonon mechanism of pairing. More recently discovered organic superconductors and high-Tc superconductors clearly show a lot of anomalies in their behaviour which differ from the BCS theory predictions, but it has not yet been confirmed whether some nonphonon mechanisms play a role in these materials. Let us discuss briefly the main predictions of the BCS theory. There is the second-order phase transition to a new electron state with the transition temperature

where γ ~ – 1.78, ωD is the Debye energy and dimensionless parameter λ = N(0)V is the coupling constant. Here N(0) is the density of states at the Fermi level EF in the normal state of a material per unit volume

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The evolution of SC theories

given for a free electron gas by N(0) = 2p2EF(m/h 2 )3⁄2 /π2 and V is the energy of electron-electron attractive interaction. The isotope effect, i.e. the dependence of the critical temperature on the isotope mass, is naturally explained by the BCS theory. That is, because ωD∝ Μ −1/2 where M is the atomic mass, it follows from (A1.0.77) that Tc ∝ M-1/2. Taking as typical values hωD = 100 K and λ = 0.3 we obtain an estimate for the critical temperature Tc~ –~ 4K—the correct order of magnitude for low-Tc superconductors. The BCS formula (A1.0.77) is valid only for small values of the coupling constant λ << 1 (weak electron-phonon interaction), whereas even for some elements like Pb, Hg and Nb this is not the case and λ ≥ 1. In some compounds the electron-phonon interaction is even stronger. For the case of strong electron-phonon interaction the theory was generalized by Eliashberg (1960) (the strong-coupling theory). Based on this theory McMillan (1968) has constructed a very useful empirical equation for Tc

where ΘD is the Debye temperature and µ* is the effective dimensionless Coulomb repulsion parameter. Usually 0 < µ* < 0.2, being ~ – 0.1 for most superconductors. The parameter λ in this theory is the dimensionless electron-phonon coupling parameter which is directly related to the electron-phonon interaction and is expressed through the parameters of a crystal potential of a given material. The McMillan expression is approximate but provides a rather good quantitative description for all known low-Tc materials when the coupling constant is not too large, λ ≤ 2. Later, Allen and Dynes (1975) considered the limit of a very strong coupling λ >> 1. A critical temperature is given then as

where 〈ω 2〉 is the mean-square average phonon energy. This expression demonstrates explicitly the possibility of increasing Tc when sufficiently strong coupling of electrons with high-frequency phonon modes exist. A practical limit to Tc within a phonon mechanism is, however, set by problems with the structural stability of the crystal lattice. Below Tc the BCS theory predicts a small energy gap in the electron excitation spectrum near EF. Physically the energy gap is one half of the minimum value of energy necessary for destroying a Cooper pair and exciting the two electrons to the normal state. At zero temperature in the weak-coupling limit the gap equals

This gap is rather small, of the order of kBTc ~ 10–4EF. The weak-coupling BCS theory predicts a universal relation between the zero-temperature gap value and Tc

In the strong-coupling limit the ratio 2∆(0)/kBTc increases above 3.52 and saturates at a value of about 11.6 at λ >> 1 (Carbotte 1990). At larger temperatures the gap decreases continuously, going to zero at T = Tc which is the manifestation of the second-order phase transition between superconducting to normal states. Near Tc the temperature dependence is approximated by ∆(0) ∝ (1 – T / Tc )1/2. An important relation between the energy gap and the previously introduced thermodynamical critical field Hc at absolute zero temperature is

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The BCS theory

25

The existence of the gap means the absence of a low-energy excitation in a superconductor, which manifests itself in the exponential temperature dependence exp(-∆ / kBT) of various physical quantities at low T. One mentioned above is the exponential temperature behaviour of the surface resistance RS. Other examples are the low-temperature specific heat, ultrasonic attenuation and tunnelling current into a superconductor, all being proportional to exp(-∆ / kBT) at low T. Tunnelling phenomena in superconductors will be considered separately in section A1.0.5. The BCS theory introduces the coherence length ξ0 as the spatial size of a Cooper pair. In the same sense as in the GL theory, this length determines the characteristic scale of spatial variation of the gap and at zero temperature is given in order of magnitude as ξ0 ∼ hυF / ∆(0). The exact expression for ξ0 depends on the relation between ξ0 and the mean free path l in a material. In a pure metal ξ0 << l (clean limit) the coherence length ξ0 is given by

In the dirty limit l << ξ0 the effective coherence length is reduced to ξ ∼pξ l , whereas the effective magnetic field penetration depth, in contrast, increases: λ ∼λLpξ0/l, where λL is the London penetration depth. The BCS theory correctly describes the surface impedance of a homogeneous superconductor Z = p4πiω/σ. Specific expressions for the complex conductivity σ depend on the relation between ξ0 and the electronic mean free path l. In the clean limit 0

and

where σn is the normal-state conductivity, f (E) = [exp(E/kBT )+1]-1is an equilibrium Fermi distribution function and

is the so-called coherence factor in the BCS theory. The above expressions determine the temperature and frequency dependence of the dynamic conductivity of a clean superconductor. The first term in equation (A1.0.84) is the contribution of thermally excited normal quasiparticles which exist only at T ≠ 0 whereas the second term is the contribution resulting from breaking of the Cooper pairs by an external field when the frequency ω > 2∆. At T = 0 there is a threshold 2∆ for excitations in a superconductor thus for frequencies ω < 2∆ there should be no intrinsic losses. At finite but low temperatures the resulting surface resistance is given by equation (A1.0.50) The penetration depth λ is related to σ2 as λ = (4πσ2ω)-1/2. In the clean limit the zero-temperature penetration depth is given by

i.e. is independent of gap parameter. The clean limit is usually referred to as the nonlocal or Pippard limit, because in most cases it corresponds to the case λ < ξ (type I superconductors). A more practical case is

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The evolution of SC theories

the one of local electrodynamics λ > ξ which mostly corresponds to impure alloys. In the dirty limit the penetration depth is given for the whole temperature range by a particularly simple expression

This temperature dependence is qualitatively similar to the two-fluid dependence ~ (1 – (T/Tc )2)-1/2 except for very low T when the BCS theory correctly predicts exponential behaviour. In addition to a strong-coupling theory, one more important extension of the BCS theory is the account of anisotropy of a Fermi surface of a superconductor (Allen and Mitrovic 1982). In an anisotropic superconductor the energy gap becomes direction dependent in momentum space and therefore the universal BCS relations described above are quantitatively modified. In particular the microwave losses at low T will be determined by exp(–∆min /kBT ) where ∆min is the lowest gap value at the Fermi surface. Larger deviations from the isotropic BCS theory take place when two or more superconducting bands cross the Fermi level (Suhl et al 1959). The anisotropy effects manifest themselves in the superconductivity of transition metals and their alloys when s and d electrons participate in a Cooper pairing. The most anisotropic superconductors among currently known materials are perovskite high-Tc superconductors. In conclusion, the essence of the BCS theory is the formation of Cooper pairs of electrons through lattice distortions (phonons). Originally formulated in the isotropic weak-coupling limit, this theory has recently been extended to the case of an arbitrary strong electron-phonon coupling and anisotropic Fermi surface. In this form it is still usually to referred as the strong-coupled BCS theory, stressing that the microscopic origin is phonon-mediated electron pairing. Thus the classification of superconductors into BCS and nonBCS is usually given in terms of a pairing mechanism. All practical low-temperature superconductors are proven to be BCS superconductors. One exception is probably the so-called heavy-fermion superconductors containing f elements α-U and La. For the perovskite high-Tc superconductors the very fact of the existence of Cooper pairs is reliably established by flux quantization with the flux quantum value corresponding to a charge 2e. However, extremely anomalous properties of these materials strongly suggest that they are very probable candidates for some nonphonon pairing mechanism. At present there is still no consensus about the nature of the pairing mechanism. A short review will be given in section A3.0.1.2. A1.0.4.2 The relation between the BCS and the GL theories In section A 1.0.3 we have considered the GL theory introduced on a purely phenomenological basis. To use it for applications one needs to be certain about its microscopic justification and limits of applicability. The microscopic derivation was done by Gor’kov (1959) who has shown that it is possible to establish the GL equations from the microscopic BCS theory and to calculate the coefficients in the GL expansion of the free energy. This provides a useful relationship between the parameters of the GL theory and the characteristics of a material. Since the work of Gor’kov a lot of work has been carried out on different extensions of the GL theory in going to lower temperatures and time-dependent fields. Prior to giving the relations for the GL parameters let us remember that they were derived from the weak-coupling BCS theory. They are modified quantitatively for the BCS theory extensions to a strong coupling and anisotropy, but still the GL theory remains valid. Moreover, the GL theory is valid even for nonBCS superconductors since it is based on the most general Landau theory for second-order phase transitions. The GL parameters α and β are given below for the clean limit ξ0 << l

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The BCS theory

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and for the dirty limit ξ0 >> l

In these limits the GL order parameters ψc and ψd are related to the BCS energy gap ∆(T) as

where ζ(3) ~ – 1.2. These relations show that in the region of applicability of the GL theory, at T ~ – Tc, the GL order parameter up to the numerical material-dependent coefficient determines the energy gap. Equations (A1.0.91) and (A1.0.92) define the magnitudes of the order parameter (energy gap). Generally, both quantities Ψ and ∆ are complex and depend on a macroscopic phase φ as: Ψ = |Ψ|0exp(iφ), ∆ = |∆|0exp(iφ). In the absence of electrical currents and magnetic fields (steady state) one can exclude the phase by assuming φ = 0. In many other situations the phase cannot be eliminated, and coherence on a macroscopic scale becomes important (e.g. the Josephson effect). The coherence length and the magnetic penetration depth in the GL theory are given in both limits by

and

where λL(0) is the zero-temperature London penetration depth. The temperature dependences of the penetration depth λ(T) in both limits coincide with the temperature dependence of the London penetration depth near Tc. The GL parameter k calculated for both clean and dirty limits from equations (A1.0.91)-(A1.0.94) as kc,d = λc,d (T)/ξc,d (T) is temperature independent near Tc

According to the last expression (A1.0.95), the GL parameter is very sensitive to impurity scattering. Therefore for most superconducting alloys with a small electron mean path l the GL parameter is large and they are type II superconductors. For intermediate values of electronic mean free path the expression for the GL parameter can be given in the following convenient form

where γ is the Sommerfeld specific heat constant, in erg cm–3 K–2, and ρ is the residual resistivity in Ω cm. On the basis of this relation, it was shown that the GL theory satisfactorily explains the change of surface energy in Ta—Nb and U—Mo alloys. The negative surface energy is not necessarily due to a short

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mean free path. Even for a pure superconductor it is possible for the coherence length to be shorter than the penetration depth. As follows from equation (A1.0.83) this is most likely for high-Tc. Among the elements, Nb and V are intrinsically type II superconductors. The newly discovered high-Tc superconductors have extremely short coherence length and therefore are intrinsically type II superconductors as well. As follows from the relations presented above, the GL theory is quantitatively correct in the vicinity of a critical temperature at T ~ - Tc, when the temperature dependence of the GL order parameter Ψ(T) coincides with that of the BCS energy gap ∆(T). In this temperature range the GL expression (A1.0.65) for k is exact. For lower temperatures the GL theory overestimates the critical field Hc(T): this theory predicts Hc(T) ~ (1–T / Tc ), whereas the BCS temperature dependence Hc(T) shown schematically in figure A1.0.1 deviates downwards from the linear behaviour 1–T / Tc as temperature decreases. The same holds for Hc1(T) and Hc2(T). Qualitative estimates with the GL theory are, however, possible at lower T. Since Gor’kov’s microscopic derivation of the GL equations, the GL theory completed by Abrikosov for type II superconductors is often referred to as the GLAG theory. A1.0.5 Tunnelling in superconductors and the Josephson effect Tunnelling measurements can yield accurate quantitative values for the gap and its temperature dependence. This technique has been applied to many conventional superconductors. Due to the nature of superconductivity macroscopic quantum coherence plays an important role in tunnelling effects. Therefore we consider separately conventional quasiparticle tunnelling and phase-coherent Josephson tunnelling. A1.0.5.1 Single-particle tunneling Single-particle tunnelling in superconductors was pioneered by Giaever (1960). According to the BCS theory, the density of states (DOS) of a superconductor is zero near the EF in the energy range E < ∆(T), while for E > ∆(T) the energy dependence of the DOS is given by

Figure A1.0.5(a) shows schematically the tunnelling process between two superconductors S1 and S2 under applied voltage V in a semiconductor-like energy representation. This is the so-called SIS tunnel junction (where I denotes an insulating layer). Solid lines denote the DOS in both superconductors; shaded regions show occupied states at nonzero temperature. The quasiparticle tunnelling current is given by the following expression

Here f1,2(E) are the Fermi distribution functions (typically considered to be in thermal equilibrium) and Rn is the normal-state (ohmic) resistance of the contact

where N1(0), N2(0) are the normal-state densities of states at the Fermi level and |T|2 is the squared tunnelling matrix element averaged over Fermi surfaces of both metals. According to equation (A1.0.98), at zero temperature T = 0 no current should flow through the contact at low bias until eV = ∆1 + ∆2. At finite temperatures current flow is due to thermal excitation of quasiparticles above the gap (shaded regions in figure A1.0.5(a)).

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Figure A1.0.5. (a) Energy diagram for the electron tunnelling process between two superconductors. (b) I—V characteristic of an SIS tunnel contact at a finite temperature.

The resulting I—V characteristic of a contact is shown in figure A1.0.5(b). The current shows the peak at the voltage corresponding to the half-difference of superconducting gaps eV = (∆1–∆2)/2. This so-called subgap current decreases exponentially at low temperatures as exp(–∆min/T) where ∆min is the smallest of two gaps ∆1, ∆2. In the particular case of two identical electrodes ∆1 = ∆2 = ∆ the subgap tunnel current is given by

for kBT << eV < ∆. In this voltage range the junction has a weak negative resistance, followed by rapid increase of current at higher bias. The current jump δ I at eV = ∆1 + ∆2 is due to singularities of the DOS according to equation (A1.0.97) and is given by

Note that this current jump is a property of an ideal tunnel junction and in practice is smeared out. The slope of the I—V curve at eV ~ – ∆1 + ∆2 as well as the ratio of the subgap resistance at eV << ∆1 + ∆2 to an ohmic resistance at eV >> ∆1 + ∆2 are the characteristics of the tunnel junction quality. For tunnelling between a normal metal N and a superconductor S (SIN tunnel junction) the I—V curve at low temperatures kBT << eV < ∆ is well approximated by the following expression

At high bias eV >> ∆1 + ∆2 the tunnelling in SIS and SIN junctions is dominated by the energy regions in both electrodes far above the gaps ∆1, ∆2, therefore the influence of superconducting singularities in the DOS becomes negligible and the I—V characteristic becomes linear with the resistance Rn determined by equation (A1.0.99).

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An important practical application of tunnelling is in the characterization of a superconducting material, e.g. in measuring the energy gap of a superconductor as a function of temperature and magnetic field, as well as in spectroscopy of the electron-phonon interaction. According to the Eliashberg strong-coupling theory, electron-phonon interaction manifests itself in the form of additional nonlinearities on I— V curves which correspond to phonon energies. This makes it possible to determine typical phonon frequencies in a superconductor as well as strength of the electron-phonon interaction. Classical results on tunnelling spectroscopy in superconductors are reviewed by McMillan and Rowell (1969). Furthermore tunnelling is used to study the proximity effect in SNIS junctions (Wolf and Arnold 1982), photon-assisted tunnelling and the lifetime of excited states. A very detailed review of tunnelling phenomena in superconductors can be found in the book of Wolf (1985). The tunnel SIS and SIN junctions considered above exhibit strongly nonlinear I—V characteristics at voltages eV -~ ∆1 + ∆2 and eV -~ ∆, respectively, thus giving the basis for many practical applications. The most important device application of the quasiparticle characteristics of a superconducting tunnel junction is as a mixing element for ultra-sensitive (sub) millimetre-wave receivers, currently used in astronomy, as well as for laboratory applications. The receivers can operate in the direct or in the heterodyne detection mode. Both modes are based on the rectification process that occurs due to the nonlinearity of the I—V characteristics. In a direct (or video) detection mode a high-frequency a.c. signal is transformed to a d.c. signal. In this broad-band mode phase and frequency information of the a.c. signal is lost. Therefore most practical receivers are at present based on heterodyne detection (mixing). The stronger the nonlinearity of the I—V curve, the more effective is the signal transformation. In a very high-quality tunnel junction with an almost vertical slope at eV ~ – ∆1 + ∆2 close to the ideal current jump given by (A1.0.101) a quantum limit can be achieved (Tucker and Feldman 1985). In this case maximum possible sensitivity corresponds to the tunnelling of a single electron as a result of absorption of a single photon. SIS mixers based on Nb tunnel junctions can operate in the quantum limit up to frequencies of 700 GHz. Another type of receiver uses bolometric devices in the visible to microwave region. Their operation is based on an increase of the electron temperature by the incoming radiation (Prober 1993). The current sensitivity is defined as R = ∆Id.c./∆Pa, where ∆Id.c. is an increase of the d.c. current under applied a.c. power ∆Pa. According to equation (A1.0.102) the sensitivity of an SIN junction is given as R = e/2kBT and becomes extremely high at low T, of the order of 5800/T A W–1. Superconducting bolometric mixers are comparable by efficiency to semiconducting Schottky diodes. Superconducting tunnel junctions are applicable for particle and high-energy resolution detection (αparticles and x-rays). The potentially high resolution of such a detector is due to the small amount of energy (typically a few meV) required to create a charge carrier in a superconductor. For example, the limiting energy resolution for a detector based on Nb equals 4 eV for 5.9 keV x-rays. The current best resolution of 36 eV was obtained using a detector with thick Al trapping layers in an Nb-based junction (Mears et al 1993). Practical devices are a subject of further intensive research. A1.0.5.2 Josephson tunneling A Josephson junction is a weak electrical contact between two superconductors. A tunnel junction is a particular case of a Josephson junction where two superconductors are separated by a tunnel barrier. More generally, Josephson junctions are classified as weak links between two superconductors (for a review see Likharev 1979). A weak link is a region which has a much lower critical current than the superconductors it joins and into which an applied magnetic field can penetrate. Different types of weak link can be realized as a narrow constriction in a superconducting film (microbridge), a point contact between two superconductors or a multilayered structure consisting of two superconductors divided by a normal metal (SNS junction). In the latter case coupling between two superconducting electrodes is due to the proximity

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effect. For all types of weak link, including Josephson tunnel junctions, the nondissipative current can flow through the link. The main features of the Josephson effect (Josephson 1962) are described in terms of the phase difference ϕ = φ1 – φ2, where φ1.2 are the phases of the macroscopic order parameters in two superconductors according to equations (A1.0.91) and (A1.0.92). The ability of a junction to sustain a d.c. supercurrent Is without generating a voltage between superconductors is called the d.c. Josephson effect. The critical current Ic is the maximum possible supercurrent through a junction. Is depends on ϕ in a 2π-periodic and in most cases sinusoidal manner

The value of the critical current Ic is typically several orders of magnitude lower than the critical current of a bulk superconductor and is determined by the type and geometry of the junction. When the current through the junction exceeds Ic , a nonzero voltage V is generated and as a result the phase difference ϕ varies with time t according to the equation

According to equations (A1.0.103) and (A1.0.104), the resulting supercurrent oscillates with the frequency νJ = 2e/h = 483.59 THz per voltage of 1 V. This fundamental relation between the frequency of Josephson oscillations in a junction and the generated voltage is known as the a.c. Josephson effect. The existence of Josephson radiation from a junction can therefore be considered as direct evidence of the Josephson effect. The first experimental detection of Josephson generation from a junction was performed by Yanson et al (1965) several years after the theoretical prediction of Josephson. Practically, the existence of the fundamental relation (A1.0.104) opens up the possibility of creating a voltage standard. The electrical characteristics of a Josephson junction are determined by the Josephson equations and by the material properties of the superconductive electrodes and the barrier layer. Barrier properties differ among the different types of weak link: tunnel junctions, SNS sandwiches, variable-thickness bridges and point contacts. A detailed microscopic formalism is required for the description of the transport characteristics of a specific junction. Complicated microscopic theories have been developed over past years, e.g. for SIS tunnel junctions, SNS proximity effect junctions and ScS (superconductor—constriction—superconductor) point contact junctions. A comprehensive description of different models can be found in the reviews by Likharev (1979, 1986). A simple and rather general derivation of the critical current of a variable thickness bridge of length L << ξ(T) was performed by Aslamasov and Larkin (1969) on the basis of the GL approach. In the vicinity of Tc the result is

where Rn is the normal resistance of a weak link. As was shown by more complicated theories, this relation holds sufficiently close to Tc not only for constrictions but essentially for all types of Josephson junction, including point contacts (Artemenko et al 1979), tunnel and SNS junctions, under the condition that the order parameter in the electrodes near the barrier equals its bulk value ∆(T). In the particular case of a Josephson tunnel junction the normal resistance Rn is given by equation (A1.0.99). The critical current depends strongly on the applied magnetic field, since that spatially modulates the phase difference across the junction. In the presence of an external magnetic field H in a barrier the supercurrent density Js(x) varies spatially along the barrier according to

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where Λ = d + 2λ(T) is the so-called magnetic thickness of the barrier and d is the distance between junction electrodes. Maximization of the total current I = ∫0W JS(x )dx with the current density determined by equation (A1.0.106) leads to the following Fraunhofer pattern for the field dependence of the critical current Ic(H )

Here W is the junction length along the x direction and Φ = HWΛ is the magnetic flux trapped in the barrier region of a junction. The critical current becomes zero each time an integer number of flux quanta penetrate into a barrier region. Equation (A1.0.107) is the diffraction pattern for a single Josephson junction. The above relations are valid only for the case of homogeneous distribution of a current in a junction. It is only possible when the junction length W is less than the so-called Josephson penetration depth λJ = (c Φ0/8π2Jc Λ). For W > λJ the phase distribution along the junction becomes strongly nonlinear and takes the form of Josephson vortices, each vortex having a flux Φ0. Such junctions are called long or distributed Josephson junctions and their dynamical properties are determined by the propagation of Josephson vortices. Potential practical applications of such junctions are their use as tunable flux-flow Josephson oscillators for the generation and detection of electromagnetic waves with an oscillation frequency controlled by a magnetic field (Pedersen and Ustinov 1995). The tunable flux-flow oscillators based on the phenomena of unidirectional and viscous flow of Josephson vortices in a long Josephson junction with high damping were successively tested near and above the gap frequency of Nb from 250 to 780 GHz (Zhang et al 1993, Koshelets et al 1993). The dynamics of short Josephson junctions with W < λJ is very conveniently described by the socalled resistively shunted junction (RSJ) model introduced by McCumber (1968) and Stewart (1968). According to this model, the total current I through the junction is given by the sum of the Josephson supercurrent Is, the quasiparticle current V / Rn and a displacement current due to the junction capacitance C. This yields a nonlinear equation for the phase ϕ

The RSJ model gives a good qualitative description of d.c. I—V characteristics for most types of weak link. The shape of the I—V characteristics is determined by the bias conditions and by the following parameters: the Josephson plasma frequency ωp = p2eI /hC defines the internal resonance in a junction; the characteristic voltage ωc = 2e/hVc ≡ (2e/h)IcRn = Rn/Lc (where Lc is the characteristic inductance of the Josephson element) determines the response of a junction to an external signal. The McCumber parameter βc = (2e/h)IcRn2C is the hysteresis parameter: the larger βc, the stronger the hysteresis in a junction. Therefore for practical purposes normal shunts are often used in order to reduce Rn and thus to avoid hysteresis. The frequency ωc defines the inverse relaxation time in a junction, i.e. the fastest pulserise times in the Josephson junctions are of the order of ωc-1. This value suggests the upper boundary for practical microwave devices based on the Josephson effect and can be as short as 10–12 s, about 1000 times faster than the switching time in silicon. The RSJ model is widely used to study the dynamics of Josephson junctions and circuits. The most simplifying assumption is the linear behaviour of quasiparticle shunting resistance, which is not fulfilled e.g. for a tunnel SIS junction. A more complicated microscopic description exists for the latter case (Werthamer 1966). According to the Werthamer theory the time-dependent tunnel current is expressed as a sum of several components, which in the case of constant voltage (large values of the McCumber parameter βc >> 1) is c

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where the time-dependent phase difference ϕ (t) is

ϕ0 being an arbitrary constant. Here Ip(V) is the supercurrent component, Iq(V) is the quasiparticle current component determined by equation (A1.0.98) and Ipq(V) is the so-called interference current component. All these current components are determined by the properties of junction electrodes. The Werthamer equations describe the junction dynamics in a fully self-consistent way. The critical current of a junction Ic is related to Ip at zero bias as Ic = Ip(V = 0) and for the case of two identical electrodes ∆1 = ∆2= ∆ is given by

This expression is usually referred to as the Ambegaokar—Baratoff formula (Ambegaokar and Baratoff 1963). Near Tc this expression coincides with the Alsamasov—Larkin result (A1.0.105). According to equation (A1.0.110), the IcRn product of a tunnel Josephson junction does not depend on the properties of a particular barrier and at T = 0 is simply related to the superconducting gap as Vc = IcRn = πΛ(0)/2e. Thus the larger the gap is, the higher would be a characteristic voltage Vc as required for most applications. In practical Nb-based tunnel junctions Vc is somewhat lowered and the current jump at eV = 2∆ is smeared out by pair breaking due to a number of physical mechanisms (Zorin et al 1979, Golubov and Kupriyanov 1988). Josephson oscillations in a d.c.-biased junction can be synchronized with an external electromagnetic field of frequency ω. Taking applied voltage in the form V + V1 cos ωt with a small amplitude V1, one can write the time-dependent Josephson current (A1.0.103) in the form

where ωJ = 2eV/h is the Josephson frequency and Jn is the nth-order Bessel function of the first kind. It follows from equation (A1.0.111) that the constant-voltage steps (‘Shapiro steps’) (Shapiro 1963) appear in the I—V characteristics at discrete voltages Vn = nhω/2e. This so-called phase-locking effect is an additional manifestation of Josephson oscillations in a junction. The amplitude of Shapiro steps, In, depends on the amplitude of the external signal V1 in an oscillating way

Therefore, a Josephson junction can be used not only as a generator of electromagnetic radiation but also as a detector of external radiation. Different detection methods are briefly described below. In addition to the existence of the diffraction pattern in a single Josephson junction (A1.0.107) there exists quantum interference between different junctions connected in parallel. A practical example is the SQUID. Consider a realization of a SQUID consisting of a superconducting ring with two Josephson junctions connected in parallel—the d.c. SQUID (Zimmerman and Silver 1966). The critical current of the system oscillates as a function of the magnetic flux Φ inside the ring

where Ic1, Ic2 are the critical currents of individual junctions. The smallness of the oscillation period, φ0, makes this device very practical for detecting and measuring extremely weak magnetic fields below 10–14 T.

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A detailed analysis of SQUIDs requires, apart from the study of single-junction properties, an analysis of all relevant inductances and noise properties of a circuit. Note that the flux Φ in equation (A1.0.113) is the total magnetic flux inside a superconducting loop and is related to the external flux fex as Φ = Φex - LIs , where L is the inductance of a ring and Is is the screening supercurrent. According to (A1.0.113), magnetic flux variation leads to a variation of critical current and therefore to a variation of the dynamical resistance when the bias current I > Ic(Φ). The magnitude of the critical current modulation depends on the ring inductance L. An optimal regime is achieved when LIc /Φ0 ~ _ 1. To reduce noise one needs to diminish the inductance L. The requirement is LIc < Φ02 / 4kBT which gives L < 10–8 H at T = 4 K. Junctions without hysteresis (βc ≤ 1) are necessary for practical SQUIDs. At present, most commercially available SQUIDs are a.c. SQUIDs consisting of a superconducting ring with a single Josephson junction connected with an a.c. circuit. The signal under study is detected by a pickup coil and is coupled inductively to the SQUID itself via an input coil. The accompanying electronics detects the change in the SQUID current. Detailed discussion of operation principles and applications of SQUIDs can be found in the books by van Duzer and Turner (1981), Barone and Paterno (1982) and Likharev (1986). SQUIDs have found a large number of practical applications. Their major practical use is as extremely sensitive magnetometers detecting small magnetic fields in biomagnetic applications (the study of magnetic properties of neurons, magnetoencephalography), in corrosion detection as well as for many laboratory purposes. Josephson junctions can be also used for sensitive detection of (sub) millimetre radiation. The detection principle is different from that in quasiparticle mixers and is based on phase locking to the a.c. Josephson oscillations. In order to take full advantage of the nonlinear a.c. Josephson oscillations the devices need to have nonhysteretic I—V curves. Such junctions can be used as detectors in several modes: direct detection, frequency selective detection and heterodyne mixing (Richards 1977). The basic principle of the recently suggested rapid-single-flux-quantum (RSFQ) logic (Likharev and Semenov 1991) is the controlling of single flux quanta with shunted βc ~ _ 1 Josephson junctions. An extremely high operation frequency of about 100 GHz of the RSFQ elements had been obtained (Kaplunenko et al 1991). Intensive further research in this field is directed at the development of more complicated logical circuits in order to create a completely new basis for a high-speed computer. References Abrikosov A A 1952 Dokl. Acad. Nauk. 86 489 Abrikosov A A 1957 Zh. Exp. Theor. Fiz. 32 1442 (Engl. Transl. 1957 Sov. Phys.—JETP 5 1174) Abrikosov A A, Gor’kov L P and Khalatnikov I M 1958 Zh. Exp. Theor. Fiz. 35 265 (Engl. Transl. 1958 Sov. Phys.—JETP 8 182) Allen P B and Dynes R C 1975 Phys. Rev. B 12 905 Allen P B and Mitrovic B 1982 Solid State Physics vol 37, ed H Ehrenreich, F Seitz and D Turnbull (New York: Academic) p 1 Ambegaokar V and Baratoff A 1963 Phys. Rev. Lett. 10 486 Aslamasov L G and Larkin A I 1969 Pis. Zh. Exp. Theor. Phys. 48 976 Artemenko S N, Volkov A F and Zaitsev A V 1979 Sov. Phys—JETP 76 1816 Bardeen J, Cooper L N, and Schrieffer J R 1957 Phys. Rev. 108 1175 Barone A and Patemo G 1982 Physics and Applications of the Josephson Effect (New York: Wiley) Carbotte J C 1990 Rev. Mod. Phys. 62 1027 Deaver B S and Fairbank W M 1961 Phys. Rev. Lett. 7 43 De Gennes P G 1964 Rev. Mod. Phys. 36 225 Deutscher G and de Gennes P G 1969 Superconductivity vol 2, ed R D Parks (New York: Dekker) p 1005 Deutscher G and Müller K A 1987 Phys. Rev. Lett. 59 1745 Doll R and Näbauer M 1961 Phys. Rev. Lett. 7 51

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References

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Eliashberg G M 1960 Zh. Exp. Theor. Fiz. 38 966 (Engl. Transl. 1960 Sov. Phys—JETP 11 696) Giaever I 1960 Phys. Rev. Lett. 5 147 Ginzburg V L and Landau L D 1950 Zh. Exp. Theor. Fiz. 20 1064 Golubov A A and Kupriyanov M Yu 1988 J. Low Temp. Phys. 70 83 Gor’kov L P 1959 Zh. Exp. Theor. Fiz. 36 1918; 37 1407 Gorier C J and Casimir H B G 1934 Physica 1 306 Josephson B D 1962 Phys. Lett. 1 251 Kamerlingh Onnes H 1991 Leiden Commun. 120b, 122b, 124c Kaplunenko V K, Filipenko L V, Khabipov M I et al 1991 IEEE Trans. Magn. MAG-27 2464 Klapwijk T M 1994 Physica B 197 481 Koshelets V P, Shchukin A V, Shitov S V and Filipenko L V 1993 IEEE Trans. Appl. Supercond. AS-3 2524 Kupriyanov M Yu and Likharev K K 1990 Sov. Phys—Usp. 33 340 Landau L D 1937 Sov. Phys—JETP 7 371 Likharev K K 1979 Rev. Mod. Phys. 51 101 Likharev K K 1986 Dynamics of Josephson Junctions and Circuits (New York: Gordon and Breach) Likharev K K and Semenov V K 1991 IEEE Trans. Appl. Supercond. 1 13 London F and London H 1935 Proc. R. Soc. A 149 71 Mattis D C and Bardeen J 1958 Phys. Rev. 111 412 McCumber D E 1968 J. Appl. Phys. 39 3113 McMillan W L 1968 Phys. Rev. 167 331 McMillan W L and Rowell J M 1969 Superconductivity vol 2, ed R D Parks (New York: Dekker) p 561 Mears C A, Labov S E, and Barfknecht A T 1993 J. Low Temp. Phys. 93 567 Meissner W and Ochsenfeld R 1933 Naturwissenschaften 21 787 Meshkovskii A and Shal’nikov A 1947 Sov. Phys—JETP 17 851 Pedersen N F and Ustinov A V 1995 Semicond. Sci. Technol. 8 389 Pippard A B 1953 Proc. R. Soc. A 216 547 Prober D E 1993 Appl. Phys. Lett. 62 2119 Richards P L 1977 Semiconductors and Semimetals vol 12, ed R K Willardson and A C Beer (New York: Academic) p 395 Shapiro S 1963 Phys. Rev. Lett. 11 80 Stewart W C 1968 Appl. Phys. Lett. 12 277 Suhl H, Matthias B T and Walker L R 1959 Phys. Rev. Lett. 3 552 Tucker J R and Feldman M J 1985 Rev. Mod. Phys. 57 1055 van Duzer T and Turner C W 1981 Principles of Superconducting Devices and Circuits (New York: Elsevier) Werthamer N R 1966 Phys. Rev. 147 255 Wolf E L 1985 Principles of Electron Spectroscopy (Oxford: Oxford University Press) Wolf E L and Arnold G B 1982 Phys. Rep. 91 31 Yanson I K, Svistunov B M, and Dmitrenko I M 1965 Zh. Exp. Theor. Phys. 47 2091 Zhang Y M, Winkler D and Claeson T 1993 Appl. Phys. Lett. 62 3195 Zimmerman J E and Silver A H 1966 Phys. Rev. 141 367 Zorin A B, Kulik I O, Likhaveo K K and Schrieffer J R 1979 Fiz. Nizk. Temp. 10 799 Further reading Abrikosov A A, Gor’kov L P and Dzyaloshinski I E 1963 Methods of Quantum Field Theory in Statistical Physics (New York: Dover) Abrikosov A A 1988 Fundamentals of the Theory of Metals (Amsterdam: North-Holland) Douglass D H (ed) 1976 Superconductivity in d- and f-Band Metals (New York: Plenum) De Gennes P G 1966 Superconductivity of Metals and Alloys (New York: Benjamin) Kresin V Z and Wolf S A 1990 Fundamentals of Superconductivity (New York: Plenum) Orlando T P and Delin K A 1991 Foundations of Applied Superconductivity (Reading, MA: Addison-Wesley) Parks R D (ed) 1969 Superconductivity (New York: Dekker) vol 1, 2

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Rose-Innes A C and Rhoderick E H 1968 Introduction to Superconductivity (Oxford: Pergamon) Schmidt V V 1982 Introduction into Physics of Superconductors (Moscow: Nauka) Schrieffer J R 1983 Theory of Superconductivity (Reading, MA: Addison-Wesley) Solymar L 1972 Superconducting Tunnelling and Applications (London: Chapman and Hall) Tinkham M 1965 Superconductivity (London: Gordon and Breach) Van Duzer T and Turner C W 1981 Principles of Superconductive Devices and Circuits (New York: Elsevier) Vonsovsky S V, Izymov Y A and Kurmaev E Z 1982 Superconductivity of Transition Metals (Berlin: Springer) Williams J E C 1970 Superconductivity and its Applications (London: Pion)

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A2 Type II superconductivity

A A Golubov

A2.0.1 Historical overview From Shubnikov’s experiments (1937) to Abrikosov’s theory (1957) Type II superconductors are characterized by the value of the Ginzburg-Landau (GL) parameter k > 1/ p 2,, i.e. by negative surface energy. Therefore the appearance of normal regions in the interior of a specimen placed in a magnetic field is energetically favourable even for H < Hc and the material should split into a fine-scale mixture of superconducting and normal regions, the arrangement being such as to give the maximum possible boundary area. Such a state is called a mixed state. Shubnikov was the first to suggest the fundamental nature of type II superconductivity (Shubnikov et al 1937). In his early experiments on alloys Shubnikov found that a specimen placed in a magnetic field does not exhibit total flux expulsion except for very low fields. The penetration field is called the lower critical field Hc 1 and is substantially smaller than the thermodynamic critical field Hc . Typical values of Hc 1 can be as small as 10–100 G, whereas Hc is typically of the order of 103 G. Figure A2.0.1 shows a typical H—T phase diagram for a type II superconductor of an ideal cylindrical shape. For weak fields H < Hc 1 there is complete flux expulsion (Meissner phase). For H > Hc 1 magnetic flux penetrates a superconductor but the penetration is incomplete. Complete penetration of a flux takes place at a much higher field Hc 2 > Hc which is called the upper critical field. The curve Hc 2(T) on the phase diagram is the line of the second-order phase transition between superconducting and normal states. This second-order transition is in contrast to the first-order phase transition of a type I superconductor placed in a magnetic field. In the field range Hc 1 < H < Hc 2 a superconductor is in a mixed state. The existence of this region of the H-T plane was first demonstrated by Shubnikov. The crossover from positive to negative surface energy at k = 1/p2 was shown in the original GL paper (Ginzburg and Landau 1950). However, until the work of Abrikosov (1952, 1957) the full consequences of a negative-surface-energy regime were not understood. Abrikosov has described the mixed state of type II superconductors as a vortex state. According to Abrikosov’s theory, the mixed state results from the penetration of magnetic vortices into a superconductor. Then Hc 1 is the field when the penetration becomes energetically favourable. Each magnetic vortex carriers the flux quantum Φ0. In the field range Hc 1 < H < Hc 2 a superconductor contains a finite density of these vortex lines. In equilibrium conditions the vortices form a regular vortex lattice. The existence of the vortex lattice was first confirmed by direct experimental observation using the so-called decoration technique (Träuble and

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Type II superconductivity

Figure A2.0.1. The phase diagram of a typical type II superconductor.

Essman 1967). Small ferromagnetic particles deposited on the surface of a superconductor reproduce the periodic magnetic structure which is due to vortices. The technical importance of type II superconductors A leading use of superconductors is to produce high magnetic fields. Therefore type II superconductors are used in most applications since they remain superconducting in much higher magnetic fields than type I superconductors. For example the type II superconductor Nb3Sn has an upper critical field of nearly 30 T at low temperatures. Despite the fact that below Hc 2 a type II superconductor is characterized by a finite resistance, i.e. by the current-voltage I-V curve, it can still carry quite a large current without completely returning to a normal state. Such a property is based on the concept of vortex pinning and is important for large-scale applications such as manufacturing high-performance superconducting wires and high-field magnets. The practical goal of material engineering is to create as many pinning centres as possible to prevent vortices from moving freely and thus to permit high currents under high magnetic fields. Conventional lowtemperature superconductors are often used in magnets at 4 K. The high-temperature superconductors are type II materials. Some of them have Hc 2 of the order of 100 T at 4 K and remain superconducting at much higher temperatures, which opens a new perspective for applications. A2.0.2 A single Abrikosov vortex A2.0.2.1 The electromagnetic region (λ) and core region (ξ). Exact solution for κ >> 1. Here we consider in some more detail the structure of a single Abrikosov vortex in a homogeneous bulk type II superconductor. Schematically the structure of one vortex line is shown in figure A2.0.2. Due to the cylindrical symmetry of a vortex line, this structure can be represented as the behaviour of the magnetic field and the order parameter as a function of distance from the vortex axis. The magnetic field is maximum near the centre of the line and exponentially decays with distance from the centre over the characteristic length λ (the penetration depth). The order parameter is reduced in a small core region of radius of the order of the coherence length ξ, therefore the vortex core can be qualitatively represented as a region of normal phase of an area ∼ ξ2. Physically, the reduction of the order parameter in the vortex

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A single Abrikosov vortex

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Figure A2.0.2. A schematic diagram of the distributions of the order parameter Ψ(ρ), the magnetic field H(ρ) and the current density J(ρ) near a single Abrikosov vortex.

core is due to large depairing currents flowing near the centre of the vortex line. For a large-k type II superconductor the electromagnetic region of the order of λ when the field is concentrated is much larger than the core region. This situation is usually referred to as the most typical one for practical type II superconductors. Mathematically the magnetic field distribution near the vortex line is most easily described in the k >> 1. In this case the second GL equation for the magnetic field H can be written in the form

where eυ is a unit vector directed along the vortex line, δ (x) is the delta-function and ρ is the distance from the core. The delta-function in the right-hand side of (A2.0.1) represents a core singularity due to a 2π phase change around the core, and the normalization factor Φ0 reflects the fact that the vortex carries exactly one magnetic flux quantum. The solution of the equation (A2.0.1) is

where K0 is the zero-order Bessel function of an imaginary argument. The asymptotic behaviour of the magnetic field for small and large distances is given by

and

The solution (A2.0.3) and (A2.0.4) demonstrate explicity the field saturation at small ρ and exponential decay at large ρ over the distance λ from the core. The supercurrent J flows circularly around the vortex and can be found from equation (A2.0.2). The dependence J(ρ) is also shown schematically in figure A2.0.2. The current density has a maximum at a distance ~λ from the vortex axis followed by exponential decay ~exp(-ρ/λL). With these solutions the vortex line energy per unit length e can be calculated using the free energy functional of the London theory (A1.0.20) and is straightforwardly give as.

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Type II superconductivity

This quantity is a vortex line tension and is important for many estimates regarding energy scales in type II superconductors. The above formula includes contributions of magnetic field and electric currents to the total energy of a vortex. An additional contribution, the so-called core energy, is given by the superconducting condensation energy within the vortex core. It can be shown that the approximation (A2.0.5) is accurate enough for large k values. Indeed, as follows from the BCS relation (A1.0.82) between Hc and ∆ at T = 0 and equation (A2.0.5), the electromagnetic contribution to the vortex line energy can be presented as ε = (π2/24)Hc2ξ 2 ln k, whereas a core contribution is approximately given by (Hc2/8π)ξ 2. Thus the ratio of the magnetic contribution to the core contribution is ~(π3/4)lnk >> l for large k. Exact numerical integration of the GL equations leads to the following expression for the total energy

where the numerical constant α ~- 0.5 represents the core contribution to the vortex energy (Hu 1972). Therefore in a large-k superconductor the energy of the vortex is mostly of magnetic origin. A2.0.2.2 The lower critical field The lower critical field Hc 1 is the magnetic field strength where the Meissner effect is destroyed and vortices start to penetrate into the bulk of a type II superconductor of cylindrical shape. The fact that the line tension ε is positive makes the penetration of a vortex energetically favourable only in a sufficiently strong magnetic field. Equilibrium penetration becomes possible at fields H determined by considering a minimum condition for the Gibbs thermodynamic potential

For H < 4πε/Φ0, G is an increasing function of the magnetic induction B in a sample, therefore the minimum of G corresponds to B = 0 (flux expulsion). On the other hand, for H > 4πε/Φ0, G is lowered by choosing some B ≠ 0, therefore there is flux penetration into a sample. Thus the lower critical Hc 1 is given by

where the core contribution is neglected. For T = 0 the relation between Hc 1 and Hc has the form Hc 1/Hc = lnk/p2k. Thus it is seen that for large k, Hc 1 can be much smaller than Hc . As seen from equations (A2.0.3) and (A2.0.8), the magnetic field in the centre of a single vortex, Hv(ρ << λ), is exactly twice Hc 1 . A2.0.2.3 A vortex near an interface and in a thin film The penetration condition is modified if one takes into account the interaction of a vortex with the boundary of a superconductor. When a vortex line is located at some distance xL parallel to the surface of the specimen, the current distribution is obtained by a method of images: one should add to the line an image of the opposite sign located at -xL. Straightforward calculation of the Gibbs potential G(H) = F(H) – ∫ (BH/4π)dV per unit length of the line with free energy F(H) determined by the London expression yields

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An equilibrium vortex lattice

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where the first term represents the interaction of the vortex with the screening currents produced by an external field, the second term describes the interaction of the vortex with its image (Hu is the vortex solution (A2.0.2)). As seen from the comparison of equations (A2.0.7) and (A2.0.9), the difference between these two terms determines the correction to the penetration field. Namely, at large field interval Hc 1 < H < Hs , where Hs = Φ0/4πλξ the surface barrier exists, thus preventing the penetration of a vortex into a superconductor. This is the so-called Bean-Livingston surface barrier (Bean and Livingston 1964). The fields Hs and Hc are of comparable magnitude and much larger than Hc 1. This effect can lead to the departures from the Meissner effect not being observed at Hc 1 but rather at higher fields of the order of Hc. In practice, the exact value of such a barrier depends on how closely the surface of the superconductor approximates the vortex penetration through a surface which has no defects of the scale of λ. A vortex in a thin film is another practical situation where vortex line structure is very different from the bulk solution (A2.0.2) (Pearl 1964). Let magnetic field be applied perpendicular to the film surface. The film thickness d is assumed to be much smaller than λ. Again there is a hard core of radius ξ in this case but the screening currents are restricted to the thickness of the film and therefore the supercurrent decays slowly as ~1/ρ 2 rather than exponentially ~exp(—ρ/λL) as it would for a vortex in a bulk superconductor. Pearl’s derivation is based on a solution of (A2.0.1) inside the film together with Maxwell’s equations in a vacuum outside the film. The current distribution is found to be

at small distances from the core, and

where the effective penetration depth λeff is given by

According to (A2.0.12), in a thin film, d << λ, the effective penetration depth λeff can be quite large, λeff >> λ. Screening of a vortex current in a thin film is much less effective than in a bulk superconductor. A2.0.3 An equilibrium vortex lattice A2.0.3.1 The interaction force between vortices and the lattice configuration Consider the simplest system of two vortices having the coordinates ρ1.2 at distance du = |ρ1–ρ2| from each other. The field distribution is given by the generalization of equation (A2.0.1)

and the solution is simply a superposition of two solutions Hu 1, Hu 2 of equation (A2.0.2). Substitution of these solutions to the free energy functional (A1.0.20) yields the interaction energy per unit length of the vortices in the form

where Hu12(du) is given by equation (A2.0.2). This energy is positive and repulsive. The last property guarantees the stability of a configuration consisting of many vortex lines. The interaction energy U12

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Type II superconductivity

decays as exp(-dυ/λ)/pdυ at large distances dυ between vortices and diverges as ln(λ/dυ) at short distances. To determine the lattice configuration in equilibrium, one should work from the Gibbs potential which can be expressed as

and minimize this with respect to B. The first term in the expression represents the individual fluxoid energies, where nL is the number of fluxoids per unit area and is related to the field B by B = nL Φ0. The second term represents the interaction energy of the i th and j th fluxoids and summation is made over all pairs of vortices. The energy minimum corresponds to the ordered vortex lattice. Of all possible ordered configurations, a triangular lattice has minimal energy. The distance dυ between neighbouring vortices in a triangular lattice (the lattice period) is given by the relation

First decoration experiments (Träuble and Essman 1967) as well as other later experimental studies using various techniques confirmed the existence of a triangular vortex lattice in equilibrium conditions. A2.0.3.2 The upper critical field The high magnetic field strength up to which the mixed state can persist is called the upper critical field Hc 2. As mentioned above, there is a second-order phase transition from the superconducting to normal state at this field. In other words, in a decreasing field, a nucleation of superconducting phase takes place at Hc 2 with the typical size of a nucleation region of the order of ξ . This nucleation problem can be solved exactly in the framework of the GL theory (Abrikosov 1952). A simple estimate for Hc 2 can be obtained from equation (A1.0.73) for the critical field of a thin superconducting film on the basis of the formal analogy between the film thickness d and the nucleation region ξ . This estimate yields Hc 2 ~ kHc . The exact result obtained on the basis of the GL equations is

This suggests that materials with a high value of k remain in the mixed state until quite strong fields are applied. Physically, Hc 2 corresponds to the onset of the overlap between the vortex cores, as can be seen directly from equations (A2.0.16) and (A2.0.17). The relation (A2.0.17) is particularly convenient for the experimental determination of the coherence length ξ (T) from Hc 2(T) measurements. It follows from equation (A2.0.17) and from the BCS theory relationship for the coherence length for a superconducting alloy ξ ~ 1/pTc, that Hc 2 ~ Tc (in a clean limit this yields Hc 2 ~ Tc2 ). Thus the upper critical field grows with Tc of a superconductor and can be of the order of 20-40 T for commercially available superconductors. For H > Hc 2 a macroscopic sample does not show flux expulsion; however, a superconducting phase still remains in a thin surface layer of the order of ξ (T). This surface superconductivity exists in an interval Hc 2 < H < Hc 3, where the so-called surface nucleation field Hc 3 ~ _ 1.69Hc 2 (de Gennes 1966). The existence of the surface superconductivity may manifest itself in measuring the resistance between two surface probes, but has minor influence on the magnetization curve of a macroscopic sample.

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An equilibrium vortex lattice

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A2.0.3.3 Reversible magnetization The ideal magnetization curve of a type II superconductor without pinning is shown schematically in figure A2.0.3. The Meissner state exists at low fields H < Hc 1, and at H = Hc 1 there is a second-order phase transition to the mixed state. The shape of the magnetization curve in the mixed state is implicitly given by the condition of a minimum of the thermodynamic potential (A2.0.15) as a function of the induction B. The vortex density nL grows in this interval from zero at H = Hc 1 up to ~ ξ -2 at H Hc 2 where the overlap of vortex cores begins and a second-order phase transition to the normal state takes place.

Figure A2.0.3. The reversible magnetization curve of a type II superconductor.

In a field slightly above Hc1 the vortex density nL is small and only nearest-neighbour interactions need to be considered in the expression for thermodynamic potential (A2.0.15) since the vortices are spaced far apart with exponential decay in the interaction energy with separation dυ >> l. Straightforward calculation of the B(H) dependence for this case leads to the result

This result leads to the conclusion that at H = Hc 1 the theoretical curve has an infinite negative slope –(∂M/∂H) = ∞. Physically, this reflects the fact that in the considered limit, du >> λ, the interaction between vortices is small and it is thus possible to form many lines in the sample just above Hc 1. In practice, however, this slope is finite due to nonequilibrium effects. To determine the structure of the vortex lattice in larger fields more complicated calculations are necessary, taking into account intervortex interactions in equation (A2.0.15) for next coordination spheres of a vortex. It can be shown that in the whole field range the triangular vortex lattice has a minimum energy. In the field range Hc 1 << H << Hc 2 the vortex density satisfies the inequality λ−2 << nL << ξ −2 and the magnetization curve is described by

where the numerical constant β ~ _ 0.23 (Matricon 1964) and the lattice period du is given by the expression (A2.0.16). This suggests that, for du ~ ξ, B is nearly equal to H and the specimen returns to the normal state. In the field range near Hc 2 the shape of the magnetization curve is given by Abrikosov (1957)

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Type II superconductivity

Note that the area under the equilibrium magnetization curve equals the free energy difference between superconducting and normal phases and therefore is related to the thermodynamic critical field Hc as

This equation yields an implicit relation between Hc 1, Hc 2 and Hc . No real material demonstrates the reversible behaviour indicated by the idealized curve shown in figure A2.0.3 because of pinning effects. Some flux pinning is practically unavoidable in type II superconductors, but this property is advantageous for applications. In practice a resulting non-reversible magnetization is very important in order to permit high currents to flow under high magnetic fields. Thus the practical goal of material manufacturing is to introduce as strong pinning centres as possible. The nonequilibrium effects and irreversible magnetization will be considered below. A2.0.4 The nonequilibrium vortex lattice A2.0.4.1 Drag force by external current In a vortex lattice the vortices are subjected to repulsive interaction forces resulting from vortex currents. In an equilibrium situation these forces exactly compensate each other and a regular triangular vortex lattice exists. The presence of external currents and crystal lattice defects is usually referred to as a nonequilibrium situation. It can be shown that the existence of an ideal equilibrium vortex lattice would result in finite resistance at any small value of external current. The drag force by an external current can be calculated by generalizing the expression for the interaction force between two vortices, equation (A2.0.14). In the latter case, one of the vortices is placed in the current distribution pattern produced by the other one and the resulting force is given by

Using Maxwell’s equation J = –(c/4π)∂H/∂x this force can be rewritten in a simple form

which shows that it is the external current flowing around a vortex which produces the drag force. This force is usually referred to as the Lorentz force. The expression (A2.0.23) applies not only to the case of two interacting vortices but to the general case of the interaction of a vortex with external transport current Jtr . This force is directed perpendicular to the current flow and results in a vortex motion in this direction. Magnetic flux motion under an external current would result in the generation of an electric field. Direct evidence of a voltage generation under flux motion was first given by Giaever (1965) in his experiment with two superimposed superconducting films separated by an isolating barrier. The voltage was observed in a secondary circuit (the superimposed film) as the current in the primary circuit (the type II plate) was increased. This leads to the conclusion that a zero-resistance state is impossible for an ideal equilibrium type II superconductor. In a real situation a zero-resistance state is possible due to pinning of vortices. Consider a vortex lattice in the presence of a uniform field H. The equilibrium thermodynamic configuration corresponds to a particular induction B(H) and a fluxoid areal density n = B/Φ. Since H is taken to be uniform, the same holds for B and n. Accordingly, the average current density J = 0. In a situation in which J is nonzero one must have a variable vortex density. The stability of a vortex configuration in the latter case is the result of a balance between two forces. The first force is a repulsive

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The nonequilibrium vortex lattice

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interaction between the vortices which acts to keep them apart, creating a uniform n. The second is the pinning force, which results from any defects in the crystal lattice and could block the free movement of the vortices under the repulsive force. Pinning forces always exist in practical superconductors and allow the formation of a static vortex density gradient. To estimate the balancing force from the pinning sites one needs to calculate the force due to the vortex density gradient. Consider an area S at flux density B and compress the vortices by keeping their number SB constant. For a system of vortices one can define a pressure P in terms of the Helmholtz free energy as P = –(∂ F/∂ S). Then the force acting in the x direction per unit volume on the vortices F is given by

The specific form of this expression depends on the shape of the magnetization curve. In materials with k >> 1, in a large-field region H >> Hc 1, the equilibrium magnetization curve is linear and therefore the force is given by

where Jy is the average current density in the y direction. This formula relates the driving force on fluxoids to magnetic field gradients in a superconductor or, equivalently, to a macroscopic current, and provides a generalization of the elementary driving force (A2.0.23) applied to a vortex from an external current. In equilibrium this force equals the effective pinning force acting on the considered volume of the flux lattice. This pinning force restrains the fluxoids from movement. Therefore for bulk type II superconductors the nonvanishing critical current is due to the pinning of the flux lines at defects, at least at sufficiently low currents. The magnitude of a critical current density Jc of vortex depinning is typically lower than the depairing current density J0

The magnitude of Jc depends on the pinning strength in a material and increases with increase of pinning force density. A2.0.4.2 Vortex pinning The origin of pinning forces is still a subject of detailed research interest. Usually it is related to the defect structure of a material. One should distinguish between the elementary pinning force at the level of an individual flux line and the bulk pinning force density. The simplest example of elementary pinning interaction between a flux line and a crystal lattice is a void which may be present due to the manufacturing process of a type II material. When a vortex passes through the void, its energy is lowered by roughly the product of the condensation energy density and the void dimensions. In practical superconductors defects which act as pinning centres include various lattice defects, nonsuperconducting precipitates, grain boundaries, dislocations, etc. The bulk pinning force density Fp is the pinning force per unit volume of a pinning centre, given as a product of the critical current density and the corresponding magnetic flux density: Fp = Jc B. It is rarely possible to sum the local pinning forces directly. The summation usually depends on the strength and distributions of the pinning centres and on the distortions they are able to produce in the vortex lattice. The statistical approach dealing with random pinning centres in a rigid array of vortices was first used by Labusch (1969). Lately, the situation has been considered where each individual pinning force is sufficiently weak in comparison to the Lorentz force applied from an external current (weak pinning). In this case the behaviour of the vortex system is described by the weak-collective-pinning theory (Larkin and

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Type II superconductivity

Ovchinnikov 1979). Importantly, it was shown that as a result of coherent summation of weak individual pinning forces the regime of sufficiently strong pinning can be achieved. Let us first consider several examples of the basic pinning interactions. The physical nature of the pinning can be made clear in some special limits by splitting up the free energy of a vortex into the terms which can be associated with the condensation energy (vortex core) and the terms which depend on the magnetic energy and the circulating currents. The examples are interaction forces with macroscopic defects like grain boundaries and voids (or nonsuperconducting precipitates). In the first case an interaction is mostly of electromagnetic nature as in the case of the Bean-Livingston barrier, whereas voids pin vortices due to the difference in the condensation energy. A practical example of electromagnetic pinning is a thin type II superconducting film in a parallel magnetic field, Hc 1 << H << Hc 2 The surface magnetization current IM = c |B – H|/4π produces an exponentially decreasing current density distribution in the interior of the sample

where M = |B – H|/4π is the magnetization of the film. The equilibrium condition for the vortex lattice is given by the balance of the Lorentz force from the magnetization current and from the transport current Itr applied to the unit volume of the vortex lattice. The maximum possible transport current corresponds to the displacement of the whole lattice to δ x ∼ a0 = (Φ0/B) 1/2 . The cross-section critical current density Jc averaged over the film is given by (Campbell et al 1968)

where d is the film thickness. An estimate for k = 103, Hc 1 = 0.01 T, Hc 2 = 10 T, B = 1 T, λ = 10–5 cm and d = 10–4 cm leads to Jc ~ 105 A cm–2. This value is about three orders of magnitude smaller than the depairing critical current density. The same magnitude of critical current density is possible in a bulk superconductor consisting of thin layers or small grains. The pinning becomes most effective when the thickness d becomes of the order of λ. For d << λ the pinning force vanishes. The same considerations hold for a granular superconductor: strongest pinning is achieved when the grain size is of the order of λ. In general, to be effective, the inhomogeneities may be on the scale of λ to produce magnetic pinning. An example is Nb3Sn in which pinning centres are grain boundaries. Consider a single spherical void (or a nonsuperconducting precipitate) of diameter d. The decrease in the condensation energy of a vortex positioned at the void is given approximately as Hc2ξ 2d/8. The resulting maximum elementary pinning force given by the energy gradient is F’p ≈ Hc2ξ d/8 and has a maximum for d ~ _ ξ . The total pinning force given by a summation of elementary forces F’p will depend on the density of defects. The resulting critical current is found from a balance between the Lorentz force per vortex unit length JcΦ0/c and is maximal for long (columnar or plane) defects of lateral size d⊥ which are parallel to the vortex. Then for d⊥ ≥ ξ the pinning permits high critical current Jc ~ J0. The same effect of pinning holds for any kind of nonsuperconducting inclusion in a matrix of a type II superconductor. An example is a Nb-Ti alloy in which under thermal treatment nonsuperconducting inclusions are formed, leading to a large increase of critical current. It is important that the introduction of pinning defects does not generally lead to a degradation of Tc . In the last example the size of defect d was assumed to be of the order of or larger than the coherence length ξ . This limit does not always hold in real type II superconductors. In the opposite limit of small defects, d << ξ , the pinning force resulting from a single defect vanishes, but the nonvanishing effective pinning force is due to the interaction of a vortex with typically a large number of weak-pinning centres.

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The nonequilibrium vortex lattice

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A successful theory for the description of random pinning is the collective pinning theory (Larkin and Ovchinnikov 1979). The central idea of the collective pinning theory is the assumption that the long-range order of the vortex lattice is destroyed by the presence of the disorder (weak-pinning centres), leaving a short-range order over some correlation length Lc . The length ni depends on the elasticity of the lattice determined by the vortex-vortex interaction and on the disorder. Each correlated volume is assumed to be pinned independently by a total pinning force. The critical current can then be estimated from the equilibrium condition between the driving Lorentz force and the total pinning force acting on this volume. The disorder strength is parametrized by γ = fp 2ni x 2 where fp is the elementary pinning force for a single defect and ni is the concentration of defects. The collective pinning length Lc is given by

where ∈0 is related to the energy of a vortex line per unit length

For a weak disorder (small γ) the collective pinning length Lc is typically much larger than the coherence length ξ , Lc >> ξ . For Lc ~ ξ the pinning should be considered as strong. In the collective pinning theory the critical current density Jc is determined by equating the total effective pinning force (γ Lc )1/2 with the Lorentz force JcΦ0Lc /c and is given by

with the depairing current density introduced above, equation (A2.0.26). The regime of weak collective pinning (large Lc >> ξ) is characterized by a large reduction of the critical current density Jc with respect to the depairing value J0. On the other hand, in the strong pinning regime, when Lc ~ ξ , the critical current density Jc achieves its maximum possible value of the order of the depairing current J0. This is the situation one needs for practical purposes in hard type II superconductors. At high magnetic fields the condensation energy decreases which leads to a corresponding decrease of Jc . The pinning behaviour of Nb3Sn is fairly well described by Kramer’s law Fp = K[Bc 2(T)]mb 1/2(1 – b)2 where K, m are constants and b = H / Hc 2(T). A2.0.4.3 Critical state at zero temperature Superconductors with strong pinning interaction are called hard superconductors. Consider a hard superconductor in an applied magnetic field H. In equilibrium the flux density would have the value B(H)/Φ0 and be the same at all points in the interior of the sample. In the metastable nonequilibrium situation considered above, equation (A2.0.25), the magnetic field gradients exist in a superconductor. These gradients are related to an average current density. The maximum possible current density is given by a maximum pinning force, which leads to the existence of the so-called critical state (Bean 1962)

where αc is a threshold pinning value which depends on a microscopic pinning interaction force Fpmax for a given volume of the flux lattice. In the critical state the flux density and the critical current density in the interior of the sample adjust themselves so that the condition (A2.0.32) is fulfilled at all points.

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To compute the penetration profile B(x) in the critical state one needs to know the dependence of the maximum pinning force Fpmax on a magnetic field B. The simplest assumption made by Bean (1962) that Fpmax is linear in B leads to a conclusion about the independence of Jy on x and to the linear decay of magnetic flux density inside a type II superconductor in a critical state. In many realistic situations the profile can be more complicated due to different dependences of Fpmax (B). According to Kim et al (1963), for NbZr and Nb3Sn alloys, Fpmax is to a good approximation field independent, which leads to a more rapid, parabolic, decay of the flux density with x. In figure A2.0.4 a typical distribution of a magnetic flux density in a slab of thickness d in a parallel field is illustrated schematically in the framework of the Bean model. The slope of the field profiles is proportional to the pinning force. Figure A2.0.5 illustrates the profiles in the case where the transport current JT flows in a slab. When the current through the slab increases, there is a crossover to a new critical state corresponding to the sign change of the initial slope in one part of the slab (in the left part of the slab for the given example). The maximum possible stable external current is called the saturation current. Thus the hard superconductors are in principle able to transmit supercurrent with a density up to the density of the screening current in the critical state.

Figure A2.0.4. The magnetic flux density distribution in a slab of thickness d. Solid line: without pinning; broken line the critical state

Figure A2.0.5. The profiles of the magnetic flux density distribution in a slab of thickness d with zero transport line: current (solid) and finite transport current (broken).

The critical state model predicts flux trapping by a superconducting sample in the process of a full magnetization cycle. Figure A2.0.6 illustrates the magnetization cycle of a slab when the external field first increases, then decreases to zero, and after that the magnetic field direction reverses and its magnitude grows again. The absolute value of the constant linear slope for the case of increasing and decreasing fields is the same and is determined by the critical current density. The corresponding magnetization cycle of a slab is shown in figure A2.0.7. Flux trapping during magnetization manifests itself in hysteresis. The area of a magnetization loop increases with increase of the pinning strength. In d.c. applications, when flux is pinned, there is no energy loss. In a.c. applications energy losses during magnetization are proportional to the area of the loop and become larger for stronger pinning in a material. Thus the larger is the critical current of a hard type II superconductor, the larger are the hysteretical losses. As a result, the hysteretical losses in hard type II superconductors at finite frequencies are much larger than those predicted by the Mattis-Bardeen theory for an ideal homogeneous superconductor and their reduction is an important practical problem. A2.0.5 The resistive state of a type II superconductor A2.0.5.1 Flux-flow resistivity When a critical current exceeds the critical value, the vortices move under balance of two forces: the Lorentz force FL (A2.0.23) and viscous drag force Fv . The latter force is due to the electric field generated in the region around a vortex core which leads to energy losses in the process of the vortex

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The resistive state of a type II superconductor

Figure A2.0.6. The profiles of the magnetic flux density distribution according to the critical state model. Solid lines: increasing magnetic field B2 > B1. Broken lines: decreasing magnetic fields B3 > B4 > B5.

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Figure A2.0.7. A typical magnetization cycle of a hard type II superconductor. Solid line: equilibrium magnetization without pinning. Broken lines: nonequilibrium magnetization. The shaded region shows the hysteresis loop.

motion and is usually represented in the form Fu = -ηυL , where η is the viscous drag coefficient and υL is the vortex velocity. The viscous drag coefficient is given by the Bardeen-Stephen (Bardeen and Stephen 1965) expression

where ρn is the normal-state resistivity of a material. Equation (A2.0.33) describes well the situation at low fields and low temperatures. The corrections to this expression were discussed by Gor’kov and Kopnin (1973) in the framework of the microscopic theory. As mentioned above, the flow of magnetic vortices under an external current leads to the generation of an electric field. This state of a type II superconductor is called the resistive state. The corresponding resistivity is called the flux-flow resistivity ρf and is given by the following simple expression

The flux-flow resistivity is a linear function of a magnetic flux and for B = Hc 2 equals the normal-state resistivity of a material. Thus the resistivity of type II superconductors in the flux-flow regime in high fields is rather large and usually is much higher than the resistivity of copper at a corresponding temperature. Therefore in practical applications this regime should be avoided. A2.0.5.2 Flux creep and current-voltage characteristics The most technologically interesting property of hard type II superconductors is their ability to carry a bulk current density with essentially no dissipation. However, a sample carrying a macroscopic transport current is in a state which is thermodynamically metastable. At finite temperatures the vortex lines will tend to move under a flux gradient by activated jumps across the pinning barriers. The latter phenomenon is known as flux creep and was first introduced by Anderson (1962) and by Anderson and Kim (1964). In the flux-flow regime, for current J > Jc , the resistivity is given by equation (A2.0.34), whereas the fluxcreep phenomenon manifests itself in the existence of a finite resistivity even in the subcritical regime

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Type II superconductivity

J < JC . As a result, a type II superconductor at a finite temperature can be generally characterized by an I—V curve. The equilibrium balance between the fluxoid pressure and pinning strength depends on thermal activation processes, which facilitate the displacements of fluxoids from their low-energy configurations at the pinning centres. This effect can be observed in a hollow tube of type II superconductor which is cooled in an applied field to obtain a uniform flux distribution throughout the specimen. When the external field is slightly changed, the field on the tube axis increases slowly in response to this change. Therefore the flux lattice does not respond immediately to an external perturbation. The time variation of the internal field is found to be logarithmic. The theory of flux creep introduces as the parameters the zero-temperature critical current, Jc(0), that would be carried without thermal activation and the observed critical current, Jc(T ). The relation between these two currents is obtained by introducing the frequency ν of oscillation of the fluxoid in a potential well of height U due to the pinning interaction and by modelling the effective well depth under the applied current as U(1 – J/Jc(0)). The diffusion of vortices is found to be due to thermal activation over the pinning barrier U (J). The resulting I—V characteristic is exponential

where E is the generated electric field, B is the magnetic induction in a specimen and d is the distance between pinning centres. The critical current determined in an experiment will depend on the lowest voltage that can be measured, Ec :

The last equation shows that Jc (T ) decreases linearly with temperature. Another consequence of the exponential I—V relationship is that the flux trapped in a specimen in a constant external field will decay logarithmically with time t . The logarithmic dependence follows from equation (A2.0.35) together with Faraday’s law for a hollow cylinder of radius r and wall thickness w : (rw/2)dJ /dt = E. For long periods the solution for the time-dependent current is

The temporal decay of the transport current is thus determined by the ratio kBT/U, which can be found experimentally by measuring the relaxation of the diamagnetic moment of a sample in the critical state. The activation energy is therefore an experimentally accessible quantity. In conventional low-Tc hard type II superconductors the exponential I-V curves were found with good accuracy in a large voltage range from 10–7 to 10–15 V (Kim et al 1963, Beasley et al 1969). The typical decay coefficients kBT / U are of the order of 10–3 which lead to very large typical waiting times. Vortex jumping from one pinning centre to another usually happens in vortex bundles because of intervortex interaction. In the classical experiments of Beasley et al (1969) this amount was estimated to be between ten and 100 fluxoids at low fields and approximately one fluxoid near Hc 2 . The exponential I-V characteristic of a hard type II superconductor (A2.0.35) suggests the existence of a nonzero resistivity at any temperature above absolute zero, ρ = ρ0 exp(–U / k BT ). This behaviour is usually referred to as thermally activated flux flow (TAFF). In practice a magnet designer’s criterion for resistivity is 10 – 6 µΩ cm under a current capacity of 105 A cm– 2, whereas the residual resistivity of copper is about 2 × 10 – 1 µΩ cm. The thermally activated resistivity is exponentially small at low temperatures, and for practical superconductors NbTi and Nb3Sn the above criterion is fulfilled at 4 K.

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References

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At higher temperatures the effect of thermal activation becomes much more pronounced. The pinning potential well U → 0 as T → Tc and thus the resistivity increases rapidly near Tc. These aspects put severe limitations on the use of high-Tc superconductors in large-scale applications. Other phenomena which are present in practical superconducting wires are the so-called thermal instabilities (Mints and Rakhmanov 1981). If in a small region of the sample the pinning energy U is slightly smaller than elsewhere, then according to equation (A2.0.35) the energy dissipation in this region will be larger than in the other regions. Thus if the thermal conductivity of a material is low this will tend to increase the local temperature. This temperature rise will, in turn, increase the vortex velocity and, therefore, the generated electric field. These thermal processes may finally result in an instability, leading to a degradation effect in superconducting solenoids. To avoid the degradation special precautions should be made such as establishing good thermal contact between the wire and a coolant or, alternatively, coating the superconductor with high-conductivity copper. The latter combination is known as a composite material. When a local flux jump happens in a composite, the effective resistance is a parallel combination of the flux-flow resistance and the low resistance of the coating, and is less than the flow resistance of the wire. When the critical current is exceeded and the superconductor goes normal, then some of the current can be carried in the copper coating, providing the stabilization of the current flow. Most practical NbTi and Nb3Sn wires are multifilamentary structures embedded in a copper or bronze matrix. At present much effort is being applied to maximize Jc of superconducting wires for large-scale applications. The field at which the M versus H curve is no longer double valued is known as the irreversibility field Hirr . In the field range between Hirr and Hc 2 thermal activation leads to flux motion, and only below Hirr does the superconductor become hard. In practical low-Tc superconductors, e.g. NbTi or Nb3Sn, this field is very close to Hc2. In high-Tc superconductors thermal activation effects cannot be neglected since Hirr is appreciably smaller than Hc2. References Abrikosov A A 1952 Dokl. Acad. Nauk. 86 489 Abrikosov A A 1957 Zh. Exp. Theor. Fiz. 32 1442 (Engl. Transl. 1957 Sov. Phys.-JETP 5 1174) Anderson P W 1962 Phys. Rev. Lett. 9 309 Anderson P W and Kim Y B 1964 Rev. Mod. Phys 36 39 Bardeen J and Stephen M J 1965 Phys. Rev. 140 1197A Bean C P 1962 Phys. Rev. Lett. 8 250 Bean C P and Livingston J D 1964 Phys. Rev. Lett. 12 14 Beasley M R, Labusch M and Webb W W 1969 Phys. Rev. 181 682 Campbell A M, Evetts J E, and Dew-Hughes D 1968 Phil. Mag. 18 313 De Gennes P G 1966 Superconductivity of Metals and Alloys (New York: Benjamin) Giaever I 1965 Phys. Rev. Lett. 15 825 Ginzburg V L and Landau L D 1950 Zh. Exp. Theor. Fiz. 20 1064 Gor’kov L P and Kopnin N B 1973 Zh. Exp. Theor. Fiz. 65 396 (Engl. Transl. 1973 Sov. Phys.- JETP 38 195) Hu C R 1972 Phys. Rev. B 6 1756 Kim Y B, Hempstead C F, and Strnad A R 1963 Phys. Rev. 129 528 Labusch R 1969 Phys. Status Solidi 32 439 Larkin A I and Ovchinnikov Yu N 1979 J. Low Temp. Phys. 34 409 Matricon J 1964 Phys. Lett. 9 289 Mints R G and Rakhmanov A L 1981 Rev. Mod. Phys. 53 551 Pearl J 1964 Appl. Phys. Lett. 5 65 Shubnikov L W, Khotkevich W I, Shepelev J D, and Ryabinin J N 1937 Sov. Phys.-JETP 7 221 Träuble H and Essmann V 1967 Phys. Lett. 24A 526

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52 Further reading

Type II superconductivity

Abrikosov A A 1988 Fundamentals of the Theory of Metals (Amsterdam: North-Holland) Blatter G, Feigelman M V, Geshkenbein V B, Larkin A I and Vinokur V M 1994 Rev. Mod. Phys. 66 1125 Campbell A M and Evetts J E 1972 Adv. Phys. 21 199 Campbell A M and Evetts J E 1972 Critical Currents in Superconductors (London: Taylor and Francis) Kim Y B and Stephen M J 1969 Superconductivity vol 2, ed R D Parks (New York: Dekker) p 1167 Parks R D (ed) 1969 Superconductivity (New York: Deker) vol 1, 2 Rose-Innes A C and Rhoderick E H 1968 Introduction to Superconductivity (Oxford: Pergamon) Schmidt V V 1982 Introduction into Physics of Superconductors (Moscow: Nauka) Tinkham M 1965 Superconductivity (London: Gordon and Beach) Williams J E C 1970 Superconductivity and its Applications (London: Pion) Yukikazu I 1994 Case Studies in Superconducting Magnets. Design and Operational Issues (New York: Plenum)

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A3 High-temperature superconductivity

A A Golubov

A3.0.1 Basic physical properties A3.0.1.1 The discovery of 90 K superconductivity and further progress Since the discovery of superconductivity in 1911 the search for superconductivity with a high transition temperature, namely above the liquid nitrogen temperature of 77 K, has been one of the most challenging tasks to physicists and material scientisis. However, before 1986, the highest transition temperature achieved was only 23.2 K for Nb3Ge (1973). As long as there were no definite guidelines to predict materials with high Tc , an empirical search for new materials was the most effective one. One of the directions taken was the search for new superconducting oxides. In 1975 superconductivity was observed in BaPb1-xBixO3 with critical temperature above 13 K. This compound is the predecessor of the current high-Tc superconductors. In 1986 Bednorz and Muller, working at the IBM laboratory in Zürich, made a remarkable discovery. To raise the Tc of superconducting oxides they carried out a series of investigations in an attempt to enhance the electron-phonon interaction. They had taken the point of view that higher Tc values might be found in materials in which the electron-phonon interaction is enhanced through the Jahn-Teller effect. In 1986, they achieved superconductivity at around 30 K in the Ba-La-Cu-O system (Bednorz and Müller 1986)—the first materials in a class of cuprates (Cu oxides). The material they used was La2CuO4 in which Ba, Sr or Ca were introduced to replace some of the La. The superconducting phase was found to crystallize in the K2NiF4 structure, which is a layered perovskite with a strongly anisotropic crystal structure. The Ba-doped material is usually written La2-xBaxCuO4 and the superconducting properties depend strongly on the doping x. A record high Tc of nearly 40 K was achieved in the material La1.85Ba0.15CuO4. Several months after the discovery of the Ba-La-Cu-O system, groups at the Universities of Alabama and Houston jointly announced the discovery of superconductivity above 77 K in the Y-Ba-Cu-O (YBCO) system (Wu et al 1987). This system was independently discovered by the Beijing group in China (Zhao et al 1987). A resistance drop starting at 93 K and completing at 80 K was detected and the Meissner effect was clearly evident below 90 K in these compounds. Consequently, superconductivity above 77 K was finally unambiguously and reproducibly achieved. Subsequent magnetic field effect measurements indicated a record-high upper critical field Hc 2 ∼ 130 T at 0 K for these compounds. Later the superconducting transition was sharpened and enhanced to between 98 K and 94 K. The identification of the phase responsible for the superconductivity led to the chemical formula YBa2Cu3O7- δ and to

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High-temperature superconductivity

the evidence for a layered structure. This crystal structure is an oxygen-defect perovskite and is very anisotropic. Cu-O planes (superconducting layers) and linear Cu-O chains along the b axis were found to exist. With the exact stoichiometry and the general structure of the superconducting phase determined, attempts were made to replace Y by the rare-earth elements to examine their role in high-temperature superconductivity. It was found that nearly all of the rare-earth elements, including magnetic rare earths like Gd, could be substituted for Y without having a significant effect on the transition temperature. Thus a new class of superconductors, ABa2Cu3O7-δ with A = Y, La, Nd, Sm, Eu, Gd, Ho, Er or Lu, with Tc above 90 K was discovered. Two exceptions are the rare earths Ce and Pr. The results show that the ‘A’ elements are used only to stabilize the so-called three-layered structure and that superconductivity must be confined to the Cu-O layers sandwiched between the Y layers. In 1988, many new compounds and classes of compounds were discovered. Notable among these were the Bi-Sr-Cu-O and the Bi-Sr-Ca-Cu-O (BSCCO) compounds, with transition temperatures up to 115 K, and the Tl-Ba-Ca-Cu-O (TBCCO) compounds, with transition temperatures up to 125 K. The general formula for the thallium compounds is TlmBa 2Can -1CunO2n +m +2 where n denotes the number of Cu atoms and m is the number of Tl atoms. Tc increases with increasing number of CuO2 planes in the elementary unit cell which is the general rule for all cuprate compounds. The Bi-based compounds have very similar structures to the Tl-based compounds. With time, the discovery of other cuprates has resulted in a large number of superconducting compounds and a maximum Tc (under pressure) of greater than 150 K (Schilling et al 1993). There are many other widely known high-Tc superconductors. In the lead-substituted TBCCO or BSCCO compounds Tl or Bi are partially substituted by Pb (Cava et al 1988), such that the chemical formula begins (TlxPb1-x…). In mercury compounds Tl is substituted by Hg (Putalin et al 1993). The resulting compound HgBa2CuO4+δ has Tc = 94 K, which is exceptionally high for a single-CuO2-layer compound. In oxycarbonates a carbonate group (CO3) is introduced in the conventional copper oxide configuration which results in the general chemical formula (Y, Ca)n(Ba, Sr)2nCu3n -1(CO3)O7n -3 (Raveau et al 1993) with typical Tc values below 77 K. Subsequent substitution of nitrates YCaBa4Cu5(NO3 )0.3(CO)0.7O11 leads to Tc values of up to 82 K. The family of high-Tc superconductors is very large. A more complete list of superconducting materials and their critical temperatures can be found in a review article by Harshman and Millis (1992). Despite high-Tc compounds having many different structures with a variety of chemical substitutions the general property is the presence of the copper oxide layers. A3.0.1.2 Unusual properties. BCS versus nonBCS superconductivity In spite of progress on the materials aspects of the phenomenon of high-temperature superconductivity, there are widely different views as to the pairing mechanism responsible for this effect. In addition to conventional electron-phonon interaction, many other explanations have been proposed, some of them being of quite an exotic nature. The normal-state properties of high-Tc superconductors are quite unusual as well. Strong anisotropies are observed, mainly caused by the nearly two-dimensional nature of the electronic properties of these materials. Besides the normal-superconducting phase transition, the new copper oxides show an unusually complex phase diagram. For example, the YBCO compounds display a wide range of behaviour, including the metallic superconductor-to-magnetic insulator change due to variation of the oxygen content. The behaviour of resistivity ρ as a function of temperature observed in high-Tc superconductors is different from that of conventional metals. Single-crystal measurements of ρ (T) in the CuO2 planes carried out for several different compounds have shown a linear behaviour over the measured temperature range, in some cases from 7 K to 700 K. This behaviour takes place only near the optimal chemical doping, i.e.

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Basic physical properties

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that corresponding to highest critical temperatures. Other anomalous normal-state properties have been observed: optical conductivity and Raman scattering data suggest a peculiar temperature-dependent charge carrier scattering rate and the Hall coefficient of YBCO exhibits a strong temperature dependence, in contrast to that of ordinary Fermi liquid metals. It is widely believed that understanding the normal-state properties of high-Tc superconductors will also shed light on the superconducting mechanism. Another important property of these superconductors is that they are related to the antiferromagnetic insulators. A typical T—x (temperature-carrier concentration) phase diagram of the electronic properties of high-Tc superconductors consists of several regions (Batlogg 1991). Near half-filling (no holes in CuO2 layers) the materials are antiferromagnetic insulators. With the increase of hole doping, the long-range magnetic order fades away and superconductivity begins to prevail at low temperature at appropriate doping levels. The superconducting transition temperature increases with doping until it reaches a maximum and then decreases with further doping and the material eventually becomes a paramagnetic metal. Shortrange antiferromagnetic order still persists into the doping region where the superconducting ground state prevails. The Fermi liquid picture is the underlying foundation of the traditional BCS theory. The unusual normal-state properties raise a question about applicability of the Fermi liquid description to high-Tc superconductors. The basic concept of the Landau theory of the Fermi liquid is that the properties of a system of fermion particles are not dramatically modified by the particle interactions, no matter how strong the interaction may be. This concept assumes a one-to-one mapping between ‘quasi-particles’ of an interacting fermion system and free particles of a non-interacting fermion system. Almost all metals are typically regarded as Fermi liquids. In high-Tc superconductors the conducting carriers are composed of oxygen 2p electrons (holes) and strongly interacting copper 3d electrons (holes), the on-site Coulomb repulsion between the 3d electrons being very strong (strong correlation effects). Given the controversy over the applicability of the Landau Fermi liquid model, much experimental and theoretical work has been carried out to test the validity of the Fermi liquid concept in the high-Tc superconductors. A very important tool to study the electronic structure is high-resolution angle-resolved photoemission spectroscopy (ARPES) (Shen and Dessau 1995). Despite the evidence of correlation effects, angle-resolved photoemission clearly demonstrated the existence of Fermi surfaces. Furthermore, the measured Fermi surfaces have similarities to those calculated theoretically by band theory (Jones and Gunnarson 1989, Andersen et al 1991, Pickett et al 1992, Cohen 1994). The key features in the electronic structure are very flat bands in the CuO2 band structure which have saddle-point behaviour and significant Fermi surface nesting. These features will have a significant impact on the physical properties, including the temperature dependence of the resistivity, the isotope effect, the Hall effect and the symmetry of the superconducting gap. The superconducting properties of cuprates are in many ways similar to those of conventional BCS superconductors. First, electron pairs in the superconducting state of the high-Tc oxides have been firmly established by flux quantization and Josephson tunnelling experiments. The Cooper pairs in high-Tc superconductors are spin singlets. The cuprates are strongly type II superconductors and the magnetic field penetrates them in the form of vortices as in conventional superconductors. The existence of the energy gap has been established by a number of advanced experimental techniques such as ARPES, electron tunnelling and infrared reflectivity. It has been shown that the maximum value of the gap exceeds the BCS value of 3.5kBTc . At the same time, the ARPES measurements on the highest-quality samples showed significant gap anisotropy with the gap minimum close to zero near some points of the Brillouin zone (Shen and Dessau 1995). The nature of the orbital structure of the Cooper pairs in high-Tc superconductors remains one of the central questions in the field. The issue of symmetry of the order parameter is discussed by Scalapino (1995). The so-called s-wave gap requires that the electron-electron interaction is attractive, while an

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High-temperature superconductivity

unconventional d-wave gap can take advantage of a repulsive interaction. Many different types of experiment have been carried out aimed at probing the symmetry of the gap function, including tunnelling, absorption of microwaves, Raman spectroscopy and Josephson junction measurements (see reviews by Dynes 1994, Van Harlingen 1995 and Goss Levi 1996). Most of them are consistent with the d-wave pairing. However, in order to distinguish between a very anisotropic s-wave gap function and a d-wave gap function phase-sensitive measurements are necessary which can only be done using Josephson junction experiments directly probing a phase difference. The outcome of the Josephson measurements in YBCO is that the gap function indeed changes its sign within a Brillouin zone (the so-called π-shift in a Josephson effect) which is consistent with the d-wave pairing (see Scalapino 1995 and references therein). However, the very fact of the sign reversal of the gap function does not necessarily prove a pure d-wave symmetry and may be a manifestation of a more complicated anisotropy related to repulsive interactions at some regions of a Fermi surface (Abrikosov 1995). In summary, the experimental evidence is that the electron—electron interactions in the cuprates within the Brillouin zone are of alternate sign and have the symmetry of the lattice; however, the key question about the origin of these interactions has not yet been answered. This issue is still the subject of hot debate. Among other anomalous superconducting properties of the cuprates are extremely high values of Tc , extremely small coherence length, nonBCS behaviour of the superconducting characteristics below Tc , e.g. the surface resistance does not decrease exponentially but rather more slowly and high residual losses are observed at low temperatures. The isotope effect in the cuprates is much weaker than in conventional superconductors. The H—T phase diagram of high-Tc superconductors is much more complicated than the conventional one. In addition to the high Tc values, the large spatial anisotropy of these materials is striking. First, c -axis anisotropy is due to the layered crystal structure. Almost all models proposed and developed for the explanation of superconductivity above 30 K are based on the low-dimensional character of the cuprates. In Y—Ba— Cu—O, structural studies show Cu—O planes and Cu—O chains in these materials. Magnetic field measurements show large anisotropy of magnetic critical fields and critical currents. In Bi—Sr—Ca—Cu—O and Tl— Ba—Ca—Cu—O compounds the anisotropy is much stronger than that in Y—Ba—Cu—O and the chains are absent. Currently it is believed, on both experimental and theoretical grounds, that superconductivity is mostly confined to the Cu—O planes, whereas the chains can still play an important role in normal transport properties. The most important ingredients of the BCS theory are the superconducting electron pair formation (Cooper pairs) and the interaction responsible for the electron pairing. Whereas the existence of spin-singlet Cooper pairs in high-Tc superconductors is firmly established, the question concerning the interaction is still open at present. The lack of consensus is due to the inability of any simple theory, like the conventional BCS theory, to explain simultaneously the many unusual properties of cuprates. A large variety of physical mechanisms of high-temperature superconductivity have been discussed in the literature, ranging from a purely phonon mechanism to a pairing due to repulsive interactions. As none of these mechanisms have been firmly identified yet, a number of selected references is given below, where more specific information can be found. The phonon mechanism of pairing in application to cuprates with account of complicated band structure has been discussed by many authors. An early review is that given by Pickett (1989). Application of the many-band generalization of the phonon pairing mechanism to cuprates was first done by Kresin and Wolf (1992). An important signature of the phonon mechanism of pairing in conventional superconductors is the existence of the isotope effect with the isotope exponent α = −21 . This exponent was found to be quite small in most of the cuprates, but this fact does not exclude the phonon mechanism since it can be attributed to anharmonic effects. There are several reasons why electron—phonon interaction in the cuprates may be strong enough to cause high Tc values (see Ginzburg and Maksimov 1992 and references therein). First, strong hybridization of d-electron states of copper and p-electron states of oxygen leads to a strong

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The phenomenology of high-Tc superconductors

57

contribution of high-frequency oxygen phonon modes to the electron—phonon interaction. Second, the low dimensionality tends to increase the electronic density of states in the Cu—O planes. Both these effects are favourable for high Tc values as given by the BCS theory. Finally, the layered structure favours an enhancement of the electron—phonon interaction due to weaker screening of Coulomb potentials. It is generally argued that if the phonon mechanism itself is not fully responsible for high Tc values, an additional nonphonon attractive interaction may contribute to the Cooper pairing. Among nonphonon mechanisms, the pairing due to exchange by acoustic plasmons and excitons has been considered. Soft acoustic plasmons are present in the cuprates due to their layered structure, whereas exciton modes may originate from nonmetallic interlayers existing between the Cu—O planes. The exciton pairing mechanism was first proposed by Little (1964) and Ginzburg (1964) (see also the reviews by Carbotte 1990, Ginzburg and Maksimov 1992, and references therein). The model based on the extended saddle-point singularities in the electron spectrum of the cuprates, weak screening of the Coulomb interaction and phonon-mediated interaction between electrons was proposed by Abrikosov (1995). The pairing mechanism based on the bipolaron formation due to strong phonon or spin interactions was discussed by Micnas et al (1990). This model essentially relies on nonFermi liquid behaviour in the normal state and considers superconductivity as a Bose condensation phenomenon. This assumption, however, seems inconsistent with the a bove-cited ARPES data and band-structure calculations which have demonstrated the existence of the Fermi surface in the cuprates. A number of unconventional pairing mechanisms not involving electron—phonon interaction were proposed for the cuprates. Most of them employ the idea that the magnetic correlations which manifest themselves in the aforementioned T—x phase diagrams play the key role in the Cooper pairing. Several phenomenological and microscopic approaches have been developed quite recently in the context of purely electronic Hubbard-type models which focus on the role of antiferromagnetic spin fluctuations. The minimal model describing hole motion in the Cu—O plane is the t —J model (Anderson 1987). The spin fluctuation exchange is then considered as a pairing mechanism. It is beyond the framework of the present introductory chapters to discuss the physical assumptions and the outcomes of the spin-fluctuation model or other quite complicated theories. An interested reader can find the theoretical details and extensive references to original publications in recent reviews by Dagotto (1994), Kampf (1994) and Scalapino (1995). In conclusion, while in conventional low-Tc superconductors remarkable progress has been achieved and the phonon pairing mechanism has been established in agreement with BCS theory, at present there is no general consensus on the proper theory for high-Tc superconductors. Such a theory must account for the results of recent phase-sensitive experiments (Van Harlingen 1995, Goss Levi 1996) which indicate a sign change in the gap function that is considered as strong evidence for d-wave pairing and a nonphonon mechanism. Further theoretical and experimental work is needed to identify the mechanism of superconductivity in high-Tc superconductors which may have great practical impact on the search for new high-Tc materials. A3.0.2 The phenomenology of high-Tc superconductors In the absence of a microscopic theory of high-temperature superconductivity the only successful descriptions are phenomenological ones such as the London theory or the Ginzburg—Landau (GL) theory. The anisotropic GL theory is a generalization of the conventional GL approach to include anisotropic materials. This is done by replacing the electronic mass by an effective-mass tensor. For an orthorhombic or tetragonal crystal, as in the case of high Tc , the situation is described by the principal values of the effective mass along the a, b and c axes, ma , mb , and mc respectively. The mass ratios yield the corresponding anisotropy coefficients.

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The penetration depths λa , b , c and the coherence lengths ξ a , b , c can be written in terms of ma , b , c as λa , b , c = λpma , b , c and ξa,b,c = ξ / pa,b,c. Here the masses are normalized according to mambmc = 1 and the average values λ and ξ are defined as λ = (λa λb λc )1/3, ξ = (ξa ξb ξc )1/3 and the GL parameter is defined as k = λ/ξ . The critical fields can be expressed through the corresponding values of ξa , b , c and λa , b , c. Neglecting possible anisotropy in the ab plane, i.e. for the case of tetragonal symmetry, one can use the symbols parallel and perpendicular for the field directed along the c axis, H || , and along the ab plane, H⊥. Then the lower critical fields are given by

and the uppper critical fields are

Here ξ|| is the coherence length in the ab plane and is of the order of magnitude of 20 Å (2 nm) for most high-Tc superconductors, whereas the coherence length in the c direction, ξ⊥, is much smaller, of the order of a few ångströms. The anisotropy coefficient γ is usually defined as γ = λ||/λ⊥ and is approximately 5 for YBCO. Therefore a simple way to model the situation in YBCO is to use the anisotropic GL theory as described above. BSCCO and TBCCO are even more anisotropic than YBCO. These compounds have coherence length ξ⊥ much smaller than the distance between neighbouring Cu—O planes and an anisotropy coefficient as large as 103. Therefore one can speak about almost two-dimensional superconductivity in BSCCO and TBCCO. This situation is quite opposite to that in the classical low-temperature superconductors which typically exhibit coherence regions extending over several lattice constants. A successful theory that treated a crossover from two-dimensional to three-dimensional behaviour was developed by Lawrence and Doniach (1972) (see also Klemm et al 1975). The Lawrence—Doniach model assumes the GL equations in each layer whereas the current perpendicular to the layers is due to tunnelling. This yields a discrete set of coupled GL equations (one for each layer), the coupling constant being the parameter of the model. An important outcome of the Lawrence—Doniach model is that the perpendicular current is the Josephson current provided the interlayer coupling is weak. For a strong interlayer coupling the results of the anisotropic London theory are recovered. The experimental manifestation of extremely strong anisotropy in Bi2Sr2CaCu2O8 was the discovery of the so-called intrinsic Josephson effect in this compound (Kleiner et al 1992): there exist Josephson weak links from a Cu—O double layer to its neighbouring double layers. The structure of the vortex lines in layered high-Tc superconductors with weak Josephson interaction between layers is very peculiar. As was predicted theoretically by Artemenko and Kruglov (1990), Buzdin and Feinberg (1990) and Clem (1991), the vortex line is a stack of two-dimensional ‘pancake’ vortices in different layers. A weakness of attractive interaction between the pancakes from different layers results in a strong reduction of the shear modulus of the vortex lattice along the layers as well as a strong influence from thermal fluctuations. The existence of pancake vortices has been established experimentally. With the high-Tc superconductors the qualitatively new regime in phenomenology of type II superconductors can be assessed. An extensive review of the present status of the theory is given by

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Blatter et al (1994). The macroscopic magnetic properties of the cuprates are very different from those of conventional type II superconductors. The new key elements are thermal fluctuations which become very important at temperatures close to Tc, strong anisotropy and weak random pinning. The pinning properties are consequences both of the small coherence length ξ|| and of the fact that the pinning centres in high-Tc materials are mainly provided by point defects, e.g. oxygen vacancies. The randomness of the pinning is due to disorder in oxygen vacancy positions that may arise from slight deviations from the complicated stoichiometry and depends strongly on doping. For magnetic field parallel to the Cu—O planes, additional intrinsic pinning exists due to the interaction of the vortex lines with the periodic potential created by the planes. For large-scale applications of these superconductors the H—T phase diagram must be understood. In the presence of pinning and thermal fluctuations, the phase diagram of a high-Tc superconductor is much more complicated than that for a conventional type II superconductor. An important new feature is the existence of new vortex phases: a vortex glass and a vortex liquid. Below a certain temperature, the vortex system will freeze into a vortex glass phase in which resistivity is exponentially small. Above this temperature the vortex system is in a vortex liquid state (the flux-flow state) with a resistance of the order of the normal-state resistance. Another feature, closely related to the vortex glass—vortex liquid transition, is the experimentally observed irreversibility line. This line provides a boundary between reversible and irreversible magnetic behaviour of a superconductor. In conventional superconductors, the vortex liquid phase is confined to a very narrow region near Hc2, with the irreversibility line essentially coinciding with Hc2(T). Enhanced thermal fluctuations, smaller coherence length and large anisotropy of the cuprates lead to an observable vortex liquid region in these materials. The phenomenological H—T phase diagrams for an anisotropic highTc superconductor, as proposed by Blatter et al (1994), are shown schematically in figure A3.0.1 (without pinning) and figure A3.0.2 (with pinning). There are two types of transition, namely vortex lattice—vortex liquid (without pinning) and vortex glass-vortex liquid (with pinning). Each of these transitions has two corresponding melting lines, the high-field and the low-field ones, which are essentially the irreversibility lines.

Figure A3.0.1. Phenomenological phase diagram for an anisotropic high-Tc superconductor including the effects of thermal fluctuations. The solid lines separating the vortex lattice and vortex liquid regions are the melting lines (Blatter et al 1994).

Figure A3.0.2. Phenomenological phase diagram for an anisotropic high-Tc superconductor including the effects of thermal fluctuations and weak disorder (pinning) (Blatter et al 1994).

As seen from figure A3.0.2, the most important region of high temperatures and high magnetic fields from the practical point of view is the vortex liquid region. However, this region is subdivided into regions

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of pinned liquid and of unpinned liquid. In the first region, close to the melting line, the barriers U against vortex motion are still large in comparison with the thermal energy, U » kBT, and the vortex liquid is in the pinned regime. Closer to the upper critical field Hc2(T) the barriers are small, U ≤ kBT, and the vortex liquid cannot be pinned. Thus, a truly superconducting state with essentially zero resistivity exists only in the vortex glass regime, at temperatures below the melting point. At higher temperatures, above the melting point, the system first enters the thermally assisted flux-flow regime (TAFF) with activated behaviour of resistivity ρ ≈ (ρf l /A ) exp(—U/kBT ), with A « 1 and ρf l = ρn(B/Hc 2 ) being the flux-flow resistivity. Then, upon crossover to the unpinned vortex liquid state, the system is in the flux-flow regime with large resistance ρf l which is of the order of magnitude of the normal-state resistance ρn . A3.0.3 Potential applications For large-scale applications large currents in superconducting wires and cables are required in environments where the magnetic field is strong. The potential advantage of high-Tc superconductors is that superconductivity is achieved above 77 K where liquid nitrogen may be used as a coolant. It is important that the high-Tc superconductors are type II materials with extremely high Hc 2 values. Potential applications include wires and superconducting magnets, magnetically levitated trains, etc. However, up to now it has been extremely difficult to fabricate bulk polycrystalline samples that have technologically significant critical current densities at fields above Hc 1 The main reason for critical current degradation in bulk high-Tc samples is the grain boundaries. Because of the very short coherence length, the grain boundaries act like very weak Josephson junctions, strongly attenuating the maximum supercurrent that can be transported across the boundary. The problem can be avoided by growth of highly textured samples and oriented thin films. The currently achieved record critical current densities in Y—Ba—Cu—O at 77 K are of the order of 107 A cm– 2 in thin films and 105 A cm– 2 in bulk polycrystalline samples. The upper critical fields for Y—Ba—Cu—O are of the order of 102–103 T depending on the direction of the magnetic field relative to the ab planes. Another practical problem arises from poor mechanical properties of high-Tc superconductors. These materials are rather brittle metals and are not as ductile as Nb—Ti or even as Nb3Sn which are currently used for wire fabrication. Nevertheless high-Tc superconductors can be synthesized in thin films. Methods of making wires and tapes out of high-Tc superconductors are in the early stages at present. For practical applications the flux-flow regime must be avoided and flux creep minimized in order to have a low-loss superconductor. This is highly relevant to the matter of carrying high currents in substantial magnetic fields at 77 K. As discussed above, the state with essentially zero resistivity exists in highTc superconductors only in the vortex glass regime, at temperatures below the melting point, whereas at higher temperatures the system first enters the TAFF regime. In YBCO the vortex glass temperature of 77 K corresponds to a field of about 4 T which provides a restriction on the practical use of this material. In the more anisotropic BSCCO and TBCCO the flux lattice melting temperature goes down to about 30 K. In order to increase this temperature, considerable efforts are being directed towards flux-pinning enhancement by the creation of strong-pinning centres. Due to the two-dimensional nature of vortices in these materials the pinning defects should represent the columns perpendicular to the layers. Such columnar defects are at present produced, for example, by irradiation of BSCCO and TBCCO samples with high-energy ions. Many aspects of the high-temperature and high-field behaviour for different highTc materials, which are of great practical importance, are at present the subject of intensive experimental and theoretical study. For reports on recent progress and prospects for large-scale applications of high-Tc superconductors see Daley and Sheahen (1992), Mukai (1990) and Lubkin (1996). Specific quantum properties of superconductors generally valid at 77 K can be used for electronic applications. Very promising is the use of high-Tc superconductors in passive microwave devices such as

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transmission lines and high-quality resonators. Among active devices the best known examples include SQUIDs and detectors based on Josephson and quasiparticle tunnelling. Since the discovery of high-Tc superconductors in 1986 a worldwide search for high-quality, reliable and reproducible high-Tc Josephson junctions has been initiated. As a result a large variety of such devices are already available (Kupriyanov and Likharev 1990, Braginski 1991). They include microbridges structured in epitaxial YBCO thin films (Anlage et al 1991, de Nivelle et al 1993). A widely used junction type is the so-called grain-boundary junction based on the suppression of superconducting properties at the boundary between two separate grains in a superconducting film (Koch et al 1987, Dimos et al 1988). Promising from the point of view of reproducibility and integration into complex circuits are highTc superconductor Josephson junctions with artificial barriers deposited over thin films. Noble metal barriers were used to produce SNS junctions (Forrester et al 1991, Dilorio et al 1991). Fully epitaxial junctions with oxide barriers chemically and structurally compatible with superconducting electrodes are most promising from the application point of view (Gao et al 1990, Laibowitz et al 1990). An example of the barrier material in YBCO-based junctions is nonsuperconducting PrBa2Cu3O7-δ . Besides PBCO, many other materials have already been applied worldwide in the search for an optimal match combined with good electrical characteristics for various applications. Hetero-epitaxial multilayered Josephson junctions provide an interesting possibility to tune transport characteristics over several orders of magnitude. High-temperature superconductivity is at present evolving from a research area into a commercial industry. However, the practical use of high-Tc superconductors is more difficult than was expected at the time of their discovery in 1986–87, and to take full advantage of superconductivity at 77 K many fundamental and technological problems remain to be solved. References Abrikosov A A 1995 Phys. Rev. B 51 11 955 Andersen O K, Lichtenstein A I, Rodriquez O et al 1991 Physica C 185–189 147 Anderson P W 1987 Science 235 1196 Anlage S M, Langley B W, Deutscher G, Halbritter J and Beasley M R 1991 Phys. Rev. B 44 9764 Artemenko S N and Kruglov AN 1990 Phys. Lett. A 143 485 Battlog B 1991 Phys. Today 44 44 Bednorz J G and Müller K A 1986 Z. Phys. B 64 189 Blatter G, Feigelman M V, Geshkenbein V B, Larkin A I and Vinokur V M 1994 Rev. Mod. Phys. 66 1125 Braginski A I 1991 Physica C 185–189 391 Buzdin A and Feinberg D 1990 J. Physique 51 1971 Carbotte J C 1990 Rev. Mod. Phys. 62 1027 Cava R J et al 1988 Nature 336 211 Clem J R 1991 Phys. Rev. B 43 7837 Cohen E P 1994 Comput. Phys. 8 34 Dagotto E 1994 Rev. Mod. Phys. 66 763 Daley J G and Sheahen T P 1992 Proc. American Power Conf. (Chicago) de Nivelle M J M E, Gerritsma G J and Rogalla H 1993 Phys. Rev. Lett. 70 1525 Deutscher G and Müller K A 1987 Phys. Rev. Lett. 59 1745 Dilorio M S, Yoshizumi S, Yang K Y, Zhang J and Maung M 1991 Appl Phys. Lett. 58 2552 Dimos D, Chaudhari P, Mannhart J and Legoues F K 1988 Phys. Rev. Lett. 61 219 Dynes R C 1994 Solid State Commun. 92 53 Forrester M G, Talvacchio J, Giaevaler J R, Rooks M and Lindquist J 1991 IEEE Trans. Magn. MAG-27 1098 Gao J, Aarnink W A M, Gerritsma G J and Rogalla H 1990 Physica C 171 126 Ginzburg V L 1964 Phys. Lett. 13 101 Ginzburg V L and Maksimov E G 1992 Superconductivity 5 1543 Goss Levi B 1996 Phys. Today 49 19

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Harshman D R and Millis A P 1992 Phys. Rev. B 45 10684 Jones R O and Gunnarson O 1989 Rev. Mod. Phys. 61 689 Kampf A P 1994 Phys. Rep. 249 219 Kleiner R, Steinmeyer F, Kunkel G and Müller P 1992 Phys. Rev. Lett. 68 2394 Klemm R A, Luther A and Beasley M R 1975 Phys. Rev. B 12 877 Koch R H, Umbach C P, Clark G J, Chaudhari P and Laibowitz R B 1987 Appl. Phys. Lett. 51 200 Kresin V Z and Wolf S A 1992 Phys. Rev. B 46 6458 Laibowitz R B, Koch R H, Gupta A, Koren G, Gallagher W J, Foglietti V, Oh B and Viggiano J M 1990 Appl. Phys. Lett. 56 686 Lawrence W E and Doniach S 1972 Proc 12th Int. Conf. on Low-Temperature Physics (Kyoto) p 361 Little W A 1964 Phys. Rev. 134 A1516 Lubkin G B 1996 Phys. Today 49 48 Micnas R, Ranninger J and Robaszkiewicz S 1990 Rev. Mod. Phys. 62 113 Mukai H 1990 Proc. Third Int. Symp. Superconductivity (Sendai) Pickett W E 1989 Rev. Mod. Phys. 61 433 Pickett W E, Krakauer H, Cohen R E and Singh D J 1992 Science 255 46 Putilin S N, Antipov E V, Chmaissem O and Marezio M 1993 Nature 362 226 Raveau B, Huve M, Maignan A, Hervieu M, Michel C, Domenges B and Martin C 1993 Physica C 209 163 Scalapino D J 1995 Phys. Rep. 250 331 Schilling A, Cantoni M, Guo J D and Ott H R 1993 Nature 363 56 Shen × Z and Dessau D S 1995 Phys. Rep. 253 1 Van Harlingen D J 1995 Rev. Mod. Phys. 67 515 Wu M K, Ashbum J R, Torng C J, Hor P H, Meng R L, Gao L, Huang Z J, Wang Y Q and Chu C W 1987 Phys. Rev. Lett. 58 908 Zhao Z X et al 1987 Kexue Tongbao 32 522 Further reading Bednorz J and Müller K A 1988 Nobel lectures in physics Rev. Mod. Phys. 60 585 Bednorz J G and Müller K A 1990 Earlier and Recent Aspects of Superconductivity (Berlin: Springer) Burns G 1992 High Temperature Superconductivity: an Introduction (New York: Academic) Ginzberg D M (ed) 1990 Physical Properties of High-Temperature Superconductors (Singapore: World Scientific) Ginzburg V L and Kirzhnitz D (ed) 1982 Theory of High-Temperature Superconductivity (New York: Consultance Bureau) Kresin V Z and Wolf S A 1990 Fundamentals of Superconductivity (New York: Plenum) Kresin V Z and Wolf S A (ed) 1990 Novel Superconductivity (New York: Plenum) Lynn J W (ed) 1990 High Temperature Superconductivity (New York: Wiley) Phillips J C 1989 Physics of High-Tc Superconductors (New York: Academic) Sheahen T P 1994 Introduction to High-Temperature Superconductivity (New York: Plenum) Vonsovsky S V, Izymov Y A and Kurmaev E Z 1982 Superconductivity of Transition Metals (Berlin: Springer)

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B1 Field distributions in superconductors

A M Campbell

B1.0.1 Introduction The behaviour of magnetic fields in superconductors differs in some ways from that in normal metals. It is, however, simpler and more intuitively obvious than in iron magnetic circuits. There is a close parallel with the induction of eddy currents in copper in that the distribution of eddy currents is qualitatively similar to that of the supercurrents. The difference is that copper exerts a viscous force on magnetic flux, while a superconductor exerts a frictional force more like the pinning force on domain walls in ferromagnets, with which there is also a parallel. Perhaps the most important concept is the Bean model. This takes the critical current density as a measurable property and determines a wide range of properties from it. Most problems can be described either in terms of magnetic flux or induced currents, and it is helpful to use both concepts. The simplest geometry is a slab in a parallel applied field. If this is made of copper the application of a field induces currents which flow up one side and down the other to exclude the field. As these die away the field penetrates to the centre. If the resistivity is zero the time taken for currents to die away will become infinite and the field is permanently excluded. In this case the superconductor behaves like a perfect diamagnet with screening currents flowing in a surface layer of about a micrometre and in some circumstances the analogy with a perfect diamagnet can be useful in calculating external fields. However for most purposes it is better to treat the material as one with macroscopic currents flowing in a nonmagnetic medium, so that there is no distinction between B and µ0H. This field exclusion requires energy B02/2µ 0 and the energy difference between normal and superconducting states is very small. Above a certain low field, Bc 1, the field begins to enter the superconductor in the form of flux vortices. These can be thought of as small cylinders of normal material surrounded by supercurrents, and they can be imaged by Bitter patterns, scanning tunnelling microscopy and magnetic force microscopy. Each contains one quantum of flux and, although this sounds obscure, they are effectively Faraday’s lines of force. Anyone used to picturing magnetic fields in terms of the movement of flux will already be familiar with how superconducting flux lines move. The complete field exclusion implies a surface current and as soon as the vortices can enter they try to move to the centre under the action of the Lorentz force, or equivalently magnetic pressure. This is resisted by any inhomogeneities which act as pinning centres resisting the motion of flux lines with a frictional force. These allow a maximum current density to be reached before the flux moves and this critical current density is perhaps the most important parameter of the superconducting state.

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B1.0.2 Properties of flux vortices Flux vortices enter or leave from the surface of the sample, they repel each other and their density gives the flux density. In a homogeneous material the flux lines move freely so that in equilibrium the flux density is uniform and the flux vortices form a hexagonal array. Since each unit cell must contain one flux quantum, h/2e, the spacing, a, is related to B by Ba2 p3/2 = h/2e. In the earth’s field they are about six micrometres apart and at 1 T the spacing is 45 nm. Whenever they move an electric field is generated (E = B × υ ) where υ is the flux line velocity. Any inhomogeneities, such as dislocations or precipitates, exert a pinning force on the flux lines. This means that as they enter a density gradient is built up so that the magnetic pressure is balanced by the pinning force. The current distribution and fields in a slab are shown in figure B 1.0.1.

Figure B1.0.1. A field applied to a slab generates supercurrents with a density Jc until the field is screened from the interior. The electric field only exists while the field is increasing.

This is the flux picture. An equivalent picture can be given in terms of induced currents. A gradient in the vortex density implies a current, dB/dx = ±µ0 J, and the Lorentz force B × J is balanced by the pinning force. Although less physically obvious than the flux picture, the expression in terms of currents is the more general since it can be written in vector form for geometries other than slabs. While the local Lorentz force is less than the pinning force the flux lines are stationary and E = 0. Once this critical value of J is exceeded the flux lines begin to move and we have an electric field. The resulting resistivity is highly nonlinear and can be comparable to that of the normal state, although it should be emphasized that the material does not go normal when the critical current is exceeded. It remains superconducting but enters the resistive flux flow state. The zero-resistance state can be thought of as a region in a graph defined by axes representing critical currents, fields and temperatures. The further away from any limiting value the higher the critical current. B1.0.3 The Bean model This is the model used to calculate field distributions in superconductors (Bean 1962, Kamper 1962, London 1963). To start with we assume that the critical current density is independent of magnetic field. This is a poor approximation at low fields, but a very good one at high fields where the external field is much larger than the self-field so that the field across the superconductor does not vary much. In terms of currents the Bean model states that any change in the external field induces surface currents which tend to screen the field change from the interior. The maximum current density at any point is Jc so these currents spread into the material from the surface until either the field change has been screened out, or the currents meet those induced from the opposite surface. An equivalent statement in terms of flux says that flux moves in from the surface but is held up by pinning centres until a critical flux density gradient (more strictly curl B ) determined by the pinning centres is reached. This is termed the ‘critical state’.

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In its most general form the field is the solution of

Figure B1.0.2 shows the flux distribution in a slab for the following sequence: (a) a field B0 is applied and the current penetrates to a distance d such that µ0 Jc d = B0 (b) the field is raised to a high value (c) it is lowered slightly (d) the field is lowered to zero.

Figure B1.0.2. The field distribution for a series of applied fields.

After this process flux is trapped in the superconductor, corresponding to the residual field in a ferromagnet. Although there may be no net current into a magnet wire, equal and opposite currents can flow up and down each wire, generating a residual field. For a given volume of superconductor the trapped flux is proportional to the thickness, so fine filaments minimize the residual field in a magnet. For this geometry the magnetization is equal to the difference between internal and external fields, but see section B1.0.4 for a more detailed discussion of magnetization. If instead of applying a field we apply a transport current the Bean model is equally applicable. The current generates external fields which penetrate from the surface in the same way as if the field had been supplied by an external solenoid. Figure B1.0.3 shows the field distribution and currents for slab geometry for an applied field and an equivalent applied transport current. (In fact this would only apply well away from the ends of a slab carrying a current but the principle applies to other shapes.) The addition of a large steady external field makes no difference to these arguments which can be applied to any changes in the field. The current distributions will be the same, although the value of Jc will be reduced and the field lines look quite different. Combinations of field and current are a straightforward extension of the diagrams above. It is only necessary to find the surface field for the relevant combination and allow it to penetrate the appropriate distance (but see section B1.0.5.5 for flux distributions in current-carrying strips).

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Figure B1.0.3. The Bean model applied to screening currents and transport currents in a slab.

Figure B1.0.4. The field distribution including surface currents.

B1.0.3.1 Surface currents There are two types of surface current. The first is a ‘reversible’ or Meissner current which arises from the fact that if there is no pinning the equilibrium flux density in the material is slightly below the external value, the difference being that given by the Abrikosov theory. Hence the boundary condition on B is that it is always this amount lower than the external field whether the field is increasing or decreasing. The second type of surface current is caused by pinning of flux lines to the surface and is an irreversible effect, since the direction of the current depends on whether the field is increasing or decreasing. It usually produces a magnetisation which is comparable with, but smaller than, the reversible magnetization. The field distribution for the two types of surface current is shown schematically in figures B1.0.4(a) (surface pinning) and B1.0.4(b) (Meissner). The difference is made clear by reducing the field from a high value. The irreversible pinning currents reverse while the equilibrium Meissner currents remain in the same direction. The effects are small at fields well above the lower critical field and can usually be ignored except in small samples. B1.0.3.2 Field-dependent Jc At low fields the dependence of Jc on B becomes important. In one dimension this is easy to deal with, although the algebra can become a bit messy and we need an empirical expression for the variation of Jc

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Magnetization

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with B. Common expressions are Jc ∝ B – 1 and Jc ∝ (B0 + B )–1 (where in this case B0 is a material parameter rather than the external field). The critical state equation is now dB/dx = µ0Jc(B) so that if Jc = 2α/B and the external field is B0 the solution for an increasing field is B2 = B02 – α x . Figure B 1.0.5 illustrates the field distributions if a field is first increased and then decreased. However in what follows it will still be assumed that Jc is a constant independent of B .

Figure B1.0.5. Field variation with position if Jc is proportional to B

–1

.

B1.0.4 Magnetization Provided no current crosses the sample boundaries it is possible to define the magnetic moment of the sample. The most general expression for the magnetic moment of a body is 2-1 ∫ r x j dV where j is the local current density and r the radius vector. For most purposes it is easier to draw the current as a series of loops and integrate the moment of each, which is the current times the area. To make comparisons with conventional magnets easier it is usual to divide the moment by the volume and call the result the magnetization M. However, it should be realized that this is not a local magnetization as can be defined in a ferromagnetic material, since the currents which give rise to it flow on a macroscopic scale as opposed to atomic dipole currents. We have B = µ0H inside the material just as we would for eddy currents in copper. Outside the sample this type of magnetization is not distinguishable from that of a conventional magnet. For example, if we have a round cylinder fully penetrated, the moment per unit length is ∫ Jcπr2dr = -13 Jcπa3. Then M = -13 Jca. It can be seen that the magnetization is proportional to the size of the sample in contrast to that of a magnetic material. We can show that it is also equal to the difference between the external field and the average internal field for shapes with zero demagnetizing factor, i.e. rods with a uniform cross-section in a parallel field. We consider a single cross-section and draw the current streamlines. We take an arbitrary radius vector r drawn from the centre to the edge at r = a. Then d B/d r = µ0 J(r) . We define an area S(r) as the area enclosed by the streamline at r (figure B1.0.6). Then the total moment is

The first term is the flux in the sample space due to the external field and the second the flux after the sample is inserted. Thus the magnetic moment is given by the difference in flux caused by the sample currents.

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Figure B1.0.6. A cross-section of a rod with a current induced by a field parallel to the axis.

Field distributions in superconductors

Figure B1.0.7. Flux lines in a film carrying a current.

B1.0.5 Other shapes Samples with finite demagnetizing factors present problems in finding analytic solutions except in high fields. We cover some particular cases. B1.0.5.1 Curved flux lines So far all the field distributions have involved straight flux lines and the driving force is the pressure gradient due to the gradient in B . If the flux lines are curved the line tension must be taken into account, but consideration of the thermodynamics involved shows that the driving force is still B × J where µ 0 J = curl B (a discussion of how we can define H so that this equation can be put in its conventional form is given by Campbell and Evetts (1972)). The critical state penetrates from all surfaces and the general problem is to find a contour inside the sample such that if we fill the space between the contour and the sample surface with a current density Jc the total field inside the contour is zero. This is the basis of a number of numerical methods of solving for the critical state (Navarro and Campbell 1991, Pang et al 1981, Zenkevitch et al 1980). The extreme case of a thin film carrying a current in a field is shown in figure B1.0.7. B is nearly uniform and the force on the flux lines, and hence the film, is due to the line tension rather than the density gradient. B1.0.5.2 Full penetration In high fields any sample will be completely filled with currents of a density Jc . These will flow in opposite directions, meeting at the electric centre which is usually obvious from symmetry. The field needed for full penetration can be found by determining the field at the centre of the sample when it is filled with currents Jc . This is the maximum field it can screen. An important example is a flat disc with the field parallel to the axis. This is the form of many samples of melt-processed YBCO which can be used as permanent magnets.

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If a disc of radius a and thickness d is filled with a current density Jc flowing round the axis the field at the top surface can be found by integrating the field due to a short solenoid. This gives

which reduces to -21µ0Jca for thick discs. B1.0.5.3 Cylinders in a transverse field This is a very common configuration but there is no general analytic solution. At low fields a good approximation is to assume perfect diamagnetism, so that the surface field is 2B0sinθ. The critical state then spreads a small distance into the wire sufficient to screen the local surface field (figure B1.0.8) i.e. a distance 2B0sinθ/µ0 Jc . Zenkevitch et al (1980) have published a numerical solution for higher fields.

Figure B1.0.8. Critical state in a transverse cylinder.

Figure B1.0.9. A loop of wire in an applied field.

These symmetrical distributions only occur if the cylinder is isolated. If we put a superconducting loop in an applied external field a current is induced so that no net flux is enclosed by the loop. Since the field close to the wire surface must approximate to that of an isolated wire, the negative flux at the inside edge is balanced by an equal amount of positive flux through the centre of the loop. If the field is removed these flux lines can meet and annihilate leaving zero field, without any flux having to cross the superconductor (figure B1.0.9). B1.0.5.4 Currents in round wires We can solve the critical state equation analytically in this case (London 1963)

The solution is

Here a is the radius and B0 is the surface field, which is determined by the total current. If Jc is constant the field gradient for small penetrations is µ0Jc and at the critical current it is 2-1 µ0Jc but in general it is nonlinear (figure B1.0.10).

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Figure B1.0.10. The field distribution in a round wire carrying a current. The flux gradient is not uniform for constant Jc. The shielded region is a concentric cylinder.

Figure B1.0.11. In an elliptical wire carrying a transport current the critical state penetrates to a concentric ellipse.

Figure B1.0.12. In a very thin strip the edges are saturated, but current is carried at all points along the width. This is just an extreme case of figure B1.0.11.

Figure Bl.0.13. The field lines around a strip carrying a current.

B1.0.5.5 Strips Many high-Tc superconductors come in the form of strips or flat plates. In these cases the critical state penetrates mainly from the edges. For the case of an elliptical wire carrying a current it was shown by Norris that the critical state penetrates as a series of concentric ellipses (figure B1.0.11). There is no equivalent analytic result for an applied field. However, the case of a long strip of zero thickness can be solved both for transport currents and magnetization (Brandt and Indenbom 1993, Norris 1971, Zeldov et al 1994). Figure B1.0.12 shows how the current density in an ellipse projects into the current density in a very thin strip. If we define a critical current per unit width the edges will carry this critical current while the remainder of the current is carried on the two surfaces at a lower density. A reasonable approximation can be found by splitting the strip into a series of square wires. The field lines are shown in figure B1.0.13 and it can be seen that the field is essentially normal to the strip so that in an anisotropic material it is the value of Jc for fields normal to the planes which is the most important.

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B1.0.6 Rotational fields Rotational fields, such as appear in electrical machines, can be dealt with in a similar way. For large fields the internal flux density is closely equal to the external flux density and rotates with it. Figure B1.0.14 shows the situation in a round wire when the field is along the x axis. The flux lines are pulled past the pinning centres generating axial currents +Jc for positive x and —Jc for negative x. Along x = 0 the electric field is zero and the magnetic moment is the fully penetrated value, perpendicular to the applied field. At low fields the arguments of section B1.0.5.3 can be used. The critical state penetrates to a concentric circle, but contains two opposite current densities which meet in a similar contour to the critical state boundary of figure B1.0.15. Numerical solutions are given by Pang et al (1981). These diagrams apply to long cylinders perpendicular to the field in which the field penetrates from the circumference.

Figure B1.0.14. Fully penetrated rotational flux.

Figure B1.0.15. Flux penetration in a low rotational field.

B1.0.7 Field cooling (the Meissner effect) The effects of cooling a superconductor in a magnetic field are complex. It is true that the discovery by Meissner and Oschenfeldt that a type I superconductor expelled field as it was cooled introduced a new phenomenon to the physics of superconductors and is a very important experiment. However, in type II superconductors the Meissner effect is likely to be small for two reasons. Firstly, if the applied field is significantly larger than Hc 1 at the lowest temperature measured, the equilibrium magnetization is much less than perfect diamagnetism. Secondly, the inevitable presence of pinning centres prevents the flux from moving to the surface, even if thermodynamics suggests it should. In order to observe a Meissner effect three conditions are necessary. The applied field must be low, the pinning must be low, and the sample must be small so that the flux can reach the surface. Since it is also easy to confuse diamagnetism with a reduction in paramagnetism, statements which can be found in the literature that the presence of a Meissner effect is the best test of superconductivity are wrong. Most practical superconductors do not show a significant Meissner effect. The best test is the shape of the magnetization curve as a function of applied field which is completely unambiguous. However, since many people make measurements as a function of temperature the principles are described here. More details are given by Campbell et al (1991). We need to take into account the difference between the applied field and the equilibrium flux density at the surface, an effect which is illustrated in figure B1.0.16 but which is usually small.

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Figure B1.0.16. The flux density above and slightly below T c.

Above Tc the field is uniform. As the temperature is lowered the equilibrium flux density of the superconductor decreases so that flux tries to move out. Since the process is resisted by pinning centres a critical state builds up towards the centre, the penetration depending on Jc . If the external field is above Hc 1 the surface current is small so that little flux escapes. Since the values of Hc 1 and Jc are functions of temperature the magnetization of a sample cooled in a field is difficult to predict and provides little useful information. B1.0.8 Longitudinal currents All the geometries considered so far have the currents perpendicular to the field. Currents with a component parallel to the field present severe problems which are largely unsolved. If a wire is placed parallel to an external field and a current is passed along the wire the flux lines might be expected to form coaxial helices. A longitudinal voltage implies that flux is moving continuously to the centre, but if the flux lines remain intact this implies a continuous increase in axial flux. Since this is impossible the flux lines must be cutting each other and the current distribution is determined by the ease of cutting (Campbell and Evetts 1972). The helical array is also unstable, as in a plasma, and so the wire can break up into domains along its length (Campbell 1980, Cave and Evetts 1978, Irie et al 1975). Some success has been achieved with a model which uses a constant Jc and a constant angle of flux cutting (Perez-Gonzales and Clem 1985), but in general any situation with a component of current parallel to the field may give unexpected results. However, in high-Tc superconductors these force-free effects are relatively small, and in BSCCO almost completely absent. B1.0.9 Granular superconductors This term was initially used to describe small grains of a metal such as aluminium separated by an oxide layer which formed a Josephson junction. The grains were very small so that the material behaved like a homogeneous superconductor with the usual parameters, but the values of these parameters were determined by the strength of the junctions rather than the aluminium parameters. Oxide superconductors are also granular in that the grain boundaries act as weak links, but the grains are much larger than the length scales of the superconductor so that the material must be treated as a composite. We have two critical current densities, an intergrain Jc which is small, and an intragrain Jc which is large. The Bean model must be applied separately to the two systems. When a field is applied it first penetrates between the grain boundaries, with minimal penetration into the grains. At higher fields the grain boundary currents are saturated, or driven normal, and the field penetrates the grains as if they were completely isolated. The granular nature of these materials is shown up very clearly in a.c. flux

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profile experiments in which a sharp change of gradient is seen as the intergrain currents saturate and flux begins to penetrate the grains (Küpfer et al 1989). B1.0.9.1 Length scales The fact that currents are flowing on a smaller scale than the complete sample makes it difficult to deduce current densities from magnetization measurements. This is because the magnetization is Jca where a is the radius of the current loops so that the length scale of the dominant currents must be known. Although the intragrain currents are always larger, they flow on a smaller length scale so may not dominate the magnetization. There are two ways of resolving this problem, neither of which is entirely satisfactory. One is to reduce the size of the sample without damaging it. If the magnetization reduces in proportion to the sample size then the currents are flowing on the scale of the sample. The other, due to Angadi et al (1991), does not require the sample to be cut up, but needs a sample of large demagnetizing factor. If the field is increased to full penetration and then reversed the initial slope is approximately that of the Meissner state so that the demagnetizing factor of the dominant magnetization can be determined. Since this is unity minus the aspect ratio, the size of the current loops can be determined from the thickness. This is an important nondestructive technique which can distinguish between currents flowing on the scale of the sample and those flowing within grains. The accuracy is limited by interactions between grains, and the reversible movement of vortices in potential wells. B1.0.9.2 Superconductors as permanent magnets Conventional superconductors came in the form of long wires which can be wound into coils. Trapping fields in cylinders and lumps was not possible because the low specific heat at 4.2 K made the fields unstable due to flux jumps. However, at 77 K large trapped fields are stable and the highest fields generated by high-Tc superconductors at 77 K have been achieved by using single-grain discs of YBCO. These are magnetized either by applying double the full penetration field, or by cooling in the penetration field. On removing the field we have what is essentially a permanent magnet. Fields of over 1 T have been generated by these discs at 77 K (Chen et al 1993) with much higher fields at lower temperatures. This is a field in which rapid progress is being made and trapped fields of 5 T at 77 K are likely to be achieved in the near future. There are the following differences from permanent magnets. The current is a bulk, relatively uniform, current rather than the very high-density surface current due to atomic dipoles. Since they are transport currents it is possible to make a hole down the centre of the sample and generate a field in air, as in a solenoid. Demagnetizing effects are also different. If the disc is thin reverse fields at the edges will induce reverse currents and so reduce the magnetization. If an external field is applied reverse currents will penetrate from the edge so that there is an effective coercive force comparable to the trapped field. Both the effects can be calculated in more detail from the Bean model. B1.0.9.3 Superconductors in iron circuits The relatively low current densities in bulk high-Tc superconductors at 77 K mean that some applications may use iron circuits to boost the magnetic field trapped in the superconductor. Two possible geometries are illustrated in figure B1.0.17. In figure B1.0.17(a) the superconductor is in the form of a hollow cylinder round the yoke of the iron circuit. For this to produce a field in the gap it is necessary for the superconductor to carry a transport current round the iron, so a granular material is not suitable. However,

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Figure B1.0.17. Bulk superconductor in an iron circuit.

so long as the iron does not saturate, the field in the superconductor is only the leakage field, and so is much less than that in the gap. The critical current density will be close to the zero-field value. In figure B1.0.17(b) the superconductor is used like a permanent magnet so that granular large-grained YBCO can be used. The gap field is

where d is the grain size. Provided the gap is much thinner than the superconductor the field is equal to the field at the centre of a long cylinder of the superconductor and therefore more than double that provided at the top face of an isolated disc. However, the benefits will be limited by the saturation of iron and flux leakage. In both cases the field can be increased by tapering the pole pieces. References Angadi M A, Caplin A D, Laverty J R and Shen Z X 1991 Current carrying length scale in superconductors Physica C 177 479–86 Bean C P 1962 Magnetisation of hard superconductors Phys. Rev. Lett. 8 250–3 Brandt E H and Indenbom M 1993 Type II superconducting strips with current in a perpendicular magnetic field Phys. Rev. B 48 12893–906 Campbell A M 1980 The stability of force free configurations in type II superconductors Helv. Phys. Acta 53 Campbell A M, Blunt F J, Johnson J D and Freeman P A 1991 The quantitative determination of percentage superconductor in a new compound Cryogenics 31 732–7 Campbell A M and Evetts J E 1972 Critical currents in superconductors Adv. Phys. 21 199 Cave J R and Evetts J E 1978 Static electric potential structures on the surface of a type II superconductor in the flux flow state Phil. Mag. B 37 111–8 Chen In—Gann, Liu Jianxiong, Ren Yanru, Weinstein R, Koslowski G and Oberly C E 1993 Quasipermanent magnets of high temperature superconductors Appl. Phys. Lett. 62 3366–8 Irie F, Izaki T and Yamafuji K 1975 Flux flow like state of a PbTl rod in a longitudinal field IEEE Trans. Magn. MAG-11 332–5 Kamper R A 1962 AC loss in superconducting lead bismuth Phys. Lett. 2 290–4 Küpfer H, Apfelstedt I, Flükiger R, Keller C, Meier—Hirmer R, Runtsch B, Turowski A, Wiech U and Wolf T 1989 Intergrain junctions in YBCO ceramics and single crystals Cryogenics 29 268 London H 1963 Alternating current losses in superconductors of the second kind Phys. Lett. 6 162–5 Navarro R and Campbell L J 1991 Magnetic profiles of high Tc superconducting granules, three dimensional critical state model approximation Phys. Rev. B 44 10 146–56

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Norris W T 1971 Calculation of hysteresis losses in hard superconductors: polygonal section conductors J. Phys. D: Appl. Phys. 4 1358–64 Pang C Y, Campbell A M and McLaren P G 1981 Losses in Nb/Ti multifilamentary composites when exposed to transverse alternating and rotating fields IEEE Trans. Magn. MAG-17 134–7 Perez-Gonzalez A and Clem J R 1985 Magnetic response of type II superconductors subjected to large amplitude parallel rotating magnetic fields Phys. Rev. B 31 7048–58 Zeldov E, Clem J R, McElfresh M and Darwin M 1994 Magnetisation and transport currents in thin superconducting films Phys. Rev. B 49 9802–22 Zenkevitch V B, Romanyuk A S and Zheltov V V 1980 Losses in composite superconductors at high levels of magnetic field Cryogenics 20 703–10

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B2 Current distribution in superconductors

S Takács

B2.0.1 Introduction The current distribution in superconductors and superconducting structures (multifilamentary conductors or strands, subcables, cables, braids) is generally connected with the field distribution (see chapter B1). At low frequencies and below the thermodynamic critical magnetic field Hc m in type I superconductors or below the lower critical magnetic field Hc 1 in type II superconductors, only the (‘pure’ superconducting) Meissner shielding currents are induced when the applied magnetic and/or electric field are changed. These currents flow without resistivity and are determined by the geometry of the superconductor and the applied magnetic field only. The electric field in the superconductor is strictly zero. However, there can be some basic differences when a magnetic field and current are applied to the superconductor. These differences are already well known for type I superconductors, where on applying the current a very complicated structure appears, the so-called intermediate state. We illustrate this briefly in the next section. In type II superconductors, the situation is different above Hc 1 where the lattice of quantized flux lines (vortices) forms the so-called mixed state. The description of the inhomogeneous type II superconductors within the framework of the critical state model (CSM) is usually sufficient for nearly all electromagnetic properties, mainly for hysteresis losses (section B4.2). However, the precise calculations can be performed only for very simple—or simplified—cases and structures, and generally only for assuming Jc to be field and position independent. Some examples (round structures in a field or current) are given in chapter B1. Generally, many calculations are possible by computers only. In addition to the induced currents in the superconductors which appear with changing magnetic field, coupling currents are induced between superconductors in composite structures, if these currents are forced to flow through some nonsuperconducting regions. As the current paths for these currents are very complicated, computer calculations are often used too. The ‘extreme’ case is the network model for cables (chapter B5). We would like to show some features of the current distribution in simple structures and then focus our attention on more complicated examples, mainly one-layer cables. We derive the diffusion equations for them leading to some analytic results, which show the general features more clearly than the computer results of the network models. In many cables, these coupling currents cause the most important contribution to the losses with changing magnetic and/or electric fields. In addition, the nonequivalent positions of the current paths in the composites, as well as the inhomogeneities in the cable structures (different critical current densities in the strands, different distances and resistivities between them, etc) cause considerable imbalance in the

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current distributions which can have very long time constants for their decay in time. All these effects lead to increased losses (which can be localized in a small volume of the cable, as we shall see further below) which can influence the stability (chapter B3) of the whole structure and cause the quenching of the superconducting magnet (chapter C3). The very long decay times for the coupling currents can even be critical for magnet designs requiring very precise field distribution, as in some accelerator magnets (chapter B5). If the current distribution is known all important physical and technical quantities of cables and magnets can be derived. The most important of the quantities seem to be the losses caused when the applied field changes and the directly connected quantities (stability and quench). However, for the reasons mentioned above, the inhomogeneous current distribution in some cables and the very long time constants for decaying these currents created new research topics which were also of interest to magnet designers. The understanding of these problems is necessary for designing and producing large reliable magnets without encountering unexpected troubles caused by increased losses, premature quench and field distortions due to long-living supercurrents. B2.0.2 Currents in normal conductors In normal conductors, the current distribution is determined by Maxwell’s equation

and the material equation jn = E/ρn , where ρn is the normal-state resistivity. In a homogeneous material, inserting B = curl A we have in the usual gauge ρn jn = -∂ A /∂ t. These equations determine the penetration of the magnetic flux into a normal conductor, as well as into different loops consisting of superconducting and normal parts in superconducting composites (strands, cables, etc). The corresponding losses are of ohmic nature and the loss density is given by jn • E . In particular, when the current I is applied to a nonmagnetic wire with radius a, the applied field inside (Bi ) and outside (Be ) the wire is given by

At higher frequencies, when the induced currents create a magnetic field comparable with the applied one, the fields are partially shielded from the conductor and they penetrate effectively to a distance δ = (2ρn /µ0ω)1/ 2 only (the skin depth), where ω is the circular frequency of the applied field. B2.0.3 Currents in type I and type II superconductors B2.0.3.1 Type I superconductors In the framework of the London and the two-fluid models, there are different contributions to the current at changing magnetic fields and/or currents: the super, normal and displacement currents. Their absolute values are related by

where Λ = m/ns e 2 (ns is the density of superconducting electrons, e and m their charge and mass, σ the normal-state conductivity and ω the frequency of the field). This means that for frequencies below 1012 Hz or wavelengths above 1 mm, the normal and displacement currents can be neglected.

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Then, the magnetic field penetrates into type I superconductors below the thermodynamic critical magnetic field Hcm only to the penetration depth λ = (Λ/µ0 )1/2 from the surface of the specimen. For a superconducting half-space, the field induction B and current density Js inside the superconductor are given by the London equations

The connection of the supercurrent with vector potential A is different from that for the normal current, namely Js = −A/Λ. The solutions for a half-space are

where B0 is the field induction at the surface. Therefore, the superconducting shielding currents are flowing effectively in the distance λ, too. As a result of the demagnetizing factor n, the critical field for destroying the ‘pure’ superconducting state is smaller and given by (1 – n) Bc m . Between (1 – n) Bc m and Bc m , the sample is in the so-called intermediate state, consisting of irregularly changing normal and superconducting regions. If the current is applied to a superconducting cylinder of radius a, the sample remains superconducting with currents flowing in the distance λ at the surface, until the resulting magnetic field at the surface of the cylinder is below the field Bc m : Ic = 2π aBc m /µ0 (Silsbee’s rule). For higher currents, the cylinder is not transformed into the normal state (the current is not able to create a field exceeding Bc m everywhere) and it is not possible to carry current with a smaller radius (the field outside this region is decreasing, i.e. it would be below Bc m ). Again, the resulting state is in the intermediate state with very complicated structure which is still not fully understood. The mixed superconducting and normal regions are inside a cylinder of radius a’ = az [1 –(1 – 1/z 2 )1/ 2 ], where z = I / Ic , whereas a normal sheath is on the surface between a’ and a. The total resistance of the cylinder has a jump at reaching the critical current (figure B2.0.1). This jump depends on the model used: the most simple two-fluid model (London 1950, Shoenberg 1952) gives R/Rn = [1 + (1–1/z 2 )1/2 ]/2. The experiments clearly show this jump up to

Figure B2.0.1. The scaled resistance R /Rn of a type I superconductor cylinder with scaled applied current I/ Ic.

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about 0.8/Rn (Rn is the normal state resistance), as well as its current dependence up to very high current values (Baird and Mukherjee 1967). The real phase boundaries, made visible by decoration experiments with fine ferromagnetic particles, are very complicated, as are the electromagnetic properties of the wires above Ic . With these experiments, one should take extra care not to allow any temperature increase of the specimen. B2.0.3.2 Type II superconductors (a) Critical currents The magnetic flux penetrates into type II superconductors in the form of quantized flux lines (FLs). The microscopic currents, connected with them, flow essentially at a length λ around their core; however, the maximum—of the order of the depairing current density—is reached at a distance comparable with the coherence length ξ . The macroscopic currents are the result of the spatial gradients in the density of FLs or due to their curvature (see chapter B1). These spatial gradients are, of course, possible only due to surfaces and/or some inhomogeneities in the volume of the superconductors, called pinning centres. They can compensate the Lorentz force FL = [Jc × B ] acting on the flux line lattice. The resulting volume pinning force Fp on the flux-line lattice then determines the ‘ideal’ critical current density Jc given by the condition Fp = FL . This ‘ideal’ value of Jc cannot be determined exactly, as, due to some excitations of the FLs or flux bundles (Anderson and Kim 1964), the flux-line lattice (or part of it) moves far below the Jc leading to dissipation and voltage (so-called flux creep). There are different models for including this effect, the best known of them in the exponential form (Anderson and Kim 1964) and the power-like form for the current—voltage characteristics

The determination of Jc is therefore somewhat arbitrary, depending on the choice of the corresponding electric field E0 at this current density. Generally, the most widely used criterion of E0 = 1 µV cm–1 is well suited for practical superconducting samples, although some resistance criteria can be more useful for magnet applications. The voltage criterion can also be misleading for some high-temperature superconductors. For classical (low-temperature) superconductors, the values of n and α are large: the CSM is therefore adequate, stating that the currents in regions of changing B (or finite electric field E) are given by ± J c , being zero otherwise. The general validity of the appropriate but simple CSM is mainly, but not only, due to the fact that the induced currents are usually not considerably larger than Jc. However, even very simple examples, like a superconducting cylinder in a transverse applied field, allow only a few numerical solutions (Zenkevitch et al 1980). There are only a few papers discussing the a.c. losses and stability in superconducting composites including more or less realistic E—J characteristics (see e.g. Maccioni and Turck 1991). Another effect influencing the E—J characteristics of type II superconductors is the flux flow due to the viscous motion of the flux lines. However, this can also be neglected in all practically used classical superconductors (Nb—Ti, Nb3Sn, V3Ga). Then, the current distribution can be derived from the field distribution directly, and vice versa. Both approaches can be seen as complementary. Some field and current distributions are given in chapter B1. We add some other types for field-dependent volume Jc in addition to the surface currents (Clem 1979) in figure B2.0.2. In figure B2.0.3, we illustrate the ‘common’ action of applied fields and applied currents. For simplicity, we use the slab geometry with Jc = constant and now neglect the possible existence of surface currents. The mean values of the magnetic induction in the sample and the magnetization are given in figure B2.0.4. Comparing the hysteresis for an a.c. field of amplitude close to Bp with and without d.c. current, we see that the hysteresis losses are strongly increased when current is applied for field amplitudes below the penetration field Bp (the multiplication factor is about three), which is very different from the case when the field amplitude is

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Figure B2.0.2. The field and current distribution in a superconducting slab with surface current and Kim-like volume critical current density Jc ∼ 1/(B + B0 ): (a) after field increase; (b) after field reversal with reversible surface current; (c) after field reversal with irreversible surface current.

much higher than Bp (figure B2.0.4). For Bm >> Bp , the hysteresis losses are about 1.8 times smaller with current applied than without the current. (b) Currents in superconducting composites To decrease the hysteresis losses, the superconductors should have small cross-sections (Bean 1964, Takács and Campbell 1987, see also chapter B4.2). Another reason for using thin superconducting filaments is to prevent large flux jumps which can cause a premature transition into the normal state (section B3.2). In order to guarantee the stability of the composite, the matrix between the corresponding filaments should be a normal metal. For the untwisted filaments, the induced voltage between the filaments is proportional to the included area between them, hence it is proportional to the conductor length l . The total losses are then proportional to l 2/ρ and the loss density ~ l/ρ, where ρ in the following sections is the resistivity of the matrix between the filaments or strands. To avoid reducing the stability of the conductor, the only way to reduce these coupling currents and the corresponding coupling losses is to twist the filaments. The same is true for cables. In all superconducting composites, the flux changes in each possible current loops induce an electric field according to Maxwell’s equation

Due to the twisting and/or transposing of the filaments, strands and subcables, there are many closed areas in the plane perpendicular to the applied field, where flux changes occur. As the resistance between any superconducting filaments and strands is nonzero, this causes a current j = E/ρe f f , where ρe f f is the effective resistivity for the currents in the loop. The currents between the superconductors have therefore a resistive nature, leading to coupling losses between different structures. These currents between the filaments in strands and the corresponding losses are extensively treated in section B4.3. We would like to point out that the losses created in the strands between the filaments are sometimes called the eddy current losses and the coupling losses are those losses between the strands and subcables only. On the other hand, the eddy current losses often include the losses created in the nonsuperconducting parts of the cables (including the casing, etc), as well as in the cryostat and the supporting structure. For most structures, it is very difficult to assure a complete transposition of all filaments, strands, etc. Then, the loops have different inductances and, at field changes, different induced currents. Generally, the currents are shared among the strands so that the actual net flux linked by each strand is equal (Turck

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Figure B2.0.3. The field in a type II superconducting slab with changing applied field with (full lines) and without (broken lines) applied current. The different positions of the field change are marked at the top of the figure. The applied current is I = 2Ic /3. (a) The field amplitude is Bm = 2Bp /3, where Bp is the penetration field of the slab; the current induces the maximum field BJ = ±Bm on both sides of the slab. (b) The same field penetrates the slab without applied current equally from both sides of the slab. (c) The slab without applied current at applied field exceeding 2Bp . (d) The same as in case (c), but with applied current 2Ic /3.

1974). The same is true when the transport current is applied to such conductors. The uneven current distribution in the latter case can be extended to regions many times larger than the twist or cable pitch (Faivre and Turck 1981). This nonuniformity of the current distribution in the otherwise identical strands

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Figure B2.0.4. The mean value of the penetrated field B and the magnetization M corresponding to figure B2.0.3 (the notation is also the same as in figure B2.0.3).

can easily lead to the increase of the current in some strands above their critical current, being thus the source of instability and quench. Such a situation often arises at the ends of magnets close to the normal (or more recently high-Tc ) connections. Thus, the additional losses can lead to some instabilities and even quench of the conductors and coils. Another possible consequence, together with the inhomogeneities in the field and current distribution, as well as in the inhomogeneities of the structure (in dimensions of the wires, strands, in contact resistances, etc) is the possible spatial and temporal distortion of the resulting magnetic field. Although this effect is usually negligible for most magnet designs, it can be serious for applications requiring very precise magnetic field distribution, as in some magnets for accelerators (see chapter B5 for the LHC project). The strongest effects connected with unequal and inhomogeneous current distribution are expected at a.c. conditions and at fast ramping of the magnetic field and/or applied current. However, they can be important in steady-state conditions, too. To this case belongs also the energizing of the magnetic system (as after the increase of the applied current considerable induced currents can flow), as well as the maintenance of the magnetic field at a constant level (as in magnets for magnetic resonance spectroscopy). A useful tool for reducing these effects seems to be the ‘overcharging’ and subsequent decreasing of the current above the desired level (Cesnak and Kokavec 1977, Kwasnitza and Widmer 1991). The partial cycling procedure can be repeated many times thus obtaining the required field asymptotically. B2.0.3.3 High-Tc superconductors The high-Tc superconductors are essentially anisotropic type II superconductors. The current distribution in the superconductor itself is analogous to the case for classical low-temperature superconductors, but can differ considerably in very anisotropic superconductors, such as BiSrCaCuO tapes and wires. As an example, we give the force free effects. As mentioned already in chapter B1, these effects seem to be

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negligible, therefore for determining the critical current density, only the field component perpendicular to the transport current should be important. Due to the increased temperatures and smaller pinning forces, fluctuations play an important role in high-Tc superconductors, mainly through flux creep. At lower Jc , even the flux-flow effects can be considerable. There are some attempts to explain the electromagnetic properties by a linearized diffusion equation with the diffusivity DF F = ρn B/µ0Bc2 and DT A F F = (ρn B/µ0Bc 2 ) exp(−U / kT) in the flux-flow and thermally assisted flux-flow regime, respectively (Brandt 1992, Campbell 1991, Coffey and Clem 1991). Here, Bc 2 is the upper critical magnetic field and ρn the normal-state resistivity. However, one can show that the validity of the linear approach should be very restricted. If we take only the flux flow into account, neglecting flux creep and pinning, the corresponding diffusion equation (Takács and Gömöry 1993) has a field-dependent diffusion coefficient (for simplicity, again in the one-dimensional case)

This nonlinear diffusion equation leads to results very different from the linear case, e.g. to a nonlogarithmic field and current relaxation. The coupling currents in superconducting composites from high-Tc superconductors should be analogous to those in classical superconductors. There will be, of course, some basic differences caused by the use of higher temperatures and the different technologies used for preparing the composites. At higher temperatures, the problem of stability and quench is much less severe. On the other hand, the use of a matrix between the high-Tc superconductors is very restricted for technological reasons (easy diffusion of different atoms from the matrix, which can have a detrimental effect on superconductor properties of high-Tc superconductors). At present, it seems that the good conductor Ag, used until now as the sheath as well as the matrix in BSCCO samples and being very favourable for the stability of the composite, should be replaced by some Ag alloy, to decrease the conductivity and thus the coupling losses. This could be one of the main tasks for a.c. applications of high-Tc superconductor composites. B2.0.4 Current distribution in multifilamentary strands and cables B2.0.4.1 Relation between currents in strands and cables For calculating the current distribution in superconducting composites, the starting point is often the general scheme of Morgan (1973), as given below, and the approach of Ries (1977). The most compact theory of electromagnetic properties of twisted superconductors is, however, the anisotropic continuum model (Carr 1983). The currents in the strands (whether of round or rectangular type—see section B4.3) are not very different from the currents in round cables. In the latter case, the currents are better defined due to the simpler (usually a one-layer) structure, without the complications of parameters such as the filling factor and the possible resistive barriers (section B4.3). If we take a normal layer core with strands on its surface, the induced currents in the strands due to a transverse applied field are closed by the resistive matrix, in just the same way as between the filaments in the strands. The current density in the matrix is constant in the simple round geometry (figure B2.0.5) and the longitudinal current in the strands (in the direction of the cable axis) has a cosine-like distribution. Usually, these currents are assumed to flow on some effective surface given by the dimensions of the cable (see, e.g. Kwasnitza and Clerc 1993). Such an approach can be very approximate, mainly for cables with a narrow space between the strands, like the cable-in-conduit conductors (CICCs). In addition, the field only penetrates into a small effective volume of such cables, at least at low values of ∂B/∂t. Therefore, the demagnetization effects should be much more pronounced in such cable types (Bruzzone and Kwasnitza 1987, Takács and Yamamoto 1995, see also section B7.5). In addition, one generally assumes that these currents flow in the direction of the cable axis, instead of in

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Figure B2.0.5. The current distribution J along a round one-layer cable and the corresponding transverse currents jx = constant through the normal-conducting matrix. The currents induced in the strands have a cosine-like distribution, creating a homogeneous field opposed to the applied field. The contribution of the currents jx to the field is usually neglected.

the real direction of the strands. Although there are some exceptions, calculations are generally possible in the low-frequency limit only (Ciazynski et al 1993). At higher frequencies, the models are extremely complicated and the results can only be approximate. B2.0.4.2 Currents in cables (a) Currents between adjacent strands In cables, currents are also induced between adjacent strands. However, these are only important for cable structures with an insulating central layer, as they prevent the current flow between opposite strands. Both types of induced current—between adjacent and between opposite strands—can be seen in figure B2.0.6. As usual, we neglect the difference between the coordinate y running with the strands and the coordinate y’ = y[1 - (δ / l0 )2 ] 1/2, running with the cable axis. Here δ equals the cable width b or the half-perimeter πR for the flat and round cable respectively. This assumption is well justified for the cabling pitch l0 >> (b, R). In addition, the distance d between the strands should be small with respect to l0. This is, of course, very well satisfied for cables, mainly those with tightly wound strands, e.g. also for CICCs. We enumerate the strands by n = 1, 2,…, N and introduce αn = 2πn/N. For not very high applied field changes B• e = ∂Βe / ∂t, we can assume that the strands are not penetrated and the currents J in the strands are pure surface currents. The induced currents between the strands are then very simple (figure B2.0.7) and we have jx = ∂ J/∂y. As the induced voltage is Ux = ρ jxd , we obtain for the currents the diffusion equation

where d is the thickness and S the area for the loops, and the diffusion coefficient D is given by the inductance L of the loop, consisting of two strands:

This result can be obtained from the general integro-differential equation for the currents in any loop with close strands (Takács 1984). The first term on the right-hand side of equation (B2.0.1) is the result of the flux change due to the induced currents, whereas the second one is due to the external field. The current loops are electromagnetically coupled, therefore we have a set of equations of the type (B2.0.1) with added terms µ0Mn k /2πρ, where Mn k is the mutual inductance of loops n and k, n, k = 1, 2, …, N (figure B2.0.8). However, these equations can be decoupled so that one obtains a set of N identical diffusion equations with effective inductance Le f f , depending on the geometrical parameters of the cable

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Figure B2.0.6. The currents induced between neighbouring strands on a cable.

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Figure B2.0.7. The induced voltage ∆Up between neighbouring and ∆Ut between opposite strands on a flat cable of width b. The distance d between the superconducting currents can be very small for tightly wound strands, as in CICCs. The cabling pitch is l0.

only. For harmonically changing homogeneous applied fields on round cables, Be = be exp(iωt), the second term in (B2.0.1) is given by B• ecos(2πy/l0–αn )/ρ = iωbe exp(iωt ) cos(u − αn )/ρ, as the field direction with respect to the surface is changing sinusoidally. For flat cables, the sinusoidal function is to be replaced by the spatial derivation of the step function (Takács 1996) and we have

Namely, the field is changing its direction (with respect to the surface between the strands) as the strands are transferred from the upper to the lower side of the cable and vice versa. Equation (B2.0.1) can be solved for different conditions (finite samples, finite region of applied fields, periodically changing a.c. field, etc). Then, all the important electromagnetic properties of the cable can be calculated, e.g. the

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Figure B2.0.8. As a result of the electromagnetic coupling, the induced currents between all strands contribute to the flux change in the area between two strands (dark sections). Only some strands are illustrated (broken lines). For flat cables, the induced currents on the opposite side are also important (e.g. numbers 11 and –11, etc), as their distances are comparable with those on the same side of the cable (numbers 1 and –1, etc). By considering all loops, one obtains an effective inductance for the currents on the whole cable.

coupling losses, as

Here, we illustrate the results for the simple case of an a.c. field sinusoidally changing along the cable axis (Takács 1982). For harmonically changing external field

the diffusion equation is given by

with υ = iwbe/ρ, w = 2π/l0, p1.2 = 1 ± 1/k, g = Dw (l0/2π)2. The solutions are

Using equation (B2.0.2), the losses P compared with the losses Po = N υ2 / 4w2 in a spatially homogenous field h/ p2 (mean square value of the sinusoidal field) are given by

Hence, the losses are strongly decreased for exactly k = 1: P = P0/4. This is an effect of the infinite sample only. For finite sample lengths, the minimum should disappear (see the explanation

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below). At small difference of k close to unity, the losses strongly increase due to the field inhomogeneity (figure B2.0.9). The explanation for this surprising effect can be easily seen from the equations (B2.0.3). The spatially changing field induces two components of currents with frequencies proportional to the sum (1 + 1/k) and the difference (1–1/k) of the wavelength of the field lm = kl0 compared with the periodicity of the cable l0. For k = 1, only the first component changes along the strands, the second one corresponds to a constant current: J(2) = υ sin αn /2iDω. Therefore, no transverse current is induced due to this component and the coupling losses decrease. This was the first solution indicating the existence of long-living supercurrents in the strands. As mentioned above, these currents in the neighbouring strands are different and they have to close somewhere at the ends for finite samples. One expects then only a maximum of losses approaching lm = l 0.

Figure B2.0.9. The losses P between adjacent strands for a cabe in a sinusoidally changing a.c. field, compared with the losses in a homogeneous field, P0, as a function of the ratio between the wavelength of the field change lm and the cabling pitch l0 (k = lm /l0 ).

On the other hand, if the difference between the two lengths is small, the second component in the transverse current has a very high frequency and it changes its direction many times along one cabling pitch. This leads to a considerable enhancement of the coupling losses, as the experiments show (Marken et al 1991, Sumption et al 1993). The same is true for other (periodic or nonperiodic) changes of the applied field: if any of the Fourier components has a characteristic wavelength approaching l0, the losses can be increased to a large extent. This should be a warning for all magnet designers for the cases where the applied field changes considerably in some places (magnets for fusion devices, generators, superconducting magnetic energy storage (SMES), etc). In the case of remarkable additional losses, one should be very careful and should improve the cooling conditions at these parts, e.g. by introducing additional cooling channels, etc. (b) Currents between opposite strands The induced currents between the strands at the opposite sides of the cable (‘diamonds’) and the related properties have been calculated in many papers (Campbell 1982, Kwasnitza and Clerc 1993, Ries 1977, Sytnikov et al 1989, Takács 1992). The currents are different in the ‘transverse’ applied field (with respect to the wide side of the cable, also called the ‘face on’ configuration) and the ‘parallel’ field (or ‘edge on’) (Campbell 1982, Kwasnitza and Clerc 1993, Sytnikov et al 1989, Sumption et al 1995). Many of the calculations, mainly in inhomogeneous fields, were made by computers (Akhmetov et al 1995, Hartman et al 1987, van de Klundert 1991, Verweij et al 1995). For analytic calculations, the diffusion equations were derived in the framework of Carr’s continuum model (Carr 1973, Carr and Kovachev 1995, Sumption

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and Collings 1994) or in analogy with the currents for the adjacent strands (Takács et al 1994, 1995). We show first of all, that the continuum-like models, based on the voltage between the crossing points of the strands, are identical (Takács 1997) with the basic assumptions of the network models. The principles of the network model are also based on the scheme of Morgan (1973) where Faraday’s equations for all loops between crossing strands are applied with four resistance points for circuits in the inner part and three at the edges of the cable (figure B2.0.10). We take as an example N = 6. The enumeration of N – 1 = 5 rows and the columns k of crossing points allows us to take the flux changes in these loops with respect to the left corner (i, k) of the loops. The induced voltage is always given by Ei,k = ji,k ρi,k. Then, the flux changes Φ• = ∂Φ /∂t in the small diamonds are given by Φ• 1,k = E1,k + E1,k+1 -E2,k, Φ• 2,k= E2,k + E2,k+1-E3,k+1 -E1,k+1, Φ• 3,k+1 E3,k+1 - E3,k+2 - E4,k+1 = + E2,k+1 Φ• 4,k+1 = E4,k+1 + E4,k+2 - E5,k+2 - E3,k+2, Φ• 5,2,+k = E 5,k +2 +E5, 5+3 - E4,k+2. The induced currents at the crossing points of two strands (shaded area) are given by the area between them and the resistivities at the crossing points only, such that Φ•5 = E1,k + E5,k+3 This is just the starting point of any quasi-continuum theory.

Figure B2.0.10. The circuit model is based on the crossing points of all strands. Instead of all crossing points in elementary loops (marked with small circles) within the circuit model, the continuum-like models consider the current flowing in the strands superconductively, until they cross the same strand for the next time (marked with squares for two neighbouring strands). As shown in the text, both models are equivalent: the resulting voltage between the crossing points (large diamond) of two strands is given by the sum of voltages between all included crossing points.

We can now determine the current distribution across a flat cable. As we know, it has a cosine-like form for round structures, with constant resistive current in the field direction (section B4.3). For a flat cable, one has to calculate the area between the loops crossing each other at different points of the cable (figure B2.0.11). The maximum area is for strands crossing at the centre of the cable (largest diamond in figure B2.0.11), given approximately by l0b/4 and changing as F ~ 1 – (2x/b)2, where x = 0 is the cable axis. Hence, jx ~ 1 – (2x/b)2 and the current along the cable is J ~ (2x/b)[1–(2x/b)2/3]. This is to compare with the round cable of radius b/2 where J ~ (2x/b) and jx = constant.

Figure B2.0.11. The areas for the flux change with respect to different crossing points of stands on the cable. As the area F1 between the crossing points (1, 1’ ) is maximum, the induced voltage is maximum at the cable centre, U0. The changing area F2 between points (2, 2’) for x ≠ 0 leads to the voltage ∆Ut = U0[1–(2x/b)2 ].

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(c) Cables with an insulating central layer As the area of flux change is much larger between the strands at the opposite sides of the cable, the corresponding losses are much higher than between the adjacent strands, if the electric resistance between the strands is the same. There are a number of cable structures where this condition is true, e.g. in almost all CICC cables and the cable for the helical coil of the Large Helical Device (LHD) programme (Takács et al 1995). The ratio of both losses for a flat cable is approximately given by (Takács 1992)

where c is the layer between the strands and b the cable width. Sometimes, c is erroneously taken as the ‘thickness of the cable’, but this assumption is wrong (see below). The ratio in (B2.0.5) can be many orders of magnitude. In spite of the technological disadvantages, one was therefore forced to return to cable structures which include some poor conducting or even insulating layer (Capton, Stabrite, etc) to reduce the expected a.c. losses (Sumption et al 1995). However, it is not possible to prevent the transverse currents completely. There will always be some layer, at least close to the cable edges, enabling the current flow between the opposite strands (figure B2.0.12). These currents lead to ‘circular’ losses being generally of the same order of magnitude as the losses between the adjacent strands (Takács 1992) and depending strongly on the contact resistance of strands close to the edges. As these circulating currents are maximum close to the edges, the local decrease of the electric resistivity at these regions (e.g. fabrication of the cable) can increase the circular losses considerably.

Figure B2.0.12. The currents induced between the opposite strands by the voltage ∆Ut cannot be fully prevented: in spite of the central insulating layer some current can flow ‘around the corner’ (paths 1 and 2). We call these currents and the corresponding losses ‘circular’.

(d) Finite samples Finite samples in homogeneous fields As a result of the large cabling pitch of many cables for high currents (some tens of centimetres), it is often difficult to ensure nearly the same conditions for the measured samples as there would be in the magnet winding (e.g. a spatially homogeneous applied field or a magnetic field with given spatial distribution). Due to the end effects, the induced currents in ‘short’ samples (comparable with the cabling pitch) can be very different from those in very long pieces used in magnets. At the end of the sample, the current along the cable axis is always zero. The solutions for currents between adjacent strands in finite samples at low frequencies are simple, due to the simple boundary condition J = 0 at the ends, i.e. y = ±l/2 for a symmetric a case. The solution of (B2.0.1) is then

where γ = β l/2, m = l/lo, β = (i - 1) (Dω/2)1/2 . The loss density compared with an infinite sample is then (Ries and Takács 1981)

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The results are given in figure B2.0.13. The loss density in a cable with length above one cabling pitch is nearly the same as for the infinite samples. This result is very important for the measurements of cables with large cabling pitch. Namely, one can restrict the measurements to about one cabling pitch to obtain the same results as in very long cables. This result was confirmed by measurements (Kwasnitza and Bruzzone 1986) and also approximately by computer calculation (Hartman et al 1987, Verweij and ten Kate 1993, and chapter B5).

Figure B2.0.13. The losses in finite samples between adjacent strands (full curve) and between strands on the opposite side of the cable (broken curve), compared with the corresponding losses in the infinite sample. The results for lengths l < l0 are in the latter case only approximate.

However, there are some differences in the currents between opposite strands. For these currents, both the effective area for the flux change and the effective resistance of the current loops is changed close to the cable ends (Takács et al 1995). The complicated nature of these currents allows the calculations to be performed again with some approximations only. The general result is

where C (ω) decreases from about +0.5 at zero frequency to –1.5 at ω τ ≈ 5 (τ is the time constant of the cable). At approximately ωτ ≈ 1, we obtain C (ω) ≈ 0, therefore there seems to be no size effect in the losses. Because one of the methods of determining the time constant is from the maximum of the losses per cycle (close to ω τ = 1), this is an important factor for experimentalists, too. For measuring the a.c. losses appropriate for cables in the magnets, it is sufficient to have samples of length about twice the cabling pitch. Cables in inhomogeneous fields The behaviour of cables in spatially inhomogeneous a.c. fields seems to be much more important. The losses between adjacent strands were calculated for cables partially in applied fields a long time ago (Ries and Takács 1981, see section B2.0.4.2(a)). The results for the currents and losses between opposite strands are very analogous (Takács 1996) and they are supported by other methods, too (Akhmetov et al 1995, Carr and Kovachev 1995, Verweij et al 1995). The effective inductance per length is Le f f ≈ π for round structures and Le ff ≈ 2π for flat cables (Takács and Yamamoto 1994). The results of the analytic calculations include important consequences for three related properties of superconducting composites: the a.c. losses, the time constants and the current distribution.

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We give here the results for the case when a long cable is exposed to a transverse applied field on a smaller length l’ comparable with the cabling pitch l0. The current distribution, the losses and the time constant for a superposed spatially homogeneous a.c. field can be treated in the usual way (Campbell 1982, Kwasnitza 1977, Wilson 1983). The time constants connected with the additional currents can be very long. For a flat cable of length l and thickness c between the strands we have

compared with the time constant connected with the spatially homogenous field

For the symmetric case, these values are independent of the length of the field region (see chapter B5 for computer results in asymmetric cases). The first time constant, τ1, determines approximately the position of the maximum of the losses per cycle (figure B2.0.14), whereas the second one, τ0, determines the slope of this function at zero frequency and its value is nearly the same as for a homogeneous field. Taking the field region to be about one cabling pitch, the losses are nearly doubled at low frequencies. As can be seen in figure B2.0.14, the field inhomogeneity causes the same factor loss ratio in the whole frequency region, if the field is applied on the length of one cabling pitch. The maximum is at ωτ1 and has a skin-effect-like character (Kwasnitza and Clerc 1993, Takács and Yamamoto 1994) instead of the form ωτ/(1 + ω 2τ 2 ) (Campbell 1982). The differences with respect to the homogeneous field become smaller with increasing field region. However, the contribution to the total losses is nearly the same for each ‘field step’.

Figure B2.0.14. The frequency dependence of loss density per cycle W/f in a flat cable with a length l’ = l0 (full curve), l’ = 3l0 (dotted curve) and l’ = 5l0 (broken curve) with m1 = ω t1, = µ0τ1W/4πh 2τ0 f V in an applied field. The latter curves are already very close to the losses for homogeneous applied field. The total length of the cable is l = 15l0.

Close to the loss maximum ωτ1 ≈ 1, the additional total losses are enhanced for λ > 2l0 by a factor of approximately (1 + 0.4l0/λ). The ‘additional’ length 0.4l0 for the loss generation is very close to the doubled value of the diffusion length ld = lo/π p2ωτ1,, because 2ld ≈ 0.45l0 at ωτ1 ≈ 1. As expected, the additional losses have such a value, as if the magnetic field were penetrating the field-free region effectively up to a length ld on both sides of the field region. On the other hand, the value of τ1 can be orders of magnitudes higher than τ0 for longer samples. We thus have an excellent example of the ‘decoupling’ of the time constant from the actual a.c. losses, where the loss density increases about twice only. This is very important for experimental research, as

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both methods mentioned above are used for determining the time constant of cables from measured a.c. losses (Campbell 1982, Takács 1996). Also for currents between adjacent strands, the additional losses between the opposite strands of the cable are concentrated in a small volume. This is demonstrated very well by considering the currents between two strands going from ‘centre to centre’ and ‘edge to edge’ in the field region (figure B2.0.15). Due to the local nature of the additional losses in inhomogeneous fields, the warning in section B2.0.4.2(a) is valid for the coupling losses between opposite strands, too. The inhomogeneities of other cable properties, e.g. the contact resistance and/or contact area between the strands, can be treated in an analogous way (chapter B5).

Figure B2.0.15. The spatial distribution of the coupling losses along the cable and the corresponding currents between them, when the transverse field is applied on finite length only (here, lm = l0 ) with u = 2πy/l0. The broken curve corresponds to the area where the magnetic field acts from centre to centre (a) with α = 0. The currents in the field region are very close to the situation when the applied field is spatially constant. The full curve represents the enhanced losses between two strands with α = ±π, for which the field is acting from edge to edge (b). One can see clearly that the current through the matrix increases considerably at some places in the field region, leading locally to very strongly increased coupling losses. The contribution of the field-free region (|u| > π) can be neglected in the low-frequency limit.

The current distributions in the individual strands and the corresponding currents between them are quite complex (Takács 1996, 1997). However, they show a twofold periodicity: one of them given by the cabling pitch l0 and the other one by ≈ 2π(ρc/µ0ωl )1/2. We refer the reader to the detailed results of the computer calculations and their important consequences for some magnet configurations given in chapter B5. B2.0.5 Comments The current distribution in superconductors and superconducting structures can be very complex. The solutions of the diffusion equations are known for very simple or simplified cases only. The computer

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calculations support these results and indicate some special effects, too. Nevertheless, there are some general features which we have tried to show here. (i)

The behaviours of superconducting composites, especially of cables, in spatially homogeneous and in inhomogeneous a.c. fields are generally different. (ii) In inhomogeneous fields, the ‘enormous’ increase of the time constant of the induced currents, confirmed also by computer calculations and measurements, is not inevitably connected with a strong increase of the additional coupling losses. Generally, the time constant determined from the slope of the loss density per cycle as a function of the frequency is different from that determined from the maximum of this function, the latter being length dependent. (iii) The additional losses in inhomogeneous applied fields are generated locally in small volumes of the cables, therefore special care has to be taken in magnet structures where this could possibly be detrimental to the stability of the cable. There are, of course, many problems that could not be treated in detail. The most important of them are: the differences in changing the field orientation (Kwasnitza and Clerc 1993, Sumption et al 1995, Sytnikov et al 1989), including the longitudinal fields (Campbell and Evetts 1972, Fukui et al 1994) and the demagnetization effects (Bruzzone and Kwasnitza 1987, Campbell 1982, Kwasnitza and Clerc 1993, see also chapter B7). Another important point is the calculation of the real volumes in which the resistive currents in cables are flowing. This includes also the most important problem connected with this point, namely the contact surface and the contact resistance of the strands (Sumption et al 1995). Therefore, many calculations and a.c. loss evaluations taking this volume as the total volume of the cable (sometimes including the casing and other structures around the ‘superconducting parts’) can be seen as very approximate and sometimes even misleading. This is true mainly when comparing different contributions to the a.c. losses generated in different effective volumes. Acknowledgment The author acknowledges the partial support by the Slovak grant agency VEGA. References Akhmetov A A, Kuroda K, Ono K and Takeo M 1995 Eddy currents in flat two-layer superconducting cables Cryogenics 35 495–54 Anderson P W and Kim Y B 1964 Hard superconductivity: theory of the motion of Abrikosov flux lines Rev. Mod. Phys. 36 39–43 Baird D C and Mukherjee B K 1967 Destruction of superconductivity by a current Phys. Lett. 25A 137–9 Bean C P 1964 Magnetization of high-field superconductors Rev. Mod. Phys. 36 31–9 Brandt E H 1992 Penetration of magnetic a.c. fields into type II superconductors Physica C 195 1–4 Bruzzone P and Kwasnitza K 1987 Influence of magnet winding geometry on coupling losses of multifilament superconductors Cryogenics 27 539–44 Campbell A M 1982 A general treatment of losses in multifilamentary superconductors Cryogenics 22 3–16 —1991 The susceptibility of superconductors near the reversibility line Proc. Int. Symp. on AC Superconductors (Bratislava: VEDA) pp 182–7 Campbell A M and Evetts J E 1972 Critical currents in superconductors Adv. Phys. 21 199–391 Carr W J Jr 1983 AC Loss and Macroscopic Theory of Superconductors (New York: Gordon & Breach) Carr W J Jr and Kovachev V T 1995 Interstrand eddy current losses in Rutherford cable Cryogenics 35 529–34 Cesnak L and Kokavec J 1977 Magnetic field stability of superconducting magnets Cryogenics 17 107–10 Ciazynski D, Turck B, Duchateau J L and Meuris C 1993 AC losses and current distribution in 40 kA NbTi and Nb3Sn superconductors for NET/ITER IEEE Trans. Appl. Supercond. AS-3 594–601

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Clem J R 1979 Theory of ac losses in type-II superconductors with a field-dependent surface barrier J. Appl. Phys. 50 3518–30 Coffey M W and Clem J R 1991 Magnetic field dependence on RF surface impedance IEEE Trans. Magn. MAG-27 2136–9 Faivre D and Turck B 1981 Current sharing in an insulated multistrand cable in transient and steady conditions IEEE Trans. Magn. MAG-17 1048–51 Fukui S, Hlásnik I, Tsukamoto O, Amemiya N, Polák M and Kottman P 1994 Electric field and losses at AC self field mode in MF composites IEEE Trans. Magn. MAG-30 2411–7 Hartman R A, Rem P C and van de Klundert L J M 1987 Numerical solutions of the current distribution in superconducting cables IEEE Trans. Magn. MAG-23 1584–7 Kwasnitza K 1977 Scaling law for the ac losses of multifilament superconductors Cryogenics 17 616–9 Kwasnitza K and Bruzzone P 1986 Measurement of end effects in the coupling losses of multifilamentary superconductors Proc. ICEC-11 (London: Butterworth) pp 741–4 Kwasnitza K and Clerc St 1993 AC losses of superconducting high-Tc multifilament Bi-2223/Ag sheathed tapes in perpendicular magnetic fields Physica C 233 423–35 Kwasnitza K and Widmer Ch 1991 On the reduction of flux creep in superconducting accelerator magnets IEEE Trans. Magn. MAG-27 2515–7 London F 1950 Superfluids vol 1 (New York: Wiley) Maccioni P and Turck B 1991 Influence of copper location on stability of composites made of superconducting filaments in a highly resistive matrix Cryogenics 31 738–48 Marken K R, Markworth A J, Sumption M D, Collings E W and Scanlan R M 1991 Eddy-current effects in twisted and wound SSC strands IEEE Trans. Magn. MAG-27 1791–5 Morgan G H 1973 Eddy currents in flat metal-filled superconducting braids J. Appl. Phys. 44 3319–22 Ries G 1977 AC losses in multifilamentary superconductors at technical frequencies IEEE Trans. Magn. MAG-13 524–7 Ries G and Takács S 1981 Coupling losses in finite length of superconducting cables and in long cables partially in magnetic fields IEEE Trans. Magn. MAG-17 2281–4 Shoenberg D 1952 Superconductivity (Cambridge: Cambridge University Press) Sumption M D and Collings E W 1994 Influence of cable and twist pitch interactions on eddy currents in multifilamentary strands calculated using an anisotropic continuum model Adv. Cryogen. Eng. Mater. A 40 579–86 Sumption M D, Marken K R and Collings E W 1993 Enhanced static magnetization and creep in fine-filamentary and SSC-prototype strands via helical cabling geometry enhanced proximity effects IEEE Trans. Appl. Supercond. AS-3 751–6 Sumption M D, ten Kate H H J, Scanlan R M and Collings E W 1995 Contact resistance and cable loss measurements of coated strands and cables wound from them IEEE Trans. Appl. Supercond. AS-5 692–6 Sytnikov V E, Svalov G G, Akopov S G and Peshkov I B 1989 Coupling losses in superconducting transposed conductors located in changing magnetic fields Cryogenics 29 926–30 Takács S 1982 Coupling losses in cables in spatially changing ac fields Cryogenics 22 661–6 —1984 Coupling losses of finite superconducting cables Cryogenics 24 237–42 —1992 Coupling losses in inhomogeneous cores of superconducting cables Cryogenics 22 258–64 —1996 AC losses and time constants of flat superconducting cables in inhomogeneous magnetic fields Supercond. Sci. Technol. 9 137–40 —1997 Current distribution and coupling losses in superconducting cables being partially in magnetic fields IEEE Trans. Appl. Supercond. AS-7 at press Takács S and Campbell A M 1987 Hysteresis losses in superconductors with very fine filaments Supercond. Sci. Technol. 1 53–6 Takács S and Gömöry F 1993 A.c. susceptibility of melt-processed high Tc superconductors Cryogenics 33 133–7 Takács S, Kaneko H and Yamamoto J 1994 Time constants of normal metals and superconductors at different ramp rates during a cycle Cryogenics 34 679–84 Takács S and Yamamoto J 1994 Time constants of flat superconducting cables at low and high frequencies Cryogenics 34 571–4

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Takács S, Yanagi N and Yamamoto J 1995 Size effect in AC losses of superconducting cables IEEE Trans. Appl. Supercond. AS-5 2–6 Turck B 1974 Influence of a transverse conductance on current sharing in a two-layer superconducting cable Cryogenics 14 448–52 van de Klundert L J M 1991 A.c. stability and a.c. loss in composite superconductors Cryogenics 31 612–8 Verweij A P and ten Kate H H J 1993 Coupling currents in Rutherford cables under time varying conditions IEEE Trans. Appl. Supercond. AS-3 146–9 Verweij A P, den Ouden A, Sachse B and ten Kate H H J 1994 The effect of transverse pressure on the interstrand coupling loss of Rutherford type of cables Adv. Cryogen. Eng. Mater. A 40 521–7 Wilson M 1983 Superconducting Magnets (Oxford: Clarendon) Zenkevitch V B, Romanyuk A S and Zheltov V V 1980 Losses in composite superconductors at high levels of magnetic fields Cryogenics 20 703–10

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B3.1 Normal zone in composites R G Mints

The stability of a current-carrying multifilamentary superconducting composite against flux jumping, i.e. against small perturbations of temperature, electrical and magnetic fields, etc, prevents spontaneous superconducting-to-normal transitions. However, these transitions can be initiated by a sufficiently strong disturbance producing local heat pulses resulting in a normal-zone seed. This mechanism of current-carrying capacity reduction was the reason why the very first superconducting magnets did not work and it was revealed by later experiments with the superconducting composites based on type II superconductors with high values of superconducting current density. The dynamics of an initial normal-zone seed is determined by the Joule self-heating of this resistive domain which either shrinks or expands depending on the value of the transport current, the parameters and the geometry of the composite superconductor, the cooling conditions, etc. The idea of cryogenic stabilization is to prevent the normal-zone propagation by increasing the amount of normal metal in the cross-section of the composite superconductor. Usually, this normal metal is either commercial copper or commercial aluminium, or a certain combination of both. The resistivity of these metals is two to three orders of magnitude less than the normal-state resistivity of a type II superconductor with a high critical current density. As a result, if the superconducting-to-normal transition occurs, most of the current flows out of the superconductor to the normal metal. An increase of the normal-metal cross-section area leads to a decrease of the Joule self-heating power and thus to a decrease of the stationary temperature of the composite superconductor. If this self-maintained temperature is less than the critical temperature the superconducting state recovers. In this section we consider the main principle of the cryogenic stabilization of a current-carrying composite superconductor. We deal with the uniform temperature distributions, the dynamics of an initial normal seed, the normal-zone propagation velocity and the quench energy for a heat pulse. We derive the cryogenic stabilization criteria using common models to approach the superconducting-to-normal transition and heat transfer to the coolant.

B3.1.1 Thermal multistability in a current-carrying composite Let us consider a superconducting-to-normal transition in a current-carrying composite superconductor (Altov el al 1973, Gurevich and Mints 1987, Wilson 1983, Dresner 1995). The physical origin of this nonequilibrium phase transition is the Joule self-heating in the normal state. In this section we treat the case where the temperature is uniform along the composite superconductor (z axes). We assume that the superconducting filaments are embedded in a normal-metal matrix and the multifilamentary area is in thermal and electrical contact with a normal-metal stabilizer, as shown schematically in figure B3.1.1.

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Figure B3.1.1. Cross-sectional schematic diagram of a composite superconductor.

The uniformity of the temperature distribution in the composite superconductor cross-section is determined by the Biot parameter, Bi

where A and P are the cross-section and the cooling perimeter of the composite superconductor, h is the heat transfer coefficient to the coolant and k⊥ is the transverse thermal conductivity averaged over the cross-section of the composite superconductor. The value of k⊥ is relatively high, namely, in a composite superconductor with a copper or aluminium matrix k⊥ is of the order of 10–103 W m–1 K–1. At the same time the heat transfer coefficient h is relatively low, namely, even for a pool helium bath h < hmax ≈ 7 × 103 W m– 2 K– 1. In many cases of practical interest the value of h is a few orders of magnitude less than hmax due to the presence of different insulation layers. We estimate the value Bi ≈ 5 × 10– 3 « 1, using the data A ≈ 3 × 10– 6 m2, P ≈ 6 × 10– 3 m, h ≈ 103 W m– 2 K– 1 and k⊥ ≈ 102 W m– 1 K– 1. Thus, in most cases of practical interest the Biot parameter appears to be much less than unity and with an accuracy of Bi << 1 the temperature of a composite superconductor can be treated to be uniform in the cross-section. Let us now consider the steady normal state for a composite superconductor carrying the transport current, I. The temperature of this steady state, Tn , is determined by the thermal balance between the Joule self-heating and the heat flux to the coolant with the temperature T0, i.e.

where ρ (T) is the longitudinal resistivity averaged over the cross-section. This self-maintained normal state exists if the temperature Tn (I) is bigger than the critical temperature, Tc . Thus, the equation

determines the minimum normal-zone existence current, Im. It follows from equations (B3.1.2) and (B3.1.3) that the value of Im is given by the formula

The minimum normal-zone existence current Im can be higher or lower than the critical current, Ic = Ic(T0), depending on the parameters of the composite superconductor and the cooling conditions. If Im < Ic , then a self-maintained normal state exists in the transport current region Im < I < Ic . In this case

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a sufficiently strong disturbance can result in a superconducting-to-normal transition. If Im > Ic , then the power of the Joule self-heating is less than is necessary to maintain the normal state in a current-carrying composite superconductor. In this case the superconducting state will recover after the disturbance causing a superconducting-to-normal transition is over. Thus, the cryogenic stabilization criterion is given by the relation Ic < Im (Stekly 1965). We introduce now the dimensionless parameter, α, characterizing the ratio of the Joule self-heating and the heat flux to the coolant

where Jc = Jc(T) is the critical current density. Using equations (B3.1.4) and (B3.1.5) we find that

and thus the cryogenic stabilization criterion Ic < Im takes the form α < 1. The dimensionless parameter α is usually referred to as the Stekly parameter (Stekly 1965). We estimate α ≈ 30 using the data typical for the composite superconductors based on the Nb-Ti alloys with a matrix-to-superconductor ratio of the order of unity, ρ ≈ 3 × 10– 10 Ω m, Jc(Tc) ≈ 109 A m– 2, A ≈ 3 × 10– 6 m2, Tc − T0 ≈ 5 K, P ≈ 6 × 10– 3 m and we take for the heat transfer coefficient the value h ≈ 103 W m– 2 K– 1. It follows, thus, from equation (B3.1.6) that for these data the minimum normal-zone existence current Im ≈ 0.18 Ic << Ic . Therefore, the superconducting-to-normal transition in a current-carrying composite superconductor can occur at I « Ic . Let us consider the Joule self-heating of a composite superconductor in more detail. To do this we use the heat diffusion equation

where C(r ) and k (r ,T) are the heat capacity and heat conductivity, Q = J ⋅ E is the Joule heat power, J is the current density, E is the electrical field. We have to add to equation (B3.1.7) the thermal boundary condition

where n is the unit vector normal to the cooling surface. Integrating equation (B3.1.7) over the cross-section of the composite superconductor and using equation (B3.1.8) we find the heat diffusion equation describing the temperature distribution, T(z ), along the sample in the following form

where C = xnCn + xsCs , k = xnkn + xsks , xn and xs are the volume fractions of the superconductor and the normal metal (xn + xs = 1) and Cn , Cs , kn and ks are the heat capacity and longitudinal heat conductivity of the normal metal and the superconductor.

and

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are the Joule heating power and the heat flux to the coolant normalized per unit volume. While deriving equation (B3.1.9) we take into account that Bi « 1 and, therefore, the temperature is uniform in the cross-section. The heat flux, q(T) = h(T) (T − T0), to liquid helium is shown in figure B3.1.2 as a function of the temperature difference, ∆T, between the sample and the coolant for T0 = 4.2 K, standard atmospheric pressure and different values of the thickness of the insulation layer, di . It is seen from figure B3.1.2 that a special singularity, i.e. a sharp drop of q at a certain ∆T, is typical for these curves. It is associated with the transition from nuclear boiling to film boiling on the sample surface. The insulation layer reduces the heat flux to the coolant and shifts the singularity to higher values of ∆T. If the thickness di is sufficiently large, then the dependence of q on ∆T is mainly determined by the thermal resistance of the insulation. In this case the singularity associated with the transition from nuclear boiling to film boiling is less significant. Therefore, in many cases we can approximate the heat flux to the coolant by considering a constant heat transfer coefficient h, which significantly simplifies analytical calculations.

Figure B3.1.2. The heat flux to liquid helium for T0 = 4.2 K and different thickness of insulation layer (cellulose): (a) di = 0, (b) di = 7 × 10– 6 m, (c) di = 1.3 × 10– 5 m (Maddock et al 1969).

Let us now consider the temperature dependence of the Joule heating power Q(T) arising in a superconducting composite when the transport current Iexceeds the critical current Ic(T). A transition to the resistive state occurs at the temperature Tr (I ) determined by the equation

In most cases of practical interest the critical current Ic(T) is linear in temperature and can be written in the form

It follows then from equation (B3.1.13) that the resistive transition temperature Tr(I) is given by the formula

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The Joule heating power Q(T) depends on the current-voltage (I—V) curve of the composite superconductor, i.e. on the dependence of the electrical field E on the current density J. Usually, two models are considered for the relation between E and J. The first takes into account the current sharing between the superconducting filaments and the normal-metal stabilizer. This current sharing occurs if the temperature is in the interval Tr < T < Tc and results in the dependence

where the resistivity ρ = constant. Using equation (B3.1.15) we find that the Joule heating power has the form

This dependence is often referred to as the resistive model. equation (B3.1.16) in figure B3.1.3(a).

We show the function Q(T) given by

Figure B3.13. The temperature dependence of the Joule heating power Q(T) for: (a) the resistive model, (b) the stepwise model.

The second dependence for the I—V curve neglects the current sharing. It reads

where the resistivity ρ = constant. Using equation (B3.1.17) we find that the Joule heating power has the following stepwise form

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This dependence is often referred to as the stepwise model. We show the function Q(T) given by equation (B3.1.18) in figure B3.1.3(b). Let us now consider the uniform stationary states of a composite superconductor carrying a transport current. The temperature of these states, Tu , is given by the heat balance equation Q(Tu) = W(Tu). A graphical solution of this equation is shown in figure B3.1.4 for the resistive model and three different current values. The intersection points of the curves Q(T) and W(T ) correspond to different uniform stationary states. There is one solution of the heat balance equation for I < Im . It corresponds to the superconducting state with Tu = Ts = T0 . There are three solutions of the heat balance equation for Im < I. They correspond to the superconducting (Tu = Ts = T0), resistive (Tu = Ti ) and normal (Tu = Tn ) states. Note that the temperatures Ti and Tn become equal for I = Im.

Figure B3.1.4. A graphical solution of the heat balance equation for the resistive model. The three curves Q1(T), Q2(T) and Q3(T) correspond to the three different current values, namely, I < Im, I = Im and I > Im .

The stability of the uniform stationary states in a current-carrying composite superconductor is determined by the development of small temperature perturbations δ Tu << Tu . Let us take δ Tu in the form δ Tu = δ T0 exp(– γ t + ikz ), where δ T0 , γ and k are constants and i is the imaginary unit. It follows then from the linearized heat diffusion equation equation (B3.1.9) that

The least stable mode corresponds to the biggest value of γ (k), i.e. to k = 0. Thus, a uniform stationary state of a current-carrying composite superconductor with the temperature Tu is stable if γ (0) < 0 and the stability criterion takes the form

This equation has a simple geometrical meaning, namely, the slope of the curve W (T) is bigger than the slope of the curve Q (T) for the intersection points corresponding to the temperatures of the stable uniform stationary states. In particular, the superconducting and normal states are stable and the resistive state is unstable for the case of the resistive model shown in figure B3.1.4 (the curve corresponding to I > Im ). Thus, at least two stable uniform stationary states exist if the minimum normal-zone existence current Im is less than the critical current Ic . Note that the number of these states can be bigger than two

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if the nuclear-to-film boiling transition and the temperature dependence of the normal-metal resistivity are considered. This thermal multistability arising in a current-carrying composite superconductor in the current region Im ≤ I ≤ Ic results in a many-value I—V curve and in a variety of hysteretic effects (Altov et al 1973, Gurevich and Mints 1987). In particular, the superconducting-to-normal transition occurs at different current values in the two cases: (a) when the current is increasing from I < Im to I > Ic and (b) when the current is decreasing from I > Ic to I < Im . B3.1.2 Quench propagation in a current-carrying composite It follows from the previous section that both the superconducting and the normal states are stable against small temperature perturbations if the current is in the region Im < I < Ic . However, the superconductingto-normal transition can occur as a result of a substantially strong initial temperature perturbation. In most cases of practical interest a normal domain is the ‘most dangerous’ perturbation of that type. A typical temperature distribution T = T(z ) for a normal domain is shown schematically in figure B3.1.5. It has a region in the normal state, where T(z) > Tc , two regions in the resistive state, where Tr < T(z ) < Tc , and two regions in the superconducting state, where T0 < T (z ) < Tr. The volution process of an initial normal domain determines whether the superconducting state recovers or the entire current-carrying composite superconductor switches to the normal state.

Figure B3.1.5. The temperature distribution corresponding to a normal domain: (N) normal-state region, (R) resistivestate regions, (S) superconducting-state regions.

In this section we consider the steady increase or decrease of the length of a sufficiently long normal zone, i.e. we treat the case where the length of the normal zone, D(t) >> L, where L is the width of the interface between the stable normal and superconducting states (the NS interface). The temperature of the normal state, Tn , is determined by the heat balance equation Q (Tn ) = W (Tn ) and the temperature of the superconducting state, Ts , is equal to the coolant temperature, T0 , everywhere except in the vicinity of the NS interfaces. Since D(t) >> L, then both NS interfaces are moving with a constant velocity v independently of each other (Altov et al 1973, Dresner 1995, Gurevich and Mints 1987, Wilson 1983). Thus, to find the value of v we treat the uniform motion of a single NS interface in an infinitely long current-carrying composite superconductor. The temperature distribution T (z , t) corresponding to the NS interface depends on ξ = z — vt, i.e. T (z , t ) = T(ξ). The function T(ξ) has the form shown in

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figure B3.1.6 and is described by the heat diffusion equation (B3.1.9) that takes the form

The boundary conditions for equation (B3.1.21) are given by the relations

Note that for these boundary conditions the value of v is positive if the normal state is replacing the superconducting state and the value of v is negative if the superconducting state is replacing the normal state.

Figure B3.1.6. The temperature distribution corresponding to the NS interface.

To find the NS interface velocity v we multiply equation (B3.1.22) by k dT/dz and integrate it over ξ from −∞ to ∞. The final expression for v takes the form (Maddock et al 1969)

where

The denominator in equation (B3.1.23) is positive for all current values and thus the sign of v coincides with the sign of the numerator. The dependence of S on I is a monotonically increasing function of I with a negative minimum value S (0). Therefore, if S (Ic ) is positive, then S(I) is equal to zero for a certain current Ip . The current Ip is usually referred to as the minimum normal-zone propagation current. It follows from equation (B3.1.23) that v < 0 if Im < I < Ip and v > 0 if Ip < I < Ic . A set of typical

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dependences of the NS interface velocity v on the current I are shown in figure B3.1.7 for different values of the Stekly parameter α (the definitions of the characteristic velocity vh , and the dimensionless current i are given by equations (B3.1.33) and (B3.1.34) later).

Figure B3.1.7. The current dependences of the NS interface boundary velocity for different values of α calculated by means of equations (B3.1.48) and (B3.1.49).

A moving NS interface switches the current-carrying composite superconductor from the stable superconducting state with temperature Ts = T0 to the stable normal state with the temperature Tn and vice versa depending on the relation between I and Ip . The value of v is negative for I < Ip , i.e. the superconducting state is replacing the normal state. Therefore, the normal state is metastable in the current range Im < I < Ip . The value of v is positive if Ip < I, i.e. the normal state is replacing the superconducting state. Therefore, the superconducting state is metastable in the current range Im < I Ip . It follows from equation (B3.1.24) that the value of Ip is determined by the equation

It takes the form (Maddock et al 1969)

if we ignore the dependence of the heat conductivity k on the temperature T in the region T0 < T < Tn(Ip ). The relation given by equation (B3.1.26) has a simple graphical interpretation shown in figure B3.1.8 and is often referred to as the equal-area theorem.

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Figure B3.1.8. An illustration of the equal-area theorem. The two equal areas are shown as filled areas.

To calculate the minimum normal-zone propagation current Ip it is necessary to know the dependences k(T), W(T) and Q (T). Usually, the functions k(T), W(T) and Q (T) are quite complicated. As a result explicit analytic expressions for Ip are known only for a few simple models. Let us consider, in particular, the resistive model for Q (T) and a constant heat transfer coefficient h. In this case the value of Ip is given by the formula (Keilin et al 1967)

It follows from equations (B3.1.6) and (B3.1.27) that Im < Ip < Ic for α > 1 and Im = Ip = Ic for α = 1. Therefore, for this model normal-zone propagation is possible only for α > 1. It follows from equation (B3.1.27) that the value of Ip decreases with the increase of α and in the region α >> 1 the dependence Ip(α) is given by the formula

Let us treat the current dependence of the NS interface velocity in more detail. The denominator of the general expression given by equation (B3.1.23) depends on the temperature distribution and thus on the value of v. As a result, in most cases of practical interest it is necessary to perform numerical simulations of the heat diffusion equation in order to calculate the NS interface velocity. Analytical formulae for the dependence v (I) can be derived only for some simple models. Nevertheless, these formulae are useful for a qualitative analysis of the effect of different parameters on the NS interface velocity. We consider now the stepwise model for the Joule heating power Q(T) and the temperature-independent heat conductivity k, heat capacity C and heat transfer coefficient h in order to derive an analytical expression for the dependence v (I). In this case the heat diffusion equation equation (B3.1.21) takes the form

where the prime is for differentiation over ξ ,

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is the characteristic space scale,

is the dimensionless temperature,

is the dimenisonles velocity,

is the characteristic NS interfacce velocity scale,

is the dimensionless current, η(x) is the step function, i.e.

and

is the dimensionless resistive transition temperature. We estimate the values of L ≈ 1.1 × 10– 2 m and vh ≈ 2.8 m s– 1, using the data h ≈ 103 W m– 2 K– 1, C ≈ 8 × 103 J m– 3 K– 1, k ≈ 250 W m– 1 K– 1, and A/P ≈ 5 × 10– 4 m. The solution of equation (B3.1.29) describing the NS interface has the form

where

and the current dependence of the dimensionless velocity u(i) is given by the equation (Keilin et al 1967, Gurevich and Mints 1987)

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It follows from equation (B3.1.39) that the NS interface velocity u(i) is a monotonically increasing function of the current i and the Stekly parameter α. The value of u(i) equals zero for the minimum normal-zone propagation current ip , where ip is determined by the equation

In particular, it follows from equation (B3.1.40) that for α » 1 the value of ip << 1 and ip ≈ p2/α. This result coincides with equation (B3.1.28) derived for the resistive model. Let us note that the dependence u(i) given by equation (B3.1.39) has two divergence points specific only for the stepwise model. Namely, the value of the NS interface velocity u → ∞ if the current tends to the critical current, i.e. if i → ic = 1, then θr → 0 and it follows from equation (B3.1.39) that u → ∞. The second divergence arises at the minimum normal-zone existence current im . The value of im is given by the heat balance equation

It follows from equations (B3.1.39) and (B3.1.41) that the NS interface velocity u → −∞ if i → im . In the limit of the adiabatic cooling conditions, i.e. for h → 0, the value of α → ∞ and the NS interface velocity v(i) is determined by the formula

where

We estimate the value of the characteristic velocity v0 ≈ 15 m s– 1, using the data Jc ≈ 109 A m– 2, C ≈ 8 × 103 J m– 3 K– 1, k ≈ 250 W m–1 K–1, ρ ≈ 3 × 10– 10 Ω m and Tc – T0 ≈ 5 K. Note that a typical value of v0 is an order of magnitude higher than a typical value of vh . In the case when the critical current is linear in temperature the resistive transition temperature θr (i) is equal to 1–i and it follows from equations (B3.1.39) and (B3.1.42) that

and

A more realistic approach to the problem of the NS interface propagation is based on the resistive model for the Joule heating power Q (T) and the temperature-independent heat conductivity k, heat capacity C and heat transfer coefficient h. It results in a transcendental equation determining the dependence v (i). This equation can be solved only numerically (Altov et al 1973). Let us mention that in this approach the function u(i) is finite for the entire current interval im ≤ i ≤ 1 and

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Dresner (1979) suggested an analytical expression approximating the NS interface velocity v (i) for the resistive model in the entire current region im < i < 1. This formula reads

and

The accuracy of equations (B3.1.48) and (B3.1.49) is within 1% over the current range im < i < 1. In figure B3.1.7 we show the dependences u(i) calculated by means of equations (B3.1.48) and (B3.1.49) for different values of α. In figure B3.1.9 are shown the experimental data for the NS interface propagation velocity as a function of the current for different values of the external magnetic field and the fit of these data by the theoretical curves (Dresner 1979).

Figure B3.1.9. The experimental data for the NS interface propagation velocity v as a function of the current I for different values of the external magnetic field Be = 2, 3 and 4 T and the fit of these data by the theoretical curves (Dresner 1979).

The quench propagation velocity is determined by the Joule heating power in the vicinity of the NS interface. During the transition from the superconducting to the normal state, the current is redistributed from the superconductor to the stabilizer. This redistribution occurs in two phases and affects the quench propagation in composite superconductors with a significant amount of the stabilizer located outside the multifilamentary area. First, the current is expelled from the superconducting filaments to the normal metal in the multifilamentary area. Second, the current diffuses into the stabilizer outside the multifilamentary area. If the interfilament spacing is small, the first phase is very fast. In contrast, if most of the stabilizer is

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located outside the multifilamentary area, the second phase can be relatively long. As a result, in the vicinity of the NS interface the current may remain confined in a small fraction of the stabilizer around the multifilamentary area. It increases the local value of the Joule heating power, leading to high quench propagation velocity (Mints et al 1993) as well as to a new phenomenon, namely, to stable propagating normal domains (Pfotenhauer et al 1991, Kupferman et al 1991). B3.1.3 Quench energy It follows from the previous section that the superconducting state in a current-carrying composite superconductor is metastable if Ip < I < Ic . This means that the superconducting state is stable against small perturbations and at the same time the superconducting-to-normal transition can occur if the initial perturbation results in a normal domain nucleation. In this section we consider the minimum energy, εc , of a perturbation initiating quench propagation in a current-carrying composite superconductor. The value of εc is an important characteristic of the cryogenic stabilization (Altov et al 1973, Dresner 1995, Gurevich and Mints 1987, Wilson 1983). In most cases of practical interest a local heat pulse is the ‘most common’ and also the most unstable perturbation that results in an expanding normal domain. Let us consider a local heat pulse with the heating power Qp(z, t). We determine the quench energy εc as the minimum value of the energy of a disturbance that results in an infinitely expanding normal domain, i.e.

To find the value of εc it is necessary to treat the normal-zone dynamics using the heat diffusion equation

with the initial and boundary conditions corresponding to an expanding or shrinking normal domain

The quench energy εc is a function of many parameters characterizing the heat pulse, the composite superconductor and the cooling conditions. In particular, the value of εc strongly depends on the heating power Qp(z ,t ) temporal width, tp , and space width, lp . The quench energy is also a function of the heat capacity C (T), the heat conductivity k(T), the Joule heating power Q (T) and the heat flux to the coolant W (T) temperature dependences. As a result, to calculate εc for a current-carrying composite superconductor is a very complicated problem. In general, numerical simulations are necessary in order to find the value of εc even for simple models. Let us first consider a qualitative approach to the quench energy problem based on the concept of a minimum propagating zone (Martinelli and Wipf 1972, Wilson and Iwasa 1978). To introduce this approach, we note that the heat diffusion equation equation (B3.1.21) has a solution describing a stationary (v = 0) normal domain for I > Ip . This normal domain is unstable against small perturbations in the regime of a constant current. It will either expand or shrink when created and, therefore, it will be referred to as a critical normal seed or a minimum propagating zone.

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The temperature distribution Td(ξ) corresponding to a critical normal seed has a form similar to the one shown in figure B3.1.5. The function Td(ξ) satisfies the boundary conditions

and is described by the formula

where

and Tm is the maximum temperature in the critical normal seed. The value of Tm depends on the current and is determined by the equation

The enthalpy of a critical nomral seed is equal to

The maximum temperature Tm in the minimum propagating zone is of the order of the critical temperature Tc . Therefore, a rough estimate of εd is given by

where D(I) is the length of the normal region in the critical normal seed. The minimum propagating zone concept estimates the quench energy εc by the value of the enthalpy εd . As will be shown below, the quench energy εc can be lower or greater than εd . However, the value of εd is a convenient characteristic of the minimum energy of a local heat pulse resulting in a quench propagation, i.e. for the quench energy εc . Therefore, it is frequently used as a rough estimate of εc for practical purposes. We consider now the stepwise model for the Joule heating power Q(T) and the temperatureindependent heat conductivity k, heat capacity C and heat transfer coefficient h in order to derive an analytical expression for the dependence εc (I). In this case the solution of the heat diffusion equation equation (B3.1.29) corresponding to a minimum propagating zone has the form

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where

is the length of the normal region, and the maximum dimensionless temperature in the critical normal seed θm(i) is equal to

Using equations (B3.1.57), (B3.1.59)–(B3.1.61) we find the enthalpy of the critical normal seed in the form

where

We estimate the value of εe ≈ 3 × 10– 3 J, using the data C ≈ 8 × 103 J m– 3 K– 1, (Tc – T0 ) ≈ 5 K, A ≈ 3 × 10– 6 m2 and L ≈ 2.5 × 10– 2 m. The dependence εd (i) given by equation (B3.1.62) is shown in figure B3.1.10 by the broken line for a critical current linear in temperature. In this case the resistive transition temperature θr(i ) is equal to 1–i and

Figure B3.1.10. The current dependences of the ratios εd /εe and εc /εe calculated by means of equations (B3.1.64) and (B3.1.76) for α = 24: (1) εd /εe , (2) εc /εe .

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A similar but a rather cumbersome expression can be derived taking the resistive model for the Joule heating power Q(T) (Gurevich and Mints 1987). The value of εd is equal to zero at the critical current and it follows from equation (B3.1.64) that

The minimum normal-zone propagation current ip is a divergence point for the function εd(i ). It follows from equations (B3.1.40) and (B3.1.64) that

In the limiting case of adiabatic cooling conditions, i.e. for h → 0, the enthalpy εd becomes infinitely large as in the absence of the heat flux to the coolant the length of the stationary critical normal seed D(I) tends to infinity. Using equation (B3.1.64) we find that if the heat transfer coefficient → 0, then the value of εd ∝ h–1/2 → ∞. The ‘most dangerous’ factors for stability are the ‘fast’ heating pulses with temporal width τp much less than the typical temperature relaxation time τ for a composite superconductor at given cooling conditions. An increase of τp results in an increase of the quench energy εc as a fraction of εc is transferred to the coolant over the time τp. Let us now treat the quench energy εc for an extremely ‘fast’ (τp << τ) pointlike heat pulse with the heating power

where ε0 is the total energy of the pulse. We use also the stepwise model for the Joule heating power Q(T) and temperature-independent heat conductivity k, heat capacity C and heat transfer coefficient h. In this case the heat diffusion equation takes a form similar to equation (B3.1.29), namely

where the dot is for differentiation over t, and

is the characteristic timescale. We estimate the value of τ ≈ 4 × 10– 3 s using the data C ≈ 8 × 103 J m– 3 K– 1, A/P ≈ 5 × 10- 4 m and h ≈ 103 W m– 2 K– 1. The quench energy is the minimum value of the energy ε0 for the heat pulse initiating an expanding normal domain. This normal domain corresponds to a solution of equation (B3.1.68) with the length of the normal region D(t) infinitely increasing with time. We show schematically the time dependence D(t) in figure B3.1.11 for two different values of ε0, namely, for ε0 < εc and ε0 > εc The expanding or shrinking normal domains correspond to the solutions of equation (B3.1.68) that satisfy the initial and boundary conditions given by

It follows thus from equations (B3.1.68) and (B3.1.70) that these solutions θ(z, t) depend on the two dimensionless parameters

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Figure B3.1.11. The schematic time dependence of the length of the normal region D(t ) corresponding to: (1) ε0 <εc , (2) ε0 > εc .

and

The superconducting state is metastable if ip < i < 1, i.e. if the parameter γ is from the interval 0 < γ < 0.5. Therefore, a quench can be initiated by a local heat pulse if 0 < γ < 0.5 and q0 ≥ qc(γ), where

To determine the dependence qc(γ) it is necessary to perform numerical simulations of equation (B3.1.68). The results of these calculations show that within 3% the function can be approximated by the expression (Gurevich et al 1989)

and thus the quench energy εc is given by the formula

The dependence εc(i) calculated by means of equation (B3.1.73) and (B3.1.75) is shown in figure B3.1.10 by the solid line for a critical current linear in temperature. In this case the resistive transition temperature θr(i) is equal to 1 – i and

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The value of εc is equal to zero at the critical current and it follows from equation (B3.1.76) that

The minimum normal-zone propagation current ip is a divergence point for the function εc (i ). It follows from equations (B3.1.76) and (B3.1.40) that

Note that, in contrast to the enthalpy εd the quench energy εc is finite in the limiting case of the adiabatic cooling conditions, i.e. for h → 0. Comparing equations (B3.1.62) and (B3.1.75) we find that the quench energy of a ‘fast’ pointlike heat pulse εc can be greater or less than the enthalpy of a critical normal seed εd . Thus, for almost the entire current interval Ip < I < Ic , we find εc << εd . At the same time for the region I—Ip << Ip the reverse is valid, εc >> εd , as εd ∝ ln( I—Ip ) and ε c ∝ (I—Ip )–1/2. Note that in the limiting case of high values of the Stekly parameter α , i.e. for α » 1, the enthalpy εd is greater than the quench energy εc in the entire region Ip < I < Ic except for a very narrow section near the minimum normal-zone propagation current Ip . The quench energy εc has also been treated in detail for the resistive model for the Joule heating power Q(T) (Dresner 1985). Let us now briefly summarize the dependence of the quench energy on the temporal width τp and the space width lp of the heating pulse (Gurevich and Mints 1987). The value of εc is proportional to τp for the case of a ‘slow’ pointlike pulse, i.e. for τ << τp and lp << L. In particular, for the stepwise model for the Joule heating Q(T) the quench energy is given by

A typical dependence of εc on τp is shown in figure B3.1.12.

Figure B3.1.12. The dependence εc(τp ) for a composite superconductor under different cooling conditions: ×, vacuum; ο, liquid helium (Schmidt 1978).

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The value of εc is proportional to lp for the case of a ‘fast’ and spatially ‘wide’ pulse, i.e. for τp << τ and L << lp . In particular, for the stepwise model for the Joule heating Q(T) the quench energy is given by

A quantitative comparison of the calculated and measured values of the quench energy is rather difficult because of insufficient information on the temperature dependence of the heat flux to the coolant W(T). As an illustration, in figure B3.1.13 are shown the results of numerical simulation (Chen and Purcell 1978) with the experimental data (Wilson and Iwasa 1978). Although the calculation yielded qualitatively the same curve εc = εc (I) as the experiment, the discrepancy is sensitive to the detailed form of the temperature dependence of W(T).

Figure B3.1.13. Comparison of experimental data for ec (•, o) with the results of two numerical calculations (∆, ) in which different functions W(T) were used in the range of the nucleus-to-film boiling transition. The solid line plots the enthalpy εd (Chen and Purcell 1978).

B3.1.4 Summary To summarize, under certain conditions a current-carrying composite superconductor exhibits thermal multistability. At least two states exist for the same current value, namely, a superconducting and a normal state. The superconducting state has the temperature of the coolant; the temperature of the normal state is self-maintained by the Joule self-heating. The superconducting state and the normal state are both stable against small perturbations. At the same time, depending on the current, one of these two states is metastable, i.e. it is unstable against strong perturbations. The normal state is metastable for low currents and the superconducting state is metastable for high currents. The borderline is given by the minimum normal-zone propagation current, which is an important characteristic of cryogenic stabilization. A sufficiently strong perturbation results in an infinitely expanding normal domain, i.e. in a quench propagation along the current-carrying composite superconductor, if the current is higher than the minimum normal-zone propagation current. The value of the minimum normal-zone propagation current depends on the characteristics of the composite superconductor and on the cooling conditions. The higher the normal-state resistivity of the composite superconductor is, the lower is the minimum normal-zone propagation current, and the better the cooling conditions are, the higher is the minimum normal-zone propagation current.

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The quench propagation velocity, i.e. the velocity of the interface between the normal state and the superconducting state, increases with increasing current. It is equal to zero at the minimum normal-zone propagation current and reaches its maximum value at the critical current. The lower the cooling intensity is the higher is the quench propagation velocity. A strong perturbation of the superconducting state initiates a quench propagation along a currentcarrying composite superconductor if the energy of this perturbation is higher than the quench energy. The value of the quench energy is an important characteristic of cryogenic stabilization. It depends on the current and on the characteristics of the heat pulse, the conductor and the cooling conditions. The ‘most dangerous’ factors for cryogenic stabilization are the local perturbations. The smaller the temporal width and the spatial width of a local perturbation are the lower is the quench energy. Its value decreases with an increase of the current, reaching its maximum at the minimum normal-zone propagation current and being zero at the critical current. An increase of the cooling intensity increases the quench energy. References Altov V A, Zenkevitch V B, Kremlev M G and Sytchev V V 1973 Stabilization of Superconducting Magnet Systems (New York: Plenum) Chen W Y and Purcell I R 1978 J. Appl. Phys. 49 3546–53 Dresner L 1985 IEEE Trans. Magn. MAG-21 392–5 Dresner L 1979 IEEE Trans. Magn. MAG-15 328–30 Dresner L 1995 Stability of Superconductors (New York: Plenum) Gurevich A V and Mints R G 1987 Rev. Mod. Phys. 59 941–99 Gurevich A V, Mints R G and Pukhov A A 1989 Cryogenics 29 188–90 Keilin V E, Klimenko E Yu, Kremlev M G, and Samoilov N B 1967 Les Champs Magnetique Intenses (Paris: SNRS) p 231 Kupferman R, Mints R G and Ben-Jacob E 1991 J. Appl. Phys. 70 7484–91 Maddock B J, James G B and Norris W T 1969 Cryogenics 9 261–73 Martinelli A P and Wipf S L 1972 Proc 1972 Applied Superconductivity Conf. (New York: IEEE) pp 325–30 Mints R G, Ogitsu T and Devred A 1993 Cryogenics 33 449–53 Pfotenhauer J M, Abdelsalam M K, Bodker F, Huttelstone D, Jiang Z, Lokken O D, Scherbarth D, Tao B and Yu D 1991 IEEE Trans. Magn. MAG-27 1704–7 Schmidt C 1978 Cryogenics 18 605–9 Stekly Z J J 1965 Adv. Cryogen. Eng. 8 585–600 Wilson M N 1983 Superconducting Magnets (Oxford: Oxford University Press) Wilson M N and Iwasa Y 1978 Cryogenics 18 17–25

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B3.2 Flux-jump instability R G Mints

One of the reasons for the possible reduction in the current-carrying capacity of multifilamentary composite superconducting wires is the thermomagnetic instability, i.e. the magnetic flux jumping. This instability against small perturbations of temperature and the electric and magnetic fields was revealed by the very first experiments with type II superconductors with high values of superconducting current density. Under certain conditions the magnetic flux jumping is accompanied by a strong heating of the superconductor, resulting in a superconducting-to-normal transition and in a subsequent quench propagation. In a multifilamentary composite superconductor the thermomagnetic instability can occur at two levels. In the first, ‘local’, level the magnetic flux jumping occurs in one or simultaneously in several superconducting filaments. In the second, ‘global’, level the thermomagnetic instability develops in the entire cross-section of a multifilamentary composite superconductor at once. In this case the magnetic flux jumping arises as a result of the coupling between the temperature and the electric and magnetic field perturbations in the superconducting filaments and those in the normal-metal matrix. In this section we consider thermomagnetic instability in current-carrying multifilamentary composite superconductors. We derive the criteria for the stability of the superconducting state in both ‘local’ and ‘global’ levels. We apply these criteria to calculate the current-carrying capacities of a superconducting filament and a multifilamentary superconducting wire. B3.2.1 The origin of magnetic flux jumping Superconductivity in type II superconductors exists if the temperature, T, is less than the critical temperature, Tc, the magnetic field, B, is less than the upper critical magnetic field, Bc2 , and the current density, J, is less than the critical current density, Jc . The value of Jc depends on T and B and the surface J = Jc (T, B) separates the normal and the superconducting states in the three-dimensional J T B space (Campbell and Evetts 1972, Lynton 1969, Tinkham 1975). However, it was shown experimentally that, even if the necessary conditions T < Tc , B < Bc 2 and J < Jc are, satisfied, in many cases of practical interest the maximum value of the superconducting current in a wire, Im , is less or much less than the critical current, Ic = Jc A , where A is the cross-sectional area of the wire. One of the possible reasons for this reduction in the current-carrying capacity is the thermomagnetic instability, i.e. the magnetic flux jumping (Altov et al 1973, Mints and Rakhmanov 1981, Wilson 1983, Wipf 1991). Let us first consider qualitatively how the thermomagnetic instability arises and manifests itself. In order to do this we treat the stability of the magnetic flux distribution inside a superconducting slab with thickness 2b. We assume that the external magnetic field, Be , is parallel to the surface of the slab. Suppose that, initially, the magnetic field is zero both outside and inside the superconductor. The external magnetic field is then increased and the magnetic flux enters the slab. This magnetic flux motion induces an electric field and a superconducting current that screens the external magnetic field. As a result,

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Figure B3.2.1. The magnetic field space distribution corresponding to the temperatures T = T0 (solid line) and T = T0 + δ T0 > T0 (broken line). the magnetic field penetrates only a certain surface layer with a thickness l, i.e. B(x , t) = 0 if |x| < b–l, as shown in figure B3.2.1. The space distribution of the magnetic flux inside the slab is given by the system of Maxwell equations. It follows from these equations that the slope of the curve B(x) is proportional to the current density, J, i.e.

Thus, to proceed with the consideration of the magnetic flux space distribution and its dynamics we have to know the dependence of the screening current density J on the temperature and the magnetic and electric fields. In general, in the case of a superconductor with a high value of Jc, the current density J is parallel to the electric field E and depends on the temperature and the electric and magnetic fields. The dependence of J on T, B and E can be presented in the form J = Jc (T, B ) + Jn (T, B, E )

(B3.2.2)

where Jn(T, B, E) is the resistive component of the current density caused by the flux motion. In this section we treat the flux-jumping instability in superconductors with a high value of the critical current density, i.e. it is assumed that Jn << Jc . The dependence of the current density J on the electric field E is discussed in more detail in appendix A at the end of this section. The important point to note here is that although the resistive current density, Jn , is much less than the critical current density, Jc , the presence of Jn is of great importance in the analysis of the stability of the superconducting state as it involves the magnetic flux dynamics. The fact that the ratio Jn /Jc is small enabled Bean and London to formulate the concept of the critical state in type II superconductors (Bean 1962, 1964, London 1963). According to this concept, a superconductor responds to any stimulation that gives rise to an electric field by establishing a critical state in which the current density (whenever it is other than zero) is equal to Jc (T, B). We now use the critical-state concept in order to give a qualitative description of the origin of the magnetic flux jumping in superconductors. Let us also assume that the critical current density Jc is independent of the magnetic field, i.e. that Jc = Jc (T). In this approximation, the critical state model is usually referred to as Bean’s model of the critical state.

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Thus, in the mainframe of Bean’s model the space distribution of the magnetic flux is given by the equation

where the penetration l is equal to

and we assume that Be < µ0 Jcb . The dependence B(x) corresponding to a certain initial temperature T0 is shown in figure B3.2.1 by a solid line. Let us suppose that the temperature of the slab is increased by a small perturbation δ T0 arising due to a certain initial heat release δ Q 0. The critical current density Jc (T) is a decreasing function of temperature. Thus, at T = T0 + δ T0 the density of the superconducting screening current of the external magnetic field is less than at T = T0. This reduction of the screening current leads to a rise of the magnetic flux inside the superconductor. We show in figure B3.2.1 by the broken line the space distribution of the magnetic field B(x) corresponding to the temperature T0 + δ T0. The motion of the magnetic flux into the slab, which occurs as a result of the temperature perturbation δ T0, induces an electric field perturbation δ E0. The induced electric field δ E0 in the presence of Jc causes an additional heat release δ Q 1, an additional temperature rise δ T1 and, consequently, an additional reduction of the superconducting screening current density Jc . This process will repeat itself as shown in the diagram in figure B3.2.2. It results in an avalanche-type increase of the temperature and the magnetic flux in the superconductor under conditions where δ T1 > δ T0 , i.e. under these conditions the superconducting state becomes unstable.

Figure B3.2.2. The thermomagnetic instability development diagram.

Thus, the thermomagnetic instability involves a coupled development of the temperature and the electromagnetic field perturbations. Usually this process is accompanied by intensive heating and a rapid rise of the magnetic flux in the superconductor. This is why the thermomagnetic instability is often referred to as magnetic flux jumping. B3.2.2 A qualitative consideration of magnetic flux jumping The propagation of heat and magnetic flux is characterized by the thermal, Dt , and the magnetic, Dm , diffusivities, where

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κ is the heat conductivity, C is the heat capacity per unit volume and

is the differential conductivity determined by the resistive current. Let us now introduce the following dimensionless parameter that is important in understanding the dynamics of the development of the thermomagnetic instability:

The parameter τ determines the ratio of the time constant of the magnetic flux diffusion

and the time constant of the heat-flux diffusion

where L is the characteristic space scale of the problem (the penetration depth l, the radius of the filament r0 , etc). It follows from equations (B3.2.9) and (B3.2.10) that we also can present τ as

We estimate now the value of the parameter τ for a thermomagnetic instability developing in a ‘local’ and in a ‘global’ level, i.e. in a superconducting filament and in an entire cross-section of a multifilamentary composite superconductor at once. Let us first consider the thermomagnetic instability in a ‘local’ level, i.e. that arising in one or a few superconducting filaments. We assume also that the background electric field E is high, meaning that E > Ef and thus the flux jumping occurs in the flux-flow regime (for details see appendix A in this section). As E > Ef , then the conductivity σ is independent of E and is equal to the flux-flow conductivity σf . We estimate the value τ ≈ 5 × 10– 4 << 1 using the data κ ≈ 10–1 W m–1 K–1, σf ≈ 5 × 106 Ω–1 m–1 and C ≈ 103 J m–3 typical for superconducting Nb—Ti alloys at the temperature T0 ≈ 4.2 K and magnetic field B ≈ 10–1 T. Thus, the development of thermomagnetic instability in the flux-flow regime at the ‘local’ level corresponds to the limiting case where the magnetic flux diffusion is much faster than the heat flux diffusion, i.e. to the case where τ << 1. As we consider the stability of the superconducting state against small perturbations of the temperature δ T and the electric field δ E we assume that

and

where tj is the time constant of the development of a flux jump.

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Thermomagnetic instability involves two coupled processes, namely, the thermal and the magnetic flux diffusion. Therefore, magnetic flux jumping is slower than the faster of these two diffusion processes and at the same time faster than the slower of them. In the limiting case tm << tt , it means that tm << tj << tt , and the magnetic flux propagation is accompanied by adiabatic heating of the superconductor, i.e. we can ignore the redistribution of heat flux on a timescale of the order of tj . The approach to the stability of the current-carrying superconducting state based on this assumption is referred to as the adiabatic approximation (Hancox 1965). Let us now estimate the value of the parameter τ for a thermomagnetic instability developing in a ‘global’ level, i.e. in the entire cross-section of a multifilamentary composite superconductor. An important feature of this process is the coupling between the perturbations of the temperature and the electric and the magnetic fields arising in the superconducting filaments and in the normal-metal matrix. We assume that the number N of the filaments in the cross-section of the multifilamentary composite superconductor is large, i.e. N >> 1. In this case the electric and the magnetic fields and the temperature vary little on a space scale of the order of the interfilament spacing and the superconducting filament diameter. It enables us to consider the development of the thermomagnetic instability after a preliminary averaging of the physical characteristics of the multifilamentary composite superconductor over the cross-section of the wire. In this approach a heterogeneous medium with a complex internal structure is treated as a homogeneous medium with some effective parameter values (Carr 1983). The background electric field in a multifilamentary composite superconductor is low and in most cases of practical interest it corresponds to the flux creep region E < Ef. We estimate τ ≈ 105 >> 1 using the data κ ≈ 10 W m–1, K–1, σ ≈ 1013 Ω–1 m–1 and C ≈ 103 J m–3 typical for the commercial Nb—Ti-based multifilamentary composite superconductors with a Cu matrix at the temperature T0 ≈ 4.2 K and E ≈ 10–5 V m–1. This background electric field is induced, in particular, by a transport current I(t) increasing with the ramp rate I• ≈ 102 A s–1. Thus, the development of thermomagnetic instability in the ‘global’ level corresponds to the limiting case where the heat diffusion is much faster than the magnetic flux diffusion, i.e. to the case where τ >> 1. As tm >> tt , then the same reasoning as above results in the relation tt << tj << tm between the time constants tm , tt , and tj . This means that in a multifilamentary composite superconductor the process of the development of thermomagnetic instability is much faster than the process of magnetic flux diffusion. Therefore, we can ignore the magnetic field and the current density redistribution on a timescale of the order of tj . The approach to the stability of the current-carrying superconducting state based on this assumption is referred to as the dynamic approximation (Hart 1969). Ignorance of the magnetic flux diffusion means that the thermomagnetic instability in the ‘global’ level arises in the background of a frozen-in magnetic flux. Therefore, on a timescale of the order of tj the current density J stays constant despite the increasing perturbations in the temperature, δ T, and electric field, δ E, and

where δ Jc ∝ δ T and δ Jn ∝ δ E. Thus, in the dynamic approximation the arising electric field δ E is not related to the global motion of the magnetic flux. All that happens is a local current redistribution between the superconducting and resistive components. B3.2.3 Flux jumping in the ‘local’ level: the stability criterion In the ‘local’ level the development of thermomagnetic instability corresponds to the adiabatic approximation (τ << 1). We may then ignore the redistribution of heat during the magnetic flux jumping, i.e. ignore the thermal conductivity of the superconductor. The heat diffusion equation then yields

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Substituting equation (B3.2.15) into the Maxwell equation

we obtain that with tm /tj << 1 the electric field perturbation δ E is described by

We have to add to equation (B3.2.17) the electrodynamic boundary conditions. In general, to determine these conditions it is necessary to find the electric field E0(r, t) outside the superconductor and match it on the sample surface with the field Ei (r, t ) inside the superconductor. The solution of this ‘external’ problem (E0(r, t )) as well as of the ‘internal’ problem (Ei(r, t )) is a sum of the background and the perturbation terms. Matching the latter we determine the boundary conditions for the electric field δ E. However, in many cases of practical interest due to the symmetry of the problem it is not necessary to calculate the electric field E0(r, t ) in order to find the electrodynamic boundary conditions. Thus, it follows from equation (B3.2.15) that in the adiabatic approximation the stability of the superconducting state is lost, i.e. δ T• > 0, if equation (B3.2.17) has a nontrivial solution, which satisfies the electrodynamic boundary conditions. Note that the stability criteria thus obtained are independent of the cooling of the superconductor. The accuracy of this result is given by the adiabatic approximation ignoring the heat flux redistribution. The physics of the superconducting-state stability criterion in a ‘local’ level, i.e. for a current-carrying superconducting filament, can be understood from the diagram shown in figure B3.2.2. Let us suppose that an initial temperature perturbation δ T0 arises. It results in a critical current reduction δ Jc = –|∂Jc /∂T|δ T0 < 0. The decrease of the superconducting current induces an electric field perturbation δ E0 ∝ |∂Jc /∂T|. The occurrence of δ E0 is accompanied by an additional heat release, δ Q1, an additional temperature rise, δ T1, and, consequently, an additional reduction of the superconducting current density Jc . The lower the value of δ T1 , the higher is the superconducting-state stability. As there is no heat redistribution in the adiabatic approximation, it follows then from equation (B3.2.15) that the additional temperature rise is equal to δ T1 ∝ Jc|∂Jc /∂T|/C. Thus, the superconducting-state stability in the ‘local’ level increases with an increase of the heat capacity and with a decrease of the critical current density. In particular, it is well known that in the temperature range T < Tc the value of C ∝ T 3 and the value of Jc ∝ (Tc –T). So, an increase of the temperature of the superconductor leads to an increase in the heat capacity and to a decrease in the critical current density and consequently to an increase in the superconducting-state stability in the ‘local’ level. We demonstrate now how to use equation (B3.2.17) in order to solve problems of practical interest. B3.2.3.1 Superconducting filament stability We consider now the current-carrying capacity, i.e. the maximum superconducting current, Im , of a filament that has the form of a circular cylinder with radius r0 . We suppose that the mechanism controlling the value of Im is the magnetic flux jumping in the flux-flow regime. We assume that the external magnetic field Be appears in this problem only as a parameter determining the value of the critical current density Jc . Let us suppose that initially there is no superconducting current in the filament. In this case, the transport current progressively penetrates the filament from the outside in and the region with J = Jc has the form of a ring, as shown in figure B3.2.3. The outer radius of this ring coincides with the radius of the filament r0 . The inner radius of this ring, ri , depends on the current and in the Bean critical-state model

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Figure B3.2.3. The current distribution in a superconducting filament.

is given by the formula

The superconducting state becomes unstable if equation (B3.2.17) has a nonzero solution δ E. In the geometry of the problem under consideration δ E is parallel to the filament axis and depends only on the variable r. Thus equation (B3.2.17) takes the form

where we introduce the characteristic space scale rc as

Let us now find the boundary conditions for equation (B3.2.19). We assume that the transport current value is determined by a certain external source and is constant. This means that the magnetic field is constant on the filament surface, i.e. B•(r0 ) = 0. The Maxwell equation curl E = —B• then leads to the first of the two necessary boundary conditions in the form

In the range 0 < r < ri , there is no current and thus the electric field is zero. So, using the continuity of δ E(r) we find the second boundary condition in the form

The solution of equation (B3.2.19) is provided by the following linear combination of the zero-order Bessel functions

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where C1 and C2 are certain constants. Substituting equation (B3.2.23) into the boundary conditions given by equations (B3.2.21) and (B3.2.22) we find a linear system of equations to determine C1 and C2. Demanding that the determinant of this system is zero, we find the following transcendental equation to determine the maximum current Im

where

The dependence of Im on the ratio r0/rc is shown in figure B3.2.4 (curve 1). It follows from the numerical solution of equation (B3.2.24) that the maximum current Im is always less than the critical current Ic . However, as seen from figure B3.2.4, the increase of Im with a decrease in r0 becomes very small if r0 < rc . In particular, if the filament radius r0 is less than rc , then the difference between the maximum current Im and the critical current Ic is less than 0.5%. Thus, within this accuracy we can consider that there is no current-carrying capacity reduction for the superconducting filaments with r0 ≤ rc . We estimate the value rc ≈ 1.6 × 10–5 m using the data C ≈ 103 J m–3, Jc ≈ 3 × 109 A m–2 and |∂Jc /∂T| ≈ 109 A m–2 K–1 typical for the superconducting Nb-Ti alloys at temperature T0 ≈ 4.2 K and magnetic field B ≈ 5 T.

Figure B3.2.4. The dependence of the ratio Im /Ic on the ratio r0/rc for a superconducting filament (curve 1) and for a superconducting filament coated with a layer of a normal metal with thickness d > dc (curve 2).

B3.2.3.2 Coated superconducting filament stability We consider now the current-carrying capacity of a superconducting filament coated with a normal-metal layer with a certain thickness d. In the flux-flow regime the flux jumping in a superconductor develops much faster than the thermal diffusion. On the other hand, the electromagnetic field induced in the process of magnetic flux jumping will penetrate the normal metal only to a depth of the order of the skin depth δs . We can estimate the value of δs as

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where σn is the conductivity of the coating. If the thickness of the normal-metal layer is bigger than δs , i.e. d > δs , it slows down the magnetic flux propagation and prevents it from entering or leaving the sample at a rate corresponding to the rate of the development of thermomagnetic instability. As a result the superconducting-state stability increases but only up to a certain limit. The existence of this limit as well as the peculiarities of magnetic flux jumping in a coated superconducting filament can be understood from the following reasoning. In the above discussion we assumed that the magnetic flux enters or leaves the superconductor in the process of the development of thermomagnetic instability. However, magnetic flux jumping can only occur due to the magnetic flux redistribution in the superconducting filament. In this case, which corresponds to a normal layer with a thickness that is bigger than a certain critical value (d > dc ≈ δs ), the thermomagnetic instability develops in two stages. In the first stage, there is an initial fast redistribution of the magnetic flux in the superconducting filament. This process develops with a time constant of the order of tj << tt . In the second stage, the magnetic flux slowly enters or leaves the superconducting filament. The time constant of this process is determined by the magnetic flux diffusivity in the coating. These two stages in the development of a flux jump exist due to the fact that the conductivity in the flux-flow regime, σf , is much lower than the conductivity of the normal layer, σn . As a result the criterion of the superconducting-state stability is practically independent of the normal-metal coating thickness if d > dc . To find the value of dc it is necessary to consider the dynamics of the development of thermomagnetic instability. We reproduce here only the result, without dwelling on the details of a cumbersome calculation, namely, it was shown (Mints and Rakhmanov 1981) that

We estimate dc /r0 ≈ 10–1 using the data κ ≈ 10–1 W m–1 K–1, σn ≈ 5 × 109 Ω–1 m–1 and C ≈ 103 J m–3 typical for superconducting Nb—Ti alloys and commercial Cu at the temperature T0 ≈ 4.2 K. Let us now consider the current-carrying capacity of a superconducting filament coated with a normal-metal layer of thickness d > dc . In this geometry the superconducting state becomes unstable if equation (B3.2.17) has a solution δ E, where δ E is parallel to the filament axis. This perturbation δ E depends only on the variable r and is described by the solution of equation (B3.2.19) matching the boundary condition given by equation (B3.2.22) at r = ri . The thermomagnetic instability for d > dc depends only on the redistribution of the magnetic flux in the superconducting filament. This means that the electric field perturbation δ E is equal to zero on the surface of the filament, i.e.

Note that in accordance with the Maxwell equations this boundary condition for equation (B3.2.19) is equivalent to the frozen-in magnetic flux in the superconducting filament. Substituting equation (B3.2.23) into the boundary conditions given by equations (B3.2.22) and (B3.2.28) and demanding that the determinant of the appropriate linear system is zero, we obtain the following transcendental equation to determine the maximum current Im

The dependence of Im on the ratio r0/rc is shown in figure B3.2.4 (curve 2). It follows from the solution of equation (B3.2.29) that the maximum current Im is less than the critical current Ic if r0 > 2.4 rc and is equal to Ic if r0 < 2.4 rc . However, as seen from figure B3.2.4, the increase of Im with decrease of r0 becomes very small if r0 < 3.3 rc . In particular, if the filament radius r0 is less than 3 rc, then the

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difference between the maximum current Im and the critical current Ic is less than 0.5%. Thus, within this accuracy we can consider that there is no reduction in the current-carrying capacity for superconducting filaments with r0 ≤ rmax = 3 rc and a normal-metal coating with thickness d > dc . B3.2.4 Flux jumping in the ‘global’ level: stability criterion Thermomagnetic instability in a multifilamentary composite superconductor can occur in two levels. In the first, ‘local’, level magnetic flux jumping occurs in one or simultaneously in several superconducting filaments. In the second, ‘global’, level the thermomagnetic instability develops in the entire cross-section of a multifilamentary composite superconductor at once. Estimates, which can be readily made, show that the thickness d of the layer of normal metal surrounding every superconducting filament in a multifilamentary composite is practically always bigger than dc . In the case d > dc the adiabatic magnetic flux jumping develops in different filaments independently of one another. The current-carrying superconducting state is then stable in each of the filaments if r0 < rmax ≈ 3 rc . Note that the stability criterion in the ‘local’ level, i.e. the inequality r0 < rmax , is only the necessary condition for the stability in a multifilamentary composite superconductor. Let us now consider the thermomagnetic instability arising in a multifilamentary composite superconductor in the ‘global’ level. We assume that the number N of the superconducting filaments in the cross-section of the composite is large. Electric and magnetic fields and temperature then vary little on a space scale of the order of the interfilament spacing and the diameter of the superconducting filaments. This enables us to investigate the thermomagnetic instability in multifilamentary composite superconductors after preliminary averaging of their physical characteristics. Consequently, if N >> 1, a heterogeneous superconductor may be regarded as a homogeneous medium with some effective values of the parameters (Carr 1983). In the ‘global’ level the thermomagnetic instability develops in the background of a frozen-in magnetic flux, this corresponds to the dynamic approximation (τ >> 1). The physics of the superconducting-state stability criterion in a current-carrying multifilamentary superconducting wire can be then understood from the diagram shown in figure B3.2.5.

Figure B3.2.5. Stability diagram.

Let us suppose that a temperature perturbation δ T > 0 arises. This temperature increase, δ T, causes a decrease in the superconducting current. To keep the superconducting state stable, i.e. to keep the total current at the same level, an electric field perturbation δ E arises. The additional electric field δ E causes an additional heat release δ Q• ∝ δ E, which is the ‘price’ for keeping the total current at the same level. The superconducting state is stable if the additional heat release δ Q• can be removed to the coolant by the additional heat flux δ W ∝ δ T resulting from the temperature perturbation δ T. Thus the

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superconducting-state stability criterion has the form

The additional heat release per unit length, δ Q• , is given by the integral of J area A of the wire

•

δE over the cross-section

where J is the current density. The additional heat flux per unit length, δ W, is given by the integral of h δ T over the cooling perimeter P of the wire

where h is the heat transfer coefficient to the coolant. Using equations (B3.2.30), (B3.2.31) and (B3.2.32) we find the superconducting-state stability criterion in the form

where δ E is the longitudinal (parallel to the filaments) electric field perturbation. To derive the explicit form of the stability criterion we have to find the relation between δ T and δ E. To do this, we follow the idea illustrated by the diagram shown in figure B3.2.5. We calculate the decrease of the current density δ J– resulting from the temperature perturbation δ T and the increase of the current density δ J+ resulting from the electric field perturbation δ E. If the superconducting state is stable in the ‘global’ level, then the total current density stays constant, i.e.

In the superconducting state, J ≈ Jc , where Jc = Jc(T) is the critical current density. Thus, the decrease of J due to the temperature perturbation δ T is given by

The increase of the current density due to the electric field perturbation δ E can be written as

where

is the differential conductivity determined by the slope of the current-voltage (I—V) characteristics and E is the longitudinal (parallel to the filaments) background electric field. The dependence of J on E is strongly nonlinear for multifilamentary composite superconductors with high critical current density (for details see appendix A at the end of this section). To find the explicit relation between δ E and δ T we have to know the explicit expression for the conductivity σ (E), i.e. I—V characteristics of a multifilamentary superconductor.

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Combining equation (B3.2.36) and the expression for σ given by equation (B3.2.A4), we find

It follows from equations (B3.2.34), (B3.2.35) and (B3.2.38) that

Equations (B3.2.A4) and (B3.2.39) allow us to understand the effect of the background electric field E on the superconducting-state stability. It follows from equation (B3.2.A4) that low electric field E results in high differential conductivity σ ∝ 1/E. High conductivity σ leads to low electric field perturbation δ E ∝ 1/σ ∝ E. The smaller δ E is, the less ‘costly’ it is to remove the additional heat release. Thus, the lower the background electric field is the more stable is the superconducting state. Substituting equation (B3.2.39) into equation (B3.2.33) we find the superconducting-state stability criterion in the form

Let us now assume that the physical properties of the multifilamentary composite superconductor are uniform. In particular, this assumption means that we neglect the dependence of Jc and J1 on the local magnetic field. If Jc , J1, |∂Jc /∂T| and h are uniform we can take them out from the integrals in equation (B3.2.40). The nonuniformity (over the wire cross-sectional area) of temperature perturbation δ T is determined by the Biot parameter

where κ is the heat conductivity. Using for estimations A/P ≈ 2 × 10–4 m and κ ≈ 102 W m–1 K–1, we get Bi ≈ 2 × 10–6 h (the value of h is given here in W m–2 K–1). The heat transfer coefficient h strongly depends on the cooling conditions and the electrical insulation of the wire. For most cases of practical interest we can evaluate h < 103 W m–2 K–1 and therefore Bi < 2 × 10–3 << 1. The temperature perturbation δ T is almost uniform over the cross-sectional area of the wire if Bi << 1. This means that we can take δ T out of the integrals in equation (B3.2.40). Finally we find by means of equation (B3.2.40) the superconducting-state stability criterion in a multifilamentary composite superconductor in the form (Mints and Rakhmanov 1982, 1988)

This criterion means that the longitudinal background electric field E averaged over the cross-sectional area of the wire has to be less than a certain critical value Emax. The electric field E depends on the transport current I(t), the transport current ramp rate I•(t), the magnetic field ramp rate B• a(t) and the geometry of the multifilamentary area in the wire. At the same time, the critical field Emax is a function of the physical properties, geometry and cooling conditions of the wire. Thus, the maximum value of the superconducting current Im depends on all these parameters. B3.2.5 The current-carrying capacity of a wire We demonstrate now how to use the superconducting-state stability criterion given by equation (B3.2.42) in order to solve problems of practical interest.

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We consider the current-carrying capacity, i.e. the maximum superconducting current, Im , of a superconducting multifilamentary wire carrying an increasing transport current I(t). We suppose that the mechanism controlling the value of Im is the instability of the superconducting state against small perturbations of the electric field δ E and the temperature δ T. We consider the case where initially the transport current is equal to zero and there is no residual superconducting current in the filaments. To calculate the current Im using the stability criterion given by equation (B3.2.42) we have to know the longitudinal (along the superconducting filaments) background electric field E averaged over the cross-sectional area of the wire, i.e. the value of 〈E〉. We find the expression for 〈E〉 as a function of the current I and the current ramp rate I• in appendix B at the end of this section (see equation (B3.2.B6)). Substituting equation (B3.2.B6) into equation (B3.2.42) we obtain the following equation determining the maximum superconducting current Im (Mints and Rakhmanov 1982, 1988)

where the dimensionless current im is given by

Let us rewrite equation (B3.2.43) in the final form

introducing the charateristic current ramp rate •Iq as

The value of I•q is a function of the cooling conditions (h), the size (R) and the parameters of the I–V characteristics (J1 , Jc ) of the wire. Note that the temperature dependence of I•q is mainly determined by the function h(T) because the dependence of ratio J1/Jc and the derivative |∂Jc /∂T| on the temperature are assumed to be weak. It thus follows from equation (B3.2.45) that the maximum superconducting current Im is a function of one dimensionless parameter I•q /I•. We show the dependence of im on I•q /I• in figure B3.2.6. It is seen from figure B3.2.6 that the value of the maximum superconducting current, Im , tends to the critical current Ic for low current ramp rates, i.e. for I• ≤ I•q . A considerable reduction of the current-carrying capacity occurs if the current ramp rate is high, i.e. for I• >> I•q. In this range of current ramp rate the value of the maximum superconducting current Im is much less than the critical current Ic . Let us estimate the value of I•q using the data characteristic for Nb—Ti—based commercial multifilamentary composite superconducting wire. To do this we approximate the derivative |∂Jc /∂T| as

where T0 is the composite temperature. Substituting equation (B3.2.47) into equation (B3.2.46) we find for I•q the following expression

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Figure B3.2.6. The dependence of the dimensionless maximum current im on the ratio Using now the data h ≈ 102 W m–2 K–1, Tc –T0 ≈ 3 K, R ≈ 5 × 10–4 m, Jc ≈ 109 A m–2 and J1/Jc ≈ 0.03 we calculate I•q ≈ 360 A s–1 . Explicit analytical formulae for the dependence of the maximum superconducting current Im on the current ramp rate I• can be found in two limiting cases, i.e. for I• >> Iq and I• << •Iq . For high current ramp rates, i.e. for I• >> •Iq , we have im << 1. Expanding equation (B3.2.45) in series we find that

For low current ramp rates, i.e. for I• << Iq , we have 1–im << 1. As a result the dependence of Im on I•/I•q takes the form It follows from equation (B3.2.50) that, in particular, the difference between the maximum superconducting current Im and the critical current Ic is less than 1% if I• < 0.28 I•q .

B3.2.6 Summary To summarize, under certain conditions the thermomagnetic instability of the superconducting state determines the current-carrying capacity of a multifilamentary composite superconductor. This instability involves a coupled development of the perturbations of the temperature and magnetic and electric fields and manifests itself as an intensive heating and a rapid rise of the magnetic flux in the superconductor, resulting in a superconducting-to-normal transition. The thermomagnetic instability can occur in two levels. In the first, ‘local’, level the magnetic flux jumping occurs in one or simultaneously in several superconducting filaments. In the second, ‘global’, level the thermomagnetic instability arises in the entire multifilamentary composite superconductor at once. The superconducting-state stability level depends on the development of the coupled perturbations of the temperature and the magnetic and electric fields. The faster the magnetic flux motion is the higher is the power of the heat release and the lower is the superconducting-state stability. The rate of the magnetic flux diffusion is determined by the differential conductivity of the superconductor. The higher

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this conductivity is the lower is the magnetic flux propagation rate. In a superconductor the differential conductivity strongly depends on the background electric field. The decrease of this field results in a rapid increase of the differential conductivity. In general, the superconducting state is the least stable in the ‘local’ level. In this case the magnetic flux jumping in a current-carrying superconducting filament arises in the background of the flux-flow regime. The flux-flow conductivity is very low, leading to a very fast magnetic flux propagation rate and adiabatic heating in the process of the development of thermomagnetic instability. The criterion of superconducting-state stability in the ‘local’ level is the necessary condition for the high current-carrying capacity of a multifilamentary composite superconductor. In the ‘global’ level the thermomagnetic instability arises in the background of a frozen-in magnetic flux, low electric field and extremely high differential conductivity. The balance between the heating and the cooling then results in a self-consistent superconducting-state stability criterion determining the currentcarrying capacity of a multifilamentary composite superconducting wire. The maximum superconducting current in this case depends on the current ramp rate, the geometry of the problem, parameters of the superconducting filaments and the normal-metal matrix and the cooling conditions. Appendix A Current-voltage characteristics of a superconductor In general, in the case of a superconductor with a high critical current density Jc the dependence J(T, B, E) can be presented in the form

where Jn is the resistive component of the current density caused by the flux motion and Jn(T, B, E) << Jc (T, B). The strong nonlinearity of the dependence of Jn on E in weak electric fields was demonstrated in the very first studies of type II superconductors (Kim et al 1963, 1964). These and subsequent experiments showed that in a certain field range E < Ef the resistive current density Jn with a high accuracy is proportional to ln(E/E0), where E0 is an arbitrary constant. It follows from the relation Jn ∝ ln(E/E0 ) that the value of Jn equals zero at the field E = E0. Thus, assuming a certain value of E0 , we determine the critical current density as Jc = J (E0). The replacement of E0 by an arbitrary quantity E~0 redefines Jc and leaves the I—V characteristics unchanged. It is usually assumed that the critical current density corresponds to the electric field E = E0 = 10–4 V m–1. The value of Ef is from the interval 10–3–10–5 V m–1 for most of the type II superconductors. In the electric field range E > Ef the resistive current density is proportional to E and

where the flux-flow conductivity σf is inversely proportional to the magnetic field B. This regime is referred to as flux flow (Huebener 1979, Tinkham 1975). It is observed experimentally in the case of intensively cooled thin superconducting films and filaments when the difference between the conductor and coolant temperatures is small despite Joule heating. In superconductors with high critical current density the flux-flow conductivity σf is low. In particular, for the superconducting Nb—Ti alloys at T ≈ 4.2 K and B ≈ 5 T the value of σf is in the interval 106-107 Ω–1 m–1. In a multifilamentary composite superconductor the electric field corresponds to the range E < Ef in most cases of practical interest and the I–V characteristic takes the following form

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where J1 = J1(T, B) and J1(T, B) << Jc (T, B). In particular, for the commercial Nb—Ti—based multifilamentary composite superconductors the dependence of the ratio J1/ Jc on temperature and magnetic field is assumed to be weak, Jc ≈ 109 A m-2 and J1 ≈ 3 × 107 A m-2 , i.e. J1/ Jc ≈ 0.0.3 << 1. The logarithmic I-V characteristic given by equation (B3.2.A3) is illustrated by the two graphs shown in figure B3.2.A1.

Figure B3.2.A1. I—V characteristics for a Nb—Ti alloy (Polak et al 1973).

It follows from equation (B3.2.A3) that the differential conductivity of a multifilamentary superconductor is equal to

Let us estimate the value of σ for a Nb—Ti-based multifilamentary superconducting wire carrying an increasing transport current I(t) with ramp rate I• ≈ 102 A s–1 , corresponding to an electric field

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E ≈ 10–5 V m–1. Using the data J1 = 0.03 and Jc ≈ 3 × 107 A m– 2 we find that σ ≈ 1013 Ω–1 m–1. This value of the differential conductivity σ exceeds the conductivity of commercial copper by approximately three orders of magnitude. Let us note that the I—V characteristics of a multifilamentary superconductor are often presented as

where n >> 1, and the dependence of n on T and B is assumed to be weak. In the case of n >> 1, equations (B3.2.A3) and (B3.2.A5) are equivalent. To show this we rewrite equation (B3.2.A5) in the following way

and expand it in series keeping the first two terms

Comparing equations (B3.2.A3), (B3.2.A5) and (B3.2.A7) we see that both presentations of the I—V characteristics are equivalent and the relation between J1 and n has the form

Thus, it follows from equation (B3.2.A3) that the value of the differential conductivity σ of a multifilamentary composite superconductor is given by equation (B3.2.A4) and σ ∝ E –1. Appendix B Background electric field: current-carrying wire Let us consider a superconducting multifilamentary composite wire carrying an increasing transport current I(t). Suppose that initially there are no superconducting currents in the wire. In this case, the transport current progressively penetrates the wire from the outside in and the region with J ≠ 0 has the form of a ring, as shown in figure B3.2.B1. The outer radius of this ring coincides with the radius of the wire R. The inner radius of this ring, Ri , depends on the current. In the mainframe of Bean’s critical state model, i.e. when Jc = Jc (T), the value of Ri , is given by the following formula

where the dimensionless current i is defined as

The longitudinal background electric field E induced by the varying transport current I(t) in the region Ri < r < R can be calculated by means of the Maxwell equations

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Figure B3.2.B1. The current distribution in a superconducting wire.

with the boundary conditions

Here Bϕ is the ϕ component of the magnetic field induced by the current flowing in the saturated region. It follows from equations (B3.2.B3) and (B3.2.B4) that

Thus, the time-dependent transport current I(t) results in a longitudinal background electric field that is proportional to the current ramp rate, i.e. E ∝ µ0I•. Using equation (B3.2.B5) we find for the value of E averaged over the cross-sectional area of the wire the following expression

References Altov V A, Zenkevitch V B, Kremlev M G and Sytchev V V 1973 Stabilization of Superconducting Magnet Systems (New York: Plenum) Bean C P 1962 Magnetization of hard superconductors Phys. Rev. Lett. 8 250–3 Bean C P 1964 Magnetization of high-field superconductors Rev. Mod. Phys. 36 31–9 Campbell A M and Evetts J E 1972 Critical Currents in Superconductors (London: Taylor and Francis) Carr W J 1983 AC Loss and Macroscopic Theory of Superconductors (New York: Gordon and Breach) Hancox R 1965 Stabilization against flux jumping in sintered Nb3Sn Phys. Lett. 16 208–92 Hart H R 1969 Dynamic stability against flux jumps J. Appl. Phys. 40 2085 Huebener R 1979 Magnetic Flux Structures in Superconductors (Berlin: Springer) Kim Y B, Hempstead C F and Strnad A R 1963 Flux creep in hard superconductors Phys. Rev. 131 2486–95 Kim Y B, Hempstead C F and Strnad A R 1964 Resistive states of hard superconductors Rev. Mod. Phys. 36 43–5 London H 1963 Alternating current losses in superconductors of the second kind Phys. Lett. 6 162–5 Lynton E A 1969 Superconductivity (London: Methuen) Mints R G and Rakhmanov A L 1981 Critical state stability in type II superconductors and superconducting–nomalmetal composites Rev. Mod. Phys. 53 551–92

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Mints R G and Rakhmanov A L 1982 Current–voltage characteristics and superconducting state stability J. Phys. D: Appl. Phys. 15 2297–306 Mints R G and Rakhmanov A L 1988 The current-carrying capacity of twisted multifilamentary superconducting composites J. Phys. D: Appl. Phys. 20 826–30 Polak M, Hlasnik I and Krempasky L 1973 Voltage—current characteristics of Nb—Ti and Nb3Sn superconductors in flux creep region Cryogenics 13 702–11 Tinkham M 1975 Introduction to Superconductivity (New York: McGraw-Hill) Wilson M N 1983 Superconducting Magnets (Oxford: Oxford University Press) Wipf S L 1991 Review of stability in high temperature superconductors with emphasis on flux jumping Cryogenics 31 936–48

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B3.3 Practical stability design L Bottura

B3.3.1 Introduction Magnets built in the 1960s out of the newly available superconducting strands had their first quench much before reaching short-sample values, which greatly disappointed the constructors. In addition, magnets trained towards higher currents, i.e. the maximum current that could be reached increased quench after quench, slowly approaching a plateau which could be still below the expected current-carrying limit of the cable. This happened in spite of the success in the development of pure superconducting materials (mainly Nb—Zr, Nb—Ti and Nb3Sn, in the form of tapes or large monofilamentary strands). The situation has been illustrated by Chester (1967) in an excellent review article on the status of the development of superconducting magnets: …the development of superconducting solenoids and magnets has been far from straightforward, mainly because the behaviour of the materials in coils frequently did not accord with the behaviour of short samples. … The large number of coils …wound from Nb—Zr and Nb—Ti wire, and …Nb3Sn, revealed several intriguing and very frustrating characteristics of these materials in magnets. The premature quenches were originally thought to have originated from bad spots in the wires or cables, and thus to be attributed to bad homogeneity in the quality of the superconductor. This idea produced the concept of degradation of the conductor performance. Although training clearly showed that a physical degradation could not be responsible for the bad performance, the misleading name remained as an inheritance of the misty understanding. Particularly puzzling was the fact that the degradation depended on the coil construction and on its geometry. Quoting Chester (1967) again: The prediction of the degraded current for any new shape or size of coil proved to be impossible and, for a time, the development of coils passed through a very speculative and empirical phase. A principle not yet understood at the time was that of the stability of the cable with respect to external disturbances (see the detailed discussion in sections B3.1 and B3.4). Insufficient stability and large external disturbances were the key issues in the failure of the early experiments on superconducting magnets. It has since been understood that a superconducting magnet is always subject to a series of energy inputs of very different nature, timescale and magnitude, the so-called disturbance spectrum (Wilson 1983), which can potentially drive a portion of the magnet into the normal state and trigger a quench. Mechanical energy release through small and local movements of a wire, friction, cracking of the bonded insulation or similar events are generally at the lower end of the energy deposition magnitude and timescale, with typical energy density of the order of 1-10 mJ cm-3 of wire and typical times are generally below 1 ms. Larger energy is involved with massive conductors or winding displacements. At the upper end of the scale are electromagnetic energy releases, associated, for example, with flux jumps or a.c. losses in

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a pulsed magnetic field. The typical energy density involved can range from some millijoules per cubic centimetre (e.g. for a flux jump) to hundreds of millijoules per cubic centimetre for large field changes in the range of 1 T. The timescale of the energy deposition in this case is governed by the dynamic of the magnetic field, and can extend to times of about 1 s, up to quasi-steady-state conditions. Superconductor stability, the balance of heat input, heat removal through cooling and conduction, and possibly a limited heat sink, must take into account the details of the disturbance spectrum. The historical path that led to the understanding of this concept is an optimal way to learn the interplay of the different phenomena. In this chapter we will emphasize the concepts of the disturbance spectrum and stability through a series of examples. Because of their historical importance, we will start with an example on the energy input and the stability condition associated with adiabatic flux jumps, followed by an example on a cryostable conductor (the Big European Bubble Chamber (BEBC) magnet at CERN). Finally we will give examples of high-performance and large magnets where the stability margin has been successfully balanced against the disturbance spectrum (a typical magnetic resonance imaging (MRI) magnet, the Euratom Large Coil Task (LCT) coil and the toroidal field magnet of Tore Supra). For the key concepts and a detailed treatment of flux-jump instabilities and superconductor stability we will refer extensively to sections B3.1, B3.2, B3.4 and C3. B3.3.2 Flux jumps As discussed in section B3.2, a small heat input into a superconductor immersed in a magnetic field causes a decrease of the critical current density Jc through the temperature increase (all technical superconductors have a negative Jc(T) slope). In adiabatic conditions, the magnetization of the superconductor (proportional to the current density) also decreases, resulting in a penetration of the external magnetic field in the superconducting bulk. A part of the energy stored in the magnetization profile is dissipated resistively within the superconductor. The energy release caused by the decrease of the magnetization can be sufficient to cause an irreversible transition of the wire to the normal state—a flux jump. This phenomenon can be the triggering event for a complete magnet quench, as discussed in sections B3.1 and C3. Because the magnetization, and thus the energy stored, is proportional to the size of superconducting filaments and tapes, flux jumps were a major problem at times when superconducting material technology did not allow the production of fine subdivisions of the superconductor in the wire. In fact, flux jumps were among the first perturbations recognized to be responsible for performance degradation, and thus they were intensely studied in the late 1960s and early 1970s (see for instance the review of the Rutherford Laboratory Superconducting Applications Group (1970)), leading to one of the first quantifications of the idea of a disturbance spectrum. Because of this, the early considerations of stability were often interleaved with flux-jump theory. Fine subdivision in small filaments is the most obvious cure for flux jumping. On one hand it reduces the energy dissipated and therefore it eases the so-called adiabatic stabilization. On the other hand, fine subdivision means an increase of the superconductor surface, making it easier to remove efficiently the heat generated by the flux penetration in the bulk superconductor, the so-called dynamic stabilization. Nowadays, as will be demonstrated later on, flux jumps are no longer a problem in the standard production of low-temperature superconducting wires (typically based on Nb—Ti and Nb3Sn materials). Flux jumps may still play some role in high-temperature superconductors operated at low temperatures (around and below 20 K), although the steadily improving technology is quickly making this statement obsolete. B3.3.2.1 Flux jumps in a superconducting slab—adiabatic stability limit To demonstrate the order of magnitude of the energy release of flux jumps and its associated stability limits we take the classical example of a superconducting slab of thickness 2a in the x direction, immersed in

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an external field Bext directed along the y direction. We assume that the slab has a thermal diffusion time constant much larger than the magnetic diffusion time constant, i.e. using the nomenclature of section B3.2 we make an adiabatic approximation. It is instructive to go in detail through the derivation of the fluxjump stability limit in this simple case. To do this we compute the magnetization energy release in the slab due to a decrease of the critical current density δ Jc , caused by an arbitrarily small energy input, during a finite time interval δt, that has increased the superconductor temperature by an amount δ T. We then relate it to the additional temperature increase in the superconducting slab δ T’ due to the magnetization energy release and we find the condition under which a flux jump does not lead to an unstable situation, namely when δ T’ < δ T. We assume that the slab carries no transport current and that its initial current density Jc is independent of the field. In addition we assume that the critical current falls linearly with temperature so that the temperature change δ T produces a critical current density change δJc given by

where Tc and Top are respectively the critical and initial operating temperatures. The initial field profile in the slab, shown in figure B3.3.1, is symmetric with respect to the symmetry plane at x = 0. A small change in the critical current density δ Jc (in the z direction) causes a change in the local magnetic flux density in the slab δ B(x) (in the y direction) given by

and the electric field E(x) (in the z direction) associated with the flux density change δ B(x) in the time interval δt can be computed from the change in the flux linked as follows

Figure B3.3.1. The geometry used for the calculation of flux-jump energy release in a superconducting slab of thickness 2a immersed in a uniform external magnetic field Bext, applied in the y direction, parallel to the slab boundary. The initial field profile (solid line) is flattened by a decrease of the critical current density δ Jc (dashed line). The local change in magnetic flux density δ B(x) is indicated.

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Following this power generation the average slab temperature will increase further by δ T’ according to the adiabatic heat balance

where C is the volumetric heat capacity of the superconductor. If now we make use of the linear Jc(T approximation we can express the additional temperature increase δ T’ as a function of the initial increase δT

and finally the condition for a flux jump to be stable is that δ T’ < δ T, or, introducing the strand stability parameter β

The condition above is a rough estimate, but it is useful to give trends and orders of magnitude. We note firstly that the parameter β can be made small for small values of Jc and of the slab balf-thickness a, and large values of the volumetric heat capacity C and of the temperature margin Tc - Top. Of course it is not useful produce a superconductor with a small critical current density. Typical heat capacities for low-temperature superconductors are around 5000 J m-3 K-1, and typical temperature margins are of the order of 3-10 K. Both are marginally affected by material characteristics. Therefore the most obvious and viable solution to flux-jump instability is fine subdivision of the superconductor, i.e. a reduction of a. To give an order of magnitude, taking for Jc a typical low-field value of 5 × 109 A m-2, the typical maximum slab half-thickness stable against flux jumps is of the order of 50 µm. In the case of a superconducting filament, we can estimate therefore that the typical maximum filament diameter stable against flux jumps is of the order of 100 µm. This estimate is in fact conservative and has been proven to be a sufficient condition for flux-jump stabilization. The presence of a low-resistivity matrix in the wire tends to slow the magnetic field diffusion and enlarge the timescale of the energy deposition, so that surface cooling becomes important. Theories that take both magnetic and thermal diffusion into account are usually called dynamic and tend to relax the requirements on the maximum filament size. As discussed in section B3.2, the presence of a transport current in a superconducting wire with filament size smaller than the stability limit given above has an effect approaching a high fraction of the critical current. This situation generally arises only at high operating fields, i.e. when the filament magnetization is much decreased and therefore flux-jump instability is no longer an issue. Section B3.2 gives a general treatment of the influence of flux-jump instability on the current-carrying capacity of a superconductor. Present Nb—Ti- and Nb3Sn-based wires are fabricated by subdividing the superconductor into filaments embedded in a low-resistance matrix such as copper, forming the so-called multifilamentary wire. Typical filament sizes are intentionally below the flux-jump stability limit. In fact, in several cases today the filament diameter in multifilamentary wires is not dictated by flux-jump considerations, but rather by requirements on the magnetization of the strands, affecting the hysteresis loss (i.e. for pulsed operation)

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and causing a distortion of the magnetic field produced (i.e. for high-field-quality applications like magnetic resonance or particle accelerators). Filament sizes well below 1 µm have been achieved in Nb—Ti wires for a.c. applications (50 Hz), and 5 µm is a typical target for large-scale production of Nb— Ti for particle accelerator magnets. Normal production gives typical filament sizes in the range of 50–100 µm, a trade-off between manufacturing costs and magnetic stability. In the case of Nb3Sn the filament size is also determined by issues inherent to the manufacturing process. Nb3Sn is obtained by chemical reaction of Nb filaments with Sn. Tin is present in localized or distributed sources within the wire before reaction, and must diffuse into the Nb. This is a slow process, and it is therefore necessary to start with small Nb filaments, of the order of 1 µm, to achieve uniform diffusion in practical times. Filaments in Nb3Sn after reaction are rarely circular, and tend to cluster into larger units. Typical effective filament diameters currently achieved, e.g. for nuclear fusion applications, are in the range of 5–30 µ m. A mention of wire twisting is finally necessary in conjunction with flux-jump instabilities. An untwisted multifilamentary wire is in fact just as potentially prone to flux jumping as a large superconducting filament. We can arrive at a simple qualitative explanation of this fact by imagining the effect of a field change normal to an untwisted wire. Shielding currents would flow along the parallel filaments, without resistance, and cross over from filament to filament through the resistive matrix. Magnets are built out of long lengths of wire, and thus the resistance involved in the crossover would be small. Therefore the filaments would act in the untwisted wire in a coupled manner, shielding collectively the magnetic field changes inside the wire. This situation is analogous to the one arising in a single superconducting filament, where, however, the transverse wire dimension now replaces the filament size. In conclusion, in an untwisted wire we would have completely lost the advantage of subdivision. Luckily, we can decouple the filaments by twisting the wire. The effect of twisting is to reverse the electric field at each half pitch. By tight twisting, the build-up of the shielding currents, and the associated magnetization, can be limited and collective flux jumps avoided. B3.3.3 Cryostability Early superconducting coils had a wide spectrum of large perturbations, either because the strands and tapes used were prone to flux jumping, or because the mechanics was not sufficient to avoid small movements, slips, insulation cracks and the associated energy release during energization. A small and localized normal zone had, in addition, no chance to recover, because the Joule heating of the superconducting material in the normal state was extremely high and therefore the coils quenched prematurely. Based on this observation, Krantowitz and Stekly (1965) and Stekly and Zar (1965) added a high-electrical-conductivity shunt backing the superconductor and a pure copper stabilizer, and exposed this material to a liquidhelium bath at constant temperature†. Cryogenic stabilization, or cryostability, was achieved when the steadystate composite temperature that would be attained with the full operating current flowing in the stabilizer was below the critical temperature of the superconductor. In this case the initial normal zone, caused by an arbitrary energy source, would shrink and eventually disappear. The cryostability condition, treated in detail in section B3.1, can be best understood by comparing the heat generated by a normal zone in a superconducting length to the heat removal at its surface. The power generated per unit length of normal zone by a wire carrying a current I and at a temperature T is given by (see section B3.1)

† Note that a parallel shunt with a high electrical conductivity is needed also for the protection of the magnet (see chapter C3) which would otherwise be damaged by excess temperature or voltage during the quench.

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Figure B3.3.2. Examples of heat generation (full curves) and heat removal (broken curves) curves in a normal zone. The heat generation increases linearly between the current-sharing and critical temperatures (Tc s and Tc respectively). The heat removal has large values in the nucleate boiling regime and drops in the film boiling regime (the solid line is the unstable transition between the regimes). The three curves of heat generation correspond to: (a) a fully cryostable wire; (b) a wire cryostable for small perturbations; (c) a noncryostable wire.

where A is the stabilizer cross-section and ρ its resistivity, To p is the initial temperature, equal to the temperature of the helium bath cooling the wire, Ic is the wire critical current and finally Tc and Tc s are respectively the critical and current-sharing temperatures. In the case of linear dependence of the critical current on temperature, the generation increases linearly above Tc s up to its maximum at Tc † (see figure B3.3.2). Note how the maximum heat generation depends on the stabilizer cross-section and on the operating current I. On the other hand the heat removal (per unit length) at the surface of the wire is given in general terms by

where p is the wetted perimeter and h is the heat transfer coefficient to the helium. This last depends strongly on the cooling conditions. In the case of boiling helium h has initially high values (typically 1000–10 000 W m–2 K–1) in the nucleate boiling regime and drops to a minimum of the order of 500–1000 W m–2 K–1 at the onset of film boiling. The cryostability condition formulated above is fulfilled when under any conditions the heat removal is larger than the heat generation, i.e. when the maximum possible heat generation is smaller than the minimum possible heat removal. This situation is fulfilled by curve (a) in figure B3.3.2, and is equivalent to the well known criterion for cryostability, expressed using the Stekly coefficient α (Stekly and Zar 1965)

† The stabilizer resistivity is generally constant over the temperature range of liguid helium. For copper, as an example, this is true up to approximately 20 K. Thereafter it increases quickly with temperature (see Reed (1983)), and the heat generation can therefore grow substantially.

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where we take for h the minimum value along the boiling curve. Note, as evident from figure B3.3.2, that intermediate stability conditions at higher heat generation could exist. An example is given by curve (b) in figure B3.3.2, which shows that under small perturbations (small temperature increase) the heat removal is still larger than the generation. A conductor operating in this condition would therefore recover from sufficiently small energy inputs, but it would be unstable for large enough energy depositions. B3.3.3.1 The BEBC magnet Cryostable magnets were among the first to be built soon after the formulation of this principle in the early 1970s. A dramatic example was the BEBC at CERN (Haebel and Wittgenstein 1970), a 4.7 m bore split solenoid with a 0.5 m gap producing a maximum field in its centre of 3.5 T, corresponding to a maximum field at the conductor of 5.1 T, and storing an energy of 800 MJ. Each coil was wound in 20 pancakes out of a flat monolithic conductor with a thickness of 3 mm and a width of 61 mm (see figure B3.3.3). This conductor was itself a composite containing 200 untwisted Nb—Ti filaments with a diameter of about 200 µm in an OFHC copper matrix. The conductor had a total Nb—Ti area of approximately 6.5 mm2 and a copper cross-section of about 176.5 mm2. The nominal operating current of the conductor was 5700 A, corresponding to an operating current density in the composite of about 30 A mm– 2. Adjacent conductors in a pancake were separated by insulation and by a copper spacer that allowed helium to wet the outer surface of the composite. If we assume that only one broad face of the composite was wetted (the other face being pressed against the insulation and a reinforcing steel strip), we have a wetted perimeter of 61 mm. For boiling helium cooling, at 4.2 K, we can take a minimum heat transfer coefficient of the order of 1000 W m–2 K–1. In a field of 5 T copper has an electrical resistivity of approximately 4 × 10–10 Ω m, while Nb— Ti has a critical temperature of the order of 7 K. With these values the Stekly parameter α can be calculated to be approximately 0.5, i.e. the conductor operated largely in the cryostable regime. Indeed the BEBC coil could be energized up to the operating current without problems, in spite of the fact that the Nb—Ti filaments were larger than our previous estimate of the flux-jump stability limit. In fact, because the filaments were not twisted in the composite, even larger magnetization was associated with the currents that flowed in the electromagnetically coupled filaments, excited by the field ramp. Owing to the cryostable operating regime it was possible to suppress the large magnetization produced by these coupling currents using heaters that temporarily quenched the conductor. The conductor recovered as soon as the heaters were switched off; a rather bizarre use of cryostability.

Figure B3.3.3. A schematic view of the BEBC conductor assembly.

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B3.3.4 Disturbance spectrum versus stability margin At present fully cryostable magnets are rarely the preferred designer choice. In an efficient magnet design the cable operating current density must be kept high to make the magnet cross-section as small as possible. For a specified field or stored energy, and thus a given magnetomotive force, a maximum current density results in a decreased material and production cost. As we have shown previously a cryostable magnet needs a large amount of copper stabilizer—compared with the amount of superconductor—and a large amount of helium to provide an ideally infinite heat sink. Therefore a cryostable magnet has an intrinsically low operating current density. On the other hand, cryostability implies that the conductor is stable against any disturbance spectrum, independent of the magnet details and operating mode. In reality the variety of conductor designs and of magnet winding techniques, together with the variety of operating requirements, result in a wide range of possible disturbance spectra. A cryostable conductor design is therefore, in general, excessively safe. Indeed, at present most magnets designed and built are not cryostable at the operating point, but they can still be operated reliably. The common feature of these magnets is that their stability margin is above the disturbance spectrum experienced during operation. The first step in a sound design is thus to estimate the envelope of the perturbations that will be experienced. Afterwards the conductor can be designed to accommodate these perturbations by means of a sufficiently large stability margin. Note that this process can imply iterations as the disturbance spectrum can depend on the conductor and coil design itself. Depending on the energy release dominating the disturbance spectrum, different stabilization principles can be used. A magnet operated in a steady-state mode, with a tightly packed winding, affected by small mechanical disturbances localized in time and space (e.g. in the case of fully impregnated windings) may rely on the heat sink provided by the small enthalpy margin of the superconductor and stabilizer themselves, which is of the order of 1 mJ cm– 3. This is a so-called adiabatic winding. To stabilize larger perturbations additional heat sinks may be necessary. A very efficient heat sink at cryogenic temperatures is helium. The enthalpy difference in supercritical helium for a 2 K temperature margin is of the order of 1000 mJ cm– 3, i.e. three orders of magnitude higher than for a superconducting wire. Bringing helium in close contact with the conductor thus increases its stability margin, provided that the heat transfer at the wetted surface is efficient in the timescale of the energy deposition considered. Magnets with small amounts of added helium (or other heat sinks) are called quasi-adiabatic as they would in any case behave adiabatically for a fast enough timescale. The stability margin can be made larger by increasing the heat sink (e.g. by helium amount) and its efficiency in absorbing heat inputs (i.e. the heat transfer). This is typically the way followed to stabilize large, pulsed magnets which are designed for use in energy storage or thermonuclear fusion applications. The disturbance spectrum is dominated in these cases by electromagnetic energy coupling through a.c. losses, which are generally much larger than the enthalpy margin of the superconducting wire itself. Several options are possible to increase the helium amount and the heat transfer. In a force-flow conductor, for instance, the helium flows in channels inside the conductor, and the strands are subdivided to increase their wetted perimeter and improve turbulent heat transfer (see section B3.4 and chapter B6 for more details on this concept). Another option is to use superfluid helium, which has an exceedingly high heat transfer rate, in close contact with the wire. In any case the superconducting cable is in a metastable situation, namely it can be quenched by a large enough energy input. The art consists in reaching the desired stability margin for reliable operation with maximum operating current density. Here we will give some examples of magnets designed with this goal in mind and successfully operated. B3.3.4.1 MRI magnets One of the most widespread and well known applications of superconductivity is in magnets for MRI. These magnets are solenoids with a very good field homogeneity and large bore, as large as 1 m in diameter

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and 1.5 m in length, to allow full-body scans of human beings. Typical field levels in the bore of the solenoid are at present in the 1 T range. For cost and maintenance reasons these magnets are built with high operating current densities, and with little or no cryogen in the winding pack. They are essentially adiabatic, and therefore they must be carefully designed to avoid training and quenches. An MRI magnet winding pack is subdivided into a series of thin coaxial, possibly nested solenoids that produce the field and correct for winding and geometrical errors. To obtain a good field homogeneity the winding geometry must be controlled and maintained tightly. In addition the contribution from the magnetization of the superconductor must be minimized. Because of these requirements MRI magnets are generally wound from single wires with medium-sized superconducting filaments. They are impregnated so that the winding pack forms a single rigid unit and the wires are constrained in position. Cooling is indirect, by conduction through the winding pack. The thin winding pack allows heat removal under a small temperature gradient. The magnet is wound with Nb—Ti wire with a high copper:Nb—Ti ratio, of the order of five to ten, mostly for protection reasons because of the large inductance of the coil. Wires for MRI magnets are produced with round or rectangular cross-section (to ease winding), and have external dimensions of the order of 1–2 mm. The Nb—Ti filaments have a typical diameter of 50 µ m. At low field they are delivered with a guaranteed critical current density in the Nb—Ti cross-section in the range of 5000 A mm– 2. These must be compared with operating current densities in the range of 100 A mm– 2 in the strand, i.e. of the order of 500–1000 A mm– 2 referred to the Nb—Ti cross-section. We see at once that MRI magnets are designed with large operating margins to increase their reliability. Still, additional care is necessary. We can appreciate this by estimating the energy release due to hypothetical wire motion, and comparing it to the stability margin. Operating at a strand current density mentioned above of 100 A mm– 2 in a 2 T field (an estimate of the typical peak field in the winding pack) results in a Lorentz force density in the wire of

This force is reacted against the other wires in the winding pack. However, even in a tightly packed winding the wire can move small distances, because of the geometrical tolerances on the wire dimensions and the limitations on the control of the winding geometry. Movements δ of 20 µ m, the typical tolerance on the wire size, are not uncommon if the winding pack is not impregnated. We can give a rough estimate of the energy release associated with such a movement by calculating the work performed by the Lorentz force when the wire is displaced

This energy is dissipated partially as mechanical friction against the other wires and partially as a resistive loss induced by the electric field on the moving wire. In the adiabatic case we can estimate the stability margin as the enthalpy of the wire from operating conditions to the current-sharing temperature. The main component in the wire is copper, which has a very low specific heat at liquid-helium temperature. Values range from 0.1 J kg–1 K–1 at 4.2 K to 0.5 J kg–1 K–1 at 8 K. If we assume a temperature margin of 4 K from operating conditions (4.2 K) to current sharing (approximately 8 K), consistent with the values quoted above of the operating and critical current densities, we have an energy margin of the wire of the order of 10 mJ cm– 3. This value is larger than our estimate of the energy release, but does not leave much contingency to cope with uncertainties in the actual temperature margin and other possible energy inputs. Resin impregnation of the winding pack, as mentioned previously, is common practice to avoid movements and thus to minimize the energy release through wire motion. Note that cracking of the impregnation resin during cool-down and energization— resins have large thermal contraction from room temperature to 4.2 K, but little tensile strength at cryogenic temperatures—can also be a source of localized energy release. This is generally avoided by filling void volumes in the winding pack with fillers and fibre cloths or ropes that increase the tensile strength.

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B3.3.4.2 The Euratom LCT coil The Euratom LCT coil was built in the framework of the LCT project, a multinational effort to demonstrate the feasibility of a toroidal field system for a thermonuclear fusion reactor (Beard et al 1988). The coil was wound in a D-shape using the two-in-hand technique in seven double pancakes. The winding pack was epoxy impregnated under vacuum and enclosed in a thick steel casing which reacted most of the electromagnetic forces. At the nominal operating current of 11400 A the maximum field produced in the winding during full-array tests in the IFSMTF test facility was 8.1 T and the stored energy was about 100 MJ. The conductor itself, shown in figure B3.3.4, was obtained using Roebel cabling with 23 rectangular Nb—Ti strands (with a copper stabilizer) around a central steel foil, and this core was encased in a steel jacket, producing a flat cable of 10 mm thickness and 40 mm width. The helium could flow between each strand within the leak-tight jacket. Each strand, 2.35 × 3.1 mm2 in size, contained 774 Nb—Ti filaments with a nominal diameter of 45 µm.

Figure B33.4. The Euratom LCT conductor. Courtesy of Forschuneszentrum Karlsruhe.

The current density in the strands was around 70 A mm– 2 in nominal conditions. This value is more than twice as high as that of the BEBC conductor described earlier, and with an operating field increase from about 5 T in the BEBC magnet to about 8 T in the Euratom LCT coil. The cooled perimeter of this complex configuration was estimated to be of the order of 165 mm, and at the nominal flow conditions the heat transfer coefficient was approximately 600 W m–2 K–1. If we calculate the Stekly coefficient for these specific conditions we obtain a value of α ≈ 4, which is much above the cryostable limit. The disturbance spectrum during the operation of a toroidal field (TF) coil in a fusion experiment is expected to be dominated by a.c. loss deposition during the field change associated with the sudden instability of the plasma column, or plasma disruption. Tests were performed on the Euratom LCT coil pulsing an external coil and producing field changes up to 0.3 T in a timescale of 0.5 s. This deposited in the conductor an energy of the order of 15 mJ cm– 3 of strand volume, without causing a quench (Beard et al 1988). Calculations and measurements showed that for heat inputs in a short timescale (0.5 ms) the stability margin was of the order of 10–30 mJ cm– 3 of strand volume (Schmidt 1984) in conditions comparable to the operating point of the cable. Over longer timescales the stability margin increased as more time was available to transfer heat to the helium. No measurements are available for the conditions of the field pulse test quoted above, but a rough estimate, considering that the stability margin scales as the square root of the timescale of the energy deposition (Schmidt 1990), results in a minimum stability

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margin of the order of 100 mJ cm– 3, well above the energy deposited by a.c. loss. Indeed, the coil never had a spontaneous quench during testing. B3.3.4.3 The Tore Supra toroidal field magnet Tore Supra (Torossian 1993) is a Tokamak built in the 1980s at the Centre d’Etudes de Cadarache (France). Its TF magnet is completely superconducting and operates in a stagnant subcooled superfluid helium bath. The TF magnet is composed of 18 circular coils, wound out of a monolithic composite conductor in 26 double pancakes. The double pancakes are separated by spacers that maintain electrical insulation but allow the free flow of helium around the conductor and ensure a helium percentage in the winding pack of the order of 50% of the conductor volume. The winding pack is kept under compression by an external steel casing which provides the tightness for the superfluid helium bath, which is maintained at approximately 1.8 K temperature and 1.3 bar (105 Pa) pressure in normal operating conditions. At the operating current of 1400 A the maximum field produced on the winding pack is 9 T, for a stored energy in the TF magnet of 610 MJ. The conductor (see figure B3.3.5) is a rectangular wire, of dimensions 2.8 × 5.6 mm2, with 11 000 Nb—Ti filaments of 23 µ m diameter in a mixed copper and Cu—Ni matrix. The nominal Nb—Ti cross-section is 4.6 mm2 and the copper cross-section is 10 mm2. At the operating conditions the current density in the wire is approximately 90 A mm– 2.

Figure B3.3.5. The Tore Supra conductor, AISA version. Courtesy of Association Euratom-CEA.

For Tore Supra the tolerance against the disturbance spectrum was formulated requiring that the conductor must be able to recover: (i) after a localized (length of the order of some millimetres) temperature excursion up to 30 K, or (ii) after a global (one full pancake) temperature excursion to 15 K, or (iii) after a plasma current disruption when the conductor is subjected to a field change of 0.6 T in 10–20 ms. Stability in superfluid helium has peculiar characteristics in comparison with the situation of a conductor wetted by boiling or supercritical normal helium. The main difference is the large heat transfer capability of superfluid helium (Van Sciver 1986). At small heat fluxes the heat transport in superfluid helium is virtually infinite and the heat transfer coefficient h from the conductor to the helium is mostly governed by the Kapitza resistance at the wetted surface, with large values in the range of several thousands of W m– 2 K– 1 (Van Sciver 1986). The picture is different for large heat fluxes. Both in steady-state and

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in transient conditions there is an upper limit to the heat flux that can be supported by superfluid helium before reaching the transition to the normal state, the so-called lambda line. This limit depends both on the helium state and on the geometry. In the case of 1 bar (105 Pa) subcooled superfluid helium, which is the operating condition of Tore Supra, a normal helium film forms at the wetted surface as soon as the peak heat flux is exceeded (Seyfert et al 1982). At the same time the heat transfer drops while the conductor temperature rises sharply. The consequence is that for small heat fluxes, e.g. deriving from mechanical energy releases, the heat removal is such that the helium heat capacity available for stabilization can be used completely. Larger energy depositions can be tolerated until the associated heat flux is below the maximum allowable value. This fact limits the available heat sink, as seen from the conductor side, to a fraction of the total helium volume. In the case of Tore Supra, calculations and experiments were performed to guarantee that the conditions given above could be satisfied. In particular a 60 m long cable was tested in conditions comparable to the operation of the TF coil (Aymar et al 1981). It was found that at the nominal operating current of 1400 A the cable was stable against a field pulse (1 T in 8 ms) comparable to the one required in the design specifications. The a.c. loss deposited by this field pulse was around 35 mJ cm– 3 of wire, and in these conditions no normal zone could be detected. The average heat flux associated with such an a.c. loss is approximately 5 kW m– 2. This value, for the geometry of the cooling channel of the Tore Supra conductor, is well below the critical heat flux limit, which can be estimated to be of the order of 100 kW m– 2 (Van Sciver 1986). As the heat flux does not limit heat transfer, practically all the helium enthalpy from the operating temperature to the lambda transition is used for stabilization. The typical helium enthalpy from 1.8 K to Tλ is of the order of 300 mJ cm– 3 of helium volume, i.e. approximately 150 mJ cm– 3 of strand volume. This last is a good estimate of the stability margin at normal operating conditions. References Aymar R, Deck C, Genevey P, Lefevre F, Leloup C, Meuris C, Palanque S, Sagniez A and Turck B 1981 Global test of the conductor for Tore Supra under actual working conditions IEEE Trans. Magn. MAG-17 2205-8 Beard D S, Klose W, Shimamoto S and Vecsey G 1988 The IEA Large Coil Task Fusion Eng. Design 7 Chester P F 1967 Superconducting magnets Rep. Prog. Phys. 30 561–614 Haebel E U and Wittgenstein F 1970 Big European Bubble Chamber (BEBC) magnet progress report Proc. 3rd Int. Conf. on Magnet Technology, DESY (Hamburg, 1970) pp 874–95 Krantowitz A R and Stekly Z J J 1965 A new principle for the construction of stabilized superconducting coils Appl. Phys. Lett. 6 56–7 Reed R P and Clark A F 1983 Materials at Low Temperature (American Society for Materials) Rutherford Laboratory Superconducting Applications Group 1970 Experimental and theoretical studies of filamentary superconducting composites J. Phys. D: Appl. Phys. 3 1517–85 Schmidt C 1984 Stability tests on the Euratom LCT conductor Cryogenics 24 653–6 — 1990 Stability of superconductors in rapidly changing magnetic fields Cryogenics 30 501–10 Seyfert P, Lafferranderie J and Claudet G 1982 Time dependent heat transport in subcooled superfluid helium Cryogenics 22 401–8 Stekly Z J J and Zar J L 1965 Stable superconducting coils IEEE Trans. Nucl. Sci. NS-12 367–72 Torossian A 1993 TF-coil system and experimental results of Tore Supra Fusion Eng. Design 20 43–53 Turck B 1991 Stability and protection of Tore Supra superconducting coils Cryogenics 31 629–33 Van Sciver S W 1986 Helium Cryogenics (Oxford: Clarendon) Wilson M N 1983 Superconducting Magnets (New York: Plenum)

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B3.4 Cable in conduits L Bottura

B3.4.1 Introduction Cable-in-conduit conductors (CICCs) are the preferred choice for magnets that must operate in a noisy electromagnetic and mechanic environment, whenever the operating conditions require a reliable and stiff design. Because of the large energy perturbation spectrum to which they are subjected they must be designed for stability, which thus becomes one of the driving issues in the selection of the cable composition and layout. Stability in CICCs is different from classical stability theory in wires and cables for three main reasons: •

the largest heat sink providing the energy margin is the helium and not the enthalpy of the strand themselves or conduction at the end of the heated length; • this heat sink is limited in amount; • finally, the helium behaves as a compressible fluid under energy inputs from the strands, implying additional feedback on the heat transfer coefficient through heating-induced flow.

As a consequence, the main issue in CICC stability is the heat transfer from the strand surface to the helium flow and the thermodynamic process in the limited helium amount. This chapter reviews the guidelines that motivated the choice of CICCs to obtain an effective and stable superconductor design for large magnets, such as those for fusion or magnetohydrodynamics (MHD) application, the particular features of the stability margin in CICCs and the models commonly used to compute and optimize the stability margin in a CICC. B3.4.2 Stability and a brief history of CICCs From the beginning of this conductor concept stability was the initial motivation for the development of CICCs. Cryostable pool boiling magnets (i.e. satisfying the Stekly criterion (Stekly and Zar 1965) are known to have low operating current density, and thus large size and high cost. There are two possible ways of improving their performance. If the spectrum of the energy perturbations has a low amplitude the conductor can be designed to operate above the cryostability limit—the main aim of this line is to decrease as much as possible the energy perturbations (motions, cracks, a.c. losses) so that the highest possible operating current can be achieved. On the other hand, large-size magnets operating in mechanical or electromagnetic noisy environments (for instance operating in rapidly changing magnetic fields or subject to significant stress cycles) require a minimum energy margin to withstand typical perturbations which cannot be absorbed adiabatically in the small heat capacity of the conductor. In this case it is necessary to

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increase the heat transfer from the superconductor to the helium, so that the cryostability limit is pushed to higher current densities. The CICC concept evolved from the internally cooled superconductors (ICSs) which had found applications in magnets of considerable size between the late 1960s and early 1970s (see in particular the work of Morpurgo (1970)). In ICSs the helium is all contained in the cooling pipe, very much like standard water-cooled copper conductors. The conductor can be wound and insulated using standard technology and the magnet is stiff both mechanically and electrically, a considerable advantage for medium- and largesize systems requiring, with the increasing stored energy, high discharge voltages. Control of the heat transfer and cooling conditions is achieved using supercritical helium, thus avoiding the uncertainties related to a flowing two-phase fluid. A major drawback of this concept, however, was the fact that in order to achieve good heat transfer (and thus stability and high operating current density) in the early ICS layouts the helium would theoretically have to flow at astronomic flow rates. The advantage of the increase of the wetted perimeter obtained by subdivision of the strands was already clear at the beginning of the development of ICSs (Chester 1967). Hoenig and coworkers (Hoenig et al 1975, 1976, Hoenig and Montgomery 1975) and Dresner and coworker (Dresner 1977, 1980, Dresner and Lue 1977) developed models for the local recovery of ICSs after a sudden perturbation, where they found that for a given stability margin the mass flow required would be proportional to the 1.5th power of the hydraulic diameter. This consideration finally brought Hoenig and coworkers (Hoenig et al 1975, Hoenig and Montgomery 1975) to present the first CICC prototype idea, shown in figure B3.4.1.

Figure B3.4.1. The original concept of the CICC, as presented by Hoenig et al (1975). Reproduced from Hoenig et al (1975) by permission of Servizio Documentazione CRE-ENEA Frascati.

Although many variants have been considered, the basic CICC geometry has changed little since. A bundle conductor is obtained by cabling superconducting strands, with a typical diameter in the millimetre range, in several stages. The bundle is then jacketed, i.e. inserted into a helium-tight conduit which provides structural support. Supercritical helium flows in the conduit within the interstitial spaces of the cable. With the cable void fractions of about 30 to 40% commonly achieved, the channels have an effective hydraulic diameter of the order of the strand diameter, while the wetted surface is proportional to the product of the strand diameter and the number of strands. The small hydraulic diameter ensures a high turbulence, while the large wetted surface achieves high heat transfer, so that their combination gives the known excellent heat transfer properties. Strictly speaking, although it can satisfy the Stekly criterion (see later) a CICC cannot be considered as cryostable, because the amount of helium available for its stabilization (which represents the dominant heat capacity) is in any case limited to the volume in the local cross-section. The consequence is that a large enough energy input will always cause a quench, a behaviour that Dresner (1980, 1984) defines as

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metastable. The uncertain quantity is the magnitude of the minimum energy input producing a quench under a particular operating condition. This parameter, identified by Hoenig et al (1975), is fundamental to the design of a CICC. It is called here the stability margin ∆E and it is usually measured as an energy per unit strand volume ( traditionally in units of mJ cm– 3 ). In its original definition the energy input was thought to happen suddenly, and initial experiments and theory concentrated on this assumption. Throughout this chapter we will use the same definition of the stability margin, extending it to an arbitrary energy deposition timescale. Finally, in spite of the fact that the cryostability concept does not apply to CICCs, we will see in the next section that the Stekly criterion, in its original form of a power balance at the strand surface, still plays a fundamental role in its stability. B3.4.3 Experimental results and the interpretation of the stability margin in CICCs Measurements of the stability margin of CICCs started early in their history (Hoenig 1980a, 1980b, Hoenig et al 1979, Hoenig and Montgomery 1977, Miller et al 1979). The original idea of reducing the necessary flow in order to obtain the desired stability margin was frustrated as soon as the first experimental data were obtained: the stability margin was largely independent of the operating mass flow, as was recognized by Hoenig and coworkers (Hoenig 1980a, Hoenig et al 1979) (see the results reported in figure B3.4.2), and soon duplicated by Miller et al (1979). This results indicated that the heat transfer at the wetted surface of the strands during a temperature excursion was only weakly correlated to the steady-state mass flow and the associated boundary layer. In later experiments, Lue and coworkers (Lue et al 1980, Lue and Miller 1981) observed multiple stability regions as a function both of the operating current and of the operating mass flow (a typical stability margin showing the dual-behaviour curve is shown in figure B3.4.3).

Figure B3.4.2. Stability margin of a NbTi and a Nb3Sn CICC as a function of the steady-state helium flow, measured by Hoenig et al (1979). Reproduced from Hoenig et al (1979) by permission of IEEE.

As discussed by Dresner (1981) and Hoenig (1980a), during a strong thermal transient the heat transfer coefficient h at the strand surface changes mainly for two reasons (see also appendix A at the end of this section): (a) thermal diffusion in the boundary layer (a new thermal boundary layer is developed and thus h increases compared to the steady-state value), and (b) induced flow (Dresner 1979) in the heated compressible helium (associated with increased turbulence and thus again an increase in h). The concurrence of these two effects explains the weak dependence of ∆E on the steady mass flow and (at least qualitatively) the multivalued stability behaviour for different pulse powers.

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Figure B3.4.3. Stability margin of an NbTi CICC as a function of the operating current, measured by Lue and Miller (1981). The experiment was performed on a single triplex CICC of 3.8 m length (Lsample ), with a strand diameter of 1 mm (φw ), under zero imposed flow (υH e ) at a helium pressure of 5 bar (pa b s ). The background field was 6 T (B), and resistive heating took place in 16.7 ms (τh ). Reproduced from Lue and Miller (1981) by permission of IEEE.

Typical behaviour of the stability margin in CICCs was found in measurements as a function of the operating current (see the vast number of data presented by Ando et al (1986), Lottin and Miller (1983), Lue and Miller (1982), Miller (1985), Miller et al (1980) and Minervini et al (1985)). Such behaviour is shown schematically in figure B3.4.4. For sufficiently low operating current a region with a high stability margin, named by Schultz and Minervini (1985) as the well cooled regime of operation, is observed. In this regime the stability margin is comparable to the total heat capacity available in the local cross-section of the CICC, including both the strands of material and helium, between the operating temperature To p and the current-sharing temperature Tc s . At increasing currents, a fall in the stability margin to low values, the ill cooled regime, is found. In this regime the stability margin is lower than in the well cooled regime by one to two orders of magnitude and depends on the type and duration of the energy perturbation. The transition among the two regimes was identified by Dresner (1981) to be at a limiting operating current Il i m

where Ac u and ρc u are the stabilizer cross-section and its resistivity, ρw is the wetted perimeter of the cable, h the heat transfer coefficient and Tc and To p the critical and operating temperatures. The above definition of the limiting current Il i m can be derived by equating the Joule heat generation to the removal at the strand surface, and is therefore fully equivalent to the Stekly criterion (Stekly and Zar 1965). Equation (B3.4.1) sets a condition necessary for recovery: the heat transfer from the strand to the helium must be larger than the Joule heat generation. This condition is satisfied for operating currents below Il i m , i.e. in the well cooled regime. On the other hand, above Il i m in the ill cooled regime, a normal zone will always generate more heat than it can exchange to the helium, and therefore no recovery will be possible once the strand temperature is above Tc s .

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Figure B3.4.4. Schematic behaviour of the stability margin (circles) as a function of the cable operating current.

This explains the behaviour of the energy margin below and above Il i m In the well cooled regime the recovery is possible as long as the helium temperature is below Tc s . Therefore the energy margin is of the order of the total heat sunk in the cable cross-section between the operating temperature To p and Tc s , including obviously the helium. In the ill cooled regime an unstable situation is reached as soon as the strands are current sharing, and therefore the energy margin is of the order of the heat capacity of the strands between To p and Tc s plus the energy that can be transferred to the helium during the pulse. In practical cases, the heat capacity of the helium in the cross-section of a CICC is the dominant heat sink by two orders of magnitude and more, and this explains the fall in the stability margin above Il i m . The transition between the well cooled and ill cooled regimes happens in reality as a gradual fall from the maximum heat sink values to the lower limit (Miller 1985). Defining the limiting fraction il i m of the critical current Ic as il i m = Il i m /Ic , the typical extension of this fall is of the order of (il i m )1/2. An intuitive explanation of this fall can be given using again the power balance at the strand surface. For the derivation of equation (B3.4.1) it was assumed that the helium has a constant temperature Top. In reality, during the transient, the helium temperature must increase as energy is absorbed, so that the power balance is displaced, i.e. power can be transferred only under a reduced temperature difference between the strand and helium. Two limiting cases can be defined. The first is the ideal condition of helium at constant temperature, giving the limiting current of equation (B3.4.1), for which, however, the energy absorption in the helium is negligible. Operation at (and above) Il i m is necessarily associated with a stability margin at the lower limit—the ill cooled value. The second limiting case is found when the Joule heat production can be removed even when the helium temperature has increased up to Tc s . This second case is obtained for a current of (and below)

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which we call the lower limiting current in analogy to equation (B3.4.1) and due to the fact that I limlow is always smaller than Il i m . For operation at (and below) I limlow the full heat sink can be used for stabilization and the stability margin is at the upper limit—the well cooled value. Between the two values Il i m and I low the stability margin falls gradually, sometimes showing the multiple stability region in the vicinity of Il lim The multiple stability region extends over a small region, which is not interesting for the safe design im of a stable CICC, therefore this feature is usually neglected. The dependence of the stability margin on the background field B is obviously explained by the influence on the critical and current-sharing temperatures. A higher B causes a drop both in the limiting current (through a decrease of Tc and increase of η) and in the energy margin (through a decrease in Tc s ). Therefore, as expected, ∆E drops as the field increases. An interesting feature, however, is that the limiting current only decreases with Tc1/2, i.e. with a dependence on B weaker than that of the critical current. At large enough B we will always have Il i m larger than Ic and the cable will reach and the cable will reach the critical current in well cooled conditions. The stability margin depends on the duration of the heating pulse, as shown experimentally by Miller et al (1979), and reported in figure B3.4.5. A change in the heating duration for a given energy input corresponds to a change in the energy deposition power. In the well cooled regime, i.e. for low operating currents in figure B3.4.5, the heat balance at the end of the pulse is in any case favourable to recovery, and therefore the energy margin does not show any significant dependence on the pulse length. When the conductor is in the ill cooled regime, the power removal capability is limited. For short heating pulse durations the heating power increases and the conductor reaches Tc s faster than for lower powers, corresponding to longer heating durations. Therefore the energy margin increases at increasing pulse length until it becomes comparable to the total heat capacity (as in the well cooled regime). This effect is partially balanced for very fast pulses, because the heat transfer coefficient can exhibit very high values (see appendix A at the end of this section) which could shift the well cooled/ill cooled transition at higher transport currents, and thus in principle higher energy margins should be expected in this range. However, the high input powers in this duration range tend to heat the conductor above 20 K, in a temperature range where the stabilizer resistivity grows quickly and the power balance is thus strongly influenced. This effect

Figure B3.4.5. The dependence of the stability margin for a CICC (indicated on this plot as ∆H ) on the heating timescale (τh), as measured by Miller et al (1979). The parameters varied in the experiment, indicated in the inset, are the transport current in the sample Is , the helium flow velocity υHe and the helium pressure p. Reproduced from Miller et al (1979) by permission of IEEE.

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causes the saturation of the energy margin for extremely fast pulses (well below 1 ms duration). The dependence on the remaining operating conditions, typically the operating temperature and pressure, is not easily quantified. The reason is that the helium heat capacity in the vicinity of the usual regimes of operation (operating pressure po p of the order of 3 to 10 bar and operating temperature To p around 4 to 6 K) varies strongly with both po p and To p . This affects both the heat sink and the heat transfer coefficient (through its transient components). An increasing temperature margin under constant operating pressure gives a higher ∆E. But a simultaneous variation of po p and To p , under a constant temperature margin, can produce a large variation (typically of the order of a factor of two in the range given above) in ∆E (Miller 1985). A mention must be made of the case where the operating point is in the superfluid helium (HeII) range. The main difference to operation in HeI is the high heat transfer capability associated with superfluid helium. The presence of HeII has thus two effects: firstly the power balance at the strand surface is drastically changed, being displaced towards the well cooled condition. In addition a significant heat flux leaks at the end of the heated region, thus making available a larger heat sink than the volume strictly contained in the heated region only. As an example, Lottin and Miller (1983) measured the stability margin of a 2 m long conductor in an operating temperature range from 1.8 to 4.2 K. For this length the end effects are small, so that the experiment is a good way of showing the influence of the surface heat transfer. The stability margin in the case of HeII operation behaves at low current in a way similar to what would be expected in the case of HeI operation. In fact, at low current, the current-sharing and critical temperatures are well above the transition temperature Tλ from HeII to HeI (around 2 K). Heating of the strands up to current sharing implies that the surrounding helium undergoes the HeII to HeI phase transition, and the stability margin is thus governed by heat transfer in HeI. At the ill cooled transition, however, the stability margin shows a peculiar behaviour. Owing to the large heat transfer capability in HeII, the power balance at the strand surface remains favourable for recovery as long as the wetting helium is in the HeII phase. Therefore, to a first approximation, the full heat sink between the initial operating point and the transition temperature Tλ is still available at levels of the operating current at which the conductor would have changed to being ill cooled for operation in HeI. In other words, the conductor can still be considered as well cooled for temperature excursions up to Tλ. As the helium undergoes a phase transition at the temperature Tλ, the available heat sink is significant, of the order of 200 mJ cm– 3 of the helium volume. At increasing current, finally, the power balance can again become unfavourable, as soon as the heat flux limits in HeII are reached. There the final transition to the ill cooled regime of operation takes place. B3.4.4 Calculation of the stability margin The calculation of the stability margin in a CICC is a difficult task, involving accurate computation of compressible helium flow and heat diffusion in a complex geometry. For practical purposes several simplified models have been developed. These models make extensive use of the experimental evidence discussed in the previous section as a basis to introduce and justify several simplifications. Here we review the basic features of the most commonly used calculation methods, from the most complex models (still based on a one-dimensional (ID) approximation) to the simpler energy balance. Supercritical helium is the most common operating condition for CICCs, therefore we concentrate on the single-phase normal fluid. A word of caution must be given on the physical accuracy of the models discussed here. In principle a CICC has an extremely complex flow geometry that, combined with nonhomogeneous thermal loads, can lead to strong local flow effects. So far no detailed modelling of local behaviour has been done, although this is the subject of on-going experiments on stability and heat transfer. The present approach

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is to average local behaviours into effective heat transfer correlations, which will then be interpreted and used with simpler, but still cumbersome, 1D modelling. 1D models With a typical hydraulic diameter in the millimetre range, the overall helium flow in a CICC can be expected to be 1D with a good approximation over flow lengths of the order of 1 m. As the helium flows generally in a turbulent regime, the helium temperature is nearly uniform in the cross-section of the CICC. Therefore the temperature gradients in the cable cross-section reduce to those across the strand, and are negligibly small. The current distribution is also assumed to be uniform in the strands. In well designed CICCs the current can redistribute over typical lengths of the order of some centimetres in times of the order of and below 1 ms. In this case the heat generation in the CICC cross-section during current sharing is also uniform. This is not the case for CICCs with insulated strands or high transverse resistance, where the current redistribution can take several seconds over lengths of several metres. In this case a homogenized treatment is not appropriate and the stability margin is actually strongly degraded. We will therefore drop this case in the following treatment. As the stability transients are fast compared with the thermal diffusivity of the conduit materials (e.g. steel) the conduit contribution to the energy balance is also neglected. These assumptions lead to a much simplified 1D model of the CICC, where two constituents are identified: the helium and the strands. Both are at uniform, but distinct temperatures. The compressible flow equations in the helium (mass, momentum and energy balances) are written including wall friction, which is modelled using a turbulent friction factor. Strand and helium exchange heat at the wetted surface, and the thermal coupling is usually modelled using correlations for the heat transfer coefficient h as discussed in appendix A at the end of this section. The system is then described by the equations

where ρ is the helium density, p its pressure and THe its temperature, ν is the flow velocity, f the friction factor and AHe and Dh are the helium cross-section and the hydraulic diameter respectively. The total specific energy ∈ is defined as the sum of the internal specific energy i and the kinetic specific energy, i.e.

The term on the right-hand side of equation (B3.4.4) represents the thermal coupling of strands (at temperature Ts t ) and helium at the wetted perimeter pw with a heat transfer h. In equation (B3.4.5) for the heat diffusion in the strand, we defined additionally an average strand volumetric heat capacity Cs t and heat conductivity Ks t , the strand area As t and the external and Joule heat sources (per unit strand length) • • q′E and q′Joule respectively. The Joule heating can be computed once the critical current dependence on xt the temperature Ic (T) is known. An accurate calculation of q•′Joule is necessary to describe properly the recovery phase. The most general procedure consists in splitting the current in the strands into a portion

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carried by the superconductor Ic and a portion shared by the stabilizer Io p — Ic . This second produces an electric field E given by

and the Joule heating term can then be obtained as

Equation (B3.4.6) is general in nature, and in the case where Ic ( T ) is linear it reduces to the well known linear dependence of q•′Joule as a function of T between Tc s and Tc. To understand the features of the 1D model, it is important to compare the spectrum of the timescales of the phenomena. The first timescale of interest is that of the external heating, for which the dominating mechanisms in CICCs are mechanical energy release (movements, slips) or a.c. losses. The timescale for the energy release τe can be very short for a mechanical perturbation (of the order of a millisecond and faster), while in the case of a.c. losses the fastest energy deposition is limited by the largest time constant of the coupling currents in the cable, generally ranging from some milliseconds to some tens of milliseconds and longer. The heating of the strands over a length L causes a heating-induced helium flow transient. Within the 1D approximation, and assuming a uniform heating over the length L, the induced flow can only be established on a timescale τs longer than the time needed for the propagation of the sound waves in the heated region. With an isentropic sound speed c, the characteristic time τs is of the order of

In reality, as in CICCs the flow is mostly governed by the friction force and a significant induced velocity is established with slower rate. Combining the equations of the model, and assuming that inertia can be neglected, it is possible to obtain a nonlinear diffusion equation for pressure. The linearized pressure diffusivity α in this equation is given by

and the characteristic time τp needed for the establishment of the pressure profile and the associated induced flow is

Finally, the heated strands are coupled thermally to the helium. When the heating power drops, i.e. during recovery, the temperature difference between the two evolves with a characteristic time τr given by

Taking a typical heated length in the range of 1 to 10 m and helium at 4.5 K and 5 bar we have that τs is of the order of 4 to 40 ms, τp is between 6 and 600 ms and finally τr is below 1 ms. Assuming, as discussed previously, a characteristic energy input time τe ranging between 1 and 100 ms, we can conclude that the influence of the global flow changes is significant during the transient only for very short normal zones or very long energy input times. In any case the temperature recovery is a local process, as it happens on a timescale much shorter than all others. On this timescale only local flow changes, e.g. those induced by the complex nature of the flow passage or nonuniform heating, are possible. These local phenomena,

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which are associated with a typical length of the order of the hydraulic diameter Dh , can govern local heat transfer, but cannot be easily quantified. The 1D model introduced above is widely used for detailed calculations of the stability margin, and has the advantage that it can be directly adopted to follow the evolution of the normal zone when the energy input is large enough and the coil quenches. The only significant modification in this case is the need to take into account the additional heat capacity of the conduit material. This modification is straightforward and consists of adding the temperature diffusion equation to the system. Several versions of the 1D model have been implemented at research laboratories, differing mainly in the solution procedure adopted. In all cases the search for the stability margin is the virtual analogue of the experimental technique (see appendix B), i.e. a trial-and-error procedure on the energy input. Because of the level of fine detail, even within the simplification of the 1D assumption, this model gives the possibility of wide parametric analysis. Its main drawback is that, in dealing with largely different timescales, it is slow and not easy to handle. In conjunction with 1D modelling, it is worth mentioning here the adjustments necessary in the case of superfluid helium. As discussed previously, HeII is characterized by a large heat transport capability, appearing in practice as a modification of the heat transfer coefficient (governed by Kapitza conductance, see appendix A) and a nonlinear conduction term in the energy balance. For short heated lengths the second term can significantly influence the power balance, resulting in increased stability through end conduction. Details on modelling and an analytical treatment of this contribution in the case of short heated zones is presented by Dresner (1987). Zero-dd imensional models As discussed in the frame of the 1D treatment, the recovery process in large magnets, often subject to energy inputs over long lengths, is a matter of local heat transfer and heat capacity. Therefore an obvious simplification of the 1D model consists in neglecting the length effects and assuming that locally a zero-dimensional (0D) balance between heat production and accumulation holds. Still maintaining the fundamental distinction between strand and helium temperature, it is possible to write this 0D balance as follows

where the helium volumetric heat capacity CHe has been introduced. The main attraction of this model is its simplicity; it can be solved efficiently and used routinely. It is accurate in describing the local energy balance on the timescale of recovery, but some care must be taken in the selection of the parameters in order to match the features of the 1D flow which have been neglected. The first parameter is the volumetric helium heat capacity. As known from thermodynamics, the volumetric heat capacity in a compressible fluid depends on the process assumed. Two exact cases can be identified: a process at constant volume, where we have CHe = ρ Cυ , or the case of constant pressure, where we have CHe = ρ Cp . The proper selection depends on the comparison of the characteristic times identified in the previous section. In a transient where the flow characteristic times are much larger than the heating and recovery time (i.e. for a long heated length or fast heating pulse) the process will be at constant volume. Approximate constant-pressure conditions will be found when the flow characteristic times are much smaller than the heating time (short heated zone or long pulse). Clearly, the real process will be in between these two extremes. As discussed by Dresner (1995), the fact that the pressure changes during the recovery process can result in some cases, namely when the pressure decreases significantly before the recovery has taken place, in an apparent increased heat sink compared to the two cases considered above. Usually, however, the increase in the heat sink is small, in the range of some 10%, and a constant-pressure process is in practice the one associated with the largest heat sink (see case study 1 at the end of this section for a quantification of the differences between the two processes).

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The second parameter which requires care is the heat transfer coefficient, changing in time during the transient. While the boundary layer formation and the associated diffusive component of the heat transfer coefficient can be approximated in a local treatment as a variable thermal resistance between strands and helium, the heating-induced flow and its effect on stability are not amenable to local treatment. An average value for this component is a reasonable choice, but the actual modelling is to a large extent left to empiricism. This is in fact one of the research areas on the stability margin in CICCs. As a final remark, we note that the 0D model can also be used for the analysis of quench, adding the contributions of other heat capacities in the CICC (i.e. the conduit). Due to its local nature, however, it can give information only on the value of the hot-spot temperature. Energy balance A final simplification of the model can be achieved by substituting for the time-dependent power balances of the 0D model a time-independent energy and power balance in the cross-section. This method gives a rough estimate of the stability margin in the well and ill cooled regimes (called here ∆Ew c and ∆Ei c ) based on the available heat capacities and the location of the well cooled/ill cooled boundary (neglecting the dual-stability region), and has the advantage of producing easily applicable design criteria for the selection of the cable layout (see also next section). We introduce the maximum heat sink in the cable cross-section (referred to the unit strand volume) ∆Em a x

and, from what has been said previously, in the well cooled regime we will then have that

i.e. the energy margin is at most equal to the available heat sink up to Tc s , and in general smaller than ∆Em a x . A first reason is that during the heat pulse τe and the recovery time τr the Joule heat generated by the current-sharing strand uses up the available heat capacity in a measure that can be approximated, referred again to the strand volume, by

The Joule heat contribution increases at increasing operating current and increasing energy deposition time, although the above approximation tends to give only an upper limit and overestimates the real contribution (the strands are assumed fully normal for the whole transient). Still, for fast and most common heating pulses (typically in the 1 to 10 ms range) the term above is small. Furthermore, the helium temperature increases during the transient as its heat capacity is used to absorb the heating from the strands. As discussed in section B3.4.3, this decreases the temperature difference between the strand and helium, and changes the power balance at the strand surface. The effect is that only below the lower limiting current Ilimlow can the full temperature margin be used, as the helium temperature can increase up to Tc s and recovery is still possible. In these conditions, we have that ∆Ew c ≈ ∆Em a x (neglecting the Joule heating contribution). Although an analytic treatment of both contributions could be possible, the uncertainties in the definition of the limiting current, mostly related to the value of the heat transfer coefficient, make a more detailed analysis cumbersome and not very useful. Generally a conservative value is assumed for h, placing the estimated (upper) limiting current below the real, expected value, and in fact close to the lower limiting current. It is then justified to neglect the gradual fall of ∆E, and to take

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For operation in the ill cooled regime at and above Il i m the energy margin can be approximated as the sum of the strand heat capacity up to Tc s and the energy transferred from the strand to the helium during the heat pulse (Schmidt 1990)

where the second term on the right-hand side is an approximation of the energy transferred to the helium under the assumption that the strands rise instantaneously to Tc s and the helium temperature To p does not change significantly. For short energy pulses the use of equation (B3.4.11) shows that generally ∆Ei c « ∆Em a x . The energy margin given by equation (B3.4.11) tends to increase when the energy deposition time τe increases, which is consistent with the experimental results quoted earlier. For very long pulses the power input in the strand can be transferred to the helium without a significant temperature difference. At the limit of long pulse times (e.g. in the second timescale and longer) the whole heat capacity is used again and we have that ∆Ei c ≈ ∆Em a x . In any case the value of the maximum heat sink ∆Em a x of equation (B3.4.9) remains the absolute upper limit of the stability margin. In summary, equations (B3.4.10) and (B3.4.11) give the estimated energy margin respectively below and above the limiting current of equation (B3.4.1). B3.4.5 Stability-optimized CICCs CICCs are used for magnets of large size and considerable cost investment. Therefore it is advantageous to have a procedure to design a CICC satisfying a set of given design criteria and resulting in a minimum cost of the magnet. Once the magnet geometry is fixed, the magnet cost is to a first approximation proportional to the inverse of the cable space current density Jo p in the CICC, where the cable space is the cross-section of the CICC bounded by the conduit, i.e. the area of the cable including strands, helium, barriers and excluding the structural material. One of the requirements in large magnets (e.g. for fusion applications) is that the CICC should have a minimum energy margin ∆E in order to withstand the external and intrinsic perturbations related to the operation. The main parameters to be selected are the material fractions in the cable space. Therefore in order to define the optimum design, it is useful to work in terms the following fractions of materials in the cable space area Ac s (so that the optimization becomes independent of the actual cable size): copper fraction: fc u = Ac u /Ac s noncopper fraction (An c is the noncopper area): fn c = An c /Ac s helium fraction: fHe = AHe /Ac s where we have that (by definition of cable space)

The following relations for the wetted perimeter and the hydraulic diameter of the cable can be easily derived

where N is the number of strands in the cable, d the strand diameter and Kp is a factor smaller than one that takes into acount the reduction of the wetted perimeter due to cable compaction.

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Although an optimization process can involve any of the models for the stability calculation illustrated in the previous section, the use of the energy balance offers the advantage of clarity and the possibility of an analytical treatment (Dresner 1991, Bottura 1993). It is actually found, exploring the parametric influence of the copper fraction, that the optimum is given by a cable designed for operation with the desired energy margin at the limiting current of equation (B3.4.1) (see also case study 2). It is advantageous then to express the limiting current in terms of cable space current density Jo p (using the material fractions definition and equations (B3.4.12) and (B3.4.13)):

Similarly, using the property that at (and below) the limiting current we have that ∆Ew c ≈ ∆Em a x , neglecting the heat capacity of the metals and assuming constant helium heat capacity and linear Jc (T), we can write a condition on the cable space current density in order to achieve the desired energy margin ∆E

where Jc is the noncopper critical current density (at the given operating point). Now the optimal cable will be such that Jlopim = Jop∆Ε , where the amount of copper is just sufficient to operate at the limiting current and that of the superconductor gives exactly the temperature margin requested. This condition can be used to compute either analytically, or by a numerical scan (including in equations (B3.4.14) and (B3.4.15) more realistic nonlinearity in the properties of the materials), the optimal copper fraction in the cable. This optimal selection can be used to generate optimized cable designs as a function of void (helium) fraction, operating condition, superconducting material, etc. The numerical scan has the advantage that additional conditions, e.g. on the protection of the magnet, can be inserted as limits on Jo p , thus selecting cables which satisfy the design constraints. This design procedure gives an approximate starting point. Obviously more sophisticated analysis is necessary once the design point is set in order to refine the selections. Case study 2 shows an example of CICC optimization. B3.4.6 Research directions As might be clear after the remarks in the previous sections, stability depends in a synergistic manner on the d.c. and a.c. operating conditions of the cable in the coil (Painter et al 1992, Tada et al 1989, Takahashi et al 1993). This is the main direction of the actual research in the field of CICC stability. In particular, in view of the applications to pulsed magnets, the interaction of stability, current distribution and a.c. losses in the cable is one of the main topics. The so-called ramp-rate limit of operation for pulsed magnets (Painter et al 1992, Takahashi et al 1993) (a decrease in the maximum achievable current at increasing field change rate, illustrated in figure B3.4.6) is an outstanding example of this synergistic interaction. The appearance of such a phenomenon, explained so far in terms of nonuniform current distribution and a degradation of the stability margin of the cable, has alerted us to the difference between d.c. stability, with constant operating current and background field, and a.c. stability of the cable. However, while d.c. operating conditions are easier to produce and simulate, a.c. stability is difficult to measure and poses some basic problems in the interpretation of the data. The simulation and prediction of a.c. stability are therefore areas of immense interest in the field of transient electromagnetics in superconductors and thermohydraulics. On the other hand, the picture of d.c. stability is also not fully consistent. The behaviour of the heat transfer coefficient to helium in the complex structure of a CICC is of fundamental relevance for the design. Experiments aimed at investigating the magnitude of the heat transfer coefficient distinguishing local turbulences from the contribution of the heating-induced helium

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Figure B3.4.6. Ramp-rate limitation in the US-DPC coil during pulsed tests (Painter et al 1992). The maximum current reached (indicated by Iq in the case of quench or Im in the case of no quench) is plotted against the time needed for the linear ramp-up (indicated by tq in the case of quench or tm in the case of no quench). Reproduced from Painter et al (1992) by permission of Technology and Engineering Division MIT Plasma Fusion Center.

expulsion are under way (Bottura et al 1995). Finally, the interaction of temperature transients with cable magnetization has been proven to result in an improved tolerance of the CICCs to fast magnetic field pulses, owing to a self-limiting mechanism in the energy deposited by a.c. losses when the strand starts

Figure B3.4.7. Helium heat capacity per unit volume for a constant-pressure and a constant-density process, plotted as a function of the temperature in the relevant range, taking the initial pressure as a parameter. Note that in the constant-density case the pressure will change (increase) as temperature increases.

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current sharing. Initiated by mere considerations of flow velocity and pumping power requests, the field of CICC stability is still expanding. As briefly demonstrated here the research direction is towards experiments on coupled effects, involving extremely complex interactions with cable losses, current distribution and local fluid flow dynamics. Case study 1 Heat sink provided by the helium We give here a quantification of the effect of the constant-pressure or constant-volume assumption and the typical orders of magnitude for the helium heat sink for a CICC. In figure B3.4.7 we plot the volumetric heat capacity of helium (obtained using standard interpolation tables of helium properties) in the two extreme cases as a function of temperature, taking the initial pressure p0 as a parameter. A constantpressure process is characterized by a marked peak in the heat capacity, crossing the pseudo-critical line, followed by a sharp drop as the density decreases. Generally the volumetric heat capacity at constant pressure (in the typical temperature range of operation of NbTi- and Nb3Sn-based CICCs) is larger than that at constant volume. Figure B3.4.8 shows the total heat sink provided by the helium in the two conditions, computed as the temperature integral of the volumetric heat capacity, and converted into the common units for the measurement of the stability margin, i.e. mJ cm- 3 (in this case referred to the helium volume). Again we note here the larger heat sink associated with a constant-pressure process in all relevant conditions. For a typical cable with a void fraction of 40% (fHe = 0.4; fC u + fn c = 0.6), operating at 4.5 K and with a temperature margin of 2 K (Tc s = 6.5 K) the constant-pressure process would provide a maximum heat sink of about 1250 mJ cm–3 (of strand volume) at p0 of 3 bar, 1000 mJ cm– 3 at 5 bar and 800 mJ cm–3 at 10 bar. For a constant-density process the heat sink is nearly independent of pressure and is approximately

Figure B3.4.8. Heat sink provided by the helium (per unit helium volume) in a constant-pressure and a constantdensity process. The values are defined up to a constant, arbitrarily set to 0 at 4 K. The difference between two points in a curve gives the heat sink between the corresponding temperatures (referred to the unit helium volume). An example of heat sink calculation is given for operation at 4.5 K and 3 bar, with 2 K temperature margin and constant pressure assumption.

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470 mJ cm- 3. To obtain these values the curves of figure B3.4.8 have been used, taking the difference of the 6.5 K and 4.5 K points (2 K temperature margin) and correcting the result by the volume ratio of helium to strands in the cable space (0.4/0.6). The metal heat capacity has been neglected throughout. Obviously a conservative choice for the calculation of the stability margin then assumes, in any case, a constant-density process. Case study 2 Optimization of a CICC for fusion application We show here a practical selection of cable parameters for a CICC for fusion application using equations (B3.4.14) and (B3.4.15). The parameters chosen for the general properties of strand and cable are given in table B3.4.1. The optimization is performed assuming a void fraction (of 40%) and computing the limiting values of the cable space current density Jo p for different values of the copper fraction. As shown in figure B3.4.9, an increase in the copper fraction results in a higher value of the limiting cable space current density Jlopim but also in a decrease in the noncopper (superconductor) fraction and thus a lower allowable Jop∆Ε (in order to achieve the desired energy margin ∆E ). The optimal selection is obtained, in the particular case analysed here, with a copper fraction of about 41%, corresponding to a strand copper:noncopper ratio of approximately 2.2, resulting in an optimal cable space operating current

TableB3.4.1. Parameter selection for the optimization study.

Figure B3.4.9. Optimal selection of cable fractions for a fusion CICC according to the parameter selection of table B3.4.1. The optimal point is indicated at the intersection of the two curves.

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density around 70 A mm- 2. Appendix A Transient heat transfer to supercritical helium The main issue in heat transfer to a flow of supercritical helium is fast transients for their relevance to stability. A strong variation of the transient heat transfer was demonstrated by Giarratano and Steward (1983) and Bloem (1986) in dedicated measurements on short test sections (see figure B3.4.A1). The experiments showed an early initial peak in the heat transfer coefficient below 1 ms. At later times, in the range of some milliseconds to about a hundred milliseconds, the initial peak decreased approximately with the inverse of the square root of time. This behaviour could be explained in terms of the diffusion of heat in the thermal boundary layer. Using the analytical solution of diffusion in a semi-infinite body (the helium) due to a heat flux step at the surface, the effective heat transfer coefficient can be computed as (Bloem 1986)

where KHe is the heat conductivity of helium. The expression above is shown to fit properly the experimental data for times longer than a millisecond and until the thermal boundary layer is fully developed. At earlier times equation (B3.4.A1) would tend to predict an exceedingly high heat transfer coefficient, consistent with the assumptions of the analytical calculation. In reality the early values of h are found to be limited by the Kapitza resistance (Krafft 1986) at the contact surface of the strand, which gives a significant contribution only when the transient heat transfer coefficient is of the order or larger than 104 W m- 2 K (or in the case that the wetting helium is in the superfluid state). At later times, usually around 10 to 100 ms, the thermal boundary layer is fully developed and the steady—state value of h is approached. Its value appears to be well approximated by a correlation of the Dittus—Boelter form, as shown by Yaskin et al (1997) and Giarratano et al (1971). A best fit of the available data is obtained with the following expression (neglecting corrections due to large temperature gradients at the wetted surface)

Figure B3.4.A1. Transient heat transfer coefficient in supercritical helium, measured by Bloem (1986). Reproduced from Bloem (1986) by permission of Butterworth—Heinemann Journals, Elsevier Science Ltd.

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Stability of SC wires and cables An empirical expression for the heat transfer during the transient can be obtained as follows

giving a good agreement with the experimental results, and showing how for short pulses the heat transfer coefficient only depends on the helium state and not on the flow conditions. During the flow of transients generated by the heating-induced flow the two processes are combined, i.e. the boundary layer changes in thickness during the thermal diffusion process. Experimental measurements in these conditions, and in particular on transient heat transfer over long lengths, pose some major problems and results are so far not available. This issue is important, as increased turbulence in the flow can contribute to the stability margin. It is not clear whether the phenomenon has a local nature or depends on the heated length and the timescales involved in the establishment of the helium expulsion from the normal zone. Because of its relevance, this issue is the object of on-going stability experiments aimed at investigating the effect of the heated length on the stability margin. Appendix B Stability measurement techniques in CICCs The CICCs are low-loss, high-stability conductors, designed for operation in highly demanding environments. As a consequence the measurement of their stability margin is not an easy matter. The basic measurement technique for the stability margin in CICCs consists of a trial-and-error procedure, where an increasing energy is deposited in the cable and the normal-zone shrinkage or growth is monitored by means of the sample voltage signal. The measurement apparatus consists of a cryostat suitable for supercritical helium, force-flow cooling operation, a magnet providing the background field and the sample properly instrumented and provided with electric and hydraulic connections in order to feed the current and the helium flow. The sample is usually in vacuum or in an He-bath (thermally decoupled through a thick ground insulation). In order to achieve relevant measurements, the sample dimensions (i.e. the cable cross-section and its length) can be considerable. Typical sample operating currents are in the kA range, with enclosed volumes of the order of one to several ×10– 3 m3 for a sample length of some metres to some tens of metres. The measurement conditions for d.c. stability are generally set at stabilized initial operating current, background field, temperature and mass flow in the sample. Alternative measurements techniques, aimed at reproducing more closely the operation of a CICC in an a.c. magnet, make use of fast background field or operating current ramps. These last measurement conditions involve the synergistic phenomena of current distribution, a.c. losses and stability. The main sources of instabilities in CICCs are a.c. losses, due to external field changes, or cable motions. Both mechanisms generate heat in the strands. Therefore a relevant measurement of the stability margin requires that the heating device deposits the energy directly in the strands. This problem is not trivial, as resistive heaters either embedded in the cable or attached to the CICC conduit would tend to heat the helium directly (or through the jacket) and therefore could give apparently very high stability margin results (always close to the helium heat capacity). The preferred heating methods make use either of field changes on the sample or of the Joule heating during brief pulses of the operating current above the critical current level. The first method, rapid field variations, induces a.c. losses in the strands, thus reproducing a typical operating condition for a CICC for a.c. magnets. Two a.c. field configurations are commonly used, longitudinal or transverse inductive heaters. The longitudinal field variation is generated by a coil wound around the conduit. The advantage is that the volume enclosed, and thus the magnetic energy which is necessary to supply the system, is small (i.e. a small power supply can be used). Longitudinal heaters must be short for practical reasons (typically several centimetres) and thus they are well suited for stability analysis of zones heated over a short length. The main disadvantage is that the parallel field loss of a CICC is small, so that high frequencies are needed in order to reach the a.c. loss power necessary to deposit

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sufficient energy in the cable. At frequencies in the kHz range the shielding effects from the conduit and the eddy currents in the strands can become appreciable—sometimes the dominating loss—and the distribution of the energy input must be carefully examined. Transverse field changes require a field variation over a volume comparable to the sample volume, and thus they imply a large stored magnetic energy (requiring large power supplies). This type of heating, however, is more efficient than the longitudinal one as the transverse a.c. loss of CICCs is generally larger than the parallel component. The energy can be deposited uniformly over long lengths of cable, so that this type of heater is well suited for the study of lengths heated over a long length. The typical frequency range for the heating time with transverse inductive heaters is from 100 to 10 Hz. Unless the sample is wound noninductively, transverse field changes tend to change the sample current during the pulse. Therefore to ease the interpretation of the measurements it is good practice to wind the sample noninductively or to use a sample power supply capable to compensate for the inductive voltage (e.g. using series inductances). Both the transverse and the longitudinal heaters require a power supply capable of pulsed operation. This can be achieved by means of a capacitor bank, operating in free resonance with the inductive heater, or wave-form controllers to produce the desired pulse shape. In the case of superconducting pulsed coils a fast discharge can be used to generate exponential field changes. The main difficulty when using inductive heaters is the energy calibration, i.e. correlating the field variation with the energy deposited in the strand. Using dedicated a.c. loss samples for this calibration is not fully satisfactory, as the CICC cable can be heavily deformed in the process of winding a sample, with a significant change in the cable coupling loss (the main loss contribution for transverse field changes). On the other hand an in situ calibration is a delicate matter, whether using electrical methods or calorimetric methods. The most successful calibration techniques are, so far, based on the use of compensated pickup coils or on the adiabatic heating of the cable under small field pulses. With the first method the energy deposited in each shot is computed as for a.c. loss experiments, measuring the average magnetization of the sample and integrating the area of the magnetization loop. In the second technique, the sample is evacuated and the superconductor is used as a thermometer (carrying a small current) detecting a temperature increase up to Tc under the field pulse. The energy deposition is deduced from the heat capacity of the cable between the initial and final temperatures. This is obviously a one-time calibration, which cannot be performed for each pulse. Under any energy input the strand temperature increases, eventually above current sharing. In fact this can limit the amount of heat that can be deposited in the strand by field changes, because above the current-sharing temperature the a.c. loss drops to the pure eddy current loss in the stabilizer matrix, usually several orders of magnitude lower than the loss due to coupling currents and hysteresis. Such behaviour complicates the calibration of the energy deposition, and favours the use of pickup coils for the on-line measurement of the energy deposited at each pulse (the magnetization drops when the superconductor is heated above Tc s ). This fact also gives a practical lower limit for the heating time τe . For a given energy, the heating power is inversely proportional to τe and this, also taking into account the transient dependence of the heat transfer coefficient, tends to produce temperature increases in the strands proportional to τe-1/2 , so that for heating pulses in the range or shorter than typically 1 ms the a.c. loss calibration is strongly affected. A solution to the above problem and an alternative heating method is to use Joule heating shots generated by pulsing the operating current above the critical temperature Tc for the heating time. The heat that is deposited in the stabilizer can be computed by integrating the product of the voltage and current signals at the terminals (or along the length). In this case, therefore, the calibration is relatively easy. The drawback of Joule heating shots is that the measurement of the energy input at the terminals of the power supply includes also the contribution of the strand Joule heating that is present whenever the strand temperature rises above Tc s . It is not possible to distinguish between these two contributions, and therefore the measurement is not fully consistent with the definition of ∆E given here and generally accepted (i.e.

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the external heating causing a quench). For a.c. stability measurements there is, so far, no clear experimental procedure. The main reason is that the operating space is enlarged by the addition of current and field waveforms, and the stability margin is a function of the operating point and of the history of its establishment. All (d.c. and a.c.) measurement methods described here generate inductive voltages on the sample. Therefore compensated voltage signals must be used, as is standard practice for a.c. loss measurements. Co-wound voltage taps, cancelling the inductive signal, are used to monitor the normal zone. Temperature measurements on stability samples are usually limited to the outer conduit surface (apart from inlet and outlet connections) as the perforation of the conduit and insertion of sensors in the flow channel would affect the experiment by changing the flow conditions and is not practical from the manufacturing point of view. For the same reason pressure and flow measurements are limited to the inlet and outlet of the sample. In a.c. stability experiments the cable current distribution is usually measured by means of transverse voltage taps (i.e. across the conductor width), pickup coils and Hall generators placed around the cable. References Ando T, Nishi M, Takahashi Y, Yoshida K and Shimamoto S 1986 Investigation of stability in cable-in-conduit condutors with heat pulse duration of 0.1 to 1 ms Proc. 11th Int. Cryogenic Engineering Conf. (Berlin, 1976) p 756 Bloem W B 1986 Transient heat transfer to a forced flow of supercritical helium at 4.2 K Cryogenics 26 300–8 Bottura L 1993 Stability, protection and ac loss of cable-in-conduit conductors—a designer’s approach Fus. Eng. Des. 20 351–62 Bottura L, Ciazynski D, Duchateau J L and Martinez A 1995 Stability experiments on long lengths of CICC’s IEEE Trans. Appl. Supercond. AS-5 576–9 Chester P F 1967 Superconducting magnets Rep. Prog. Phys. 30 561–614 Dresner L 1977 Stability-optimized, force-cooled, multifilamentary superconductors IEEE Trans. Magn. MAG-13 670 Dresner L 1979 Heating induced flows in cable-in-conduit conductors Cryogenics 19 653 Dresner L 1980 Stability of internally cooled superconductors: a review Cryogenics 20 558 Dresner L 1981 Parametric study on the stability margin of cable-in-conduit superconductors: theory IEEE Trans. Magn. MAG-17 753 Dresner L 1984 Superconductor stability 1983: a review Cryogenics 24 283 Dresner L 1987 A rapid, semiempirical method of calculating the stability margins of superconductors cooled with subcooled He-II IEEE Trans. Magn. MAG-23 918–21 Dresner L 1991 Rational design of high-current cable-in-conduit superconductors IAEA-TECDOC-594 pp 149–63 Dresner L 1995 Stability of Superconductors (New York: Plenum) Dresner L and Lue J W 1977 Design of forced-cooled conductors for large fusion magnets Proc. 7th Symp. on Engineering Problems of Fusion Research (Knoxville, 1977) vol 1, p 703 Giarratano P J, Arp V D and Smith R V 1971 Forced convection heat transfer to supercritical helium Cryogenics 11 385–93 Giarratano P J and Steward W G 1983 Transient forced convection heat transfer to helium during a step in heat flux Trans. ASME 105 350–7 Hoenig M O 1980a Internally cooled cabled superconductors—part I Cryogenics 20 373–89 Hoenig M O 1980b Internally cooled cabled superconductors—part II Cryogenics 20 427–34 Hoenig M O, Iwasa Y and Montgomery D B 1975 Supercritical-helium cooled ‘bundle conductors’ and their application to large superconducting magnets Proc. 5th Magnet Technology Conf. (Frascati, 1975) p 519 Hoenig M 0, Iwasa Y, Montgomery D B and Bejan A 1976 Supercritical helium cooled cabled, superconducting hollow conductors for large high field magnets Proc. 6th Int. Cryogenic Engineering Conf. (Grenoble, 1976) p 310 Hoenig M O and Montgomery D B 1975 Dense supercritical helium cooled superconductor for large high field stabilized magnets IEEE Trans. Magn. MAG-11 569

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Hoenig M O and Montgomery D B 1977 Cryostability experiments of force cooled superconductors Proc. 7th Symp. on Engineering Problems of Fusion Research (Knoxville, 1977) vol 1, p 780 Hoenig M O, Montgomery D B and Waldman S J 1979 Cryostability in force cooled superconducting cables IEEE Trans. Magn. MAG-15 792 Krafft G 1986 Heat transfer below 10 Cryogenic Engineering ed B A Hands (New York: Academic) pp 171-92 Lottin J C and Miller J R 1983 Stability of internally cooled superconductors in the temperature range 1.8 to 4.2 K IEEE Trans. Magn. MAG-19 439 Lue J W and Miller J R 1981 Parametric study of the stability margin of cable-in-conduit superconductors: experiment IEEE Trans. Magn. MAG-17 757 Lue J W and Miller J R 1982 Performance of an internally cooled superconducting solenoid Adv. Cryogen. Eng. 27 227 Lue J W, Miller J R and Dresner L 1980 Stability of cable-in-conduit superconductors J. Appl. Phys. 51 772 Miller J R 1985 Empirical investigations of factors affecting the stability of cable-in-conduit superconductors Cryogenics 25 552 Miller J R, Lue J W, Shen S S and Dresner L 1980 Stability measurements of a large Nb3Sn force-cooled conductor Adv. Cryogen. Eng. 26 654 Miller J R, Lue J W, Shen S S and Lottin J C 1979 Measurements of stability of cabled superconductors cooled by flowing supercritical helium IEEE Trans. Magn. MAG-15 351 Minervini J V, Steeves M M and Hoenig M O 1985 Experimental determination of stability margin in a 27 strand bronze matrix, Nb3Sn cable-in-conduit conductor IEEE Trans. Magn. MAG-21 339 Morpurgo M 1970 The Design of the Superconducting Magnet for the ‘Omega’ Project, Particle Accelerators vol 1 (New York: Gordon and Breach) Painter T A et al 1992 Test data from the US-demonstration poloidal coil experiment, Plasma Fusion Center MIT Report PFC/RR-92-1 Schmidt C 1990 Stability of superconductors in rapidly changing magnetic fields Cryogenics 30 501 Schultz J H and Minervini J V 1985 Sensitivity of energy margin and cost figures of internally cooled cabled superconductors (ICCS) to parametric variations in conductor design Proc. 9th Magnet Technology Conf. (Zurich, 1985) pp 643–6 Stekly Z J J and Zar J L 1965 Stable superconducting coils IEEE Trans. Nucl. Sci. 12 367 Tada E et al 1989 Downstream effect on stability in cable-in-conduit superconductor Cryogenics 29 830 Takahashi Y et al 1993 Experimental results of stability and current sharing of NbTi cable-in-conduit conductors for the poloidal field coils IEEE Trans. Appl. Supercond. AS-3 610–3 Yaskin L A, Jones M C, Yeroshenko V M, Giavatano P J and Arp V D 1977 A correlation for heat transfer to supercritical helium in turbulent flow in small channels Cryogenics 17 549–52

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B4.1 Introduction to a.c. losses A M Campbell

A superconductor only has strictly zero resistance under d.c. conditions (and not always then). Whenever a changing magnetic field penetrates a superconductor an electric field is created which causes a loss. The mechanisms are varied but losses are generally found from classical electromagnetism, with only a little input from the microscopic theory of superconductors. We can divide the losses into the following broad regimes. Firstly at high frequencies (> 100 MHz) the main loss mechanism is the acceleration of normal electrons. This is greatly increased if there is any trapped flux in the sample because of both the presence of normal cores in the flux lines and viscous drag on the flux lines as they oscillate. This loss is not discussed further in this section which is directed towards high-current applications. Secondly there are low losses in the Meissner state where the only flux penetration is in the London penetration depth or at asperities. This is relevant to low-Tc cables but it depends very much on surface roughness and is difficult to quantify. Details can be found in the article by Melville (1971). Near the irreversibility line flux creep and viscous drag leads to losses in a field. This linear regime is covered in a recent review (Golovsky et al 1996). The main content of this section concerns the third regime which is that of large amplitudes and low frequencies in the critical state. This is the regime of power engineering, although similar effects can occur at high powers in microwave components if the critical current is exceeded locally at the edges of films. In practice the losses are divided into two interdependent components. The first is the hysteresis of the superconducting material, the second is due to eddy currents in the conducting matrix which surrounds it. Most applications involve magnets, and the primary loss is that of the conductor in the field due to the rest of the coil. The loss due to current in the conductor is secondary since the field it produces on its own is relatively low. However, quite different considerations apply to power cables in which the self-field of the conductor is the major factor. This leads to a second division between regimes in which the magnetic field is larger than that needed to penetrate to the centre of the superconductor, and those where the field is confined to the surface. B4.1.1 Conductor development The development of practical superconducting wires is a striking example of the advantages to be obtained by combining the expertise of physicists, materials scientists, engineers and industrialists. In the early 1960s, soon after the discovery of NbTi and Nb3Sn, there were some who thought that the potential of these compounds could never be realized because of thermal instability and flux jumps. The solution to this problem proved to be the subdivision of the superconductor into fine filaments, surrounded by a copper matrix which served both to carry away heat and to spread currents between filaments if one was damaged. However, this procedure introduced a new problem in the form of eddy currents in the copper matrix when an external field was applied. If we have straight superconducting filaments in a block of copper the flux cannot cross them and must diffuse from the ends. If the wire is a kilometre long this

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will take 200 years during which there is continuous dissipation. This problem was solved by twisting the filaments of the wire so that the effective diffusion length was the twist pitch. In electrical engineering terms the twisted wire will have a very low mutual inductance with the coil providing the external field so that induced currents are low. This technique does not, however, lower the losses due to a transport current, which would require transposition of the filaments. In this way the highly successful ‘Rutherford’ wire was developed and in succeeding years the technology has become more sophisticated in several ways. For large magnets high-current cables are needed so that the individual wires must be bonded together and driven in parallel. Insulating them with a resin can cause incomplete current sharing because of different inductances and this makes cooling difficult. It is therefore common to solder cables or leave them in only mechanical contact. However, the performance of the magnet is then difficult to calculate since we have interactions between cable pitch and wire pitch, with uncertain resistivities between the wires. It is problems of this type which have prevented some large-scale magnets fulfilling their expectations. The subdivision brought additional bonuses in the form of reduced a.c. losses and trapped fields, both of which are proportional to the filament diameter. In recent years it has become possible to produce filaments of under 10 nm in diameter which allows superconductors to be used at 50 Hz for the first time. At such small diameters the spacing becomes comparable with the coherence length so that the proximity effect allows supercurrents to flow through the copper between the filaments. To prevent this copper alloys are used which break up Cooper pairs and prevent the wire behaving as a monolithic system. The materials science required is extremely sophisticated as can be seen from the fact that the NbTi starts off with a diameter of 1 cm and a length of 1 m and ends up after drawing, repacking in copper tubes and heat treatment with a package of 10 000 filaments, each with a diameter of 10 nm and an effective length of 1012 m or ten times the distance from the earth to the sun. The prospect of doing this kind of thing to Nb3Sn might seem remote since this is an intermetallic compound with the ductility of Wedgewood china. Nevertheless even this was not beyond the skills of materials scientists who developed a diffusion process in which the Nb3Sn was formed at the interface between niobium and bronze, after the whole composite had been drawn down in a ductile state. Even more complex composites have been developed in which cupronickel barriers against eddy currents and niobium diffusion barriers were added to these conductors. Most effort in high-Tc materials has quite rightly been directed at improving d.c. fields, since this is the first requirement in most applications. However, the greater cooling power and stability at nitrogen temperatures means that even at the present stage of development there are applications of high-Tc , materials at 50 Hz which were not possible in low-Tc materials. a.c. losses in these materials are in principle calculable as before, but the flat voltage-current (V-I) characteristic and the weak links at grain boundaries (granularity) mean that more parameters are required for an accurate calculation. The need for filamentary material to ensure stability disappears at nitrogen temperatures, but the need to reduce a.c. losses remains. However, much larger losses can be tolerated in nitrogen than in helium so that the optimum size depends on economic rather than scientific factors. Multifilamentary high-Tc material has been made by a number of groups and it is found to bring benefits such as greater consistency and strain resistance so that this type of material is likely to be used in practice quite independently of its loss properties. Recently American Superconductor has been able to make twisted high-Tc BSCCO tapes with a twist pitch low enough for 50 Hz operation. B4.1.2 Orders of magnitude Before going into the details of loss mechanisms it is worth stating some general principles derived from electromagnetic theory which lay down broad limits within which detailed theories must exist. The conclusions will be justified in more detail later.

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If a body is placed in an oscillating magnetic field there are two extreme possibilities. One is that the field is completely excluded, i.e. perfect diamagnetism, and the other is that it completely penetrates the sample. The exclusion of field may be due to any combination of supercurrents and eddy currents but in both cases the loss per cycle is zero since the magnetization curve is a straight line. These two lines are shown in figure 4.1.1. Intermediate cases occur when eddy currents and supercurrents are such that the external field can just penetrate to the centre of the sample. If the demagnetizing factor of the conductor is small the maximum diamagnetic moment for an amplitude B0 is -B0 so that the hysteresis curve must lie within the triangles of figure 4.1.1. Since the loss per cycle is the area of the hysteresis loop, this puts an upper limit on the loss per unit volume per cycle of B02/µ0 . This is a very useful scaling factor by which to assess losses. It can be used for transport currents less than the critical value if B0 is the self-field of the current, and in general this loss is reduced by a factor equal to the ratio of the penetration of the field to the size of the sample at lower amplitudes, and by the reciprocal of this factor at higher amplitudes. For ohmic materials the penetration is the skin depth which is independent of amplitude but dependent on frequency. In the superconductor the depth is of the form B02/µ J0 which is dependent on amplitude but independent of frequency. (For geometries other than a slab this is multiplied by a numerical factor of order one.)

Figure B4.1.1. The magnetization curves for complete, partial and low flux exclusion.

This idea of the depth of field penetration is central to loss calculations. For small penetrations the loss is proportional to surface area and decreases with increasing Jc . For fields much larger than that needed to penetrate the sample fully the loss depends on sample volume and sample size and increases with increasing Jc . The field which just penetrates to the centre of the conductor, the penetration field, is a useful parameter with which to characterize the sample. It is given by the average Jc times the sample size (in this context size means a characteristic linear dimension which is approximately the width perpendicular to the applied field). Different formulae must be used for amplitudes above and below this value. Conductors with large demagnetizing factors such as thin strips in an external field have recently become very important since many high-Tc conductors are of this shape. For ohmic eddy currents the loss per unit volume becomes very large for thin strips in a perpendicular field, but it appears that losses in superconducting wires are relatively insensitive to the shape. The relevant dimension to which the penetration is compared may be filament size, wire size, cable size or twist pitch, according to the regime of amplitude and frequency. In most practical cases losses are reduced by allowing large field penetrations, i.e. small filaments and short twist pitch. However, the case of power cables, where self-fields are low compared with external fields, uses the opposite strategy by having a large wire radius, with a hollow core, and currents on the

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surface. The general practice is to avoid periods of the order of the time constant and amplitudes of the order of the penetration field (see section B4.1.3.2) since these are the conditions for maximum loss. B4.1.3 Hysteresis losses from the Bean model The calculation of a.c. losses in superconductors is carried out from a knowledge of the electric and magnetic fields in the material under a.c. conditions. We use average fields where the average is taken over volumes larger than any features of the microstructure. However, it is useful to have at least a qualitative idea of the microscopic processes taking place so we begin with a short summary of the basic physics. B4.1.3.1 Flux lines The properties of type II superconductors, which are the superconductors of most practical importance, depend on the fact that an external field can enter them in the form of vortices or flux lines. These are lines containing one quantum of magnetic flux, which behave much as Faraday’s lines of force. An isolated line can be regarded as a small normal core containing the field surrounded by supercurrents in the superconducting material, but at practical flux densities they form a close-packed hexagonal array. They move freely in a homogeneous material, their density is proportional to the flux density and they can be pinned by inhomogeneities in the material. Their relationship to losses is much the same as that of domain walls to hysteresis in ferromagnets, that is to say we understand the losses in terms of movement of vortices and their interaction with the microstructure. However, when it comes to calculating losses we do not need to know the details of flux lines, but can start from macroscopically measured parameters, just as a magnetic loss is calculable from the B—H loop. In the case of superconductors the relevant parameter is the critical current density Jc . However, the parallel with a magnetic material must not be carried too far. In the ferromagnetic material the distribution of domains is uniform throughout the sample and we can define a local magnetization at any point in the material. In the superconductor flux lines can only enter and leave from the surface so that the magnetic properties depend on the size of the sample, and the magnetization, defined as the total magnetic moment divided by the volume, has no useful meaning on a local scale. This is an important distinction when we try to apply ideas developed for ferromagnetic materials to superconductors and it is discussed further in chapter B4.2.3. Flux pinning and losses If a superconductor is completely homogeneous flux can move freely in it. The flux lines have normal cores and when they move there is a resistive loss in the core. This means that there is a viscous drag on the flux lines, and since the loss is supplied from a power supply there must be a corresponding electric field. If a current is applied in the presence of a magnetic field the vortices move under the influence of the Lorentz force, J × B , and the moving flux generates an electric field E = B × υ. This is illustrated in figure B4.1.2. In a homogeneous material the effective resistivity due to the continuous movement of flux lines is comparable to the normal-state resistivity and the material is indistinguishable from an ohmic conductor. Most of the superconducting properties only appear if there are defects which act as pinning centres. All real materials contain defects such as grain boundaries and dislocations, which exert a force on passing flux lines (figure B4.1.3). The defects prevent the movement of flux until the Lorentz force exceeds the pinning force. This leads to a maximum current density Jc , the critical current density, at which the driving force, J × B , is equal to the pinning force. Below this current the flux is stationary and the voltage zero. Above this current the flux moves past the pinning centres and there is dissipation. Figure B4.1.4

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Figure B4.1.2. A current passed through a superconductor in a magnetic field exerts a Lorentz force on the vortices which tend to move across the superconductor causing an electric field parallel to the current.

Figure B4.1.3. A flux line has a small normal core surrounded by supercurrents. It is pinned by any change in superconducting properties.

Figure B4.1.4. The Lorentz force can be borne by pinning centres up to a limiting current density Jc . Above this there is a large differential resistivity.

Figure B4.1.5. As a flux line passes a pinning centre it dissipates the stored line energy in the unstable depinning process.

shows the voltage—current characteristic which results from this mechanism. The dotted line shows the resistivity without pinning. The flux line depins in an unstable manner and the stored energy in the line tension of the flux line is dissipated on a microscopic scale as a local viscous loss (figure B4.1.5). Unless the superconductor is in the Meissner state, which only occurs at very low magnetic fields, every change of field, whether caused by an external field or a transport current, will move the flux in the sample and give a loss. This is the main mechanism for a.c. losses and it causes a hysteresis loss since the energy dissipated in each unpinning event depends only on the stored energy in the line tension of the flux line. In general the loss per unit volume is E J but unless we are well above Jc we can assume that J = Jc so that the loss is EJc . This is consistent with a frequency-independent hysteresis loss per cycle. There is a very close parallel with losses in ferromagnets in which the hysteresis loss is caused by unpinning of domain walls, while at high frequencies an additional resistive loss is caused by eddy currents. This description ignores the effect of thermal activation on flux lines. This causes a slow rate of unpinning at currents below Jc , called flux creep, and is more significant in high-Tc materials than in conventional ones. However, the effect on a.c. losses is not usually very important. It seems reasonably clear that in commercial NbTi the pinning centres are walls of dislocations, while in Nb3Sn they are grain boundaries. The dominant pinning centres in high-Tc materials remain to

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be identified but they are probably very small and close together. The pinning forces are always strong compared with those in single-phase annealed low-Tc materials. Some low-Tc materials can be prepared with critical current densities at half Tc and half Bc 2 of less than 1 A cm– 2, but no high-Tc material can approach this degree of crystalline perfection. Low critical current densities in high-Tc materials are due to granularity at grain boundaries and the irreversibility line. The penetration field Problems can normally be assigned to one of two limits and the appropriate limit depends on the size of the external oscillating field compared to the field needed to penetrate to the centre of the sample, a useful parameter introduced in chapter B1. Flux and currents penetrate from the sample surface and the penetration field Bp is the field at which flux first reaches the centre. At this field the critical current density completely fills the sample. Expressions for the loss are different for a.c. amplitudes below and above this field, which is µ0d Jc for a slab of thickness 2d. For other geometries with constant Jc the penetration field can be found by calculating the field at the centre of the conductor when it is completely filled with a current density Jc . Oscillating amplitudes much greater than Bp penetrate the sample fully, and are fairly easy to deal with. Low amplitudes mean amplitudes much less than Bp which do not penetrate very far into the sample and losses are more difficult to calculate due to shape effects. (Although sometimes called demagnetizing effects the demagnetizing factor is only defined for a uniformly magnetized ellipsoid and even in ellipsoids the Bean model produces quite different fields from a uniform magnetization.) The electric centre It can be seen that in an external magnetic field flux enters the superconductor from the surfaces and meets in the middle. At this point flux is stationary so that the electric field is zero and this is called the ‘electric centre’ of the conductor. If a transport current is flowing which is less than the critical value the electric centre is displaced towards the edge of the superconductor, but there is still a line along the conductor along which E = 0 and no flux crosses this line. Only when the transport critical current is exceeded is there no line with zero electric field. In this case either flux moves continuously across the superconductor from one side to the other, or flux rings collapse to the centre and annihilate. In either case there is continuous flux movement in a static external current or field which requires extensions to the Bean model to include the effects of flux-line viscosity. Practical regimes There are three main regimes of practical importance. The first is where the superconductor is in a large applied field which has an oscillating component. This occurs in magnets, particularly in rotating machines. The losses can trigger a quench, even if their magnitude is otherwise acceptable. The second regime is in a cable carrying mains frequency current. This involves losses at low field strengths, but high oscillating amplitudes compared with the mean field. Finally there are the losses caused by powering up a magnet or changing its field. Here the losses limit the rate at which the magnet can be run up to its maximum field. B4.1.3.2 A simple loss calculation As an illustration we now use the Bean model to calculate the loss in a simple case. There are several methods which can be used, but in this case we use Faraday’s law of induction to find the electric field. This has one advantage over using the area of the magnetic hysteresis curve in that it can be applied when there is a transport current. We apply a steadily increasing field parallel to the surface of a slab (figure B4.1.6). We assume that the field has penetrated to the centre so that a constant current Jc is flowing throughout the slab and the internal field is rising at the same rate as the external field, B• (figure B4.1.7). We now find the electric field in the sample. The flux enters from both sides so by symmetry the electric

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Figure B4.1.6. Circulating currents are generated in a slab by a field parallel to the surface.

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Figure B4.1.7. The field inside rises at the same rate as the external field.

field is antisymmetric, as in the case of a copper slab. We apply Faraday’s law of induction to a circuit x from the central plane. The rate of increase of flux is 2 B• wx, and the electromotive force (EMF) is 2wE, where E is the electric field at x. Hence E = B• x. The local loss is E J which in this case is E Jc per unit volume, so integrating across the slab gives a mean loss per unit volume of superconductor of

We see that this is a hysteresis loss since if we change the field by ∆B the loss is

In terms of the penetration field Bp the loss is

As pointed out above, this calculation is independent of the magnitude of the applied field provided the appropriate value of Jc is used. At high applied fields the value of Jc is constant across the sample so this is therefore a very accurate calculation of the losses in a conductor subject to a change in field small compared with a large static field, which is the case for ripple fields in magnets. At low applied fields the variation of Jc with B leads to inaccuracy. Since Bp is equal to µ0 Jcd , the loss per unit volume of superconductor is proportional to the thickness of the superconductor. Since the volume of superconductor is usually determined by the field we wish to achieve, we can reduce the losses for a magnet providing a given field by subdividing the superconductor. This is one reason why fine filaments are used in magnets. We can use equation (B4.1.3) to find the loss in a periodic field of zero to peak amplitude B, provided we ignore the part of the cycle at each extreme where the reversing field direction has not penetrated to the centre. This is legitimate if the amplitude is large compared with that needed to reach the centre. For a peak amplitude of B the loss per cycle is double that of the loss between B and —B, i.e.

Since the loss is hysteretic in nature the dissipated power is proportional to the frequency.

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B4.1.4 Eddy current or coupling losses In a conventional superconducting wire the superconductor is in the form of twisted filaments embedded in copper. The solution of Maxwell’s equations in a wire containing a large number of helical superconducting filaments embedded in copper subjected to a transverse alternating field sounds like a problem only soluble numerically. In fact there is a simple analytical solution for a round wire which can be obtained either from a circuit model (Fevrier 1987, Morgan 1970) or from an average field model (Carr 1974). Since Maxwell’s equations are linear they can be averaged over any convenient scale with appropriate definitions of the material parameters and standard methods can be used to solve for the mean potential. This technique is powerful but mathematical and others have proved useful in different situations, as will be described in chapter B4.2. The geometry is indeed complex so that it is important to understand the results from a number of different points of view. The following derivation produces the main results with the minimum of algebra and will appeal to those who think in terms of flux movement. The treatment of problems in terms of the motion of flux lines is a very powerful technique not only in superconductors but also for electromagnetic problems in normal conductors. It can be misleading if there are currents flowing with a component parallel to the magnetic field but this does not occur in many practical geometries. B4.1.4.1 Derivation of coupling currents Although rotating fields are normally considered after oscillating fields, in many ways they are simpler, particularly in nonlinear materials since the material properties remain constant in a rotating field. However, here we consider a round wire of copper containing superconducting filaments, which could be type I material, so that the system is entirely linear. The filaments are twisted with a pitch p, with a positive p indicating a right-handed twist. If we cool in a transverse external field B0 and rotate the wire or the field, the flux cannot cross the filaments so it is screwed down the wire like a nut on a bolt (figure B4.1.8).

Figure B4.1.8. If the field rotates the flux must move along the axis.

If the angular velocity of the rotating field B0 is ω the axial velocity is υ = ωρ/2π. This movement of flux down the wire axis creates a perpendicular electric field B × υ = B0ωρ/2π (at low frequencies the internal field will equal the external value). The transverse current density is σ B0ωρ/2π where σ is the transverse conductivity. This current rotates 90° behind B for a right-handed twist of the filaments. The longer the pitch the faster the velocity and the larger the current density. Since the system is linear we can add a similar field rotating in the opposite direction to produce a uniform oscillating field. Adding the two rotating current densities gives an oscillating current density σ B0ωρ/2π which is antiparallel to B for a right-hand twist. This model shows that there is a self-consistent solution of Maxwell’s equations in this geometry in which the current density is uniform across the sample and antiparallel to the applied oscillating field. The

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transverse current is picked up by the filaments at the surface of the sample, while the internal filaments carry no current. The mean loss per unit volume is 1-2 σJ2 = 1-2σ(αΒοωp/2π)2 W m-3. This is the result obtained by Carr (1974). Viewed in cross-section (figure B4.1.9) the surface current varies sinusoidally round the circumference, and the twist of the filaments means that this θ current is associated with an axial current which is much larger and goes down one side of the wire and back up the other. This is the same as in a diamagnetic cylinder and produces a uniform screening field proportional to ω, which is considered in the next section.

Figure B4.1.9. In an oscillating field J is antiparallel to B0.

The movement of flux in the rotating case is obvious, but much less clear in the oscillating case. Perhaps the best way of looking at it is to consider the removal of the applied field from a figure-of-eight loop. Flux in each circular section is forced to the cross-over, turning as it goes so that when the flux from the two loops meets it is antiparallel and annihilates. Thus in the similar case of helical filaments the effective diffusion distance is the pitch length. B4.1.4.2 Time constants The time constant of a conductor is a useful combination of material parameters (Campbell 1982, Kwasnitza 1977). In calculating the coupling losses it was assumed that the field in the conductor was equal to the external field. However, the longitudinal component of the surface current in figure B4.1.9 screens the interior in the same way as in a diamagnetic cylinder. Since at low frequencies the current density is uniform the surface current varies sinusoidally and so produces a uniform internal field in opposition to the external field. This is generally true for ellipsoids and a tolerable approximation for other shapes. This allows us to draw a close connection between the losses and the time constant of the system. The field due to induced currents is proportional to B• so in a harmonic external field of peak amplitude B0 it follows that B = B0 – jωτB where τ is an unknown constant of proportionality. We shall see below that it has the physical significance of a time constant. Now the magnetization is proportional to the current density, which is proportional to B. If this constant of proportionality is β then M = − jωβ B0/(1 +jω τ )µ0. However, we know that at high frequencies the sample is perfectly diamagnetic and M = —B0/ (1 – N )µ0 where N is the demagnetizing factor. Hence β = τ / (1 – N). • The mean loss is the time average of B0M or the real part of -12B0(jωM*). This is 1-2 B02 ω2τ / µ0(1 +

ω2τ2)(1- N) W m-3

For a cylinder N = 1-2 and comparing this with the expression above it follows that the time constant is τ = °σρ 2/8π 2. This is consistent with the surface currents of figure B4.1.9. In terms of the skin depth of copper, δ, the loss per cycle at low frequencies is (B02/µ0 )(ρ 2/2δ 2 ). The transverse current and field remain uniform, but the internal field is less than B0.

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We see that apart from a dimensionless constant of order one which depends on the shape, the coupling losses are determined by a single parameter, the time constant. The maximum loss per cycle is of order B02/µ0 when ωτ is unity. It is reduced by a factor ωτ or 1/ωτ at frequencies on either side of the maximum. The behaviour is that of an LR circuit in which the resistance is the transverse resistivity per unit length, the inductance is determined by the twist pitch and the EMF is that induced by the changing field; figure B4.1.10 shows the equivalent circuit for a round wire of length l.

Figure B4.1.10. The equivalent circuit of a twisted filamentary wire, radius a, in a transverse field.

B4.1.4.3 Summary of techniques The various methods used to find the fields and losses in multifilamentary superconductors fall into two classes and both have been developed to a high degree of sophistication. The one most commonly used in single conductors is to average material properties and use field theory. This was the method proposed by Can (1974) who solved for the potential and it is used in many of the sections which follow. Although in most circumstances it is easier to solve for a scalar potential than a vector field, in filamentary wires it is often simpler to solve for the electric field (Campbell 1982). This is because uniformity in the z direction causes the z component of E to rise linearly from the electric centre independently of the presence of the superconductor and the component of E along the filaments is zero, leaving only one component to be determined. This is a useful technique for rectangular conductors. The second common method is to treat the filaments and wires as circuit elements with an inductance and resistance (Fevrier 1987, Morgan 1970). This method has generally been used to deal with cables and complete magnets, although it has also been successful with single wires. This is also used extensively in subsequent sections. B4.1.5 Further complications The derivations above give simple expressions for what are usually the main components of the loss. These are the hysteresis loss of the superconductor and the coupling loss in the copper. We now list the complications which must be taken into account if accurate predictions are to be made. B4.1.5.1 Saturation As the coupling currents increase, the currents carried back by the filaments on the surface will exceed the critical current of the outer layer of filaments. This creates a saturated layer into which the flux penetrates

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in a similar way to the penetration of a superconducting cylinder by a transverse field, although the details differ. Eventually the whole wire is saturated and the flux penetrates to the centre by cutting straight across the filaments instead of diffusing along the twist pitch. In this regime the wire behaves as a solid superconducting cylinder with a reduced effective flux flow resistivity because of the copper, and the twist has no effect on the losses. This regime is reached when ωτ B0 is larger than the penetration field of the complete wire. Figure B4.1.11 illustrates the dominant loss mechanisms at various fields and temperatures with the critical state region filled in in black. The amplitude is written as a fraction of the penetration field of the complete wire, Bp w , and the frequency is scaled by the time constant. At low frequencies and low amplitudes the coupling loss is dominant. As the amplitude increases, the hysteresis loss of the filaments, which increases as B3, becomes dominant until the penetration field of the filaments is reached. After this the filament loss varies as B and becomes less significant. If we increase the frequency instead of the amplitude the coupling currents progressively saturate the outer filaments and a critical state spreads in on the scale of the wire radius. The losses are then dominated by hysteresis as in a solid cylinder. A regime not shown is at very high frequencies where the skin depth is comparable to the wire radius and currents flow entirely on the surface.

Figure B4.1.11. The dominant loss mechanism at different amplitudes and frequencies.

B4.1.5.2 Transport currents As described in section B4.1.3.1, transport currents in the superconductor are treated with the Bean model, using the condition that the field at the surface is determined by the total current flowing and the external field. If the amplitude of the external field is large compared with that due to the transport current the effect of the current is to move the electric centre from the middle of the conductor at zero current to the edge at the critical current. The flux has to go twice as far so losses are approximately doubled. The effect of transport currents on coupling losses is quite complex. Twisting does not reduce losses since flux rings must penetrate from the circumference, and transposition is needed to obtain the same effect as twisting. No superconducting technology has combined twisting with transposition (Linz wire) but fortunately this is not really necessary for practical applications.

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B4.1.5.3 Cables If a large magnet were to be wound with a single wire the inductance would be very large and need unacceptable voltages to power it up. For this reason wires are generally wound into many-stranded cables with some electrical contact between the strands. The strands are often soldered together to give good mechanical and thermal conductivity. The equivalent circuit of figure B4.1.10 can be extended to deal with cables. Figure B4.1.12 shows the low-frequency circuit for a three-strand cable but in real magnets much more complex networks are needed to model the properties. In this figure the subscript f refers to a single wire and c to the cable. Ri f is the resistance between wires. Normally the twist pitch of the cable is much longer than that of the filaments which can lead to significant voltages across the inter-wire resistances. It is therefore important to control this resistance if unexpected losses and increased time constants are to be avoided.

Figure B4.1.12. A low-frequency equivalent circuit for a three-strand cable.

B4.1.6 High-Tc, materials The losses in high-Tc materials are identical in principle to those of low-Tc materials. Complications arise due to the weak links at grain boundaries (granularity), and the existence of low-level voltages at currents well below the standard Jc . The granularity means that we have two critical current densities, one associated with the grains and a much lower one for the grain boundaries. The grain boundary Jc is the transport current value; the Jc of the grains must be determined magnetically. For a given transport Jc the loss is in fact lowered by the presence of unpenetrated diamagnetic grains. The lack of a genuine zero resistance in many high-Tc conductors does not add significantly to the losses, but makes calculations more difficult since the critical state model may not be sufficiently accurate. Most conductors are in the form of silver-sheathed tapes, either single filament or untwisted multicore. This means that the coupling losses are in the fully saturated regime but the capability of absorbing losses is far greater in nitrogen than in helium and the costs are much less. For this reason even these higher loss levels do not seem to be a major problem in applying high-Tc materials to applications at 50 Hz. However, the final conclusion on this matter will depend on a full economic analysis of a complete system. There are problems in measuring transport losses in tapes and these are described below. The electric field in the silver sheath is approximately equal to the field in the superconductor provided the cross-sectional area is less than the square of the skin depth (Campbell 1995). This means that silver losses can be calculated by first finding the electric field in the superconductor assuming no silver is present, and

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then calculating the loss due to this field in the silver. In general, silver losses in the self-field are small but in large oscillating fields they will dominate if the wire is not twisted. Losses in the superconductor are hysteretic and agree well with the expressions based on the Bean model. References Campbell A M 1982 A general treatment of losses in multifilamentary superconductors Cryogenics 22 3–16 Campbell A M 1995 AC losses in high Tc superconductors IEEE Trans. Appl. Supercond. AS-5 682–7 Carr W J 1974 J. Appl. Phys. 45 929 Fevrier A 1987 Losses in a twisted multifilamentary superconducting composite submitted to any space and time variations to the electromagnetic surrounding Cryogenics 23 185–200 Golosovsky M, Tsindlekht M and Davidov D 1996 High frequency vortex dynamics in YBaCuO Supercond. Sci. Technol. 9 1–15 Kwasnitza K 1977 Scaling laws for the AC loss of multifilamentary superconductors Cryogenics 17 616–9 Melville P H 1971 Theory of a.c. loss in type II superconductors in the Meissner state J. Phys. C: Solid State Phys. 4 2833–48 Morgan G W 1970 Theoretical behaviour of twisted multicore superconducting wire in a time varying magnetic field J. Appl. Phys. 41 3673–9

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B4.2 Hysteresis losses in superconductors A M Campbell

B4.2.1 Introduction In chapter B4.1 the origin of losses in superconductors was described and the general principles of loss calculations established. In this section we apply these general techniques to losses in the superconducting material of a conductor in more detail. We concentrate almost entirely on the hysteresis losses at power frequencies in the vortex state since this is the regime of most practical interest. We also describe some measurement techniques. B4.2.2 General techniques B4.2.2.1 The electric field The method of chapter B4.1 which uses the electric field is of general application. It is consistent with the idea of pushing flux into the sample against the pinning forces of the material, which act as a frictional resistance. If the flux moves with velocity υ against a pinning force Fp the work done is Fp • υ. This can be written J ×B •υ=B ×υ•J =E•J since E = B × υ. Thus the electrical loss in a superconductor E • J is equivalent to a frictional force on the flux Fp • υ, just as that due to eddy currents appears as a viscous force. The method has the disadvantage that two integrations are involved. Firstly the loss per unit volume must be integrated over the sample at an arbitrary point in the cycle, and the loss must then be integrated over a complete cycle. However, it is probably the only technique easily applied to a ramped field, since methods described below, such as that using the Poynting vector, only give the loss over a complete cycle unless the stored energy in the field is included. B4.2.2.2 Magnetic hysteresis An alternative route to finding the loss in a complete cycle when no transport current is flowing is from the area of the magnetization curve (Pippard 1957). (This is also a common measurement technique.) The magnetic moment is defined in terms of current loops as described in chapter B4.1. To illustrate the use of the magnetic hysteresis we find the magnetic moment per unit volume of a fully penetrated slab of width 2d and unit surface area carrying a circulating current Jc . We take a strip x from the centre of width δx (see figure B4.2.1). The currrent per unit surface area is Jc δ x and if we ignore end effects the magnetic moment is 2xJc δ x. The total magnetic moment is the integral of this from 0 to d and the magnetization is found by dividing by the width d. The fully penetrated magnetization is therefore

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Figure B4.2.1. The current can be split into current loops and the moments summed.

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Figure B42.2. The magnetization curve is approximately rectangular.

1 -2 Jcd . On reducing the field the moment appears in the opposite direction. If we ignore effects at the field extremes when the critical state is penetrating to the centre we get a rectangular magnetization curve (figure B4.2.2) and the area is 2B0 Jc d so this is the loss per cycle per unit volume. Hence we reproduce the loss obtained from the electric field in the introduction (equation (B4.1.1)). This is a convenient method for finding the losses in shapes in which the electric field may be difficult to calculate since only the saturated magnetization is needed.

B4.2.2.3 Losses from the Poynting vector The Poynting vector can be used to give the energy flux into a sample by integration of E × H over the surface of the sample. This avoids a volume integration, but does not distinguish between energy going into the local fields and energy which is dissipated. This means that we must subtract any change in field energy from the integral to obtain the loss. However, for a periodic change the field energy does not change over a complete period so integration of the Poynting vector gives the loss. The electric field can usually be calculated from the rate of change of flux inside the sample. Only the surface fields are needed so this may be shorter than integrating the loss through the sample. (It is essentially the same as calculating the area of the B0 – M loop.) Note that if there are two sources of energy, an external magnet and a power supply providing a current, the Poynting vector must be integrated over a closed surface which includes the end cross-sections of the wire. The distribution of the work done between the external field and the current source requires detailed calculation, even when the transport current is constant. Measurement of the losses needs a search coil round the sample as well as voltage contacts, since the magnetic and transport components of the losses cannot be separated. B4.2.2.4 Losses from the flux Two techniques are described in a paper by Norris (1970). This paper contains a number of important results which will be used in what follows. The first method can be used whenever Jc is constant. If a flux φ crosses a current I the work done is φ I. If we go from +B to −B the flux crossing a current at x is 2φ where φ is the flux inside x at the peak of the cycle. The total loss per cycle is 2Jc φ integrated through the sample. Although this needs an integration over space it is only needed at the peaks of the cycle where the flux distribution is relatively simple. It is also necessary to have at least one line not crossed by any flux, i.e. the electric centre must

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not move. This excludes combinations of current and field, unless the sample is not fully penetrated, and an example of this method is given in section B4.2.3.1. A second method applies to conductors carrying a current in zero (or constant) external field, again requiring that no flux crosses one line down the conductor, i.e. that I < Ic . We return the current along a cylinder at a large radius compared with the conductor size. The voltage along the conductor per unit length is φ• , where φ is the flux between the electric centre and the outer conductor. The loss per half cycle is ∫ I dφ over the half cycle and the flux can be found from the Biot—Savart law if the current distribution is known. For fully saturated samples it is directly related to the self-inductance and the relevant integrals have been approximated in tables of self-inductance, such as those in the book by Grover (1962). This technique is used in section B4.2.4.3. B4.2.3 Particular expressions for slab geometry B4.2.3.1 Low amplitudes An important result is the loss of a slab with a low-amplitude oscillating field applied parallel to the surface. This is because when it is expressed as a surface loss per unit area it can be applied to a wide, range of shapes, provided we use the local amplitude of oscillation to find the local surface loss per unit area. Since these currents are induced by an external field they are often called screening currents, as opposed to transport currents, but as far as conditions in the superconductor are concerned there is no difference between these two types of current. The flux distribution inside one surface of the slab at a peak amplitude B is shown in figure B4.2.3. (We assume that the field has gone through more than one cycle. The loss in the first cycle is different from that of subsequent cycles.) The flux per unit area inside a point x is -12 µ0Jcx2. Hence the loss is

This can also be written as a loss per unit volume

Figure B4.2.3. The field penetration for B < Bp . We find the flux inside a point x at the extreme of the cycle.

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Figure B4.2.4. The field penetration for B > Bp .

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B4.2.3.2 Intermediate amplitudes The flux distribution for B slightly greater than Bp is shown in figure B4.2.4. The flux at peak B within a distance x from the centre is

Integrating from x = 0 to d gives a loss

B4.2.3.3 Transport currents As described in the introduction, a transport current causes a similar field distribution to an external field, except that the fields are now antisymmetric, and the currents symmetric, about the central plane (figure B4.2.5). Transport currents, like induced currents, flow on the surface to whatever depth is needed to screen the magnetic field from the interior without exceeding Jc at any point. Figure B4.2.5 shows the current and field distributions on the face of a slab carrying a current. The losses can be calculated in the same way as for an oscillating external field, but the result is not the main loss in slab geometry, since edge effects dominate the losses (see section B4.2.4.2). However, the diagrams illustrate the field distributions in round wires normal to the field across a diameter perpendicular to the field. If the local value of B at the surface is used equation (B4.2.2) gives a good approximation to the loss.

Figure B4.2.5. (a) A transport current applied to a slab. (b) The flux distribution as a function of of distance from the surface. (c) The current density.

B4.2.3.4 Combined currents and fields These have been treated in slab geometry by Hancox (1966). They apply in large external oscillating fields when the field due to the transport current is small since otherwise losses at the ends will dominate (see section B4.2.4.2). For B < Bp and B > 1-2 µ0Ι where Ι is the current per unit width in the sheet, it is just a question of adding the two surface losses. On one side of the slab the current adds 1-2 µ0Ι to the field, on the other side it subtracts from the field. If the current is constant there is no effect on the loss. If it varies in proportion to the field from equation (B4.2.2) it is

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Figure B4.2.6. The flux distribution with combined fields and currents for an external field decreasing from its peak value. In (a) the current is d.c. while in (b) the current varies in proportion to the field. The straight broken lines indicate the positions of the electric centre. In case (a) there is a net transfer of flux across the sample in a complete cycle.

If the sample is fully penetrated the flux profiles become complicated. The sequence for a d.c. current is shown in figure B4.2.6(a). The loss can be calculated from the electric field as above, but it can be seen that the electric centre moves across the sample immediately after each field peak so that methods relying on a fixed electric centre cannot be used. Even though the electric centre, the point where the vortex velocity is zero, stays within the sample, the fact that it moves through the vortex lattice means that flux crosses the sample. There is a continuous movement of flux across the sample every cycle so that both an a.c. and a d.c. voltage appear at the ends of the sample. If a wire ring carrying a persistent current is exposed to a fully penetrating a.c. field the current dies away. This apparent resistance has been termed the ‘dynamic resistance’ by Ogasawara et al (1976) who have calculated the losses in a range of circumstances. In the limit where the wire is carrying a critical d.c. current all the loss appears as a voltage along the sample, and none as a magnetic hysteresis loss as seen by a coil round the sample. If, as is more common, the field and current oscillations are in proportion, there is no d.c. voltage, but the a.c. field still affects the voltage seen by the contacts measuring the transport current. B4.2.3.5 Field-dependent critical current densities At low fields the critical current density is very field dependent, and formulae such as Jc ∝ 1/B or Jc ∝ 1/(B + B0 ) (where in this expression B0 is a material parameter rather than the external field) give a better approximation than a constant Jc . The flux profile becomes nonlinear and the algebra more complex, although the principles of the loss calculation remain the same. Losses where Jc ∝ 1/B have been considered by Hancox (1966) who concludes that the loss can be well approximated by equation (B4.2.3) using the value of the critical current density at the surface field for Jc . Additional complications at low fields are the effect of the lower critical field Hc1 and surface barriers. For surface fields below Hc1 no vortices enter and for amplitudes close to this it is only the part of the cycle above Hc1 which causes losses of the type described so far. This regime is considered in section B4.2.8. Surface barriers can be treated as a thin layer of very high Jc (Ciszek et al 1989).

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B4.2.4 Other shapes The losses in slabs have been treated in some detail since it is easy to obtain analytic expressions for a wide range of fields and currents. For other shapes of conductor the results are qualitatively similar in that the field varies in the same way with distance from the surface. However, there are few analytic solutions of the Bean model so that losses can only be found in the high-amplitude and low-amplitude limits, or by numerical methods. A smooth curve connecting the two limits will be a pretty good approximation to the losses at intermediate amplitudes. B4.2.4.1 High-amplitude oscillating fields For most shapes the current flow after complete penetration is reasonably straightforward to calculate. The loss can be found either from the magnetic moment, or from the electric field, which increases linearly with distance from the electric centre. A general expression is given below. Here are the results for a few shapes with zero transport current. Conductors parallel to the field Figure B4.2.7 shows the current distributions in two shapes of conductor. The loss can be found from the saturated magnetic moment. Figure B4.2.7(a) is a round cylinder with radius a where the flux moves radially to the centre. The moment is ∫ Jc πr 2dr and the loss is -31 B• 0aJc. Figure B4.2.7(b) is a rectangular conductor b × a (b > a). The loss is ((3b-a) / 12b) B• 0aJc. In all cases the loss is approximately 2-1 B• 0aJc W m-3 for reasons made clear in section B4.2.2.2.

Figure B4.2.7. (a) In an isotropic cylinder parallel to B flux moves in radially. (b) Currents in a rectangular conductor with an axial field.

Conductors perpendicular to the field Perhaps the most important example is a cylindrical wire of radius a transverse to an applied field. The electric centre is a plane parallel to the y axis, the flux moves in a direction parallel to the x axis (figure B4.2.8(a)) and the electric field is B• 0x. The loss is EJc integrated over the volume

The loss per unit volume is then

A similar calculation for a sphere gives

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Figure B4.2.8. (a) A cylinder transverse to a large field B. The flux moves in parallel to the x axis and the electric field is B• x. (b) An anisotropic cylinder in an axial field.

Anisotropic J c Most high-Tc materials are very anisotropic so that the pattern of current flow is different. Figure B4.2.8(b) is the current distribution in a very anisotropic cylinder in an axial field in which the high-Jc planes are parallel to the x axis. (The actual streamlines must be obtained by summing the currents shown.) The flux moves between them parallel to the x axis so that the electric centre is a plane along the y axis and the local loss is B• 0xJc, where the low Jc is the value to use. The loss per unit volume is (4/3π )B• 0rJc, which is the same as for a transverse cylinder and about 30% greater than that for an isotropic axial cylinder. The lines of current flow are arcs of circles of radius r centred at various points along the x axis. The expressions for an anisotropic rectangular conductor have been worked out by Gyorgy et al (1989); the current pattern is similar to that in figure B4.2.8(b). B4.2.4.2 Low-amplitude oscillating fields Losses in the limit of B0 << Bp are calculated by assuming perfect diamagnetism to get a first approximation to the surface field. The loss at any point on the surface is given by equation (B4.2.2) with the appropriate field at each point, and the loss is integrated around the surface to give the total. Perhaps the most important example is a round wire transverse to the applied oscillating field. The penetration of the field is illustrated in figure B4.2.9(a). Addition of a transport current increases the penetration on one side and decreases it on the other. If there is no transport current the local field round a cylinder in a field B0 is 2B0 sinθ, and the local loss per unit area is given by using this field in equation (B4.2.2) and integrating round the surface. The result for a wire of radius r is

The equivalent for a sphere is

For ellipsoids with large demagnetizing factors the algebra becomes difficult, but the same method can be used. In the extreme case of a long thin strip in a perpendicular field the result obtained by Norris (1970) for a wall across a conductor can be used. Figure B4.2.10 illustrates the field lines, and it can be seen

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Figure B4.2.9. A cylinder transverse to a field B0 has a surface field 2B0sinθ.

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Figure B4.2.10. A thin strip in a transerse field.

that there must be significant currents on the strip face inside the penetrated region to screen the parallel component of the field. The edge losses at low amplitudes in a strip of rectangular cross-section 2a × 2t (t << a) are

(here Bp = µ0 Jct ). Note that in common with other results on very thin strips the losses increase as the fourth power of the amplitude, rather than the more common third power. There are significant differences between ellipsoids with a large aspect ratio and rectangular strips. For example the low-amplitude losses in an ellipsoid of any aspect ratio vary as B3 in contrast to the B4 of a strip. The work of Norris (1970) has recently been extended by Brandt and coworkers (Brandt et al 1993, Brandt and Inderbom 1993) to cover ohmic as well as hysteretic losses in discs and strips. For strips they find the same results as Norris but give more detailed expressions for the penetration. The critical state penetrates from the edge, but only moves significantly after a certain critical field is reached. This can give the impression of a surface barrier, but is in fact a direct result of the Bean model. There are also significant currents flowing on the surface of the strip inside the penetrated region. B4.2.4.3 Self-field losses due to transport currents in various conductors The surface field of a round wire carrying a current is uniform, and at small amplitudes penetrates in the same way as it does in a slab. The field gradient is dB/dr = µ0 Jc and we can use the surface loss from equation (B4.2.3). If we have a layer of superconductor on a cylinder of radius a carrying a current I, the loss is µ0I 3/6π 2a 2Jc J/cycle/m. Since this is a real geometry, for practical conductors it is worth putting in some numbers. Suppose we coat a 5 cm diameter tube with 0.1 mm of superconductor and that the critical current density of the material is 108 A m– 2. At the critical current density the current carried is 1571 A (peak). At 50 Hz the loss is 1.3 × 10– 3 J/cycle/m or 0.065 W m–1. If the conductor were solid copper with a resistivity of 0.2 × 10– 8 Ω m the loss would be 126 W m–1, a factor of 2000 greater. Since the loss varies with the cube of the current, lowering the current carried gives the superconductor a greater advantage, as will improvements in critical current density.

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For penetrations comparable with the radius we need to solve the critical state equation curl B = µ0 Jc which for this geometry is

For constant Jc the solution is

where a is the outside radius and i the ratio of the current to the critical value. Figure B4.2.11 shows the flux penetration and profile, which is not linear even for constant Jc , except for small penetrations, and at the critical current.

Figure B4.2.11. The critical region and field profile in a round wire carrying a transport current.

The most direct way to calculate the loss is from the flux at the extremes of the field oscillation. The result is

At Ic the loss is µ0I 2c/2π J/cycle/m. Although worked out for a round wire with constant Jc this expression is of very general application. Norris has shown it works for various shapes. In BSCCO tapes the large aspect ratio means that most of the loss occurs where the field is normal to the tape, so it behaves like an isotropic material. Even the most rapid variation of Jc with B cannot increase this loss by more than a factor of four. The only situation in which the loss is reduced significantly is if the superconductor forms a thin sheet on a large cylinder. The loss is then reduced by the ratio of the thickness to the radius. At low amplitudes (B4.2.9) reduces to

This is the same result as that derived from plane geometry. Norris has calculated losses in a number of important geometries. He shows first that in elliptical conductors the critical state penetrates to a series of concentric ellipses. The result is that the expression for a round wire (equation (B4.2.9)) applies to an ellipse of any eccentricity. This apparently surprising result shows that losses due to transport currents are not very sensitive to demagnetizing factors. Norris also uses a conformal transformation to find the losses in a thin rectangular strip, at the edges of strips, in gaps in an array of strips and in an asperity. These are exactly the types of loss likely in a typical highTc cable, and he finds that the loss of a thin strip is less than that of an elliptical or round-section wire. The results can be summarized as follows. The loss at Ic for most shapes of conductor is in the range 0.4–0.6 times µ0I 2c/π J/cycle/m. For low amplitudes the loss at the edges of thin sheets is

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The losses at a gap of width g between two current-carrying strips of critical current per unit width j is

A correction to this result has recently been published by Majoros (1996). This is probably the dominant loss in high-Tc cables wound with BSCCO tape, although there is not yet much experimental corroboration. B4.2.5 Low amplitudes in fine filaments In this context low amplitudes mean amplitudes which are not large enough to build up a critical flux gradient in the superconductor rather than amplitudes small compared with the penetration field. This situation occurs when a small ripple field is imposed on a large bias field. If there is no pinning and we apply an oscillating field b sin(ωt) to a slab in a large field B0 the electric field x from the centre has an amplitude xωb. The velocity of the flux lines is E/B0 = xωb/B0 . Hence the amplitude of the oscillation is x b /B0 . If this is much smaller than the spacing between flux lines, or the size of pinning centres, the motion is linear and reversible and the Bean model does not apply. The physical picture is that the flux lines oscillate within their potential wells with too small an amplitude to escape into the next one. Note that the distance the flux lines move is essentially a classical calculation which does not depend on the size of the flux quantum. To take a specific example if we carry out a susceptibility experiment with an amplitude of 0.1 mT in an external field of 10 T in a reversible sample of radius 0.1 mm the flux moves a distance 1 nm. This will not be far enough to unpin a vortex so the Bean model cannot be used. Instead the material obeys the London equations but with an effective penetration depth, determined by the pinning strength, which can be much larger than the London penetration depth. If there are pinning centres the amplitude will be reduced even further. It is therefore very important to establish in any experiment in high fields whether we are in the small oscillation linear regime, or the large oscillation critical state regime. To characterize the material we need another parameter, the maximum distance the vortices can move reversibly, the elastic limit d. The elastic limit of vortex oscillations can vary between a few nanometres in high-Tc films to several hundred nanometres in reversible metallic superconductors, and varies with field and temperature, so that it needs experimental evidence to determine the division between regimes in any particular situation (Campbell 1971). We see that the assumption used in the Bean model, which is that currents are always either +Jc or — Jc , fails at low amplitudes in small filaments. At larger amplitudes the curve of force against displacement becomes nonlinear and irreversible. The loss in this regime can be found by assuming the shape of the force displacement curve of the flux lines, and also assuming that the curvature is due to unpinning so that the hysteresis loop is a Rayleigh loop. In the general case this will involve calculating the flux profile numerically and the problem becomes complex. However, in practice this situation usually occurs in very thin wires or small grains which are much smaller than the effective penetration depth. In this case we can assume that the flux density, and hence the vortex displacements, are to a first approximation the same as if there were no pinning forces. We can then find an analytic expression for the loss, which is much less than that predicted by the Bean model (Takacs and Campbell 1988). The amplitude is characterized by a parameter c which is the ratio of the vortex displacement at the surface (bd/B0) to the maximum reversible vortex movement or elastic limit.

The Bean model expression is then multiplied by a factor This leads to a large reduction in the losses of submicrometre filaments compared with the predictions of the critical state model.

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B4.2.6 Some general features We summarize here some general results which apply to a range of regimes. Firstly we repeat the conclusion in the introduction that saturated losses in samples of width 2a perpendicular to the field are typically 1-2 a•Bc W m-3. In solid conductors a.c. losses due to transport currents at the critical current are about 1-2 πµ0Ι c2 J/cycle/m. We justify this result in the next section. B4.2.6.1 Losses in the large penetration limit A useful case for which we can get a very general result is a long conductor carrying its critical current transverse to an applied oscillating field (figure B4.2.12). For the case of complete penetration of the field the magnetic field increases at all points at the same rate as the external field.

Figure B4.2.12. A long conductor of arbitrary shape carrying its critical current is in an external oscillating field. The electric centre is at the extreme edge and the electric field rises linearly with distance from it.

The electric centre then lies along a plane parallel to the external field at the edge of the sample and the electric field rises linearly with distance from the edge. This allows a simple calculation of the losses for a ramped field. Take the origin at the centre of gravity and the y axis parallel to the external field. The edge of the sample is at x = a. Then the electric field is B• 0(a - x) and the local loss B• 0(a - x)Jc. If the thickness of the conductor is y at x the total loss per unit volume is

This is an exact result if Jc is constant and we see that in the large amplitude limit the loss in W m– 3 is proportional to B• , to Jc and to the projected width of the superconductor perpendicular to the applied field. If the conductor is not carrying a current the integration must be carried out explicitly, but this is normally easy for most conductors. In general, at zero current, the flux has to travel about half the distance it had to travel at Ic since the electric centre is now in the middle of the conductor, so the loss is about half the value at Ic and the same general conclusions apply. This is why most of the formulae in section B4.2.3 are about 21 aJcB• 0. More precisely the loss can be written(-21 aJcB• 0/µ0)(1+αi2) where i is I / Ic and α is a number between and -12 and 1. B4.2.7 High-Tc materials The loss mechanisms in high-Tc materials are similar to those in conventional superconductors, but two features complicate matters. Perhaps the most important is the granularity which is caused by weak links

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at grain boundaries. These have a very low Jc compared with the bulk material and flux penetrates to the centre of the sample, while leaving the grains unpenetrated (figure B4.2.13(a)). Although in homgeneous superconductors the difference between B and H is the Abrikosov magnetization, which is negligible, in the case of granular materials where the grains are completely diamagnetic it is better to average fields over many grains. Provided the loss in the grains can be neglected, granular materials can be incorporated in the previous results by defining H in the material as the external field in equilibrium with the local flux density B, averaged over many grains. With this definition the transport current is J = ∇ × H and the field distributions at the edge of a slab are as illustrated in figure B4.2.13(b). (This is one of the situations in which the difference between H and B in a superconductor is significant.)

Figure B4.2.13. (a) The grains remain diamagnetic while flux penetrates between them. (b) The average flux density B is reduced by the diamagnetism. The transport current is dH/dx.

We can define an effective relative permeability µ as B/µ0H and the usual equations of electromagnetism can be used. We see that, for a given external field, B in the material is reduced by a factor µ compared with nongranular materials so that the electric field is reduced by the same factor. Hence for given Jc the granular material will have a loss lower by a factor µ. The value of µ can be measured from the differential susceptibility at amplitudes greater than that needed to penetrate the sample. It is typically about 0.8 in well sintered material. It is closely related to the intergranular volume fraction of normal material, and would be equal to it if the grains were in the form of long rods parallel to the field. The second feature of high-Tc materials is rapid flux creep. This becomes extreme at the irreversibility line where Jc goes to zero. A.c. losses are not very relevant by this stage, but in general the model of a clearly defined Jc above which the resistivity is large may require modification in high-Tc materials. The V—I characteristic can be very flat, so that the value of Jc measured at d.c. and in a short sample may be rather different from that appropriate to long lengths and high frequencies. The frequency effect only appears when the skin depth is comparable to the conductor size, but the resistivity which goes into the skin depth at low voltages is the low value associated with flux creep, so that even at power frequencies the effective Jc may be higher than the d.c. value. Using the d.c. value will overestimate losses in the low-amplitude regime, but underestimate them for full penetration. Another way of looking at the same phenomenon is to move away from the standard picture of losses as a hysteretic effect, with modifications due to flux flow resistivity, to a combination of hysteresis and resistive effects. This is essential when dealing with transport currents in wires since if there is a significant ohmic voltage at low current densities the loss per cycle goes to infinity at low frequencies as is the case in normal metals. When the a.c. amplitude rises losses in the grains become significant as the field penetrates into the grains from the grain boundaries. This is a complicated situation and the algebra becomes tedious. The papers by Takacs et al (1990, 1991) and by Müller (1991) contain references to a series of papers on

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the subject. These authors calculate waveforms and harmonics explicitly and the losses can be deduced directly. The general conclusion is that losses in high-Tc materials can be calculated to a first approximation using standard techniques, but if we want to do a detailed design for a practical system some complex numerical analysis will be needed. Experimental measurements of transport current losses are beginning to be published. First results show that the frequency dependence is intermediate between a hysteretic loss proportional to frequency and an ohmic frequency-independent loss. The balance between these two will depend very much on the amplitude and nature of the conductor. B4.2.8 Losses in the Meissner state Below the lower critical field Hc 1 there are no vortices present in the equilibrium state and the losses are extremely low at power frequencies. In practice there are two sources of loss, but they are difficult to treat theoretically. The first occurs due to asperities in the surface of the samples. Here the local demagnetizing factor can cause field concentrations so that Hc 1 is reached at the tip of the asperity and flux lines can enter at low applied fields. The second source of loss is flux trapped in the sample due either to the earth’s field, or to a previous overload. Cooling in the earth’s field will put vortices 6 µ m apart in the sample and where these emerge through the surface external fields can move them and cause losses. These losses are very low, and sample dependent, so that not very much analytical work has been done on them. More details can be found in the paper by Melville (1971) but this is an area difficult to quantify and experiments must be done on the material of interest to get reliable answers. At intermediate amplitudes the effective amplitude for loss calculations is essentially that part of the cycle where Hc1 is exceeded. The effects of Hc1 have been included in the calculations by Dunn and Hlawiczka (1968), and more will be found on high-Tc materials in the articles by Takacs et al (1990, 1991) and Müller (1991). B4.2.9 Longitudinal losses and inclined fields The critical state model applies if the local field is perpendicular to the local current. If there is a longitudinal component we need to consider the effect of force-free configurations of the field. This configuration increases Jc because the flux lines are helices of different pitch which have difficulty in crossing each other. The situation can occur not only if a current is applied parallel to a field, but also in purely magnetic measurements. For example, if a coppper cylinder has a field applied at an angle to the axis the eddy currents induced are not perpendicular to the field. Therefore in an inclined superconducting cylinder the critical current, and hence the losses, depends on the ability of flux lines to cut each other as well as pass the pinning centres. The subject is not well understood and there is no satisfactory theory in most cases. In a cylinder the flux lines form helices, but in most practical situations the pitch of the helix is large compared with the sample dimensions. In this case it is a satisfactory approximation to ignore the component of the applied field parallel to the current. As for losses in the Meissner state, experiments must be done with the materials of direct interest in the regime of interest. A detailed treatment of inclined fields has been given by Soubeyrand and Turck (1979) but this was for NbTi which has a very anisotropic Jc so that the situation is even more complex. However, experiments show that for isotropic cylinders the loss varies smoothly between the values for parallel and transverse fields.

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B4.2.10 Rotational losses In electrical machines superconductors are exposed to rotating fields. The effect of a uniform rotating field can be found in the large penetration regime by considering the work done by pulling the flux lines through the frictional force exerted by the pinning centres (figure B4.2.14). At every point the dissipation is B J c υ where υ is the local velocity perpendicular to the flux lines. The result for a round wire of radius a transverse to the field is

This is π times that for the oscillating field of the same amplitude (Harrowell 1971). For small penetrations the critical state must be solved, but numerical results (Pang et al 1981) show a similar factor occurs. In general a linear approximation, which gives a factor of two over the oscillating loss, is a small overestimate of the loss.

Figure B4.2.14. A rotating cylinder in a uniform field. An element at r, θ moves at a velocity ωr through the flux and experiences a force BJc in the x direction.

B4.2.11 Measurement of losses Measurements are of two types, calorimetric and electromagnetic. In electrical measurements a voltage or susceptibility is measured while most calorimetric methods measure the boil-off of the cryogen due to the loss. B4.2.11.1 Calorimetric methods In this technique the heat generated by the loss is measured by the increase in boil-off of the cryogenic coolant, or the rise in temperature of the wire, both of which can give an unambiguous loss measurement. For accurate results from boil-off measurements it is necessary for the loss to be considerably greater than the background boil-off. Even with a calorimeter round the sample to collect the gas this may be difficult to achieve because of the boil-off due to conduction, particularly if current leads are attached to the sample. The background boil-off is usually dependent on the level of cryogen present, and the cooling due to increased boil-off can change the conditions markedly, so the most reliable method is to use a null method in which the power to a resistor beside the sample is reduced in such a way as to keep the boil-off constant. The reduction in power is then equal to the dissipation in the superconductor. The technique is used for large volumes of material and particularly for magnet coils carrying a current, where the large inductive voltage makes electrical measurements difficult. It is generally limited

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to temperatures for which a liquid cryogen can be used, although in principle the rise in temperature of a gaseous coolant might be used. The best sensitivity is about 1 mW. Another thermal technique is to isolate the sample thermally and measure the rate of temperature rise. This has not been used much as temperature gradients and the continuously changing temperature make the results difficult to analyse. However, a modification of this method in which the sample is connected to the helium bath by a thermal resistance and the temperature rise in the steady state measured can give good results. Recently Schmidt and Specht (1990) have used the technique to measure losses with a resolution of 10 nW. This method is not usually considered suitable if losses in the presence of a transport current are required because of the heat leak down the current leads. However, Dolez et al (1996) has recently published a technique using the steady-state temperature rise of a wire which can be used to measure the losses of wires carrying a current. B4.2.11.2 Electrical methods These are convenient and sensitive provided there is only one source of power, i.e. either a current in a wire or a magnetization loss. If the power is coming from two sources, such as a solenoid and a transport current supply, it is necessary to measure both magnetization and longitudinal voltage. This is true even if one of them is not oscillating since a wire carrying a d.c. current in an oscillating magnetic field will have a voltage induced along its length with a consequent power loss from the supply (the dynamic resistance). There are always background voltages induced in the leads which must be balanced out before a measurement can be made and there can be considerable problems in doing this accurately and setting the phase to pick up the lossy component. If the losses in a field are needed the loss in zero field can be assumed to be zero and the Meissner state used to set the balance. However, if there are two sources, or zero applied field, a criterion for balance based on the expected shape of the waveform may have to be used. Calorimetric methods may be more reliable in this case. If the losses are purely magnetic the sample is placed in an oscillating field and the magnetic response measured. We can then measure the area of a hysteresis loop on the M—H curve. However, a more sensitive method is to measure the imaginary part of the susceptibility with a lock-in amplifier. For this technique it is only the out of phase component of the fundamental which gives the loss, so a tuned filter is needed to eliminate harmonics. The main problem is in balancing out signals which are not due to the sample. It is important to do the balancing in such a way that there is not a large ‘in phase’ component since if this is present a very small error in setting the phase leads to a large error in the loss. Therefore if the penetration of the signal is small the bridge should be balanced in the superconducting state and the phase set in the normal state. If the penetration is large the bridge should be balanced in the normal state and the phase set from the signal in the superconducting state. This works for losses in a field when the losses in zero field can be assumed negligible and this state used for calibration and phase setting. For zero field losses it is helpful to be able to move the sample out of the coils to set the phase. A sign that the balance is correct if the loss is hysteretic is the absence of a linear signal at low amplitudes; the lossy component of the susceptibility due to hysteresis is always proportional to at least the square of the drive signal since the sample must be driven round a Rayleigh loop. For samples carrying a current in zero applied field it is usual to measure the voltage along the wire from which the loss can be calculated. It has recently been realized that the usual four-contact method must be analysed with care since in contrast to the situation in a normal conductor the loss voltage is comparable to the inductive voltage. The voltage measured by contacts on the edge of a BSCCO tape is very different from that measured by contacts on the top surface (Ciszek et al 1994, Fukunaga et al 1994). The meaning of the reading of a voltmeter attached to contacts on a superconducting wire is illustrated in figure B4.2.15. Although the nature of the meter is irrelevant to the conclusions the use of an electrostatic

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Figure B4.2.15. The voltage measured is the rate of change of flux enclosed by the voltage leads extended to the electric cables.

meter is helpful in a thought experiment, although extremely impractical in a real one. The reading of the meter is proportional to the electric field between the plates of a capacitor. Since the impedance is high there is no current in the wires to the meter so E is zero here, as it is along any line perpendicular to the axis of the superconductor. We can now apply Faraday’s law of induction along a circuit including the meter, the axis of the wire and radial lines joining the voltage contacts to the axis. This shows that E in the meter, and hence the measured voltage, is directly proportional to the rate of change of flux included between the meter wires and the axis of the superconducting wire plus the field along the axis Ec . If we keep the current below Ic , the electric field along the central axis is zero so that the indicated voltage is the rate of change of flux beteen the meter and the axis. If we plot the field lines of a strip conductor carrying a current it is immediately obvious that over most of the strip the field is normal to the faces, so that the flux between the edges and the centre is much larger than the flux between the surface at the centre point and the centre of the wire (figure B4.2.16).

Figure B4.2.16. The field lines around a strip carrying a current.

This explains why, if we minimize inductive voltages by twisting the leads close to the wire, voltages from surface contacts are much smaller than those from edge contacts. Even if we take only the resistive component neither gives the true loss. The voltage measured from any pair of voltage taps can be calulated from the work of Norris (1971), and Campbell (1995) compares the predictions of the apparent loss based on a strip and an ellipse. Detailed formulae will be found in the articles by Fleshier et al (1995) and Zeldov et al (1994). The true loss can only be measured by taking the voltage leads out perpendicular to the wire until the field lines are circular (Campbell 1995). This is the only way of making sure that all the flux entering the superconductor is measured. It might be thought that including extra air would only produce an inductive

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component but this is not the case. The flux enters the superconductor in a hysteretic nonlinear way, and therefore all the flux travelling through the air towards the wire is affected by this lossy motion, even if it never reaches the wire. Fortunately it is only necessary to make the voltage loop two or three times the wire width to obtain satisfactory loss measurements on a tape. The situation with large cables is much more uncertain. B4.2.12 Numerical methods A number of numerical methods have been used to solve for fields in superconductors. The subject has not been developed very far, not because it is inherently difficult, but because in the low-field regime, where numerical methods are needed, the knowledge of material parameters has not been sufficient to justify accurate calculations. It was easier to measure properties directly. Also the main regime in which accurate assessment was needed, which is in electrical machines and big magnets, involved very fine filaments in a large external field so that the high-field approximation based on the Bean model is very accurate. High-Tc materials may be used in monolithic forms; they are also made as thin strip conductors, and one of the main applications may be in a.c. cables and transformers which produce low fields. For these situations numerical methods are likely to be much more important. The first technique was by Witzeling (1976) using a mutual inductance argument. This approach leading to equivalent circuits has been developed in great detail by Fevrier (1987). Norris (1971) dealt with transport currents in long conductors of arbitrary cross-section, and proposed splitting them up into polygons as in a finite-element calculation. The computation then found contours of constant vector potential at the vertices. Most subsequent methods were directed at solving the critical state, which is a necessary first step, but they were not usually extended to loss calculations. A common technique is to postulate a boundary, fill the region between the surface and the boundary with current and adjust the boundary so that the external field is cancelled inside it. This was used to find the critical state in transverse cylinders in oscillating and rotating fields (Pang et al 1981). A similar technique has been used to work out the magnetization of cubes (Navarro and Campbell 1991), which appears to be the only three-dimensional calculation. Given the amount of effort which has gone into numerical methods for conventional materials it seems sensible to make as much use of these as possible. The vector potential has greater significance in superconductors than in normal materials since it can usually be directly related to the distance moved by flux lines. This can be seen from the fact that —A• = E = Bυ where υ is the flux velocity. Since υ = dy/dt where y is the flux displacement we see that the vector potential is a direct measure of how far the flux has moved. A number of equations start with the expression for the current density

∇2 A/µµ0 = jωσA. For a For a conductor J = σ E = jωσ A so we get the eddy current equation −∇ superconductor obeying the London equations = ∇2A/µµ0 = Λ2A and for a superconductor obeying the Bean model ∇2A/µµ0 = ±Jc or zero. Thus techniques used for eddy current solutions can be readily adapted for superconductors. It is not difficult to add a field-dependent Jc , and an effective permeability to take account of granularity. By making the permeability complex it would also be possible to take into account the losses in the diamagnetic grains, but this is a rather doubtful process in such a nonlinear system. Many conventional programs split the conductor into a series of wires and use inductance calculations. This has the attraction that it is only necessary to apply the analysis to the volume of the conductor, in contrast to field methods which need to go to large distances from the conductor to get the correct boundary conditions. However, the field methods give very sparse matrices which may offset the disadvantage of

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the large number of equations. Although essentially a linear technique the inductance method can be extended to critical-state calculations by increasing currents from zero and leaving out any that reach Ic , from subsequent calculations (Witzeling 1976). Prigozhin (1996) has published an algorithm for a finite-element analysis of the critical state. All these methods involve finding the critical state, which must be derived for all points in a cycle, and then finding the losses from the electric field or the Poynting vector. An alternative which is more flexible, but much more time consuming, is to use the measured V—I characteristic of the superconductor and standard eddy current programs for a nonlinear resistivity. This is closest to the physical process and will also give the frequency dependence of the losses, while the methods above give only the hysteresis loss per cycle at low frequencies. It remains to be seen how easily standard numerical techniques can deal with the very sharp discontinuity in voltage at Jc and the problems posed by the 40:1 aspect ratio of most BSCCO tapes. However, in principle there should be no problem in applying numerical methods to a.c. losses in superconductors. We are only awaiting a sufficiently detailed design problem to justify the effort involved. It is clear that numerical methods will play an increasing role in the development of the subject into practical devices. B4.2.13 High frequencies This chapter has been concerned with power applications, which involve relatively low frequencies and high amplitudes. For these purposes the critical state model leading to a hysteresis loss proportional to the frequency is a good one. The first effect of increasing frequency is to add viscous forces on flux lines to the pinning forces. The viscous forces on flux lines are directly related to the flux flow resistivity, which is comparable to the normal-state resistivity, and we will see higher-frequency effects when the skin depth calculated from this resistivity approaches the sample size (Navarro and Campbell 1991). However, thermal activation can lead to a much lower resistivity close to Jc . This leads to a flat V—I characteristic, especially in high-Tc materials, and the apparent Jc will increase at frequencies of a few kilohertz in millimetre-sized samples. Higher frequencies are rarely met in conjunction with high magnetic fields and amplitudes. Even if vortices are present at megahertz frequencies it is unlikely that the amplitude is sufficient to cause unpinning and we are in a linear regime in which pinning effects are irrelevant. The losses can be found from a simple spring and dashpot model for vortex movement. However, more relevant for practical purposes is the extension of the losses in the Meissner state to higher frequencies. In practice losses are usually measured in the gigahertz region and contain terms dependent on the acceleration of normal-state electrons. These can be treated by a linear two-fluid model. Finally there are ultra-high frequencies comparable to the energy gap, which require the microscopic theory, and above which the material behaves as if there were no superconducting transition. References Brandt E H and Indenbom M 1993 Phys. Rev. B 48 12 893 Brandt E H, Indenbom M and Forkl A 1993 Europhys. Lett. 22 735 Campbell A M 1971 The interaction distance between flux lines and pinning centres 1971 J. Phys. C: Solid State Phys. 4 3186–98 Campbell A M 1995 AC losses in high Tc superconductors IEEE Trans. Appl. Supercond. AS-5 682–7 Ciszek M, Campbell A M and Glowacki B A 1994 Physica C 233 203-–8 Ciszek M, Teklel P and Koslowski G 1989 Influence of surface layer on the ac loss minimum in type II superconductors Supercond. Sci. Technol. 1 360-–3 Dolez P, Aubin M, Willen D, Nadi R and Cave J 1996 Calorimetric AC loss measurements of silver sheathed Bi-2223 superconducting tapes Supercond. Sci. Technol. 5 374-–8

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Dunn W I and Hlawiczka P 1968 Generalised critical state model of type II superconductors J. Phys. D: Appl. Phys. 1 1469-–76 Fevrier A 1987 Losses in a twisted multifilamentary superconducting composite submitted to any space and time variations to the electromagnetic surrounding Cryogenics 23 185–200 Fleshier S, Cronis L T, Conway G E, Malozemoff A P, McDonald J, Pe T, Clem J R, Vellego G and Metra P 1995 Measurement of the true AC power loss of Bi 2223 composite tapes using the transport technique Appl. Phys. Lett. 67 3189-–91 Fukunaga T, Maruyama S and Oota A 1994 Advances in Superconductivity (Tokyo: Springer) pp 633–6 Grover F W 1962 Inductance Calculations (New York: Dover) Gyorgi E M, Van Dover R B, Jackson K A, Schneemeyer L F and Waszczak J V Anisotropic critical currents in Ba2YCu3O7 analyzed using an extended Bean model Hancox R 1966 Calculation of ac losses in a type II superconductor Proc. IEE 113 1221-–8 Harrowell R V 1971 AC losses in rotating and reciprocating superconductors and normal conductors J. Phys. D: Appl. Phys. 4 1769–75 Majoros M 1996 Hysteretic losses at a gap in a thin sheet of hard superconductor carrying an alternating transport current Physica C 272 62–4 Melville P H 1971 Theory of a.c. loss in type II superconductors in the Meissner state J. Phys. C: Solid State Phys. 4 2833–48 Müller K-H 1991 Detailed theory of the magnetic response of high temperature ceramic superconductors Magnetic Susceptibility of Superconductors and other Spin Systems ed R A Heim et al (New York: Plenum) pp 229–50 Navarro R and Campbell L J 1991 Magnetic profiles of high Tc superconducting granules, three dimensional critical state model approximation Phys. Rev. B 44 10 146–56 Norris W T 1970 Calculation of hysteresis losses in hard superconductors carrying ac: isolated conductors and edges of thin sheets J. Phys. D: Appl. Phys. 3 489–507 Norris W T 1971 Calculation of hysteresis losses in hard superconductors: polygonal section conductors J. Phys. D: Appl. Phys. 4 1358-–64 Ogasawara T, Yasukochi K, Nose S and Sekizawa H 1976 Effective resistance of current carrying superconducting wire in oscillating magnetic fields, 1: single core composite cable Cryogenics 16 33-–8 Pang C Y, Campbell A M and McLaren P G 1981 Losses in Nb/Ti multifilamentary composites when exposed to transverse alternating and rotating fields IEEE Trans. Magn. MAG-17 134–7 Pippard A B 1957 Classical Thermodynamics (Cambridge: Cambridge University Press) Prigozhin L 1996 The Bean model in superconductivity: variational formulation and numerical solution. J. Comput. Phys. 129 190–200 Schmidt C and Specht E 1990 AC loss measurements on superconductors in the microwatt range Rev. Sci. Instrum. 61 988–92 Soubeyrand J P and Turck B 1979 IEEE Trans. Magn. MAG-15 248 Takacs S and Campbell A M 1988 Hysteresis losses in superconductors with fine filaments Supercond. Sci. Technol. 1 53–6 Takacs S, Gömöry F and Lobotka P 1990 The influence of viscous flow on ac losses of high Tc. superconductors Physica B 165/166 1399–400 Proc. LT19 Takacs S, Gömöry F and Lobodka P 1991 Frequency dependence of AC susceptibility due to the viscous motion of flux lines IEEE Trans. Magn. MAG-27 1057–60 Witzeling W 1976 Computation of screening currents in superconducting persistent current devices Cryogenics 16 29–32 Zeldov E, Clem J R, McElfresh M and Darwin M 1994 Phys. Rev. B 49 9802–22 Zenkevitch V B, Romanyuk A S and Zheltov V V 1980 Cryogenics 20 703

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B4.3 Coupling-current losses in composites and cables: analytical calculations J L Duchateau, B Turck and D Ciazynski

B4.3.1 Introduction Modem superconducting composites are made of twisted filaments in a normal-metal matrix which provides paths for currents to couple them in changing fields. These coupling-current losses may be very important and the design of the composite in relation with the magnet field cycle has to be done so as to limit them to the lowest possible value. In the early 1960s operating superconducting magnets with varying fields was a hard task. Multifilamentary composites, resistive barriers, subdivision and cables have been major milestones that have allowed the operation of a large variety of magnets with convenient ramping time or with externally changing fields. Solutions are now available which allow 10–100 s rise time in magnets for particle accelerators and 10 T s–1 poloidal field changes on the superconducting toroidal field magnets of Tokamaks. In particular, in the near future, fusion research has to face the problem of producing high-field and high-current poloidal magnets to be discharged in 30 s. Advanced solutions have to be found to meet that aim and to limit the heat load on the refrigerators associated with these machines. Extensive work has been done on this subject. In this chapter elementary materials are given to allow engineers to perform loss calculations in practical cases. More complete theories can be found in particular in the work of Carr (1983), Ciazynski (1985) and de Reuver (1985). It is to be noted that the behaviour of a superconducting composite submitted to a field variation of any orientation (neither transverse nor parallel) has not been fully treated theoretically (with convenient analytical expressions). B4.3.2 Multifilamentary composites in transverse time-changing magnetic fields This is the most common situation. In magnets such as accelerator coils or solenoids, the self-field produces field variations transverse to the conductors. B4.3.2.1 Coupling-current loss in round twisted multifilamentar y composites with a normal-metal matrix The composite is supposed to be submitted to the time variations of an external transverse uniform field Be . Assuming that the twist pitch is very long as compared with the composite radius, the twisting of the filament creates a very particular situation of interfilamentary eddy currents inside the matrix, called in this case coupling currents. They are parallel to the field and not, as normal eddy currents, perpendicular to the field.

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To evaluate these currents we can follow the demonstration of Ries (1977). As long as the external filaments are not saturated, a zero electrical field exists inside these filaments. This means that the critical current is not exceeded in these filaments, which is generally valid if the field variation is not too fast. Circulation of the electrical field along the (C) contour in figure B4.3.1 leads to

where p is the twist pitch of the composite, B• i the internal magnetic field variation, Rf the radius of the filamentary zone and V(z) the voltage at point z between the filament at radius Rf and the composite axis at a distance z from the plane x—y.

Figure B4.3.1. A multifilamentary composite in a

transverse field.

Figure B4.3.2. The electromagnetic reaction of a multifilamentary composite to a transverse field.

Within the cross-section of the composite perpendicular to the z axis, the same voltage can be related to the x coordinate (figure B4.3.2)

associated with the uniform electrical field Ex =−( p/2π )B• i and with the transverse uniform current density

where ρt is the transverse resistivity of the filamentary zone and e the thickness of the normal-conducting layer. Simultaneously this voltage generates in parallel an azimuthal current through the normal-conducting layer. If e << Rf then

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where ρn is the outer normal-conducting-layer resistivity. The current densities Jx and Jϕ are collected by the surface linear longitudinal current density IS flowing in the outer layer of filaments. A correct balance of Jx and Jϕ gives

where Is is the linear current density per metre of circumference and Is m is the maximum value of Is at ϕ = π/2. In turn, it is well known that such a sinusoidal distribution of linear current density generates a uniform magnetic field, inside the composite: ∆Bi = –µ0Is m /2 such that the resulting internal field in the bundle of filaments is

with

θ10 is the time constant of the decay of the coupling currents in time. Using known formulae from the theory of differential equations, the general solution for Bi can be written as

It can be demonstrated that the power losses due to these coupling currents can be written as

where P is the loss power per unit volume of the filamentary zone. Remark on loss measurements This time constant, θ10 , is of major importance. It is a clear physical manifestation of the coupling current and its measurement can be used in principle to characterize the losses of a composite through the time constant of the magnetization of the composite. This time constant θ10 is the time constant of the composite alone in space submitted to the magnetic field variation. To measure this time constant, it can be useful to build a sample made not of a single strand but made of several strands in order to increase the sensitivity of the measurement. The time constant of such a sample θ20 is no longer the time constant of the strand alone, but depends on the geometry of the sample. The relation between θ20 and θ10 is

where N is the demagnetization factor of the sample geometry (N = 0.5 for a single composite) (Bruzzone and Kwasnitza 1987). Loss measurements can be made either by direct calorimetric measurements or by magnetization measurements. In this second case, it can be demonstrated that

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where W is the loss per unit of composite volume during a field variation and M is the magnetization. The magnetization (as defined here) can be experimentally measured by the difference between two pickup coils. The first one, wound around the sample, records the flux internal to the composites and the second one records the external field and is wound far away from the sample so as not to be influenced by it. K is a calibration coefficient depending on the sample’s geometry and the type of pickup. Its exact calculation is difficult and its value can be accessed through an initial experiment with a known sample. In certain cases, the power is no longer related to the filamentary zone but to the composite. A new time constant θ1 can be defined in this case such that

where P is the loss power per unit volume of composite. θ1 is no longer the time constant of the decay of the coupling currents but is directly derived from it

B4.3.2.2 Coupling-current loss in a rectangular multifilamentary composite If the composite is not found but rectangular, it can be characterized by its aspect ration α

Equation (B4.3.3) is no longer valid and must be correced

The coefficients f (α) and g (α) have been given by Turck (1982a) and are presented in figure B4.3.3

B4.3.2.3 Coupling-current loss in multizone multifilamentary composites It is to be noted that, for a given field variation, the losses are highly dependent on the time constant θ1. High loss levels are connected with high values of θ1. Composites are designed to adapt the time constant to the field variation rates. Modern composites are often not simply constituted by a filamentary zone surrounded by a copper shell but are more generally represented as in figure B4.3.4 by a central zone without filaments, surrounded by a filamentary zone and two shells of different materials such as Cu—Ni and Cu. The exact treatment of this composite has been made by Turck (1979a). The Laplace equation written in cylindrical coordinates, for a composite, yields the voltage at a given point (r, ϕ)

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Figure B4.3.3. Loss correction coefficients f (α) and g (α) versus aspect ratio.

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Figure B4.3.4. A schematic diagram of a multizone multifilamentary composite.

In the two external shells, i = 1 and i = 2, the solution is given by

The boundary conditions account at each interface (between a zone i and a zone i + 1) for the continuity of the azimuthal component of the electrical field and for the continuity of the radial component of the current density

where ρ1 and ρ2 are the resistivities of zones 1 and 2 respectively and R1 is the interface radius between zones 1 and 2. Two other conditions are required to solve the problem

Along the circle of outer radius R2, the current is azimuthal: Er(R2,ϕ ) = 0. The voltage in the central zone can be derived similarly. The time constant associated with the loss power per unit of volume of composite is

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(a) Central zone θ c

where ρt is the central zone resistivity. (b) Filamentary zone

where ρt is the filamentary zone resistivity. (c) First shell (subscript 1)

where ρ1 is the first-shell resistivity and ρ2 the second-shell resistivity. (d) Second shell (subscript 2)

(e) Time constant due to eddy currents The normal eddy currents in the copper outer layer generate another time constant. The power associated with this time constant has to be added to the power resulting from the coupling time constant. The time derivative of the internal magnetic field •Bi induces in the normal metal a z component of the electric field Ez = r sin ϕ B• i. It generates normal eddy currents with a loss power density in the normal metal P2 = Ez2/ρn . The integration of this power in the different normal zones leads to the expression

where ηC u i is the copper ratio in region i and

where P is the eddy current loss power per unit volume of composite.

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B4.3.3 Evaluation of the transverse resistivity The evaluation of the transverse resistivity of the composite needs to have a good electrical model of the strand. The evaluation of the transverse resistivity is of great importance for the time constant evaluation and is generally not so easy to calculate. Significant traps have to be avoided. The classical and most common situation is represented in figure B4.3.5. The filaments are surrounded by a layer of thickness eb and of resistivity ρb . These plated filaments are immersed in a matrix of resistivity ρm . For the case where the thickness eb is very small compared with the filament radius rf , the following formula has been given by Ciazynski (1985) in good agreement with Carr’s results

where

and λ is the nonmatrix proportion in the filamentary zone. Extreme cases can be found from this formula.

This describes the classical case of copper-plated filaments in a cupro—nickel matrix, i.e. low-loss composites associated with high transverse resistivity.

This describes well the case of cupro—nickel-plated filaments in a copper matrix.

Figure B4.3.5. Typical mesh associated with the representation of a multifilamentary composite.

Equation (B4.3.13) is also of great interest for the practical calculation of the transverse resistivity of pure copper Nb—Ti composites, assuming a pure copper resistivity does not lead to values in agreement with experiments. As a matter of fact, in this case it can be shown that a resistive intermetallic interface exists between the filament and the matrix, the value of which can be estimated (Turck 1982b).

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(a) Evaluation of the copper resistivity This value depends on the magnetic field. Particular attention must be paid to the fact that experimental loss measurements are generally performed around a rather low value of magnetic field. The evaluation of the resulting time constant has to be corrected, in particular for the high-field region of the magnets. An approximate expression of the field dependence is given by: ρm = ρm0 + 0.2 × 10–10B (ρ in Ω m and B in tesla). Note the coefficient 0.2 (and not 0.45) due to the orientation of the magnetic field parallel and not transverse to the coupling currents. ρm0 is the reference value of the resistivity which at 0 T depends on the residual resistivity ratio (RRR) of the composite. Typically: ρm0 = 1.7 × 10-10 × (100/RRR). In Nb-Ti composites the RRR of copper can vary across the section of the composite. Especially if the filament spacing is very small, this may decrease the RRR by affecting the mean electron free path in this region. The time-constant calculation must take into account these various resitivities. (b) Copper-nn ickel resistivity The amount of nickel can be adjusted, depending at what level the losses have to be limited. For low-loss applications Cu-Ni30% is recommended. In table B4.3.1 the resistivity is given as a function of the amount of nickel. Table B4.3.1. Copper-nickel resistivity at 4.2 K as a function of Ni proportion (wt%).

B4.3.4 Examples of time-constant evaluation It has been seen that the time constant of coupling and eddy currents is a key parameter for a.c. loss calculation. Increasing demand for rapidly changing fields has pushed it to the front of the stage. In the past few years, the calculation of this time constant has become a step on the way to loss evaluation. Moreover this well known parameter has somewhat helped to clarify the qualification process of low-loss superconducting wires. As for the effective filament diameter which is a determining parameter of the hysteretic losses, the time constant can be specified by the manufacturers. Its value, though sometimes difficult to estimate, may be checked through simple loss experiments (Ciazynski 1985). Two ways are commonly used to do this, the calorimetric method and the magnetization method. For a given transverse-field variation, for instance with a trapezoidal shape, applied to an assembly of composites, it is possible to discriminate the hysteretic components and the coupling loss components. This second component is directly proportional to the time constant which can be thus derived. The relaxation of the magnetization after application of an abrupt step of transverse field also yields the time constant of the composite but it is somewhat affected by the shape of the assembly (see section B4.3.2.1). In the following examples, calculations are presented on composites of different structures. B4.3.4.1 Copper matrix Nb—Ti composite This composite has been manufactured by Vacuumschmelze. The filaments are located in a pure copper matrix with the following parameters:

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composite radius R2 = 445 µm twist pitch p = 46 mm number of filaments: 636 Cu/Nb—Ti ratio: 1.9 filamentary zone radius Rf = 400 µm central zone radius Rc , = 150 µm filament diameter: d ≈ 21 Am nonmatrix proportion in the filamentary region η = 0.5. In this case the time constant θ10 is very simple: it can be derived from (B4.3.9) (making Rc = 0), (B4.3.10) (making R1 = R2 ) and (B4.3.6):

If B = 1.31 T, in this case ρCu = 4.51 × 10–10 Ω m. From (B4.3.13) χ = 1.26 and ρt , = 5.06 × 10–10 Ω m. The first term of the time constant connected to the transverse resistivity gives: θ11 = 66.5 ms. The second term connected to the copper shell gives: θ12 = 7.4 ms. Therefore θ1 = θ11 + θ12 = 73.9 ms. The time constant of this composite has been measured. The experimental result is 74.2 ms in good agreement with the evaluation. B4.3.4.2 A mixed matrix Nb—Ti rectangular composite This paragraph is devoted to the composite of the superconducting magnets of Tore Supra: the French Tokamak based at the Cadarache Centre. It was first produced with a round shape then modified to the final rectangular shape. (a) Round version The structure of the composite visible in figure B4.3.6 (rectangular version) is ((Nb—Ti, Cu1, Cu— Ni1)m , Cu2 )nCu3 with the following parameters: number of filaments in a bundle n = 199 number of bundles m = 54 filament diameter: 25.7 µm thickness of Cu1 around a filament: 2.57 µm thickness of Cu—Ni1 around a copper-plated filament: 1.34 µm thickness of Cu2 around a bundle: 44.7 µm thickness of the outer copper shell: 414 µm overall diameter: 5 mm Nb—Ti: 29%, copper: 64%, Cu—Ni: 7% twist pitch: 54 mm. The composite can be represented by four concentric zones as in figure B4.3.4. The Cu—Ni barrier around the filaments has to be represented as an equivalent Cu—Ni shell for azimuthal currents, which must flow through it before having access to the copper shell. Neglecting the central zone, the time constant may be represented as the sum of three contributions

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Figure B4.3.6. The Tore Supra composite (Vacuumschmelze version).

(i) Filamentary zone θf . Neglecting the contribution of bundles, the main contribution is given by current loops circulating between peripheral bundles such as from (B4.3.5)

where δ is the Cu—Ni thickness around one filament, h the half mean path through the Cu matrix and λ is the noncopper ratio in the filamentary zone, λ = 0.65. (ii) Cu Ni shell θs1 . From (B4.3.8)

(iii) Copper shell θs 2 . From (B4.3.9)

(iv) Eddy current contribution. Mainly only the copper shell has to be considered

where ηC u f is the copper ratio in the filamentary region. Table B4.3.2 presents the main contributions to the time constant of the composite. The copper resistivity has been evaluated, assuming a magnetic field of 1 T. The experimental result of 29.5 ms can be considered as being not so far from the theoretica evaluation of 36.4 ms, taking into account the complexity of the strand.

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Table B4.3.2. Detailed contribution to the Tore Supra composite time constant.

(b) Rectangular version In fact the Tore Supra composite is a rectangular 2.8 × 5.6 mm2 composite. The time constant of this composite can be evaluated from the time constant of the round version. From (B4.3.7) θ1 = θf f (α ) + ( θs 1 + θs 2 )g(α ). For a field perpendicular to the wide face of the conductor α = 2, f (α ) = 3.8, g(α ) = 1.5 (figure B4.3.3) and thus θ1 = 92.2 ms. However, this time constant has to be considered for the magnet current ramp. Now with the field parallel to the wide face of the conductor α = 0.5, f (α) = 0.43, g(α) = 0.37 (figure B4.3.3) and thus θ1 = 13.5 ms. This time constant has to be used in the case of a plasma disruption where the field variation from the poloidal field system is orientated such as to give low losses. B4.3.4.3 Modified jelly roll (MJR) Nb3Sn composite This composite has been manufactured by Teledyne Wah Chang Albany. It has been ordered by CEA (French Atomic Commission) for fusion applications. The composite is made of 18 bundles of filaments in a copper matrix (figure B4.3.7). The thickness of bronze around the bundles is 5 µm; the thickness of copper around the bundles is 10 µm and the thickness of the outer copper shell is 55 µm. The overall diameter is 0.73 mm and the composition is Cu: 50%, nonCu: 50% with a twist pitch of 8.5 mm. As for the Tore Supra composite, the time constant may be represented as the sum of three contributions:

The Cu—Ni barrier, in this case, is replaced by the bronze barrier. In table B4.3.3 are given the main contributions to the time constant. The copper resistivity has been evaluated assuming a magnetic field of 1 T (RRR = 40). The time constant of this composite has been measured. The experimental result of 0.43 ms is in very good agreement with the evaluation. For this composite a special 5µm thick bronze barrier between the bundle and the pure copper region has been provided at the request of CEA. Without this barrier the time constant is expected to be 50% higher. B4.3.5 Losses in a composite subjected to a transverse external harmonic magnetic field The external harmonic field is

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Figure B4.3.7. A modified jelly roll Nb3Sn composite (TWCA).

Table B4.3.3. Detailed contribution to the MJR composite time constant.

B4.3.5.1 The low frequency region ω (θ10 + θ2 ) << 1 (a) No saturation of the external layer θ10 and θ2 are the two time constants presented in section B4.3.2. They must be kept separated because they are not associated with the same volume. If θ = θ10 + θ2, then

From (2), (4) and (16) we get

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The magnetization (see section B4.3.2.1) is proportional to θB• i. ϕ is the phase angle between the magnetization and the external field.

All the powers and energies calculated hereafter are presented per unit volume of composite. Let PE+C be the combined coupling and eddy current losses in the filament bundle. From (B4.3.5)

The mean associated power can be derived

and the energy per cycle

This energy is a maximum for θω = 1. The influence of frequency on the coupling losses is pointed out in figure B4.3.9(a) for: Rf = 0.9R and B0 = 1 T. The eddy current losses in the outer copper shell are

The total energy per cycle dissipated in this nonsaturated mode is W = WE + WE+C. (b) Saturation of the external layer As soon as the external layer of filaments begins to saturate, the situation is different. We follow here the simplified presentation of Turck (1979b). The influence of the transport current is neglected. The composite is presented in figure B4.3.8 in the slab approximation (half thickness R).

Figure B4.3.8. Saturation of the composite as a function of the external field.

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The condition of saturation is: (Be — Bi )m a x = θ 10 B• m a x > µ0ηf d where d is the filament diameter and ηf is the superconductor ratio in the composite. The field profile varies from -B0 to +B0 through profiles 1 and 2 as in a compact superconducting wire. The extension of the saturated zone reaches the point P associated with the value xs a t such that

The losses associated with (B4.3.17) must be calculated only in the nonsaturated zone, except for the eddy current losses of this zone. In the saturated region, additional losses Ws appear, derived from the equivalent compact superconductor subjected in the partial penetration mode to a changing field θi i m a x in amplitude. The result is

Thus it can be demonstrated that the overall energy per cycle in the composite is

B4.3.5.2 The high frequency region ωθ2 >> 1 Losses are dominated by the classical skin effect in the copper shell. The field and the losses are zero outside the skin depth. In the slab approximation:

B4.3.5.3 Coupling and eddy current as a function of frequency There is a limit for coupling losses. From (B4.3.18)

The skin effect introduces a decrease of the coupling losses. The losses due to this effect are not limited and increase as ω 1/2. The situation is presented in figure B4.3.9(b). B4.3.6 Losses in a composite submitted to an exponential external variation The external field is:

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Figure B4.3.9. (a) Influence of frequency on a twisted composite subjected to a transverse harmonic magnetic field (case without saturation and eddy currents neglected). (b) Coupling losses and eddy-current losses.

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B4.3.6.1 No saturation of the outer layer of filaments

From (B4.3.2), (B4.3.4) and (B4.3.20)

A very good approximation of the overall loss can be given by

B4.3.6.2 Saturation of the outer layer of filaments Partial saturation appears when

The eddy current and coupling losses must be calculated only in the nonsaturated region. additional losses in the saturated outer layer are

and the overall losses are

B4.3.7. Losses in a composite submitted to a low-rate ramp The external field is Be = ∆B(t /τ). If τ >> θ then

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B4.3.8 A.c. losses in cables The large magnets needed for particle accelerators or detectors are energized by high currents in order to limit to acceptable values voltages during transients. For fusion applications, 40 kA conductors are now under production. Braided cables and twisted cables try to ensure as much as possible transposition relative to the transverse fields. In any case, contradictory requirements must be met by cables dedicated for a.c. applications: (i)

twist pitch lengths small to reduce losses, but sufficiently large to ensure a good mechanica stabilityof the assembly after cabling and to avoid wire breakages during cabling;

(ii) resistive barriers between the wires composing the cable but certainly not a full insulation. Complete inslation for cables has proven to be a source of degraded performance due to the difficulty of current transfer between strands and the associated recovery problems in the case of transition to the normal state. In cables the time constants are multiple and coupled. Magnetization measurements, through a direct evaluation of the losses, usually show that there is more than one time constant representative of the losses for a large range of time. A complete analysis of coupling losses in cables is hardly possible because of the difficulty of the 3D representation. Moreover, even simple things such as the way to ensure a reliable and defined interstrand resistance are not established and depend highly on the fabrication process of the coil. Two kinds of cable can be discussed. (i) (ii)

Multistage cables: the main application is magnetic fusion. The coupling currents cross the contact surface of strands running side by side, all along their length. The contact resistance is highly dependent on the void fraction. Rutherford cables: the main application is in particle accelerators. The cable is made of two layers and the main path for coupling currents is the crossover of strands situated in the two different layers.

B4.3.8.1 Evaluation of the time constant for multistage twisted circular cables General analytical formulae usable for calculations of a.c. losses in multistage twisted circular cables have been developed in the model presented by Schild and Ciazynski (1996). They come from electromagnetic calculations (potential method) as used for coupling-loss calculations in superconducting strands. This method enables us to treat the case of any number of cabling stages theoretically. This model is valid if all the strands and successive stages running side by side inside the conductor experience a uniform contact resistance along the length of the cable. This situation is achieved in conductors where the void fraction is not too high. For steady-state regimes (i.e. when the magnetic field variation rate anywhere inside the cable is equal to the applied field variation rate), the total cable a.c. loss power per unit volume of strand can be written as follows

where B• is the applied magnetic field variation rate and θ is called the cable a.c. loss time constant. It is clear that this formula is only valid when all the coupling-current time constants are small compared with the field-variation time constant τ, that means in practice when θ << τ. Under these conditions, the cable a.c. loss time constant θ can be written as a summation of N time constants θn , each of them being associated with the contribution of each cabling stage to the total a.c.

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losses (n = 1 for the basic strand)

Each θn corresponds to the increase of a.c. losses at each new cabling stage n. In a general way we can write (for n > 1)

where pn*, ρn , vn are respectively the effective twist pitch length, the effective resistivity and the average void fraction of cabling stage n. Then we have

where pn , Rn , rn , εn are respectively the apparent twist pitch length, the outer radius, the twist radius, the contact area ratio of cabling stage n (see figure B4.3.10), and ρbeb is the product (resistivity x thickness) of the contact resistive barrier. When n = 2, r1 is the strand filamentary area radius and R1 is the strand radius.

Figure B4.3.10. A schematic view of cabling stage n .

The apparent twist pitch length pn is related to the cabling twist pitch ln but it also depends on the cabling process (percentage of torsion, manner of torsion, back-twist, etc). The following is an example. For 100% torsion of all the cabling stages

where lk , pn are algebraic values < 0 or > 0, depending on the method of cabling. For a cabling process with full back-twist, we have: pn = ln . The expression for ρn offers the advantage of making a distinction between geometrical parameters such as Rn and εn (which depend on vn ) and an electrical parameter such as ρb.

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For n = N, the coupling currents crossing the steel jacket (see figure B4.3.11) give an additional contribution to the a.c. losses

where Aj is the jacket area and Ac is the cable space area (inside the jacket) and ρs t is the steel electrical resistivity.

Figure B4.3.11. The 40 kA conductor developed at CEA as part of the European programme for magnetic fusion (40 mm × 40 mm).

By definition we put: θ *N = θN + θ ’N and we write:

which is in fact the definition of ρ *N . Also to be added are the pure eddy-current losses inside the steel jacket, but this contribution is generally negligible (< 1 ms). B4.3.8.2 Correlation between theory and experiments for an Nb3Sn 40 kA conductor A cross-section of this conductor is presented in figure B4.3.11. This conductor has been designed and tested by CEA and has been made by Dour Metal. It is very representative of the conductors actually manufactured in the framework of the ITER programme. The strand is an MJR Nb3Sn composite fabricated by Teledyne Wan Chang Albany, chrome plated as already described in section B4.3.4.3. The stainless steel bandage around the last subcable (petal) is 0.2 mm thick. The cable contains 3 × 3 × 4 × 4 × 6 = 864 strands (n = 6). The void fraction in the petals is about 30%. Detailed results have been presented by Ciazynski et al (1993). The main contribution to the time constant will be brought by the petal contribution, the ‘interpetal’ coupling currents being cut by the stainless steel wrapping. Particular attention will be given to ρ5, the effective resistivity between the four quadruplets constituting the petal. (1) Contact resistance measurement These measurements have been carried out in situ on a piece of jacketed conductor, 350 mm in length. This length corresponds to the last cabling twist pitch of the cable. The measurements have been performed at 4.2 K with no background magnetic field (B = 0 T).

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From the contact resistance measurements, we can calculate the average value of ρ5 (0.12 µ Ω m), then deduce the value of ρbeb (B4.3.22) using the geometrical value of ε5 (0.1) obtained from a cross-section examination of the cable

This value seems to correspond to the thin bronze shell surrounding the filamentary matrices of the strand (figure B4.3.7), assuming

Taking this assumption it is possible to calculate the theoretical contribution of all the stages. The result is presented in table B4.3.4. Table B4.3.4. The theoretical contribution to the time constant for the 40 kA Nb3Sn conductor.

(2) A.c. loss measurements The a.c. losses under trapezoidal field variations have been measured. It turns out that the time constant depends on the the ramp time Tm , which was quite unexpected. The value at very slow ramp (Tm = 10 s) corresponds to a value of 30 ms which is half the theoretical value. There is no clear explanation for that. From this early measurement, other experiments (Bruzzone et al 1996, Nijhuis et al 1995) have been performed in the framework of the ITER programme. For this programme the aim is to reach a value of θ of 50 ms for the conductor. The effectiveness of the chrome plating to limit the time constant by increasing the interstrand resistance has been questioned since these experiments. In spite of its capability to harden the strand and influence the contact surface, other ways have been explored such as barriers internal to the strand and less subjected to any friction abrasion under magnet operation. B4.3.8.3 Evaluation of the time constant for Rutherford cables A detailed model has been presented by Turck (1979b). In particle accelerator magnets, the coupling current can produce, in addition to power dissipation, field distortions which can seriously disturb the operation of the machine especially during the injection phase. A typical Rutherford cable is presented in figure B4.3.12. We consider slow variations of uniform field Be . Be is the field component transverse to the wide face of the conductor (dimension b). The thickness of the conductor is c

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Figure B4.3.12. A typical Rutherford cable manufactured by GEC Alsthom. Courtesy of GEC Alsthom.

(a) Intrastrand coupling current The first contribution is the so-called intrastrand coupling current. Strands used are generally made of Nb—Ti filaments embedded in a pure copper matrix. Typical time constants of about 5 ms can be met (see section B4.3.4). (b) Interstrand coupling current Two cases can be considered. (1) Time constant dominated by crossing strands.

where ρ is the equivalent resistivity of the material situated between the two strand axes belonging to the two different layers, i.e. the strand matrix, coating and eventual metallic strap between the two layers. The contact surface influences this value. (2) Time constant dominated by adjacent strands. It is the case if there is an insulating strap between the two layers

(c) Discussion of the time constant control in Rutherford cables (Devred and Ogitsu 1996) If we consider that the geometrical parameters regarding dimensions and twist pitches are imposed, the formulae presented above show that the contact resistances drive the time constants. During the mechanical assembly, large pressures are applied to the cable which result in large contact surfaces and possible low contact resistances. The solution retained for the HERA dipole magnets was to coat the strands with a thin layer of 5 wt% silver–95 wt% tin solder called Stabrite to avoid the formation of a copper oxide and ensure crossover resistances as uniform as possible. For the dipoles of LHC (a large hadron collider project at CERN) again, this question is at the forefront. Severe ramp-rate limitations have been observed on several prototype magnets, and pertubating field errors associated with long time constants have been measured (Verweij 1996). This has been explained in relation to supercurrents as a kind of coupling current which can appear if the field ramp or the contact resistance are nonuniform along the length of the conductor. In fact this situation is quite common in accelerator magnets. A possible solution could be to increase the crossover contact resistance by a stainless steel zip between the two layers.

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B4.3.9 Multifilamentary composites in axial fields B4.3.9.1 General When a multifilamentary twisted composite is submitted to an axially changing field, currents are induced in the outer layers of the helical filaments to shield the interior of the composite. These currents turn round at the ends of the sample and return in the central part of the composite. The current distributions have been investigated by several authors (Lefevre and Turck 1981, Ries and Jüngst 1976, Wilson 1983). The length of the sample plays a significant role which can be characterized by comparing the timescale t of the field change with the diffusion time τ = L2/2D where L is the half-length of the sample and D is the magnetic diffusivity (D = ρ/µ). (In a copper matrix conductor, D is usually very small, D ≈ 2 × 10– 4 m2 s–1, which means that a 1 m long sample can be considered as infinitely long for times less than 1000 s.) It can be seen (Lefevre and Turck 1981) that the average current density in the inner layers far from the ends for a field change B• is equal to

and in particular for long samples

This expression shows that in this case the current distribution is not time dependent but only field dependent. For an infinitely long conductor (or for t << τ ), during a field rise from a virgin state, the process can be summarized as follows. The current in the outermost layer increases up to the critical density ηJc which forces the neighbouring layer to carry the current and to be filled gradually up to the density ηJc and so on. During this process, the return current is carried uniformly by the inner layers with the average density 4πB/µ0 p (see figure B4.3.13). This process is very similar to the self-field effect which forces the outer layers of a composite to be filled up to critical current density during an increase of transport current. Both effects are direct consequences of a nonperfect transposition of a multifilament twisted composite. A consequence of the saturation of the outer layers is the existence of global hysteretic losses (again similar to the self-field losses). It is clear that a particular condition appears when the reverse current density also reaches the critical value nJc in which case the composite is saturated by currents flowing in both directions. The saturation of the inner layers is achieved when the field reaches the threshold Bc for which

of conversely

The threshold field Bc is only a function of the twist pitches and of the critical current density. A tighter pitch (smaller twist pitch length) leads to a smaller value of the critical threshold Bc and hence expectedly to larger hysteretic coupling losses. For ηJc = 109 A m–2 and p = 1, 10, 50 mm: Bc = 0.1, 1 and 5 T respectively.

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Figure B4.3.13. The distribution of current density associated with a longitudinal field variation.

In practical conditions the longitudinal coupling-current losses remain small as long as all layers have not been saturated at least once during a field change cycle. When no transport current is carried the losses increase dramatically when the field change exceeds the field threshold given above. In the case of a superimposed transport current the analysis is more delicate. The losses are strongly dependent on the distribution of the transport current and on the direction of the twist in respect to the direction of the changing field. Different possible patterns have to be considered. Four cases have been investigated. Type 1. The transport current is first increased in the conductor and saturates the outer layers of the composite. Then the axial field is swept so as to induce currents flowing in the same direction in the outer layers. Type 2. The transport current saturates the outer layers of composite, but now the currents are induced in the opposite direction. Type 3. The transport current is uniformly shared, with a constant average density, and then currents are induced in the same direction. Type 4. The transport current is uniformly shared, but the currents are induced in the opposite direction. The a.c. losses are the highest in type 1, principally during the first rise. A.c. losses over a cycle (decrease and second rise) are not significantly different in types 3 and 4. In practical conditions, it may be not necessary to consider all cases, and for simplicity two typical cases only can be described. (i ) For a simple field variation (pulse), use type 1 current distribution and the first rise of the changing field so as to get a conservative estimation of the losses. (ii ) In the case of a large number of cycles, the transport current is redistributed with an average current density in the composite and the filaments are not saturated. When the parallel field is applied the critical currents can flow in the outer layer in the same or in the reverse direction depending on the geometry, the pitch direction and the local field direction.

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Both situations can even occur over one full turn of the conductor of a coil. Consequently we propose for a field cycle to use type 2 distribution, in the second rise and subsequent decrease. On account of the fact that in a cycle the second rise almost immediately follows a decrease, the heat generated in the conductor by a.c. losses is deposited almost entirely during these two subsequent phases. As a result, it is correct to use for simplicity only one model for a.c. loss evaluation (type 2) and not to search carefully for which type of distribution prevails in any particular part of a coil and whether type 4 distribution should be used instead. B4.3.9.2 Expressions for the ‘coupling-current’ a.c. losses Rf :radius of the filamentary zone ηf : volume fraction of superconductor in the filamentary region p : twist pitch length Pf : loss power per unit volume of filamentary region i : ratio IT /Ic . For convenience, the field is considered as negative, changing from 0 to—Bn during a rise, and from —Bn to zero during a decrease. B is only the changing longitudinal component of the field. The instantaneous loss power at any given field value B can be written as

where b = B/Bc , i = IT /Ic . (a) Type 1 current distribution For a first rise

from 0 to -Bc

for B < -Bc

Note that for i = 0 (no transport current) the loss power varies roughly as saturation) and as B• for |B| > |Bc| (full saturation). Note also for comparison that during a first rise of type 2

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• B B2

for |B| << |Bc| (partial

Coupling-current losses in composites and cables: analytical calculations (b) Type 2 current distribution Expressions of a.c. losses are given only for the second rise.

Expressions for a second rise. B changes from 0 to — Bn . (1) Bn < (1 + i) Bc

From 0 to Bx with

From Bx to — Bn

(2) (1 + i)Bc < Bn < 2Bc

From 0 to By with

From By to—Bn

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230 (3) 2Bc ≤ Bn

From 0 to Bz with

From −2Bc to −Bn

Expressions for a field decrease. B changes from —Bn to zero. (1) Bn < (1 + i)Bc

From −Bn to zero

(2) (1 + i)Bc < Bn < 2Bc

From −Bn to zero

(3) 2Bc ≤ Bn

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From −Bn to −Bn + 2Bc

From −Bn + 2Bc to zero

Note that during the various field changes the losses vary roughly as B 2 B• when the composite is not fully saturated and as B• in the case of complete saturation. References Bruzzone P and Kwasnitza K 1987 Influence of magnet winding geometry on coupling losses of multifilament superconductors Cryogenics 27 539 Bruzzone P, Nijhuis A and Ten Kate H J 1996 Effect of Cr plating on the coupling current loss in cable in conduit conductors Proc. ICEC16 Conf. (Kitakyushu, 1996) Carr N J 1983 AC Loss and Macroscopic Theory of Superconductors (New York: Gordon and Breach) Ciazynski D 1985 Thèse de Doctoral d’Etat University Pierre et Marie Curie Ciazynski D, Turck B, Duchateau J L and Meuris C 1993 AC losses and current distribution in 40 kA NbTi and Nb3Sn conductors for NET/ITER IEEE Trans. Magn. MAG-3 594 de Reuver J L 1985 AC losses in current-carrying superconductors Thesis Twente University Devred A and Ogitsu T 1996 Influence of eddy currents in superconducting particle accelerator magnets using Rutherford type cables Proc. CERN Accelerator School. Superconductivity in Particle Accelerators ( 1995 ) ed S Turner (Geneva: CERN) p 93 Lefevre F and Turck B 1981 Experimental and theoretical investigations of losses in a multifilamentary composite subjected to transient axial fields IEEE Trans. Magn. MAG-17 958 Nijhuis A, Ten Kate H J, Bruzzone P and Bottura L 1996 Parametric study on coupling loss in subsize ITER Nb3Sn cabled specimen IEEE Trans. Magn. MAG-32 Ries G 1977 AC losses in multifilamentary superconductors at technical frequencies IEEE Trans. Magn. MAG-13 524 Ries G and Jüngst K P 1976 Filament coupling in multifilamentary superconductors in pulsed longitudinal fields Cryogenics 16 143 Schild Th and Ciazynski D 1996 A model for calculating ac losses in multistage superconducting cables Cryogenics 36 1039 Turck B 1979a Coupling losses in various outer normal layers surrounding the filament bundle of a superconducting composite J. Appl. Phys. 50 5397 Turck B 1979b Losses in superconducting filament composites under alternating changing fields Los Alamos Internal Report LA-7639-MS Turck B 1979c Energy losses in a flat transposed cable Los Alamos Internal Report LA-7635-MS Turck B 1982a Coupling losses in rectangular multifilamentary superconducting composites Cryogenics 441 Turck B 1982b Effective transverse resistivity in multifilamentary superconducting composites ICEC-9-ICMC (Kobe, 1982) (Guildford: Butterworth) Verweij A P 1996 Modelling boundary induced coupling currents in Rutherford type cables Applied Superconductivity Conf. (Pittsburg, PA, 1996) to be published Wilson M 1983 Superconducting Magnets (Oxford: Oxford University Press)

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B4.4 Numerical calculation of a.c. losses

E M J Niessen and A J M Roovers

B4.4.1 Introduction Since in the field of physics many complicated mathematical problems cannot be solved analytically, much attention has been devoted to numerical methods. This section deals with the numerical calculation of a.c. losses in superconducting wires and cables. Hysteresis losses in superconducting slabs or in filaments are not considered. We have limited ourselves to coupling losses and transport currrent losses in multifilamentary conductors. The objective of this section is to show some numerical techniques and to highlight a number of aspects that are typical for superconductors. The numerical techniques are illustrated by a few examples. The impact of numerical calculations on the process of learning to understand the electromagnetic behaviour of superconducting composites is well illustrated by the two following examples. For a long time the behaviour of superconducting composites has been studied for relatively simple cases. More practical situations, e.g. the behaviour of a current-carrying conductor subjected to an alternating magnetic field, could be evaluated in more detail using a numerical approach. The current distribution due to the combined action of a transport current and a changing transverse magnetic field is nicely shown by the calculations performed by Rem (1986). A result of those calculations is presented in figure B4.4.1. It shows a cross section of a strand. The y axis reflects the current density, where Js = 1 or Js = − 1 indicates that a filament is positively or negatively saturated respectively (see section B4.4.3.1). This figure shows the competition between the transport current in the centre and the shielding coupling currents at the wire surface. A second case in which numerical calculations have provided the means of gaining a better understanding is the one in which a conductor is subjected to an alternating magnetic field parallel to the wire axis. Niessen showed the occurrence of saturated areas (Niessen 1993). Those results had not been obtained previously by analytical methods. The electromagnetic behaviour of a superconducting composite is described by a set of nonlinear differential equations. This set of equations can only be solved analytically in a limited number of (simplified) cases. To overcome this problem various numerical methods have been applied to solve these equations in more complex situations. In general, the reasons for performing numerical calculations are threefold: (i) to find bounds for the validity of analytical approximations (for design purposes) (ii) to compare nontrivial analytical solutions with numerical data (checks on the solution methods) (iii) to be able to obtain results where no analytical results can be found. The main reasons why, in general, a.c. losses cannot be calculated analytically are:

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Figure B4.4.1. The current distribution in a multifilamentary wire subjected to the combined action of a transport current and an alternating transverse magnetic field (Rem 1986).

•

nonzero filament radius

•

nonzero applied current

•

time dependence/transient phenomena including saturation

•

calculation of the exact form of free boundaries between positive, negative and unsaturated regions.

Even the largest supercomputers are not able to solve efficiently in a general form four-dimensional space-time problems due to large processing times and high storage requirements. Therefore, in general, the dimensions of the problem have to be reduced, using symmetry arguments such as rotational symmetry (φ invariance) or z invariance. Consequently some terms in Maxwell’s equations are no longer present. The number of unknowns and the computer processing time are reduced considerably. First we will try to give an overview of the various methods used by different authors to describe the behaviour of superconductors. The first calculations of a.c. losses in superconductors dealt with superconducting slabs or filaments. The flux penetration in slabs of type II superconductors can be calculated analytically. In the case of round filaments, subjected to perpendicular magnetic fields, the flux penetration profile is more difficult to obtain. Some authors assumed a profile, e.g. an ellipse, and used parametrization to obtain the hysteresis loss (Kanbara 1982). Pang (1980) used a discretization method to calculate the a.c. losses in superconducting filaments, subjected to alternating and rotating magnetic fields. The actual shape of the flux penetration area was not predefined, but followed from his calculations. The next step was the calculation of the electromagnetic behaviour of superconducting composite conductors. The main effects on the calculations for these conductors compared with those for superconducting filaments are due to (the twist of) the filaments and the partially ohmic behaviour, which leads to complex anisotropic properties. The introduction and application of Carr’s continuum model (Carr 1974a, 1974b, 1975a, 1975b, 1977, 1983) provided a solid base for modelling multifilamentary superconductors.

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Ciazynski and Turck (1984) applied a network model to calculate the behaviour of a composite conductor. Fournet and Boyer (1976) and van Overbeeke (1986) modelled the conductor as a set of concentric shells. A more general method, based on Carr’s model, was introduced by Rem (1986) and Hartmann (1989). They used a discretization of Maxwell equations and constitutive equations on a staggered grid. This way the problem can be defined in a general way. An overview of the electromagnetic response of multifilamentary superconductors can be found in the works by Carr (1983) and van de Klundert et al (1992). In the next section the general mathematics describing the behaviour of superconducting wires will be presented. This includes Maxwell’s equations and the constitutive equations, as well as boundary conditions. Solving these equations, by means of discretization techniques, is dealt with in the subsequent sections. As an example, the response of a multifilamentary superconducting wire to an axial changing magnetic field is given. Another numerical technique is the network method. Using this method the conductor is modelled as an electrical network interconnected through resistors. Such networks obey Kirchhoff ’s laws. Using the Biot—Savart equation the response of such a network can be calculated. This method is illustrated by describing the behaviour of a superconducting cable, subjected to a perpendicular changing magnetic field, while carrying a transport current. B4.4.2 Mathematical aspects The basic elements for the description of the electromagnetic response of superconducting wires are Maxwell’s equations supplemented with a set of constitutive equations. The set of equations is completed with initial and boundary conditions. This set holds for isothermal conditions. In cases where the temperature profiles are not negligible, for instance in a superconducting wire due to dissipation, a heat diffusion equation has to be included as well. Such temperature profiles will be neglected here. The configuration that is considered is an infinitely long, round, composite superconducting, twisted, straight wire. The wire radius and twist length are denoted by R and Lp respectively. The wire consists of many filaments surrounded by normal-conducting matrix material. Besides the Cartesian (x, y, z) coordinate system, the cylindrical (r, φ, z) coordinate system is also used. The z axis is chosen parallel to the wire axis. The radial (r) and azimuthal (φ) directions are introduced in the usual way (see figure B4.4.2). Since the wire is twisted, it is useful to define a local orthogonal coordinate system, related to a filament, with one unit vector, e||, parallel to the filament direction and two transverse unit vectors, er and e⊥. The unit vector e⊥ is at a tangent to the cylinder r = constant (0 < r ≤ R). The transformation between both coordinate systems is given by

where ψ represents the local twist angle, i.e. tan ψ = β r, with

Bean’s critical-state model (Bean 1962) and Carr’s anisotropic continuum model (Carr 1975a) form the basis for analysing the response of superconducting wires. Bean introduced a theory, known as the critical-state model, which describes type II superconductors from a macroscopic point of view (Bean 1962). He used averaged quantities in Maxwell’s equations where the volume of averaging is small compared with the typical volume scales of the superconductor. Bean’s

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Figure B4.4.2. Definition of the coordinate system.

theory gives only a description of the superconducting material. For more information refer to Campbell’s chapter (chapter B1.). Carr describes the electromagnetic response in composite superconducting wires by averaged quantities, where the averaging takes place over an area containing a filament and the surrounding part of the matrix (Carr 1975a). Input for Carr’s continuum model is a current—voltage (I—V) relation for a single superconducting filament, which can be found by applying the critical-state model. Carr’s model is widely used for analysing the response of a composite wire. B4.4.3 Calculating a.c. losses using the discretization method This section deals with the general principles of calculating the response of superconducting wires using a discretization technique. Special interest is paid to specific aspects concerning the superconducting properties of the problem. The technique will be explained in detail by a worked example. The basis of the method is Carr’s anisotropic continuum model. Then the material in the wire can be divided into small pieces in which the properties are calculated. A proper limit towards still smaller pieces is justified as the material is considered to be a continuum. It is worth mentioning that, in case islands of filaments appear in the matrix, one should be careful when applying the continuum model principle. The following sections deal with the specific aspects of the discretization method. B4.4.3.1 Set of equations First of all, Maxwell’s equations read

or, explicitly, in cylindrical coordinates

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Since ∇ ⋅ (∇ × A) = 0, where A is any smooth vector, in some situations it will appear to be beneficial to replace one of the equations (B4.4.3)-(B4.4.5) by ∇ ⋅ B = r -1 (∂rr Br + ∂φBφ ) + ∂zBz = 0 or one of the equations (B4.4.6)–(B4.4.8) by ∇ ⋅ J = 0. Furthermore, it is useful to split the magnetic field B into an applied field B A and an induced field B I. B I is a perturbation field due to currents flowing inside the wire. At infinite radius, B I is zero so only B A is present. B A results from currents flowing outside the wire. The given Maxwell equations do not determine the response completely. They must be supplemented with constitutive equations and boundary conditions, so forming a complete set of equations. Considering these constitutive equations, the continuum which consists of normal-conducting matrix material and superconducting filaments has to be investigated. The matrix material has an isotropic linear electrical conductivity which is denoted by σ0. The constitutive relation describing a filament can be found by applying the critical-state model. The obtained relation is for convenience approximated by a piecewise-linear approximation that matches the derivative at zero applied electric field as well as the limiting current density at very large applied electric fields. This piecewise-linear approximation reads

with

and Rf the filament radius. The quantities Jc and |J||s | correspond to the average critical current density (conductor property) and the actual average current density parallel to the filaments. The advantage of this relation is twofold. Firstly, numerically the problem becomes much simpler as it has become piecewise-linear. Secondly, an advantage is that it is easy to refer to different regions contained in the solution: unsaturated (|E||| ≤ E0 ), negatively saturated (E|| < −E0 ) and positively saturated (E|| > E0 ). For zero filament radius this constitutive relation reduces to

independent of the direction of the applied magnetic field. Note that J ||s can be nonzero if E|| = 0. This property is chosen because in the limit for small filament radius the piecewise-linear approximation approaches this kind of behaviour. This limit of zero filament radius is important for obtaining analytical solutions. Finally, the constitutive equations for the continuum consist of an average of both normal and superconducting quantities and read

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where η is the volume fraction of the superconducting material in the wire and

Here δ denotes whether there is a thin insulating layer between the filament and the matrix (insulating layer present: δ = 1; not present: δ = − 1): see equation (B4.3.11) in the chapter written by Duchateau (chapter B4.3) for a more detailed formula. In cylindrical coordinates these constitutive equations read

here

Note that the constitutive equations are isotropic with respect to the (r, ⊥ ,||) coordinate system, and anisotropic with respect to the (r, φ , z) coordinate system. Before treating the boundary conditions, the numerical grid layout is investigated. In the interior of the wire, Maxwell’s equations and the constitutive equations have to be fulfilled. To get difference equations which are second-order consistent with these equations central discretization is used, i.e. expanding all electromagnetic quantities at the centre of the stencils into a Taylor series and making the Taylor series fulfil the difference equations upto second-order terms in the discretization steps ∆r, ∆φ, ∆z and ∆t. This way a so-called staggered space grid is obtained. There is no staggering in the time grid. For the space discretization the midpoint rule is used, while for the time discretization mostly the three-point backward method is usually used (see equation (B4.4.21)). As an example the two-dimensional r–φ grid given in figure B4.4.3 is considered. It shows the positions at which the values of the magnetic and electric field components are considered in the discretization process. Recognize that ∇ • B and ∇ × E are calculated at the same position (solid curve in figure B4.4.3), as well as ∇ × B and ∇ • J (dashed curve). The formulae for discretizing the equations using the solid basic cell as considered in figure B4.4.3 read

with B• z (i, j, k) given by the three-point backward discretization

The indices i, j, k are related to the r, φ, t coordinates, respectively. The discretizations of the equations using the dashed basic cell are similar to the discretizations given above. This way of discretizing can be characterized as an implicit method, as the values on the new time step k do not explicitly follow from old time step (k–1, k–2 etc) values, but are also determined by new time step values in neighbouring points. The basic molecules in three dimensions for both equations are given in figure B4.4.4, taken from Hartmann (1989). These molecules can be combined to a staggered three-dimensional (3D) grid

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Figure B4.4.3. An illustration of the grid in one half of the cross-section of a wire, used for the discretization of the electromagnetic field inside a wire (Nr = 2, Nφ = 3) (Rem 1986).

Figure B4.4.4. The two basic molecules for the 3D staggered grid (Hartmann 1989).

(figure B4.4.5). Notice that the staggering of these two molecules is imposed by the coupling of the two sets of Maxwell’s equations. This grid, useful for isotropic and weak anisotropic media, can also be used for superconducting media. The numerical scheme is second-order consistent where E and B are sufficiently smooth. At free boundaries, however, the consistency is only first order in the discretization steps, because then Jφ and Jz are not differentiable functions of E. This is due to the piecewise-linear approximation used. Furthermore, the time step is limited, because the initial guess (due to the iteration process, as described in section B4.4.3.4) may be too inaccurate for large time steps (Rem 1986). B4.4.3.2 Boundary conditions In this section the boundary conditions imposed on the set of equations as well as the numerical implementation are presented. Since no current can flow through the surface of a wire, Jr(R) is zero. Furthermore, the total

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Figure B4.4.5. The 3D staggered grid for the computation of the electromagnetic fields (Hartmann 1989).

magnetic field is continuous at the wire’s surface and equals the applied magnetic field at infinite radius. Here splitting the total magnetic field into an applied and induced term is useful. The applied magnetic field is known and continuous at the wire’s surface. This implies also that the induced magnetic field BI is continuous at the wire’s surface. Furthermore, it vanishes at infinite radius. The applied current IA(t) can be chosen freely. From Maxwell’s equations it follows that it prescribes a property of the induced magnetic field

Furthermore, the time dependence of the applied field B A can be chosen freely. Normally, two time dependences are considered: B A(t ) changes linearly in time or is harmonic. If B A = 0 one speaks of a self-field problem. However, the spatial dependence of B A has to satisfy the equations ∇ • B A = 0 and ∇ × B A = 0 for 0 ≤ r ≤ R , as the applied magnetic field is generated by currents flowing outside the wire. In general, the spatial dependence of the applied magnetic field can be written using Fourier expansions in φ and z (Hartmann 1989). This spatial dependence consists of four characteristic basic components, of which two components are uniform in z and two are periodic in z. In cylindrical coordinates they read

Here Lz is the period length of the magnetic field in the z direction and I0 and I1 are Bessel functions of the second kind and I’1 a derivative (Abramowitz and Stegun 1972). Notice that for small arguments I0

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and I1 can be approximated as 1 and 0 respectively. B1A and B3A have their main contribution in the x direction, while B2A and B4A have theirs in the z direction. Next the implementation of the boundary conditions on the numerical grid is treated. The boundary condition for the currents, Jr(R) = 0, can be easily implemented. The implementation of the continuity of the induced magnetic field components at the wire’s surface is not that easy. It must be stressed that there is a fundamental difference between a continuity relation and a boundary condition. A continuity relation only connects properties in different regions. A boundary condition prescribes a property. This means that, when solving the set of equations, the magnetic field in all space must be considered; not only in the interior of the wire but also in the vacuum surrounding it, because the boundary condition is stated at infinite radius, where the induced magnetic field vanishes. However, we are only interested in the solution in the interior of the wire and prefer boundary conditions on its surface. These can be found by considering that outside the wire the induced magnetic field satisfies the equations ∇ • B I = 0 and ∇ • B I = 0, as the induced magnetic field is generated by currents flowing inside the wire. The magnetic field components outside the wire can be written using double summations in φ and z over an infinite number of terms which are then matched to the magnetic field at the surface of the wire itself. In this way the boundary condition at infinite radius is replaced by a boundary condition at the wire’s surface. The result is that the equations in the interior of the wire have to be discretized and that the correct boundary conditions at the wire’s surface have to be applied. This way there is no need to go to large distances from the conductor to get the correct boundary conditions. This translation of the boundary condition from infinite radius towards the wire’s surface will now be explained in detail for two important cases: (i) a perpendicular applied a.c. magnetic field case (z-invariant problem) (ii) a parallel applied a.c. magnetic field case (φ - invariant problem). Boundary condition at r = R for the magnetic field for a φ-ii nvariant problem For a uniformly applied a.c. magnetic field in the y direction, perpendicular to the wire, Br and Bφ can be matched at the wire’s surface with the r–φ-dependent solution of the equations ∇ • B = 0 and ∇ • B = 0 outside the wire

Notice that the first term in the right-hand side of the expression for Br and Bφ corresponds to the applied magnetic field. In the expressions for Br and Bφ , the symmetry arguments Ar(r, φ – π/2) = –Ar( r, π/2–φ ) and Aφ(r, φ – π/2) = Aφ( r, π/2–φ ), with Ar and Aφ , the r and φ components of any vector field A, have already been included. Since not the specific values of the Fourier terms, but only the implemention of the correct boundary conditions for Br and Bφ , are of interest, the factors an are eliminated using the orthogonality of the sine and cosine function. This provides a relation between Br and Bφ at the wire’s surface

Here δi.j is the Kronecker delta.

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For the numerical grid refer to figure B4.4.3. With ∆φ = π/Nφ , for any (Nφ + 1) equally spaced Bφ points on the wire surface, this relation can be used Nφ times in its discretized form

The symbol B’r is introduced, because on our grid Br is not defined at the surface, but at positions half a cell dimension inside the wire. To calculate B’r it is approximated by a linear interpolation of the value Br(φj ) at a grid point near the surface and B’’r (φj ) at an imaginary point half a cell dimension outside the wire. This is valid as the magnetic field is a continuous function. Using the discrete representation of ∇ • B = 0 the imaginary point B’’ can be eliminated. Furthermore, the (Nφ + 1)th equation for Bφ(R, Nφ + 1) is

Boundary condition at r = R for the magnetic field for a φ− invariant problem The only condition for this problem on BφI (R) is prescribed by the applied current (equation (B4.4.22)). To simplify the expressions in this section the applied current is considered to be zero. Considering the characteristic applied magnetic field B4A (see equation (B4.4.26)), Br and Bz are matched at the surface of the wire with the r-z-dependent solution outside the wire (r ≥ R)

In these expressions two boundary conditions are already included. The first boundary condition is the symmetry argument ∂zBz = 0 at z = 0, which eliminates the npz terms in the expression for Bz . The second boundary condition is Bz(pz = π/2) = 0, which causes n to be odd. The functions K0 and K1 are modified Bessel functions (Abramowitz and Stegun 1972). Eliminating an and substituting r = R gives

with n odd. Here the identity

is used (Abramowitz and Stegun 1972). These integral equations describe boundary conditions on R. The numerical implementation is similar to the r–φ grid. B4.4.3.3 The stationary solution The response of the fields for the wire placed in an applied magnetic field increasing linearly in time is in general as follows. The induced currents are time dependent, causing a time-dependent induced magnetic field. This time dependence becomes less significant as time increases: there is an in time decaying effect,

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the so-called transient. For times which are long compared with the specific response times of the system, ∂tB I vanishes. This situation is referred to as the stationary case. In the case where one wants to obtain the stationary current profile one does not have to apply the method already described, but it can be found using a simpler model. Because ∂tB I is zero, ∂tB is equal to ∂tB A, which is known. The field and current patterns can then be found from ∇ × E =−∂tB A, ∇ • J = 0, and the constitutive equations. The advantage of this method is that the difficult implementation of the boundary conditions for the magnetic field at the wire’s surface is not needed. B4.4.3.4 The iteration method Due to the nonlinearity in the constitutive relation the numerical solution cannot be found directly. Therefore, the problem is linearized and iterated. The iteration process (see Hartmann 1989, Rem 1986) is fully based on the piecewise-linear relation between the parallel component of the electric field E|| and the superconducting current density J||s . The iteration process is as follows. Every grid point is labelled. A grid point is labelled to be positively saturated if the previous or initial value of E|| > E0 , negatively saturated if E|| < −E0 and unsaturated otherwise. The labelling corresponding to the constitutive equations is used when creating the system of equations describing the solution on the new time level. Then the set of linear equations is solved. With the solution the predicted labels following from E|| can be checked. This means that it is checked whether the predicted label and the label of the obtained solution are equal. If the prediction is not equal to the solution, the label is changed. However, for convergence reasons an element should not be allowed to change from a negatively saturated element into a positively saturated one or vice versa: one should only allow saturated elements to turn into unsaturated ones or vice versa. There is no mathematical proof that this iteration scheme converges but in practice it works very well. Furthermore, notice that it is most accurate to calculate both Eφ and Ez at the same position, as they determine E|| . Then it is obvious that J||s should also be calculated at that position. B4.4.3.5 Worked example In this section a simple worked example is presented with the intention of explaining the method more explicitly. The problem treated here is related to Duchateau’s chapter (chapter B4.3) on multifilamentary composites in axial fields. For didactical reasons that problem is simplified by considering the wire length to be infinite. This results in a problem with an infinite time constant indicating the essential nonstationary behaviour of the solution. The applied magnetic field is chosen to be rotationally symmetric and uniform in space

This problem is φ–z invariant. The corresponding Maxwell equations read

resulting in a radial time-dependent problem. From the φ – z invariance of the problem, it immediately follows that Br = 0 and Er = 0 (see equations (B4.4.3) and (B4.4.6)). Furthermore, the filament radius is considered to be zero. This simplifies the constitutive equation and provides a means of comparing the numerical results with the analytically obtained data. The boundary conditions read

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Boundary conditions (B4.4.38) follow from the regularity of the field components and equation (B4.4.37). Condition (B4.4.39) relates the applied current IA(t) with the electromagnetic fields (see equation (B4.4.22)). Condition (B4.4.40) follows from the fact that outside the wire there are no induced currents.

Figure B4.4.6. Explanation of the grid variables.

The equations are discretized on a staggered grid using a finite-difference scheme which is secondorder accurate in space and time. The variables in this grid are explained in figure B4.4.6. As already indicated, Eφ , Ez and J||s should be calculated at the same position. The magnetic field components should be calculated in between. As an example ∂rEz = B• φ is discretized as follows

with ∆r and ∆t the grid sizes in the radial and the time direction. Since three of the four boundary conditions are given in terms of the magnetic field, it is convenient to calculate magnetic field components at r = 0 and r = R. Then it is only necessary to calculate Eφ(0, t) using an extrapolation of two internal values

If the boundaries are not taken into account, five variables have to be calculated at every grid point (see figure B4.4.6). With N the number of grid points, 5N – 3 variables have to be determined. A set of 5N – 3 independent equations can be obtained using: the four boundary conditions (equations (B4.4.38)–(B4.4.40), N – 2 times the first two equations of equation (B4.4.37), and N – 1 times equation (B4.4.10) and the last two equations of equation (B4.4.37). Now some results of numerical calculations will be considered and compared with analytical data. For a time-harmonic applied magnetic field, BzA (t) = B0 sin ωt, the dissipation in the wire is calculated. Since the filament radius is zero, the hysteresis losses are zero. The mean coupling loss per unit length is given by

which can be rewritten using Poynting’s theorem (Jackson 1962, van de Klundert et al 1992)

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where the asterisk indicates a complex conjugate. Numerically P can be found by calculating the surface integral (B4.4.41) or the contour integral (B4.4.42). Despite the higher workload, P is calculated using the surface integral for the following reasons. (i) When using the contour integral Eφ(R) is needed, which is calculated by extrapolation from interior values. The loss is very sensitive to the phase difference between BzA and Eφ(R). The calculation of this phase difference is very inaccurate if Eφ(R) is calculated by extrapolation. (ii) The time integration of the surface integral is a summation of only positive terms, which is not the case for the time integration of the contour integral. In figure B4.4.7 the scaled loss 2π P /(ω2 B 02) is given as a function of B0. The data result from three methods: (i)

a linear analytical method which does not take saturation phenomena into account: the so-called neglect saturated currents approximation (NSCA) (Niessen 1993) (ii) an analytical method which takes saturation phenomena into account, but neglects the normal currents in the φ and z direction: neglect normal currents approximation (NNCA) (Niessen 1983) (iii) the numerical method. The grid sizes for the numerical data are: ∆r = R/50 and ∆t = 2π/200.

Figure B4.4.7. Scaled coupling losses as a function of the amplitude B0 .

Comparing the results of the three methods, it can be seen that in the case of small values of B0 there is excellent agreement between all three results because the saturated region is negligibly small. It means that the problem is linear. No saturation occurs on the numerical grid. The only loss term that contributes is σ⊥E⊥2 . The deviation between the numerical method and NSCA starts at B0 = B1 because for B0 > B1 saturation occurs in the outer grid cells. In this case the loss terms σ|| Ε 2| | and η Js|||E| also contribute to the total loss. The NSCA, however, does not take these extra losses into account. Notice that the NNCA already deviates for smaller B0 values because it does not have the threshold behaviour by which nonlinear behaviour first appears when the first outer grid cell saturates. Note that the first deviation between the numerical method and the NSCA depends on the grid size ∆r. When ∆r decreases, saturation occurs at the outer grid cell for smaller values of B0. Comparing the

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numerical method and the NNCA it can be seen that there is a significant error in the numerical result when the first outer grid cells saturate. Then the saturated region, of the order of a few grid cells, is described inaccurately on this relatively (compared with the size of the saturated region) coarse grid. In this region the NNCA predicts the loss most accurately. For an increasing amplitude of B0, the saturated regions penetrate more towards the centre of the wire, but for B0 = B2 a saturated region starts growing outwards from the centre of the wire and the loss increases even more. Then the NNCA is difficult to use and is therefore not applied. It is clear that the nonlinear effects cannot be neglected as they are an essential part of the response. In the case of very large values of B0 the term E|| J|s| can be neglected compared with the ohmic loss terms. The linear behaviour is dominant and the deviation between numerical and NSCA results decreases. It can be concluded that nonlinear effects are an essential part of the response and that the linear approximation fails to describe them. The results for the numerical method and a modified analytical approximation agree very well. B4.4.4 The network method applied to a cable Many approaches to the calculation of the characteristics of multifilamentary superconductors consider the superconductor to be a continuum. However, since cables have a finite number of strands and empty spaces in between, a discrete approach seems to be a more appropriate one. Morgan (1973) applied the network method to an unsaturated braid. Niessen (1993) applied the network method to a 29-strand braid. In this case saturation was taken into account for a cable of infinite as well as of finite length. Sytnikov et al (1989) used a one-dimensional network to study a Rutherford cable in the unsaturated mode. In this section the network method is illustrated by means of a Rutherford cable. A 3D network of node points is defined. The configuration, shown in figure B4.4.8, is taken from Niessen et al (1990). Let us consider the top view of a Rutherford cable (see figure B4.4.9). Node points are defined at the strands where the projections of the strands intersect. Furthermore, node points are defined at the edge of the conductor. In this way two types of cross-section, with node points and corresponding resistances, are defined (indicated by A and B in figure B4.4.8). It is assumed that between two node points flows either a superconducting current in a strand or a coupling current between two touching strands. The corresponding resistances are denoted by R|| and R⊥ respectively. The parallel resistance is due to the dynamic resistance and saturation effects. The electrical resistance between strands consists of the ohmic

Figure B4.4.8. Network configuration for a Rutherford cable.

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Figure B4.4.9. Top view of a Rutherford cable and the corresponding network indicating the interstrand resistances R⊥ and R||.

resistance of the matrix material, insulating layers (e.g. a Cr layer) and the contact resistance. The latter depends strongly on the mechanical load of the cable and is often dominant. The Kirchhoff laws

and

supplemented with the Biot—Savart law are equivalent to the Maxwell equations used in the case of the wire. Equation (B4.4.43) is valid in each node point of the network. Index i contains all the currents connected to a specific node point. The transport current IA through the cable can be prescribed by replacing equation (B4.4.43) in one node point by

denoting that the Ik through a cross-section of the cable add up to the transport current. The right-hand side of equation (B4.4.44) corresponds to the induced voltage due to the change of flux pointing through the surface of a mesh. Index j contains all the voltage numbers in that specific mesh. The stationary solution, i.e. ∂t B I = 0, is obtained by calculating — V i n d using ∂t B A. In the nonstationary case, —V i n d contains ∂t B A + ∂t B I, which is much more complicated. This case is equivalent to the case of the wire, where the equations ∇ • B I = 0 and ∇ × B I also had to be solved. For numerical calculations there are now two possibilities: (i) calculation of —V i n d using ∂t B A + ∂t B I

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(ii) calculation of —V i n d using ∂t B A and putting the effects of ∂t B I in the Vj of the right-hand side of equation (B4.4.44). In this way self- and mutual inductances of line elements (Grover 1946) appear in equation (B4.4.44). Basic input elements of the model are the I—V relation of a single wire and the ohmic contact resistance between wires. These equations can have three forms denoted by the indices n(ormal), u(nsaturated) and s(aturated), respectively: (i) normal contact current

(ii) unsaturated superconducting current if |Vu| ≤ V0

(iii) saturated superconducting current if |Vs| > V0

In equations (B4.4.45)–(B4.4.47) Rk is the contact resistance, Rp is the parallel resistance of the normal-conducting part of a wire element Lw e , Rq is the effective resistance of an unsaturated wire element, Ic is its critical current and V0 is the voltage across a wire element if the element is just saturated (V0 = 8Rf|B•⊥ w|Lw e /(3π). Equation (B4.4.45) describes the I—V characteristics perpendicular to the strands. Equation (B4.4.46) and equation (B4.4.47) describe the I—V characteristics parallel to the strands. The latter contains an ohmic (Rp) and a dynamic resistance part (Rq ) in the unsaturated case. Substitution of equations (B4.4.45)–(B4.4.47) in equation (B4.4.44) yields

A set of equations can be obtained by considering the proper meshes. 5N – 3 currents have to be determined in a periodic length of cable, where N is the number of strands. There are 2N superconducting currents in the strands, 2N normal currents between parallel strands and N – 3 normal coupling currents between crossing strands. Solving the equations yields the currents flowing between the node points. The coupling loss and the transport current loss can be obtained in the case of a current-carrying cable subjected to a changing magnetic field. The hysteresis loss in the superconducting material is not included here. The nonlinear behaviour of the constitutive equations implies that the solution can be found only in an iterative way, similar to the wire case. References Abramowitz M and Stegun I A 1972 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (New York: Wiley) Bean C P 1962 Magnetization of hard superconductors Phys. Rev. Lett. 8 250-3 Brechna M 1973 Superconducting Magnet Systems (Berlin: Springer) Carr W J Jr 1974a AC loss in a twisted filamentary superconducting wire. I J. Appl. Phys. 45 929-34 Carr W J Jr 1974b AC loss in a twisted filamentary superconducting wire. II J. Appl. Phys. 45 935-8 Carr W J Jr 1975a Electromagnetic theory for filamentary superconductors Phys. Rev. B 11 1547-54 Carr W J Jr 1975b Conductivity, permeability, and dielectric constant in a multifilament superconductor J. Appl. Phys. 46 4043-7 Carr W J Jr 1977 Longitudinal and transverse field losses in multifilament superconductors IEEE Trans. Magn. MAG13 192–7

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Carr W J Jr 1983 AC Loss and Macroscopic Theory of Superconductors (New York: Gordon and Breach) Ciazynski D and Turck B 1984 Theoretical and experimental study of the saturation of a superconducting composite under fast changing magnetic field Cryogenics 24 507-13 Fournet G and Boyer L 1976 External field effects on current distribution in multifilamentary composites Proc. ICEC-6 (Guildford: Buttenvorth) pp 451–3 Grover F W 1946 Induction Calculations (New York: Dover) Hartmann R A 1989 A contribution to the understanding of ac losses in composite superconductors PhD Thesis University of Twente, Enschede Jackson J D 1962 Classical Electrodynamics (New York: Wiley) Kanbara K 1982 Transverse field loss of a twisted multifilamentary round wire in windings of superconducting magnets Proc. ICEC-9 (Guildford: Butterworth) pp 715–8 Morgan G H 1973 Eddy currents in flat metal-filled superconducting braids J. Appl. Phys. 44 3319-22 Niessen E M J 1993 Continuum electromagnetics of composite superconductors PhD Thesis University of Twente, Enschede Niessen E M J, ter Avest D and van de Klundert L J M 1990 Application of the network method to superconducting cables Proc. LTEC90 (Southampton) p 17 Pang C Y 1980 Losses in type-II superconducting wires due to alternating and rotating fields PhD Thesis (MIT, Boston, MA) Rem P C 1986 Numerical models for ac superconductors PhD Thesis University of Twente, Enschede Sytnikov V E, Svalov G G, Akopov S G and Peshkov I B 1989 Coupling currents in superconducting transposed conductors located in changing magnetic fields Cryogenics 29 926-30 van de Klundert L J M, Niessen E M J and Zandbergen P J 1992 Electromagnetic response of composite superconducting wires J. Eng. Math. 26 231-65 van Overbeeke F 1986 On the application of superconductors in power transformers PhD Thesis University of Twente, Enschede

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B5 Rutherford-type cables: interstrand coupling currents

A Verweij

B5.0.1 Introduction Rutherford-type cables are often used in accelerator dipole and quadrupole magnets operating under d.c. conditions and at low field-sweep rates. During a field-sweep the cable is subject to a varying magnetic field and interstrand coupling currents (ISCCs) are generated if the strands in the cable are not insulated but in electrical contact with each other. The ISCCs generate the interstrand coupling loss (ISCL) (in addition to the hysteresis loss in the superconducting filaments and the interfilament coupling loss within the strands) which has to be compensated for by the cryogenic system. The ISCCs also cause field distortions and can affect the stability of the cable since some strand sections carry larger currents than the transport current. The main issue in this chapter is to describe, both qualitatively and quantitatively, the ISCCs and the energy loss caused by them. It will be shown how the field distortions caused by the ISCCs can be calculated and the effect on the stability will be briefly discussed. While the interfilament coupling currents are normally calculated by considering the multifilamentary superconductor to be a continuum, the ISCCs can be conveniently calculated by means of a network model in which the nodes are interconnected by contact resistances Ra between parallel strands and Rc between crossing strands. A description of a three-dimensional network model for a fully transposed Rutherford-type cable is given in section B5.0.2. In section B5.0.3 general formulae are given for the distribution of the coupling currents, their time constants and the generated loss. The increase of the time constant in stacked cables compared with that in a single cable is discussed. This enables the calculation of time constants in entire coils if the time constant of a single cable is known. In section B5.0.4 attention is paid to nonuniformities in Rc and dB/dt across the cable width, which are present in all coils. Nonuniformities in Rc and dB/dt along the length of the cable cause a nonuniform current distribution among the strands, often described in terms of ‘super (coupling) currents’ or ‘boundary-induced coupling currents’, resulting in an additional coupling loss, sinusoidally varying field distortions and a decrease of the stability of the cable. Under normal operating conditions of most magnets (especially for accelerators), wound from Rutherford-type cables, the effect of these additional coupling currents is small compared

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with the ISCCs. These currents will therefore not be discussed here but several interesting references are given in section B5.0.5. The effect of the cable length on the coupling currents is explained in section B5.0.6 and is important in order to compare the coupling loss in small cable lengths (typically smaller than one cable pitch) with that in large cables and coils. In section B5.0.7 the impact of the interstrand coupling currents is discussed with respect to the performance of magnets. Several methods for measuring the cross-contact resistance Rc (and therefore the coupling currents) will be discussed in section B5.0.8, and the main parameters that affect the Rc value are discussed. A typical measurement using the calorimetric method is illustrated in section B5.0.9 by means of an example, and it is shown how the various loss components in a cable can be distinguished. The magnitude of the ISCCs is difficult to predict since it depends not only on the geometry of the cable but also on the contact resistance between the strands, which is difficult to control. For large cables, having more than ten strands, the energy loss and field distortions caused by the ISCCs are often larger than those caused by the interfilament coupling currents unless the value of the contact resistance between the strands is enhanced by means of coatings or insulating sheets. B5.0.2 Network model of a Rutherford-type cable In 1973 Morgan suggested calculating the ISCCs by modelling a Rutherford-type cable using a network of nodes interconnected by strands and cross-contact resistances (Morgan 1973). This network model could be applied to unsaturated cables with a uniform distribution of the contact resistance Rc and fieldsweep rate dB/dt in the longitudinal direction of the cable. Since 1988 more advanced network models have been developed (Niessen el al 1990, Sytnikov et al 1989) that can also handle saturated strands in finite cable pieces as well as infinite cables. Implementation of the self- and mutual inductances between the strands made it also possible to calculate time constants (Verweij and Ten Kate 1993). This last and most advanced network model will be briefly described in this section. A detailed description is presented by Verweij (1995). The Rutherford-type cable has a width ω and a height h1 on side 1 and h2 on side 2, with h = (h1 + h2 )/2, and consists of Ns strands, having a twist pitch Lp, s . In general these cables have a small keystone angle αk = arctan[(h1–h2 )/w] in order to have a uniform coil structure and to facilitate the coil winding. The cable is modelled by a three-dimensional network of nodes interconnected by strand elements and contact resistances Ra and Rc between adjacent and crossing strands respectively (see figure B5.0.1). Note that in this chapter the values for Ra and Rc denote the resistance per contact and not an average value per twist pitch or per metre of cable. The currents Ia and Ic are the currents through Ra and Rc . The current Is t r denotes the total current in the strands, i.e. the sum of the transport current It r, s t r in the strand and the coupling current Is in the strand. The sum of the strand transport currents equals the cable transport curent It r, c a b . At both edges the strands follow a skewed path from one layer to the other. The aspect ratio α c a b of the cable is defined as the width of the cable divided by the average height of the cable, so αc a b = w/h. The aspect ratio of strands with a round cross-section is αc a b = α0 = Ns /4. In practical cables α c a b is slightly larger than ∝0 due to the cabling process. The cable is longitudinally subdivided into NB calculation bands with a length Lp, s /NS . Hence, the length of the cable is given by lc a b l e = NB Lp, s /NS . Each band, consisting of ( Ns–1) calculation cells, has (5Ns– 3 ) unknown currents, namely ( 2Ns – 1 ) currents Is l r , 2Ns currents Ia and (Ns – 1) currents Ic . The components of the field perpendicular to the small side of the cable Bx , the wide side of the cable By and the cross-section of the cable Bz are given by (see figure B5.0.1)

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Figure B5.0.1. Network model of a Rutherford-type cable and description of the angles θ and ϕ and the coordinates x , y and z , and an example of an enclosed surface A. One strand is shown with a thick bold line. Since the strands are represented by line elements, the real cross-section of the cable (w × h) corresponds to the following distances in the network model: h1 /2 and h2 /2 between the strands in the two layers and w (1 – 2/Ns ) between the strands on both edges of the cable.

Using Kirchhoff ’s laws the (5Ns – 3) equations needed to solve the currents of one band can be set up. The following symbolic notations demonstrate the implementation in the computer program. The (5Ns–3) equations consist of (2Ns – 2) equations in the nodes

(3Ns – 2) equations for a circuit (with three or four nodes in the corners)

(with dB⊥ A /dt the component of dB/dt normal to the enclosed surface A of the circuit), and one constraint

stating that all the currents flowing through the cross-section of the cable add up to the cable transport current It r, c a b . The voltage Us t r , over a strand element consists of a resistive part UR and an inductive part Ui n d . The inductive part will be discussed later. The resistive part can be expressed by

with Is t r, c r the critical current of a strand, UR.0 the voltage at this critical current and n the n-power exponent of the current—voltage ( I—V ) transition, which is usually about 10–50. This equation implies that the set of equations has to be solved iteratively. To increase the speed of the solving routine, a linear

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approximation can be used where the surplus of the current above the critical current is assumed to flow through the resistive matrix (Verweij 1995). In this case an iteration is only required if one or more of the strands in the cable become saturated. The following distinction can be made: (i) weak excitation, indicating that the total strand current Is t r in each strand element in the cable is smaller than the critical strand current Is t r, c r (ii) strong excitation, indicating that in at least one strand element the critical current is reached. In the next two sections the current distribution for these two conditions is discussed. The equations of band Nb contain not only currents of band Nb itself but also currents of band (Nb – 1) and band (Nb + 1). This implies that for the first and the last band appropriate boundary conditions are required. Two different cases can be distinguished. (i) The current distribution in all bands is the same. In this case only one band has to be calculated. (ii) The current distribution in the first band (Nb = 1) and the last band (Nb = NB ) are given. Obviously the equations for these two bands are slightly different. A cable of finite length carrying no transport current is modelled by the constraint It r, s t r = 0 in each strand. By incorporating mutual interactions between the coupling currents in the cable non-stationary situations can also be analysed. The inductive part Ui n d of the strand voltage Us t r is given by

with Mi, j the mutual inductance between the strand elements i and j (both parallel and crossing). The selfand mutual inductances of the contact resistances are disregarded since dIa /dt and dIc /dt are much smaller than dIs t r /dt. The summation has to be made over all the strand elements (2Ns – 2) of all the bands (NB ), hence N = NB(2Ns – 2). the distribution of the coupling currents is constant along the cable length (so that only one band has to be calculated), only (2Ns – 2) summations are required (Verweij 1995). In the numerical model the time derivative of the strand current is represented by the difference in current between two discrete time steps

Equation (B5.0.3) can then be rewritten so that the right-hand side is known

with ∆t = tm – tm – 1 taken so that for each time step the coupling currents decay with about 2%. The time constant is then calculated as a best fit for the first 20 time steps. B5.0.3 Weak excitation: general formulae for a cable with constant parameters First of all the ISCCs and the ISCL in a cable will be given for: (i) weak excitation (i.e. all strand currents are smaller than the critical strand current Is t r, c r ) (ii) constant resistances (across the width and along the length of the cable)

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(iii) no spatial variation of dB/dt (iv) stationary conditions (dB/dt is constant for a time much larger than the time constants involved). The set of equations (B5.0.2)–(B5.0.4) can now be solved taking UR = 0 and Ui n d = 0. The total generated power (per unit length of cable) in the contact resistances Ra and Rc is given by

with the factors 0.170, 0.125 and 0.0085 constants of proportionality that result from the numerical calculation (Verweij 1995). These equations show that the power loss in Rc is only given by the field change transverse to the cable width while the power loss in Ra is also present for fields changes dBx /dt. The power loss Pa, x is, however, a factor h 2/w 2 = 1/a 2c a b smaller than Pa, y (for dBy /dt = dBx /dt) and can therefore be disregarded. The power loss generated by a field change dBz /dt is not included in these equations as it is still another two orders of magnitude smaller than Pa, x (Verweij 1995). The current distributions Ia , Ic , and Is as a function of the position x across the cable width are given by

The factors 0.25, 0.125 and 0.0415 are constants that result from the numerical calculation (Verweij 1995). In these equations x and x’ have the discrete values

for the currents Ia and Is , and

for the currents Ic . For a field change dBx /dt currents flow through Ra and through the strands with a single amplitude the sign of which depends on the layer of the cable. The typical distributions for a field change dBy /dt are depicted in figure B5.0.2. In a coil the currents through Ra can be disregarded if

Equations (B5.0.9)–(B5.0.13) show clearly that, in the case of a field change dBy/dt, the coupling currents and power losses depend strongly on the cable width and the cable pitch, which cannot be

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Figure B5.0.2. Ia , Ic and Is , distributions across the cable width (Lp, s = 0.1 m, Rc = 1 µΩ, w = 17 mm, Ns = 26).

changed independently since the cable pitch has to be about six to ten times the width in order to have a mechanically stable cable. The geometrical parameters of the cable are usually defined by the application in which they have to be used. The possibility of changing the cable width (and hence the twist pitch) is therefore restricted, considering the quench protection and the required field. This means that Rc is the only parameter which can essentially change the magnitude of the ISCCs. In section B5.0.8 several parameters by which Rc is influenced are briefly enumerated. Equation (B5.0.13) shows that Is increases linearly with dB/dt until the voltage along the strand becomes comparable with the voltage over Rc , i.e. until the sum of the transport current and the coupling current reaches the critical current IC, s t r , which depends on the temperature and the field. The critical change of the field (in the y direction) can then be deduced from equation (B5.0.13)

In a similar way a maximum strand transport current It r, s t r, m a x can be defined given by the transport current at which the first strand becomes saturated

In section B5.0.4 the case of It r, s t r > It r, s t r, m a x will be treated in more detail. Similar equations for a field change dBx /dt can be set up using equation (B5.0.13). In a cable the current in each resistance and in each strand section has its own time constant, which can differ considerably from the average time constant, especially for nonuniform Rc distributions. Figure B5.0.3 shows the time constant spectrum of Ic and Is in a 1 m long straight cable piece. The time constants are calculated from the response to a step decrease of dBy /dt from 1 to 0 T s–1. It is clear that the time constants increase towards the centre of the cable. The average time constant τi s, c a b of a straight cable is a function of the cable geometry and the cross-contact resistance Rc . For a

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Figure B5.0.3. The distribution of the time constant τi s of the currents Ic and Is across the cable width (Lp, s = 0.1 m, Rc =1 µΩ, w =17 mm, Ns = 26).

rectangular cable (h1 = h2 ) having strands with a round cross-section (hence αc a b = α0 = Ns /4) the time constant τi s, c a b is by approximation given by (Verweij 1995)

The constant C varies between about 1.6 × 10–8 and 1.7 × 10–8 Ω s m–1 for twist angles between 10 and 25 and is slightly larger for cables with a small keystone angle and for heavily compacted cables (typically 10% larger for αc a b = 1.2α0 ). The time constant is almost independent of the cable length for lc a b > Lp, s . The stationary power (per metre of cable) and the coupling-current distribution in a stack of cables is exactly the same as in a single cable. However, each cable in a stack has its own average time constant τis,cab,i which depends on the configuration of the stack and is always larger than the time constant of a single cable. The time constant is minimum in the upper and lower cables and increases towards the centre of the stack. The increase of the average time constant τi s, s t of the stack compared with that of a single straight cable is given by (Verweij 1995)

with Nc the number of cables in the stack and C a constant which depends to a small extent on the number of strands in the cable: C increases from about 1.0 for Ns = 16 to 1.15 for Ns = 36. This formula shows that the time constant of a stack is limited to about αc a b τ i s, c a b . The coupling power Pc can be well calculated from the measured time constant of a stack even for a nonuniform Rc distribution over the separate cables (in combination with a nonuniform Rc distribution across the cable width). The factor τi s, s t /Pc varies by less than 20% even for variations of Pc greater than a factor of two (Verweij 1995). Combining equations (B5.0.10) and (B5.0.18) results in

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which can expressed per unit volume as

This equation differs by a factor of -32 from the more general expression for the interfilament coupling loss of a conductor with a rectangular cross-section subjected to a field change perpendicular to the wide side of the conductor (Campbell 1982). The difference can be attributed to the different distribution of the ISCCs in a cable compared with that of the interfilament coupling currents in a conductor. B5.0.4 Strong excitation It has been shown in section B5.0.3 that for a transport current larger than the maximum transport current It r, s t r, m a x the strand currents at one edge of the cable reach their critical value. The saturation influences the ‘normal’ distribution of Ia , Ic and Is being present at weak excitation. The difference in the steady-state distribution of the coupling currents is illustrated for a cable with Ns = 26, w = 17 mm, h = 2.25 mm, Lp, s = 120 mm, Rc = 1 µΩ and IC, s t r = 500 A. The I—V relation is taken as given in equation (B5.0.5) with n = 15 and an effective strand resistivity of 10–14 Ω m at the critical current It r, s t r = IC, s t r . Figures B5.0.4 and B5.0.5 show the distribution of Is and Ic across the cable width as a function of the transport current for dBy /dt = 0.1 T s–1. The transport current at which the first strand saturates is It r, s t r, m a x = 280 A (using equation (B5.0.17)). Beyond this current level the voltage over the strands increases sharply with increasing current (see equation (B5.0.5)) so that the voltage over Ra and Rc , will decrease, since the electromotive force remains the same (see equation (B5.0.3)). The current distributions across the cable width become slightly asymmetric, i.e. the coupling current Is decreases more on that side of the cable where It r, s t r and Is have the same sign. Therefore, the currents Ic , become asymmetric since Ic can be regarded as the derivative of Is and the currents Ia become asymmetric since the sum of the

Figure B5.0.4. The distribution of the coupling currents Is across the cable width as a function of the relative transport current.

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Figure B5.0.5. The distribution of the coupling currents Ic across the cable width as a function of the relative transport current.

coupling currents should be zero. Incorporating the self-field in the simulations would further enhance the asymmetry since the self-field is larger at the edge of the cable where It r, s t r and Is have the same sign. Figures B5.0.6 and B5.0.7 show the total power loss Pt o t and the relative decrease in the maximum strand coupling current Is, m a x as a function of the transport current at several field-sweep rates. The power loss Pt o t is equal to the sum of the coupling power loss Pc and the loss Ps generated in the strands (i.e. the product of the voltage over the strand and the current through the strand). The current Is, m a x, 0 denotes the steady-state value of Is, m a x max if the strands are not saturated (see equation (B5.0.13) with x = w/NS ). The currents Is start to decrease as soon as It r, s t r approaches It r, s t r, m a x . Beyond this current Ps starts to increase while Pc starts to decrease. The total power loss, however, remains more or less constant until the transport current approaches the critical value.

Figure B5.0.6. The total power loss Pt o t as a function of the relative transport current at four different field-sweep rates.

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Figure B5.0.7. The relative maximum coupling current Is, m a x as a function of the relative transport current at four different field-sweep rates.

B5.0.5 Cables with varying parameters across the cable width Rutherford-type cables with a small keystone angle are often used in accelerator magnets in order to have a uniform coil structure and to facilitate the winding of the coils. This keystone angle results in a gradient of Rc across the cable width since the contact area increases towards the small side. Simulations of several Rc distributions (Verweij and Ten Kate 1993), such as (i) a linear increase in Rc from one side to the other side, (ii) a small Rc in the middle and (iii) a small Rc at the edges, have shown the following. (i)

The coupling currents and power loss are to a first approximation equal to those given in equations (B5.0.9)–(B5.0.13) taking the average Rc in the middle of the cable. (ii) The ratio between the average time constant and the total coupling power τis,cab/Pc differs only slightly for different Rc distributions. A so-called zebra cable (i.e. a cable in which half of the strands have a soft metallic coating) can be used to reduce the ISCL. The coupling loss that can be obtained with this cable can be four times less than that in a cable in which all the strands have such a coating. During (de)excitation of, for example, accelerator magnets the field change dB/dt varies considerably across the cable width (since the field in the coils varies strongly). For a linear increase in dB/dt across the cable width the current distributions are to a first approximation equal to those given in equations (B5.0.11)(B5.0.13) using the dB/dt value in the centre of the cable. However, the ratio between the time constant and the coupling power is no longer constant. The above also means that a change in the sign of dB/dt across the cable width reduces the coupling loss significantly. Also the coupling currents in a cable having an Rc distribution and subject to a dB/dt distribution across the cable width can be estimated using equations (B5.0.11)–(B5.0.13) taking the Rc and dB/dt values in the centre of the cable. Longitudinal Rc and dB/dt distributions result in additional coupling currents with characteristic loop lengths much larger than the cable pitch and characteristic times much larger than τi s, c a b . It is beyond the scope of this chapter to deal with these coupling currents which are often referred to as ‘boundary-induced coupling currents’ or ‘super (coupling) currents’. In most applications these additional coupling currents will enhance the coupling power loss only slightly but result in sinusoidally varying field distortions along

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Cables of finite length

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the cable and could affect the electromagnetic stability of the cable (Akhmetov et al 1994, Krempasky and Schmidt 1995, Verweij 1995). B5.0.6 Cables of finite length Experimental loss measurements are usually performed on short pieces of cable. The loss in a cable of finite length compared with the loss of an infinitely long cable has been described for hollow round cables (Ries and Takacs 1981) as well as Rutherford-type cables (Verweij and Ten Kate 1993). Figures B5.0.8 and B5.0.9 show the average coupling loss per metre of cable, Pa and Pc, versus the scaled length of the short sample for field changes dBx /dt and dBy /dt (with Ra = Rc = 1 µΩ , Ns = 26, Lp, s = 0.1 m, w = 17 mm, h = 2.6 mm).

Figure B5.0.8. The coupling power losses Pa, x and Pa, y (for the field changes dBx /dt = 0.32 T s–1 and dBy /dt = 0.32 T s–1 ) of a piece of cable as a function of the scaled cable length.

These figures illustrate clearly that the ISCL of a short cable with a length equal to an integer times the cable pitch corresponds well to the ISCL of a long cable. For samples smaller than the cable pitch, the ISCL in a long cable has to be estimated using these figures, which can result in a large error since the magnitude of the ISCL of a short piece of cable depends quite strongly on the Rc distribution. Note that the loss components Pc, x and Pa, x can be disregarded when compared with Pc, y and Pa, y . B5.0.7 The impact of the interstrand coupling currents on the characteristics of magnets The ISCCs and the ISCL in coils wound from Rutherford-type cables can be easily calculated if the field distribution over the cross-section of the coils and the average Rc are known. Since Rc can hardly be measured within 10% accuracy equations (B5.0.9)–(B5.0.13) can be used to calculate the ISCCs and the loss in each turn taking the average dBy /dt in the centre of the cable. The total ISCL in the coil is the sum of the losses of each turn. A good approximation of the time constant in a magnet is obtained from equations (B5.0.18) and (B5.0.19). Of course, Nc (see equation (B5.0.19)) is smaller than the number of turns in a magnet, especially if the magnet consists of several blocks of conductors. As a first estimate the ratio τi s, s t /τi s, c a b is about a factor of three to five depending mainly on the number of turns in the block where most of the loss is generated. For a dipole magnet (like the one used at HERA and LHC) the turns close to the midplane generate most of the loss since the field is normal to the large face of the cable.

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Figure B5.0.9. The coupling power losses Pc, x and Pc, y (for the field changes dBx /dt = 0.32 T s–1 and dBy /dt = 0.32 T s–1 ) of a piece of cable as a function of the scaled cable length.

The maximum transport current in a strand reduces due to the ISCCs assuming that a quench will occur if the first strand becomes saturated. The maximum transport current in each turn of a coil for a field change dBy /dt can be calculated using equation (B5.0.17) taking an average Rc in the centre of the cable. For accelerator applications it is usually possible to reduce the field-sweep rate towards the end of a ramp, where the transport current is close to the critical current. In these magnets the reduction of the maximum transport current is therefore not the most important issue. Besides the generated loss and the reduction of the maximum transport current, the ISCCs also affect the field homogeneity of a magnet. Once the ISCC distribution in each turn of a magnet is known, the field distortions caused by the ISCCs can be calculated by representing the currents Is by infinitely long straight currents normal to the x—y plane. The field produced at a position z = x + iy by such a current at position r = rx + iry satisfies (according to the Biot-Savart law)

if the influence of the structure which surrounds the windings is neglected. In the aperture of the magnet, where |z| < |r|, equation (B5.0.22) can be expanded into a Taylor series as

showing clearly that all multipole components are present in the field produced by a single line current. The field errors in a magnet during the ramp can then be calculated by summation of equation (B5.0.23) over all the strands in each turn of the cross-section. In dipole magnets mainly the low odd harmonics (n = 1 and n = 3) can become important. Typical values for Rc = 10 µ Ω and a central-field-sweep rate of 0.01 T s–1 are (Devred and Ogitsu 1994, Verweij 1995): • 6 × 10– 4 T (HERA (hadron electron ring accelerator) and 6 × 10– 4 T (SSC (superconducting super collider) and LHC (large hadron collider) dipoles) for n = 1 • 0.1 × 10– 4 T (HERA dipoles), 0.2 × 10– 4 T (LHC dipoles) and 0.4 × 10– 4 T (SSC dipoles) for n = 3.

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Also low even harmonics (mainly n = 2) are present if the Rc distribution over the cross-section of the coils is strongly nonuniform. B5.0.8 The cross-contact resistance Rc In section B5.0.3 it has been shown that, for a given cable geometry, the only parameter which can essentially change the coupling currents is the cross-contact resistance Rc . The Rc values of cables are strongly influenced by: • • • •

the applied pressure the strand coating the heat treatment (e.g. for curing the pre-impregnated cable insulation) the size of the contact surface between the strands.

In order to estimate the loss, the field errors and the reduction in quench current caused by the ISCCs in the application, the Rc value of the cable has to be known and hence measured on a piece of cable. It is important that this measurement is performed on a cable which has been exposed to the same pressure and heat treatment as the cable in the application itself. In Rutherford-type cables used in accelerator magnets Rc is typically 2 µΩ (HERA dipole magnets), 1–30 µΩ (LHC dipole model magnets) and 5–40 µΩ (SSC dipole model magnets). The following three methods are usually applied to deduce Rc of a cable piece. (i) Calorimetric method. The amount of helium that evaporates if a piece of cable is subjected to a changing magnetic field is a measure of the loss and hence the Rc value of the cable. (ii) Magnetization method. The magnetization M of a piece of cable, subject to a changing magnetic field Ba , is related to the loss per cycle Q by the integral

(iii) U—I method. Two strands of a piece of cable are connected to a current supply. The voltage over any two strands of the cable can be calculated using the network model and the result is directly related to Rc . If the two strands placed on both edges of the cross-section of the cable are connected to a current supply, the voltage Ue e over these two strands is to a first approximation given by (valid for Ra ≥ Rc and lc a b ≥ Lp, s /2 ):

with IM the measurement current (in A) and Ra and Rc in µ Ω (Verweij 1995). It can be assumed that Ra = Rc for soldered cables and Ra » Rc for coated or unsoldered strands (subjected to a traverse pressure). The calorimetric and the magnetization method give the total loss in the cable, which has three main components: • • •

hysteresis loss in the filaments Qh y s t interfilament coupling loss Qi f interstrand coupling loss Qi s .

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These loss components can be separated (if Ra ≥ Rc ) by measuring the loss (per cycle) as a function of the frequency (or field-sweep rate) for field changes dBx /dt as well as dBy /dt. The hysteresis loss, which is independent of the frequency (at small frequencies), is given by the offset of the Q—dB/dt relation, the interfilament coupling loss by the slope of the curve for a field change dBx/dt and the ISCL by the slope for a field change dBy /dt corrected for the interfilament loss. This separation of loss components is illustrated in the example in section B5.0.9. The U—I method results in an Rc value which can differ from the average Rc mainly due to local nonuniformities in Rc . An estimate of Rc and hence the ISCL is therefore less accurate than that obtained from the calorimetric and the magnetization methods unless the interfilament coupling loss in the strands is large compared with the ISCL. The loss in a magnet is often measured using the calorimetric method or the electrical method. In the electrical method the magnet (or other application) is exposed to one or more field cycles. During each cycle the voltage over the magnet UM and the current through the magnet IM are measured continuously. The loss per cycle is then determined by

Note that with this method the total loss is measured, including the eddy current loss and hysteresis loss in the structure and the resistive loss in connections between superconducting parts. B5.0.9 Worked example In most magnets restrictions are given concerning the additional cryogenic power and the field distortions during ramping and hence Rc . In this worked example it is shown how Rc can be determined by performing a calorimetric loss measurement on a stack of cable pieces (Verweij et al 1994). The specifications of the cable are Ns = 26, Lp, s = 129 mm, w = 17 mm, h = 2.04/2.50 mm so that αc a b = 7.5. Assume that the cable in the magnet is subject to a transverse pressure of about 60 MPa. The cable is unsoldered and the strands are not coated. Ra is therefore of the same order or larger than Rc so that Pa can be neglected (see equations (B5.0.9) and (B5.0.10)). The sample consists of a stack of four pieces of cable (to have a large helium boil-off) each of which has a length three times the cable pitch (to avoid the finite-length effect as shown in figure B5.0.9). The total length is therefore ls a m p l e = 4 × 3 × Lp, s = 1.55 m. Figure B5.0.10 shows the total energy loss Qt o t per cycle for a triangular field change with an amplitude Bm a x = 0.40 T. Three curves are drawn: two for a field change dBy /dt (at 30 and 60 MPa) and one for a field change dBx /dt (which is to a first approximation independent of the pressure). Extrapolation of the curve for a field change dBx /dt gives Qh y s t = 1 J/cycle and dQt o t /df = dQi /df = 3.8 J s/cycle. Extrapolation of the curves for a field change dBy /dt gives dQt o t /df = dQ i f /df + dQi s /df = 263 and 533 J s/cycle (at 30 and 60 MPa). The maxima of the curves correspond f to f = 0.14 and 0.069 Hz so that τi s, s t = 1 /( 2πf ) = 1.13 and 2.31 s (at 30 and 60 MPa). The slopes of the ISCL per metre are dQi s /df = (263–3.8)/ls a m p l e = 167 and (533–3.8)/ls a m p l e = 341 J/cycle/m (at 30 and 60 MPa). These slopes should be given by (using equation (B5.0.10))

since for a triangular field change dBy /dt = 4Bm a x f. The Rc values can now be calculated and are equal to 3.2 and 1.6 µ Ω , which should result in time constants of about (using equation (B5.0.18))

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References

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Figure B5.0.10. Energy loss as a function of the frequency for triangular field changes dBx /dt and dBy /dt with amplitude Bm a x = 0.4 T.

for a single cable, and (see equation (B5.0.19))

for the stack of four pieces of cable, which correspond well with the time constants deduced from the maximum of the Q t o t – f curve. References Akhmetov A A, Devred A and Ogitsu T 1994 Periodicity of crossover currents in a Rutherford-type cable subjected to a time-dependent magnetic field J. Appl. Phys. 75 3176–83 Campbell A M 1982 A general treatment of losses in multifilamentary superconductors Cryogenics 22 3–16 Devred A and Ogitsu T 1994 Ramp-rate sensitivity of SSC dipole magnet prototypes KEK Preprint 94–156 Krempasky L and Schmidt C 1995 Theory of ‘supercurrents’ and their influence on field quality and stability of superconducting magnets J. Appl. Phys. 78 5800–10 Morgan G H 1973 Eddy currents in flat metal-filled superconducting braids J. Appl. Phys. 44 3319–22 Niessen E M J, Ter Avest D and Van de Klundert L J M 1990 Application of the network method to superconducting cables LTEC 90 Ries G and Takacs S 1981 Coupling losses in finite length of superconducting cables and in long cables partially in magnetic field IEEE Trans. Magn. MAG-17 2281 Sytnikov V E, Svalov G G, Akopov S G and Peshkov I B 1989 Coupling losses in superconducting transposed conductors located in changing magnetic fields Cryogenics 29 926–30 Verweij A P and Ten Kate H H J 1993 Coupling currents in Rutherford cables under time varying conditions IEEE Trans. Appl. Supercond. 3 146 Verweij A P, Den Ouden A, Sachse B and Ten Kate H H J 1994 The effect of transverse pressure on the inter-strand coupling loss of Rutherford type of cables Adv. Cryogen. Eng. 40 521–7 Verweij A P 1995 Electrodynamics of superconducting cables in accelerator magnets PhD Thesis University to Twente.

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Further Reading Devred and Ogitsu 1994: deals with the effects of interstrand coupling currents and power loss on the field errors and ramp-rate limitation in SSC dipole prototype magnets. Verweij 1995: gives a detailed description of all coupling currents in Rutherford-type cables and the effects of these currents and power loss on the performance of accelerator magnets, and in particular the LHC dipole magnets.

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B6 Cable-in-conduit superconductors

J-L Duchateau

B6.0.1 Introduction A new kind of superconducting conductor, using the so-called cable-in-conduit conductor (CICC) concept, is slowly emerging mainly related to fusion activity. However, it should be noted that at present no significant magnet in the world is operating using this concept. The difficulty of this technology, which has been studied for 20 years, is that it has to integrate major progresses in multiple interconnected new fields such as: • • • • • • •

cabling of a large number of strands (1000) high-current conductors (50 kA) forced-flow cryogenics Nb3Sn technology low-loss conductors, in pulsed operation high-current connections high-voltage insulation (10 kV).

Inserting the strands carrying the current of a conductor inside a conduit and cooling them by a forcedflow coolant is of course an idea commonly used in conventional electrical techniques and, for instance, in stators and rotors of generators. The aim in doing this is to achieve at the same time, by separating the two functions, a good cooling of the conductor and a high level of electrical insulation, which is wrapped around the conduit. In addition to that, the basic idea at the origin of the cable-in-conduit is to design so-called ‘well-cooled’ conductors according to the Stekly criterion (Stekly and Zar 1965), while keeping the current density, and thus the size and the cost of the magnet, at an acceptable level. This ‘dream’ of any magnet designer is not easy to solve and has always been a much discussed topic. Several kinds of answer have been historically given to that question: (i)

To accept ‘ill-cooled’ high-density conductors as is the case in high-energy physics for the hundreds of magnets of the Tevatron and of Hera (Orr 1983, Wolf 1985). The spectrum of energy release in these magnets is severely limited because of a very sophisticated mechanical structure. In these conditions, these magnets can operate very near their critical current. It is to be noted that the design current is only of the order of 10 kA, the size of these magnets is limited and external electromagnetic perturbations must be avoided.

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(ii) To design well-cooled conductors incorporating very important sections of stabilizer (copper caluminium) in detectors for high-energy physics such as BEBC (Haebel and Wittgenstein 1970) (copper to superconductor ratio of 26) which remains at present the magnet with the highest store energy ever built (800 MJ). (iii) To design well-cooled monolithic one-strand conductors at low current operating in a helium 4.2 K bath. This is the case of the thousands of magnetic resonance imaging (MRI) magnets which at operating with success throughout the world (Lesmond and Lottin 1985). In the same field the particular case of Tore Supra (600 MJ) must be pointed out. The well-cooled situation is achieve here with a large monolithic conductor (2.8 mm × 5.6 mm) operating in a superfluid 1.8 K helium bath (Equipe Tore Supra 1985). But all these particular solutions cannot be extrapolated to large future magnets and it is not surprising that researchers from the fusion field pioneered the cable-in-conduit concept in the 1970s. Magnets for fusion have to meet simultaneously several requirements, such as high currents (50 kA), high fields (13 T high voltages (10 kV), low losses and restricted space due the very high impact of size on the cost of these large fusion machines such as ITER (Montgomery et al 1994) (International Thermonuclear Experiment Reactor). None of the old concepts could of course satisfy these requirements and something new had to be invented. The solution of internally cooled superconductors (ICSs) proposed and developed by Morpurgo (1970) (see figure B6.0.1) was progress in that direction, introducing in particular forced-flow cryogenics as an alternative to the conventional immersion in a liquid helium bath. In this concept the strands of the conductor are not individually wetted by helium but embedded in the stabilizer through which the heat is transferred to the cooling pipe. However, because of the limited capability of the heat transfer coefficient in helium, the Stekly criterion was still very difficult to satisfy without large sections of stabilizer. It is to be noted that this kind of technology has nevertheless been applied with success in systems like the Piotron magnet (Horvath et al 1981) and two magnets of the Large Coil Task (LCT) (Haubenreich et al 1988), the Swiss magnet and the General Electric magnet. For these two magnets the specifications have been reached in operation in ‘ill-cooled’ conditions according to the Stekly criterion.

Figure B6.0.1. The ICS conductor developed by CERN for the Omega detector.

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Figure B6.0.2. Original CICC concept in 1975.

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B6.0.2 The cable-in-conduit history The original cable-in-conduit concept was presented by Hoenig et al in 1975 (see figure B6.0.2). The ‘well-cooled situation’ is achieved here no longer by huge amounts of stabilizer, which can remain in this case limited, but by the subdivision of the conductor into many traposed strands each of them being wetted by forced-flow supercritical helium. With such aconcept the implicit statement is that movements of strands are likely to occur but they are acceptable due to the extremely enhanced condition of the heat transfer in such a cable. In this condition, the level of energy release which is acceptable by the conductor is no longer in relation to the enthalpy of the materials (superconductor and copper) taken from the temperature of operation to the temperature of current sharing but to the energy of the helium sink which is very near the materials in the cross-section of the conductor. This helium available energy, which is discussed further, is about 500 times the enthalpy of the materials! Of course, at that time, perturbing effects such as the limited capability of increased mass flow rate to improve stability, the degrading effect of stainless steel conduits on compressive strain in Nb3Sn in the wind and react concept and the large pressure drop limiting the increase of mass flow rate in such an assembly had still to be discovered and mitigated. In this concept, to avoid any damage to the very brittle Nb3Sn, the coil is reacted after winding to limit the bend of the strands after the formation of the A15 strand. The first important magnet to demonstrate this concept was the Westinghouse coil of the LCT (Haubenreich et al 1988) (see figure B6.0.3). Introducing both the cable-in-conduit concept and Nb3Sn, this magnet was supposed to surpass the results of all the other more conventional magnets. The result was disappointing due to the presence of large resistive parts in the conductor spread over the whole winding, proving the difficulty of the wind and react method on these large magnets. It is to be noted that, in the same LCT, the Euratom magnet (Haubenreich 1988) (see figure B6.0.4) was also relevant to the cable-in-conduit concept, but was still clearly ill cooled due to the large size of the strands, each of them carefully clamped inside a stainless steel jacket. The magnet reached with success the original specifications in operation and can be considered as an intermediate very impressive step between the ICS concept and the cable-in-conduit concept.

Figure B6.0.3. Nb3Sn forced-flow conductor used in the Airco-Westinghouse LCT coil. Courtesy of Airco and Westinghouse.

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Figure B6.0.4. NbTi forced-flow conductor for the Euratom LCT coil. Reporduced from Haubenreich et al (1988) by permission of Elsevier.

Figure B6.0.5. NbTi CICC for the POLO coil. Reproduced from Bayer e t al (1944) by permission of Elsevier.

Recently three CICC magnets have been built and tested. (i) The Demonstration Poloidal Coil (DPC) U1 and U2 magnets (Okuno et al 1989)—an NbTi magnet— part of an important test stand facility for fusion. Current ramp limitations on the magnet have been observed due to the insulation of the individual strands. (ii) The DPC US magnet (Painter et al 1992)—an Nb3Sn magnet—demonstrated the capability to meet on a large conductor the same quality of critical current density as on simple strands due to the thermal expansion coefficient of Incoloy 908 which has been used as the conduit. Again ramp rate limitations have been observed. (iii) The Polo magnet (Bayer et al 1994)—again an NbTi magnet. This magnet has shown very good operation in pulsed conditions with a conductor which heralds on a smaller scale (figure B6.0.5) the future conductors of the fusion programme. The recent period has been very important and has seen the selection of the conductor for the ITER fusion programme. Different grades of this conductor will be present in ITER. One of them is presented in figure B6.0.6 as an illustration. More than 1000 t of this conductor will be needed for the ITER programme and the production has already started. The European industry is particularly important in this production, especially regarding the strand, the cabling and the jacketing of the conductor (della Corte et al 1994). This activity also involves companies from Japan, Russia and the USA. Most of the key points of this conductor have been already been tested during the manufacture by Dour metal industry (Belgium) of 20 m of a 40 kA Nb3Sn conductor developed by CEA (Cadarache) (figure B6.0.7). This conductor was successfully tested in the European test stand facility Sultan at the beginning of 1993 (Bessette et al 1992, 1994). B6.0.3 Manufacturing and design issues in cable-in-conduit superconductors It is clear that the design of the conductor has to be made in relation to the project. Such parameters as the nominal current and the nature of operation (pulsed or steady state) have an impact on the manufacture

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Figure B6.0.6. CICC for the ITER model coils. CS: central solenoid, TF: toroidal field.

and design of the conductor and on the strands constituting the conductor. The main manufacturing and design issues of this conductor can be discussed taking as an illustration the CEA 40 kA conductor which is very similar in design to the ITER (figure B6.0.7). B6.0.3.1 The conduit One of the most important features of the conduit is the internal round shape of the structure giving a natural circular vault on which the six strand bundles can find a support. The main interest of this shape must be seen from the manufacturing point of view. The conductor is manufactured in two main steps. The first step is the cabling process during which the whole cable is made and stored on mandrels. The

Figure B6.0.7. 38 mm × 38 mm Nb3Sn 40 kA CICC developed by CEA for fusion applications.

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second step is the jacketing process. Several hundred metres of jacket are formed by orbital butt welding short unit lengths of jacket (about 8 m). The vacuum and the quality of the jacket and of the weld can be inspected step by step during the making of the conduit. The cable is then pulled through the jacket. It has been demonstrated on long lengths that a very small gap of about 1 mm is sufficient to draw the cable without any lubricant, thanks to this round shape. The external shape, round or square, can be adapted and is more relevant to the magnet design itself. The jacket is then compacted on the cable by pulling it through a die or by a rolling technique. The material used for the conduit is of great importance. On one hand the use of stainless steel A316LN is recommended as it is the reference steel used until now in most magnet structures and vessels, it is easy to weld and its mechanical properties are very well established. On the other hand, the wind and react concept demands a material whose thermal contraction from the reaction temperature to 4 K matches as closely as possible the thermal contraction of Nb3Sn to avoid any additional compressive strain on Nb3Sn (see table B6.0.1). From that point of view titanium and especially Incoloy 908 are candidates for the conduit. It is to be noted that the welding procedure and the heat treatment are very difficult for Incoloy 908 with severe constraints on any oxygen content in the atmosphere during the procedure. Table B6.0.1. Thermal expansions for different materials.

B6.0.3.2 The strand The superconductor associated with the cable can be NbTi or Nb3Sn depending on the specification for the field. The nature and the content of the strand will not be discussed here, but it depends in particular on the project and whether the magnet is operated in the steady state or pulsed. Typical diameters are in the range 0.7 mm to 0.8 mm to ensure an important wetted perimeter and to limit the number of strands. The copper to noncopper ratio has to be adjusted through design criteria which will be examined in section B6.0.5. However, numbers less than unity are difficult to achieve industrially. The copper section embedded in the strand plays a role in the stability. If additional copper is needed for protection it can appear economical to include it in special less-expensive pure copper strands. Particular attention has to be paid to ensure that the transposition of the strands in that operation is not destroyed. For instance, perfect transposition seems to be achieved if one pure copper strand is inserted in a triplet but not in a quadruplet. B6.0.3.3 The chromium coating A great deal of experience exists in chromium coating of Nb3Sn. The aim of this coating is to avoid any sintering during the heat treatment which could be a source of stress accumulation or could locally

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change the good conditions of heat transfer to helium. Moreover, the chromium coating is playing a role in the contact resistance between the wires in a way which is now better understood. By hardening the wire, the chromium limits the contact area between strands (and so the coupling losses) as a function, of course, of the void fraction. This capability is not infinite in the face of the increasing pressure associated with the decrease of the void fraction. A thickness of 2 µm seems so far a good compromise to ensure a good uniform quality and an acceptable additional price (about 10% of the price of the Nb3Sn strand). The coupling currents between strands can be limited by the thin bronze layer surrounding the outermost filaments of the strands. In the case of NbTi, the chromium coating can be avoided and the limitation can be provided by a thin (10 µm) CuNi shell arranged around the filamentary zone (figure B6.0.8).

Figure B6.0.8. Cross-section of a typical NbTi composite for ITER application.

B6.0.3.4 The internal arrangement The internal arrangement of the conductor made of six multistage petals cabled around a central hole allows a mechanical stability of the strands taken inside a kind of vault. Each stage is cabled with a back twist to suppress any residual torsion introduced by the cabling. The twist pitch of a given stage is typically ten times the local diameter. The void fraction of each stage can be adjusted at typically 36% by pulling the stage through a die. Questions still remain concerning the transposition of the strands. It can be demonstrated that in such an arrangment a perfect transposition is not achieved. This small defect slightly destroys the symmetry in the inductive equations which govern the current balance between the strands, in particular during transients. This effect may be at the origin of current ramp limitations observed in large conductors. The central channel allows a significant mass flow rate to circulate on a long length with acceptable pressure drops. The presence or absence of an inner metallic conduit to support the strand vault is still under discussion. On one hand it simplifies the fabrication, avoiding any preshaping of the petal before the cabling, but on the other hand it can roughly double the pressure drop associated with a given mass flow rate and limits the range of operation of the conductor. If the metallic inner conduit exists, it must be perforated to improve the heat transfer between the two channels and avoid any thermal gradient between them. The perforation of such a conduit is not so easy to perform.A possible solution using a metallic spiral is being studied.

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B6.0.3.5 The metallic wrapping The metallic wrapping around the petals plays some part in the mechanical stability of the assembly. The large twist of the last stage (about 400 mm) could give rise to large coupling currents in the case of bare electric contact between the petals. A metallic wrapping with a thickness of 0.1 mm (80% coverage) made of stainless steel or Inconel is likely to limit these losses and will not be eroded during the pulsed operation of the magnet. The external wrapping of the whole bundle maintains the cable during the period between the fabrication of the cable and the insertion into the conduit. It acts also as a safe protection to ensure the integrity of the strands during this phase. B6.0.4 Thermohydraulics in cable-in-conduit superconductors B6.0.4.1 Limits on permanent heat load extraction in a cable-in-conduit superconductor The heat load power falling on the conduit can have several origins: heat radiation, heat conduction or internal losses in the superconductor due to field variations. This power has to be removed by the mass flow rate running inside the conduit. The first question arising concerns the mass flow rate to be taken into account. The answer is clear in the case of a single-channel system as in figure B6.0.2. To a first approximation the double channel of figure B6.0.7 can also be treated as a single channel if the temperature in the cross-section is uniform because of very good heat transfer between the two channels. In these conditions, although two physical flow speeds exist in the two channels, one rapid in the central channel and the other slow in the ring region containing the strands, one unique thermal mass flow can be considered which is the sum of the two mass flow rates proceeding at the average velocity v

where m• is the mass flow rate, A is the helium section, ρ ¯ is the mean helium density in the conduit in the range of temperature and pressure considered. The main equation governing permanent heat load extraction in a conduit is then

where Q is the heat load power on the conductor (constant value independent of time) and ∆H is the available enthalpy. A discussion on this equation has recently been presented by Katheder (1994). The main conclusions are presented here. The question is to know for a given conduit, characterized mainly by its length and its internal hydraulic diameter, the maximum heat power load which is extractable. This maximum heat load is not infinite due to the pressure and temperature boundaries which have to be respected. The upper maximum temperature can be taken at 6 K and the lower minimum pressure taken to 3 bar (3 × 105 Pa), allowing for some margin for the critical pressure (2.3 bar) to avoid the therma instability related to it. Under these conditions the only free parameter is the inlet pressure. At first glance it could appear favourable to increase the inlet pressure and so to increase m• in equation (B6.0.1), but ∆H is in fact a decreasing function of the inlet pressure due to the temperature increase produced by the decompression of the gas along the conduit in this range of temperature and pressure. Some maximum can be pointed out depending on the particular hydraulics of the cable in conduit considered.

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For a single-channel system the mass flow rate can be related to the pressure drop in the following form

where A is the helium section, P the wetted perimeter, ∆p the Pi n l e t – 3 bar, dh the hydraulic diameter, L the length of the cable in conduit and λ the friction factor. The friction factor is given as a function of the Reynolds number in figure B6.0.9 taken from chapter D11.1 for classical tubes (Moody 1944) and in figure B6.0.10 taken from the article by Katheder (1994) for typical cables in conduits. In the case of the double-channel system, assuming the same pressure drop across the two channels, it is possible to derive the mass flow rate distribution between the two channels and then the relation between the mass flow rate and the pressure drop. This work has been done on a classical cable in conduit for fusion applications and the result is presented in table B6.0.2 for a typical total mass flow rate of 20 g s–1. The pressure drop as a function of the mass flow rate is presented in figure B6.0.11. The available enthalpy as a function of the inlet pressure is presented in figure B6.0.12 assuming an outlet pressure of 3 bar. The inlet and maximum temperatures (not necessary at the outlet) are respectively 4.5 K and 6 K and the heat deposition along the conduit is supposed to be linear.

Figure B6.0.9. The friction factor as a function of the Reynolds number for tubes. Reproduced from Moody (1944) by permission of Elsevier.

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Figure B6.0.10. The friction factor as a function of the Reynolds number for different tested CICCs. Reproduced from Katheder (1994) by permission of Elsevier.

Table B6.0.2. Helium distribution in a double-channel system. L = 800 m. The roughness of the inner tube has been taken equal to 0.001.

The extractable heat load as a function of the mass flow rate is presented in figure B6.0.13. A maximum can be pointed out at about 30 g s–1. The design value for operation is lower than this. In fact another term has to be considered for the final thermal balance: it is the fluid work which has to be produced to overcome the flow resistance, that is to say to recompress the fluid at the outlet of the conduit. In large magnet systems such as ITER, the circulation of the fluid is independent of the main refrigerator. It is operated by a system of cold pumps whose industrial development has been in progress in recent years (Zahn et al 1992). The pump work can be calculated as the enthalpy variation to recompress the fluid from 3 bar to the inlet pressure in an isentropic process and is affected by the efficiency of the pump (η ≈ 0.6). This heat load must be kept at a lower level than the main heat load which has to be extracted. It increases rapidly with the mass flow rate. It is also presented in figure B6.0.13.

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Figure B6.0.11. Pressure drop as a function of mass flow rate for a typical ITER CICC.

Figure B6.0.12. Enthalpy available for a typical ITER CICC. Inlet temperature—4.5 K, maximum outlet temperature— 6 K.

Figure B6.0.13. Limitation on permanent heat load extraction for a typical CICC for ITER.

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B6.0.4.2 Transients and train effect on double-channel systems In fact many of these systems operate in pulsed conditions. The average load power over a long period must not exceed the value given in section B6.0.4.1, but this condition is not sufficient if a maximum temperature, say 6 K, must not be exceeded. The temperature increase due to a sudden heat release on the channel is

where W is the sudden heat release due to internal losses (J m–1) and CH e is the specific heat of helium (J m–3 K–1 ). Equation (B6.0.2) is in fact a very simplified presentation of what really happens in the cable. The whole treatment of the thermohydraulics is far more complicated and has to include the treatment of the perturbation wave due to this heat release with its associated effects on the mass flow rate and on the pressure. This full treatment has to take into account the real extremities outside the conduit and the real size of the hydraulic bellows or chambers which are situated there. This temperature increase must be limited and the only factor limiting this increase is, this time, not the mass flow rate but the helium section in the cable. Moreover, the time needed to recool the channel before any other heat pulse occurs is in this case a very important notion governing the design of the central solenoid of the ITER and limiting the repetition rate of the runs which can be performed on that machine. It is given by

and is the time taken by the cold wave entering the cable to replace entirely the hot helium gas accumulated during the transient. In fact due to the limited heat transfer between the two channels, the cold wavefront does not propagate with a straight front. At the end of the channel the earliest decrease of temperature will take place before trc and the complete cold situation will be established after a time greater than trc (Martinez and Turck 1993) in a kind of train effect. The time to be added to tr c is

where v1 is the fluid velocity in the annulus and v2 is the fluid velocity in the central channel

where Pm 1 is the wetted perimeter in the annulus and A1 is the helium section in the annulus

where Pm 2 is the wetted perimeter in the central channel, A2 is the helium section in the central channel, Cp is the mean specific heat of helium in the range of temperature and pressure considered and h is the effective heat transfer coefficient between the two channels. h can be considered as the superposed contribution of three terms

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where h1 is the heat transfer from fluid in annulus to the tube, e, k are the thickness and thermal conductivity of the tube and h2 is the heat transfer from fluid in the tube to the tube. A practical application has been carried out for the CICC presented in table B6.0.2 at a mass flow rate of 20 g s–1. The values used are as follows: h = 100 W m– 2 K–1 (typical value for a 1.5 mm thick inner conduit) Cp = 5000 J kg–1 ρ = 120 kg m– 3 Pm 1 = 47 mm Pm 1= 37.7 mm u2 = 1.15 m s– 1 u1 = 0.095 m s– 1 v = 0.33 m s–1 tr c = 2400 s ∆tt r a i n = 446 s. It can be seen that the real time to recool completely the conduit is not 2400 s but more likely 2886 s. One way to fight this train effect is to enhance the heat transfer between the two channels by suppressing the inner tube of stainless steel.

B6.0.5 General optimization of a cable-in-conduit conductor For the insulation of CICCs a two-stage fiberglass—Kapton component is recommended. This enables the insulation test of the magnet before impregnation thanks to the Kapton and so enables corrections to be made for any disorder as this correction whould be very difficult, even impossible, after the impregnation. In the case of Nb3Sn conductors certain people are pushing for an insulation to accept the heat treatment but this solution cannot yet be considered from an industrial point of view and the main option is always to install the insulation after reaction. The amount and the size of the outer conduit are more related to the mechanics of the magnet and will not be discussed here, keeping in mind that the nature of the conduit and the void fraction will influence the compressive strain of the Nb3Sn filaments and thus the critical properties Jn o n C u , Bc 2, Tc 2 to be considered in the further criteria. The parameters under discussion for the design of a CICC are given below. For a given field and transport current they have to be selected to give the maximum current density. The internal arrangement of the strands is not discussed here. Only a limited approach concerning the hydraulics is proposed: (i) the helium sections (ii) the noncopper section (iii) the copper section inside the superconducting strands (iv) the copper section in extra copper strands (if needed) (v) the filament effective diameter, the time constant of the conductor (vi) the mass flow rate (vii) the strand diameter. The action of these different parameters will be followed in the discussion of the different following criteria.

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B6.0.5.1 Operation temperature and temperature margin The temperature increase ∆Ta c due to a.c. losses from T0 , the inlet temperature, is determined either by equations (B6.0.1) or (B6.0.2) from section B6.0.4, depending on the nature of the thermal load of the magnet, permanent or transient. The temperature of operation is given by

∆Ta c is affected by (i), (v) and (vi) in the list above. The a.c. losses can be minimized in particular: • •

for the hysteretic part by using filaments with small effective diameters, for the coupling-current part by using a resistive metal wrapping around the last but one stage of the cable, and a sufficiently thick bronze barrier around the filament bundles of the strands in the case of Nb3Sn and an adapted CuNi shell in the case of NbTi strands.

Once this temperature of operation is calculated it is necessary from the design point of view to keep a temperature margin between the temperature of operation and the current-sharing temperature

where Tc s is the current sharing temperature. This temperature margin is necessary to provide the conditions for normal operation without any risk of transition. • • • •

It ensures stability against perturbations and mainly plasma disruption effects in the case of fusion applications. It takes into account the scattering in Jc for large quantities of superconducting material (±15% on Jc affects Tcs at 13 T by ±0.4 K). It covers the local stress concentration, inhomogeneities, local resistive effects and ‘n’ value effects for long lengths of strands. It establishes the level of heat release which is possible in the conductors with recovery of the superconducting state. This level is related to the helium enthalpy. Typically a margin of 2 K is taken in the fusion program. ∆Ts is affected essentially by (ii).

B6.0.5.2 Hot-spot temperature criterion During a coil dump triggered by a quench, the temperature of the hot spot is supposed to increase adiabatically. The maximum value, when limited to 150 K, ensures that no significant thermal stresses appear in the winding pack due to temperature inhomogeneities, in particular if some parts of the coils are still at 4 K. The maximum temperature Tm a x is given by

J, C, ρ are average current density, specific heat and resistivity of the conductor including all materials. (iii) and (iv) are particularly concerned in this criterion. It is clear that the jacket heat capacity can be taken into account as long as the diffusion length is larger than the jacket thickness. During a field decay with a time constant as long as 20 s, heat diffuses in steel over more than 10 mm between 10 K and 150 K. This can have a great impact on Tm a x as the heat capacity of steel greatly exceeds the heat capacity of copper.

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References

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B6.0.5.3 Stability and well-cooled criteria This topic has been treated extensively in chapter B3.4 and by Ciazynski and Turck (1993). The energy that can be deposited without irreversible transition strongly depends on the current and heat transfer. There is a current above which this critical energy is very limited (ill-cooled conditions). For a constant heat transfer, hc , the boundary is given by the expression

where d is the strand diameter, α is the wetted perimeter coefficient (typically -56 for a 36% void fraction) and Tc is the critical temperature. To enter into the well-cooled regime, that is to make use of a significant part of the available energy Em a x in helium (between T0 and Tc s ), implies operating at currents significantly below this limit. In these conditions only, the basic original purpose of the cable-in-conduit concept is reached. In fact this criterion applies to very narrow perturbations and the situation is more favourable for long perturbations (above 100 ms, in which case most of the energy in helium is available). However, considering that short perturbations (of a mechanical nature for instance) can be expected in CICCs, a well-cooled criterion affected by a safety coefficient β less than unity should be applied to determine the stable operating currents (Turck et al 1993)

The criterion is affected by (iii) and (vii). Remarks on the available energy in helium The available energy is related to the length over which the perturbation is deposited. While localized perturbation can give rise to a local flow acceleration (improved heat transfer) the fluid enthalpy (at constant pressure and constant mass) can hardly be expected. In fact in that case the available energy is very near the local enthalpy (isobaric). On the other hand for long heated-zones, helium is heated at almost constant volume, and only the internal energy can be used for stability. This discussion is not academic because the order of magnitude is very different depending on the thermodynamic process. The following values are related to the helium volume for a temperature excursion of typically 2 K, corresponding to the margin (see section B6.0.5.1): • • • • •

internal energy in helium 640 kJ m–3 local enthalpy ∫ δ(T) Cp(T)dT 1660 kJ m–3 enthalpy in helium 2270 kJ m–3 δ helium density Cp helium specific heat values. For comparison:

• • •

enthalpy in copper 2700 J m– 3 enthalpy in stainless steel 40 000 J m– 3 enthalpy in Nb3Sn 7400 J m– 3.

These features are very difficult to validate experimentally. Even if the ill-cooled-well-cooled transitions have been very well assessed over many experiments, the level of available energy on long conductors, typical of large magnets subjected to wide perturbations, has still to be evaluated. The main difficulty with this experiment is how to install long conductors in large-bore magnets which will produce both a high relevant background field and a field variation which will produce the energy deposition in

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the conductor itself. Heating by external resistors is generally not the best way to simulate this heat deposition. The exact evaluation of the energy deposited through accurate magnetization measurements is just as difficult a problem in such experiments. References Bayer H et al 1994 Test of the POLO model coil, a superconducting poloidal coil according to the specifications of the tokamak operation in the KfK TOSKA facility Proc. 18th Symp. on Fusion Technology (Karlsruhe, 1994) (Amsterdam: Elsevier) p 917 Bessette D et al 1992 Fabrication and test results of the 40 kA CEA conductor for NET/ITER Proc. 17th Symp. on Fusion Technology (Rome, 1992) (Amsterdam: Elsevier) p 788 Bessette D, Duchateau J L, Decool R and Turck B 1994 Qualification of a 40 kA Nb3Sn superconducting conductor for NET/ITER coils IEEE Trans. Magn. MAG-30 2038 Ciazynski D and Turck B 1993 Stability criteria and critical energy in superconducting cable in conduit conductor Cryogenics 33 1066–71 della Corte A et at 1994 Conductor fabrication for ITER model coils. Status of the EU cabling and jacketing activities Proc. 18th Symp. on Fusion Technology (Karlsruhe, 1994) (Amsterdam: Elsevier) p 885 Equipe Tore Supra 1991 A Tokamak with superconducting toroidal fields coils. Status after the first plasmas IEEE Trans. Magn. MAG-27 2057 Haebel E U and Wittgenstein F 1970 Big European bubble chamber (BEBC) magnet progress report Proc. 3rd Int. Conf on Magnet Technology (Hamburg, 1970) (Hamburg: DESY) p 874 Haubenreich et al 1988 Fusion Eng. Design 7 (special issue on the IEA Large Coil Task) Hoenig M O et al 1975 Supercritical-helium cooled bundle conductors and their applications to large superconducting magnets Proc. 5th Int. Conf. on Magnet Technology (MT-5) (Rome, 1975) (Laboratori Nazionale del CNEN) p 519 Horvath I L, Vecsey G and Zellweger J 1981 The PIOTRON at SIN. A large superconducting double torus spectrometer IEEE Trans. Magn. MAG-17 MT-7 p 1878 Katheder H 1994 Optimum thermohydraulic operation regime for cable in conduit superconductors (CICS) Cryogenics 34 (ICEC Suppl.) 595 Lesmond C and Lottin J C 1985 A 2 teslas NMR superconducting magnet Proc. Int. Conf on Magnet Technology (Zürich, 1985) (SIN Publisher) p 255 Martinez A and Turck B 1993 A supercritical helium cooling of a cable in conduit conductor with an inner tube Internal CEA Note PEM 93.18 Montgomery B, Okuno K, Torossian A, Trobhachev G and Tsiyi H 1995 Fusion Eng. Design 30 133 Moody L F 1944 Friction factor for pipe flow Trans. ASME 66 Morpurgo M 1970 Review done at CERN on superconducting coils cooled by a forced circulation of supercritical helium Proc. 3rd Int. Conf on Magnet Technology (Hamburg, 1970) (Hamburg: DESY) p 908 Okuno K et at The first experiment of the 30 kA Nb—Ti Demo Poloidal Coils (DPC U1 and U2) Proc. 11th Int. Conf on Magnet Technology (MT-11) (Amsterdam: Elsevier) p 812 On J R 1983 Status of the energy saver IEEE Trans. Magn. MAG-19 195 Painter T A et al 1992 Test data from the US-Demonstration Poloidal Coil experiment MPT and JAERI Internal Report DOE/ER154 110-1 Stekly J and Zar J L 1965 Stable superconducting coils IEEE Trans. Nucl. Sci. 12 367 Turck B, Bessette D, Ciazynski D and Duchateau J L 1993 Design methods and actual performances of conductors for the superconducting coils of Tokamaks 15th SOFE (Hyannis, MA, 1993) (Piscataway, NJ: IEEE) Wolf S 1985 The superconducting magnet system for Hera Proc. 9th Int. Conf on Magnet Technology (Zurich, 1985) (SIN Publisher) p 62 Zahn G et al 1992 Test of three different pumps for circulating HeI and Hell Cryogenics 32 (ICEC Suppl.) 100

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B7.1 Critical temperature

Julian R Cave

B7.1.1 Introduction The number of independent parameters needed to describe homogeneous superconductors can be surprisingly few. From the BCS theory four parameters suffice (Tc , Bc 2 , κ and ρn , the normal-state resistivity) and near to Tc the Ginzburg—Landau theory uses only two independent parameters—for example Tc and Bc 2 and the other properties can be derived (see the discussion by Evetts (1983)). However, in practical materials, both low- and high-temperature superconducting materials, inhomogeneity, anisotropy and granularity considerably complicate this picture and care has to be taken in in terpretation of Tc measurements. The critical temperature, Tc , of a superconducting material can be characterized by any property that changes rapidly at Tc . The most commonly used methods are the ‘resistive’ method, in which a small measuring current is passed through the sample and the transition from zero to finite resistance is detected, and the ‘inductive’ method, in which magnetization currents are induced in the sample and the resulting magnetic moment or susceptibility is measured. Each method can be realized in several ways and is sensitive in different ways to the spread of Tc in an inhomogeneous material, the sample geometry, its orientation and granularity. For sample geometries and inhomogeneity distributions that are well defined from other physical knowledge then the resistive and inductive Tc transitions used in conjunction can lead to a fine analysis of the variation of Tc within a sample. For materials, such as high-Tc materials, that are known to be granular simple extensions of the basic methods can be used to investigate this specific property. In the development of practical wires (both for liquid-helium applications and for liquid-nitrogen applications) the measurement of Tc is used as a tool in comparing the relative merits of different fabrication processes. In particular, for high-temperature superconducting wires these processes have many parameters that can be varied. In these cases quick measurements of Tc are necessary that are of relative accuracy only—more detailed absolute investigation can be made later on representative samples. As the number of independent primary parameters for the description of superconductivity is low, many scaling theories for derived properties, such as pinning and the critical current density, Jc , have been very successful. Modelling of the behaviour of a homogeneous superconductor in this way is a useful way of exploring and understanding a given measurement technique and facilitates extensions of methods for practical inhomogeneous materials. B7.1.2 Critical temperature: T c ( B ) The critical temperature, Tc( B ), is reduced in the presence of a magnetic field. The relationship between Tc , Bc 2 and Jc is shown schematically in figure B7.1.1. Near to Tc (and also near to Bc 2 , see the next section,

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Figure B7.1.1. The critical surface for superconductivity shown as a surface of constant electric field, E(B, T, J ). With an appropriate selection of the electric field criterion this surface can be used to define Tc (B ), Bc 2 (T ) and Jc (B, T ). T2*, Bc2* are extrapolated values that are more representative of ‘bulk’ properties rather than the tail of the inhomogeneity distribution.

B7.2) the inhomogeneity in a material leads to broadening of the transition from the superconducting to the normal state. For the resistive measurement the terms shown in the lower part of figure B7.1.1 are commonly used. When using such measurements to compare materials consistency of approach is necessary—a reasonable choice for J(measuring) and Tc extracted in the same way, i.e. Tc (midpoint) and ∆Tc can be more useful than just Tc (midpoint) or Tc (onset). B7.1.3 Basic measurement techniques for Tc Two frequently used techniques are the resistive measurement and the inductive measurement. For the resistive measurement, a classic four-terminal version is shown schematically in figure B7.1.2. A small DC or AC measuring current is injected into the sample and the voltage detected by sensitive electronics. A temperature controller slowly sweeps the temperature of the sample holder up and down. By using a computer the measurement can be automated. This not only saves time but also ensures more reproducible results. A typical inductive AC Tc set-up is shown in figure B7.1.3. A low-field and low-frequency AC susceptibility measurement using standard lock-in techniques (giving χ′ and χ″ , the real and imaginary components respectively), unlike its DC counterpart (for example using SQUID magnetometry), is not sensitive to the reversible magnetization in the presence of a DC bias field, B0 > Bc 1 . This is because the reversible magnetization is not hysteretic and its effect tends to cancel out over an AC cycle. For certain types of lock-in detector the voltage waveform harmonics can be retained and the phase adjusted

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Figure B7.1.2. (a) A typical laboratory-constructed resistive Tc measurement set-up and (b) a detail of the sample platform. In classic versions such as this and in more modern versions using cryocoolers the same considerations apply—good thermalization of measuring leads and sample, correction for thermal emfs (by reversing DC polarity or by using low frequency AC and a lock-in amplifier) and accurate temperature control of the sample platform (several heaters and thermometers can be used). For measurements in magnetic fields, field-insensitive thermometers are needed (carbon glass, capacitance, thin-film ceramic oxide sensors,…).

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Figure B7.1.3. A typical set-up for a laboratory constructed inductive measurement of Tc . The addition of a DC coil extends the range of possible measurements (in this case a useful trick is to wind a ripple coil consisting of two windings in opposite senses that reduce mutual inductance effects between the AC and DC coils).

to give an output proportional to the total flux entering and leaving the sample per cycle; this is useful for comparisons with calculated flux profiles. By reducing the input filter range to pass only the fundamental frequency, the AC loss can be calculated for further correlation with various models (loss = χ″ b 02π /µ0 [ J m– 3/cycle] where b0 is the AC amplitude). Obtaining both the total flux and the AC loss requires two measurements with different lock-in settings on two separate experimental runs, or continually switching between the broad-band and narrowband modes. Both these methods have drawbacks, the former requiring more time and the latter requiring the reference phase setting to also be changed. An alternative would be to use a waveform averager or averaging digital oscilloscope in broad-band mode to acquire the full voltage waveform and then to obtain the desired values by analysis. For high-Tc materials their granular nature, extended E—J curve, strong flux creep effects and the presence of irreversibility fields and temperatures means that the exact regime has to be identified in terms of the sample size in relation to the flux-flow skin depth and the normal-state skin depth (see Campbell 1991) as well as demagnetizing fields (see Goldfarb et al 1991). B7.1.4 Rapid screening for superconductivity In developing superconducting wires for applications it is often necessary to optimize processing techniques. Here, a rapid feedback of information on properties is required. Figure B7.1.4 shows a versatile (low-absolute-accuracy but high-relative-accuracy) rapid-sample-property-measuring system. Lowfield magnetization curves, AC susceptibility, V—I curves and multiple resistive Tc transitions can be obtained. In our experience, such a measurement technique has helped cope with the large number of samples that are generated in fabrication process optimization.

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Figure B7.1.4. A rapid screening system for superconductivity: (a) the basic system and inductive measurement, and (b) multiple-sample resistive measurement. With this apparatus the relative values of both resistive and inductive Tc can be scanned quickly for many samples. For example, we have developed this type of apparatus in our laboratory for the optimization of fabrication process parameters of Ag/Bi-2223 high-temperature superconducting wire. The main advantage is that high relative accuracy and speed of measurement can be achieved; the best samples are analysed on slower but more precise equipment. In addition, critical current and low-field magnetization can also be measured using this set-up in slightly different configurations (see the insets).

B7.1.5 Measurements on practical wires and cables The resistive Tc transition is sensitive to the best superconducting material present. The onset occurs when the first regions of size ~ ξ (the coherence length) become superconducting and progresses to completion when the first percolation paths along the sample are established (see figure B7.1.5). For a

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Figure B7.1.4 Continued.

Figure B7.1.5. A schematic diagram showing the difference between resistive and inductive Tc measurements in an inhomogeneous superconducting material.

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uniform distribution this means a 15–17% volume fraction in 3D and ~50% in 2D. The inductive transition, however, only begins when percolation loops of size > λ (the magnetic penetration depth) are formed and is complete when many percolation loops circulate the entire sample dimension. This transition thus reflects a more averaged property of the sample. As in any magnetization measurement, the signal is a product of a current and an area or the sum of many currents multiplied by their respective areas. This leads to ambiguity in interpretation of results for different sample geometries and phase distributions (see Hein 1986). However, if some prior knowledge of physical properties is available in certain cases this can be used to extract useful information. For example, for an Nb3Sn diffusion layer the Tc will vary smoothly from ~9 to ~18 K across its width. A modified London equation,

can be used to calculate flux penetration from the superconducting/normal interface and the Tc (x) variation across the layer can be derived (Cave and Evetts 1985). For granular and two-component superconductors (frequently encountered in high-Tc materials) both resistive and inductive transitions are broadened (see Goldfarb et al 1991, 1987). Where the nature of the granularity is known (or assumptions can be made) useful information on intragrain and intergrain Jc can be obtained (see Müller 1989, 1991, Müller et al 1994). Figure B7.1.6 shows schematically some example resistive and inductive transitions. In figure B7.1.7 data are given for a practical low-Tc multifilamentary wire and in figure B7.1.8 a derived Tc (x) profile for Nb3Sn is shown. B7.1.6 Example measurements on high-Tc superconductors High-temperature superconductors are complex oxides with typically five or more constituent elements. They can be in the form of highly anisotropic sub-micrometre thin films with Jc over 106 A cm– 2, as powder-in-tube fabricated composite wires containing partially aligned grains and a normal metal sheath with Jc from 1 to 7 × 104 A cm– 2 or as less well aligned bulk sintered artefacts such as plates and cylinders that can contain significant amounts of secondary phases and which have Jc around 103 A cm– 2. The complex microstructure of a silver—gold-alloy-sheathed BiPbSrCaCu oxide (AgAu/Bi-2223) superconducting monofilamentary wire with onset Tc around 110 K is shown in figure B7.1.9. Clearly one would not expect simple resistive and inductive Tc transitions for such a material. In general, the higher the Jc , the more uniform the material and the smaller the corresponding transition breadth. Figure B7.1.10 shows DC (SQUID) ZFC Tc transitions (ZFC, zero-field cooled; the sample is first cooled in zero field to a low temperature then slowly warmed to above Tc ) for AgAu/Bi-2223 tape samples that have undergone different heat treatments. An important difference between AC measurements and DC measurements is that the time scale over which the internal flux profiles are allowed to creep are considerably different. The DC measurement allows far more creep to take place, thus lowering the measured signal; near to Tc this affects the sensitivity of the measurement. In the AC measurement (hysteretic) flux profiles are re-established each AC cycle on a time scale of 1/f and thus far less creep takes place. B7.1.7 Analysis and modelling† AC susceptibility is sensitive to the granularity of the sample and figure B7.1.11 shows typical (lowfrequency) AC profiles. This figure suggests a method of extending the measurement to qualitatively

† Based on the discussion by Cave (1991)

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Figure B7.1.6. A comparison of normalized resistive and inductive Tc transitions for a homogeneous superconductor (top), an inhomogeneous superconductor (middle) and an example of a two-component superconductor (bottom). The inductive transition is scaled to the hypothetical case where λ = 0. The finite value of λ and its temperature dependence, for example λ(T ) ∼ λ0( 1 – t 4 )–-1/2 where t = T/Tc , lead to a reduction in the low-temperature inductive signal. In the granular case the reduction becomes more evident due to the large number of grains and the weak superconducting properties between them. ∆Tc 1 and ∆Tc 2 indicate the separate spreads in Tc for the granular and intergranular material respectively.

reveal the degree of granularity and the low-field variation of Jc, which consists of applying a DC offset field, B0, larger than the expected Bc1 (0 K). In this case it is no longer reversible surface currents causing flux exclusion, there is some flux penetration and sequences of hysteretic flux profiles are being swept out. The resulting effect is a shift of the χ′ and χ″ curves to lower temperatures. If the sample is non-granulur and has strong pinning, then this displacement is slight If, however, the sample is granular with the typical decrease of Jc with field, then the shift will be much larger, and, usually, a double-peak structure will appear in the χ″ curves. The low-temperature peak occurs as the flux penetration through the intergrain material changes from an incomplete-penetration regime (where the AC profiles do not penetrate to the sample centre) to a complete-penetration regime (where the profiles touch at the sample centre). The position of the maximum of this peak occurs approximately when the profile first touches the centre. At

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Figure B7.1.7. Scaled resistive and inductive Tc transitions for an unreacted and a partially reacted (1 h at 750 °C) commerical multifilamentary NB3Sn composite (IM1 All Bronze 3000 × 5 µm filaments) (data from Cave and Weir (1983), figure redrawn by permission of IEEE). The large spread in the resistsive transition is due to the presence of a very small amount of pre-reacted NB3Sn; this illustrates the large effect a small amount of material can have on the resistive transition.

Figure B7.1.8. The critical temperature variation across a 2 µ m thick Nb3Sn diffusion layer: Tc (x) (dashed and dotted (extrapolated)), derived from the corresponding inductive Tc transition S( T ) (solid). The position x = 0 represents the Nb/Nb3Sn interface where Tc ∼ 9 K. The curvature in the inductive transition near to Tc is caused by inhomogeneity and the resulting Tc variation (and not by λ ( T/Tc ) variation near to Tc ) and when seen is a tell-tale sign that a particular sample is inhomogeneous. This type of detailed information can be related to the superconducting phase growth mechanisms (data from Cave and Evetts (1985), figure redrawn by permission of Plenum Publishing Corporation).

the peak, the interdomain Jc , the temperature, T, and the AC ripple field, b0, can be related approximately through b0 = µ0 Jc (T, Bapplied )a′ where 2a′ is the relevant sample dimension and Bapplied = B0 (plus a small contribution from b0 ). Thus, by varying the DC and AC components, a small region of the Jc –T–Bapplied surface for the intergranular Jc can be mapped out (Cave 1991).

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Figure B7.1.9. An example microstructure of a monofilamentary silver–gold-alloy-sheathed BiPbSrCaCu oxide showing some of the many complex features typical of these materials.

Figure B7.1.10. Examples of the use of sensitive SQUID magnetometry to measure inductive Tc transitions for AgAu/Bi-2223 tape samples with different heat treatments. In this case the sample was transverse to the field to obtain not only a stronger signal but also information on the circulating macroscopic transport currents (see Müller et al 1994). Two types of information can be obtained from this type of measurement. Firstly, a large signal at low temperatures that does not diminish significantly with temperature (until near to Tc ) means strong screening currents in a well connected sample; however, a rapid decrease of signal with temperature (as seen here) is indicative of granular and inhomogeneous material—many small screening current loops give a smaller magnetization signal than a single loop circulating the entire sample. Secondly, the values of Tc (both Tc (onset) and T c* (obtained by linear extrapolation from just below Tc (onset)) show the effect of processing on the composition of the superconducting material.

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Figure B7.1.11. A schematic representation of extremal flux profiles in an AC susceptibility measurement for nongranular material and granular material with and without a DC offset field. A typical susceptibility measurement for the non-granular (solid) and the granular (dashed) materials is shown in the inset (from Cave 1991).

Popular characterization techniques which do not need current and voltage connections to be made to a sample are DC magnetometry (SQUID, VSM, integrators etc) and AC susceptibility. A drawback of these inductive techniques, which measure the averaged bulk magnetization, 〈µ0M 〉, is that the critical current density has to be calculated. Magnetization is proportional to the product of current density multiplied by a distance. An often-used expression, derived from the critical-state model, which relates magnetization to Jc , is µ0 Jc a′ = α 〈µ0M 〉 where α is a geometrical constant and a′ is the characteristic scaling length for the induced circulating transport current. A problem that arises for granular materials is that a′ may vary (depending on applied field and temperature) between the full sample dimension and the coherence length, thus making an accurate determination of Jc difficult. Three methods that help overcome this problem are to progressively thin the sample (to change the maximum scaling length possible), to vary the measurement parameters to check for consistency and to use analysis to obtain the scaling length as well as the Jc (Angadi et al 1991). A typical low-field DC susceptibility measurement generates three curves designated ZFC, FC and REM (figure B7.1.12: the susceptibility is obtained by dividing the magnetization by the applied field). The REV, ZEC, FC and REM curves were calculated numerically using the following functions for the reversible magnetization and the critical current density:

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Figure B7.1.12. (a) Calculated DC susceptibility curves REV, ZFC, FC and REM and (b) a schematic diagram of the evolution of the internal flux profiles. At T* the profile first penetrates to the sample centre. The parameters are: Tc 0 = 92 K, Bapplied = 6 mT, Bc10 = 18 mT, Bc20 = 10 T and Jc 0 = 106 A cm– 2 for a slab of width 2a = 20 µm. This example is chosen to be similar to experimental results for YBCO single crystals (from Cave 1991).

and

for Bc1 ≤ µ0H ≤ Bc2. For For this example the critical current density is given by

with Bc 1(T) = Bc 10(l – (T/Tc 0)2 ), and Bc 2(T) = Bc 20(l – (T/Tc 0 )2 ). To obtain the ZFC curve the sample is first cooled from above Tc to a low temperature in zero applied field. A small DC field (usually under 100 mT) is then applied and the magnetic moment is measured as the sample is slowly warmed up to above Tc . This curve shows how well flux is excluded from the sample. The second curve, FC (field cooled), is the magnetic moment measured whilst cooling the sample in the DC field from above Tc to low temperatures. This curve shows the Meissner effect, i.e. flux expulsion from the sample. The third curve, REM (remanent trapped flux), shows the evolution of the magnetic moment as the sample is again warmed after removal of the applied field following an FC measurement. The REM curve can approach the temperature axis more steeply than the others near to Tc (because the external boundary condition is zero applied field) giving a more accurate measurement of Tc . Further information can be obtained from the ZFC curve, which should be flat and equal to minus the applied field up to the temperature, T0, where Bc 1( T0 ) = Bapplied. If the curve is not flat and/or the experimental value of T0 is lower than expected then the sample may be granular and/or inhomogeneous. The closer the FC curve is to the temperature axis the stronger the pinning, i.e. the flux lines are being effectively retained within the sample. Strong pinning, and thus high Jc , is also reflected in a large difference in the magnetization values of the ZFC and FC curves, especially in the region from T0 up to Tc . In the case of the FC and REM curves (see the flux profiles in figure B7.1.12(b)), the flux lines near the centre of the sample are often not described by the standard critical state model with J = ± Jc or 0, but rather they are in a sub-critical state. This state can arise because the pinning strength increases as the temperature is lowered during the FC measurement, thus freezing the position of these central flux lines. The REV curve also shown in figure B7.1.12 is the reversible magnetization—both the ZFC and FC curves lie on this curve in the case of no pinning. In addition to Tc , low-field DC susceptibility measurements can give much information on pinning and the reversible component of the magnetization. However, it can be seen from the above example and discussion that the evolution of the flux profiles is complicated. Consequently, the extraction of Jc , with

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its temperature and field dependencies, is best obtained from a comparison to calculated curves, rather than using simplified expressions, such as µ0Jc a′ = α 〈µ0M〉 , which have little range of validity in this situation. B7.1.8 Conclusions The measurement of the primary properties of a superconductor, such as Tc , are important for testing theories and for optimizing fabrication processes. Commonly used techniques are resistive Tc measurements (with small DC or AC applied current) and inductive Tc measurements (low-field and low-frequency susceptibility and DC magnetization, for example SQUID and VSM magnetometers). Practical wires and cables made from both low-Tc and high-Tc superconductors can be granular, anisotropic and inhomogeneous. The measurement of Tc by more than one method can be useful in identifying physical properties. In particular, the onset of a resistive Tc measurement is very sensitive to the best material present and an inductive Tc measurement gives a more macroscopically averaged value. The use of less accurate but rapid measurements of Tc can help speed the optimization of fabrication processes, especially for high-Tc superconductors where multistage processes are common. For well defined samples, even if inhomogeneous, the basic techniques can be extended to obtain more detailed information on a fine scale. References Angadi M A, Caplin A D, Laverty J R and Shen Z ´ 1991 Non-destructive determination of the current-carrying length scale in superconducting crystals and thin films Physica C 177 479–86 Campbell AM 1991 DC magnetization and flux profile techniques Magnetic Susceptibilities of Superconductors and other Spin Systems ed R A Hein (New York: Plenum) pp 129–55 Cave J R 1991 Susceptibility and magnetization characterization of bulk high Tc superconductors Proc. 6th Int. Workshop on Critical Currents, Scaling Length and Critical Current Densities (Cambridge, 1991) Supercond. Sci. Technol. Suppl. 399–402 Cave J R and Evetts J E 1985 Critical temperature profile determination using a modified London equation for inhomogeneous superconductors J. Low Temp. Phys. 63 35–55 Cave J R and Weir C A F 1983 Cracking and layer growth in Nb3Sn bronze route material IEEE Trans. Magn. MAG-19 1120–3 Evetts J E 1983 The characterization of superconducting materials—conflicts and correlations IEEE Trans. Magn. MAG-19 1109–19 Goldfarb R B, Lelental M and Thompson C A 1991 Alternating-field susceptometry and magnetic susceptibility of superconductors Magnetic Susceptibility of Superconductors and other Spin Systems ed R A Hein, T L Franca villa and D H Liebenberg (New York: Plenum) Hein R A 1986 AC magnetic susceptibility, Meissner effect, and bulk superconductivity Phys. Rev. B 33 7539–49 Müller K-H 1989 AC susceptibility of high temperature superconductors in a critical state model Physica C 159 717–26 —1991 Detailed theory of the magnetic response of high-temperature ceramic supercondutors Magnetic Susceptibility of Superconductors and other Spin Systems ed R A Hein (New York: Plenum) pp 229–50 Müller K-H, Andrikidis C, Liu H K and Dou S X 1994 Intergranular and intragranular critical current in silver-sheathed Pb—Bi—Sr—Cu—O tapes Phys. Rev. B 50 10218–24

Further reading Ando Y, Kubota H, Sato Y and Terasaki I 1994 Linear AC magnetic response near the vortex-glass transition in single-crystalline YBa2Cu3O7 Phys. Rev. B 50 9680–3 Brandt E H 1996 Superconductors of finite thickness in a perpendicular magnetic field Phys. Rev. B 54 4246–64 Campbell A M and Evetts J E 1972 Critical Currents in Superconductors (London: Taylor and Francis)

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Cave J R 1989 Electromagnetic properties of ultra-fine filamentary superconductors Cryogenics 29 304–11 Cave J R, Mautref M, Agnoux C, Leriche A and Fevrier A 1989 Electromagnetic properties of sintered YBaCuO superconductors: critical current densities, transport currents and AC losses Cryogenics 29 341–6 Chandran M and Chaddah P 1995 Low-field AC magnetization of granular high-Tc superconductors Supercond. Sci. Technol. 8 774–8 Chen D-X, Burg J A, IEEE and Goldfarb R B 1991 Demagnetizing factors for cylinders IEEE Trans. Magn. MAG-27 3601–19 Chen Q Y 1991 AC inductive measurements: application to the studies of high-Tc superconductivity Magnetic Susceptibility of Superconductors and other Spin Systems ed R A Hein (New York: Plenum) pp 81–105 Däumling M and Larbalestier D C 1989 Critical state in disk-shaped superconductors Phys. Rev. B 40 9350–3 Dubots P and Cave J 1988 Critical currents of power-based superconducting wires Cryogenics 28 661–7 Ekin J W 1983 Four-dimensional J–B—T–ε critical surface for superconductors J. Appl. Phys. 54 303–6 Evetts J, Cahn R W and Bever M B 1992 Concise Encyclopedia of Magnetic and Superconducting Materials (Oxford: Pergamon) Evetts J E, Cave J R, Somekh R E, Stanton J P and Campbell A M 1981 Characterization of Nb3Sn diffusion layer material IEEE Trans. Magn. MAG-17 360–3 Gélinas C, Lambert P, Dubé D, Arsenault B and Cave J R 1993 Texturing of thick films on a metallic substrate Supercond. Sci. Technol. 6 368–72 Goldfarb R B, Clark A F, Braginski A I and Panson A J 1987 Evidence for two superconducting components in oxygen-annealed single-phase Y—Ba—Cu—O Cryogenics 27 475–9 Hautanen K E, Oussena M and Cave J R 1993 Detailed analysis of magnetization data and transport critical current measurements for Ag—(Bi,Pb)SrCaCuO composite tapes Cryogenics 33 326–32 Hein R A, Francavilla T L and Liebenberg D H 1991 Magnetic Susceptibility of Superconductors and Other Spin Systems (New York: Plenum) Hoare F E, Jackson L C and Kurti N 1961 Experimental Cryophysics (London: Butterworths) Larbalestier D C and Maley M P 1993 Conductors from superconductors: conventional low-temperature and new high-temperature superconducting conductors MRS Bull. August 50–6 Li Y H, Kilner J A, Dhalle M, Caplin A D, Grasso G and Flukiger R 1995 ‘Brick wall’ or ‘rail switch’ the role of low-angle ab-axis grain boundaries in critical current of BSCCO tapes Supercond. Sci. Technol. 8 764–8 Lobb C J, Tinkham M and Skocpol W J 1978 Percolation in inhomogeneous superconducting composite wires Solid State Commun. 27 1273–5 Malozemoff A P 1993 Superconducting wire gets hotter IEEE Spectrum December 26–30 Mikheenko P N and Kuzovlev Y E 1993 Inductance measurements of HTSC films with high critical currents Physica C 204 229–36 Pardo F López and de la Cruz F 1994 Low field brick wall model behaviour in ceramic Gd1Ba2Cu3O7 Physica B 194–196 2013–4 Russell B S 1959 Cryogenic Engineering (Princeton, NJ: Van Nostrand) Shaulov A, Krause J K, Dodrill B C and Wang V 1991 Harmonic susceptibilities in high temperature superconductors Lake Shore Cryotronics 12/91 M2 1–3 Suenaga M and Clark A F 1980 Filamentary A15 Superconductors (New York: Plenum) Takács S and Gömöry F 1993 AC susceptibility of melt-processed high-Tc . superconductors Cryogenics 33 133–7 Tinkham M 1975 Introduction to Superconductivity (New York: McGraw-Hill) White G K 1968 Experimental Techniques in Low-Temperature Physics 2nd edn (Bristol: Wright) Zhu J, Mester J, Lockhart J and Turneaure J 1993 Critical states in 2D disk-shaped type-II superconductors in periodic external magnetic field Physica C 212 216–22

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B7.2 Critical fields

Julian R Cave

B7.2.1 Introduction Superconductivity is characterized by several critical magnetic fields as well as the critical temperature. Superconductors destined for applications, such as wires and cables, are almost invariably type II as these materials can be used in high magnetic fields. In type II superconductors magnetic flux penetrates through the superconducting material in the form of quantized flux lines for magnetic fields greater than Bc 1 , the ‘lower critical field’ and up to Bc 2 , the ‘upper critical field’. For practical materials the value of Bc 1 is usually below 100 mT and Bc 2 is over 10 T. Useful bulk transport currents can be carried in the superconductor up to Bc 2 (T0) where T0 is the device operating temperature as long as the magnetic flux lines can be effectively pinned. In this case the measurement of Bc 2 (T) is the most important as it defines an upper working limit for the design of applications. However, as the applied field, or combination of applied field and self-field generated by the transport current, approaches Bc 2 (T0 ), the fluxline pinning mechanisms become weaker and can disappear completely; this is especially true for hightemperature superconductors. This not only has the obvious implications for device design but also affects the interpretation of certain measurement techniques for Bc 2 . The recent progress in high-temperature superconductivity has led to a class of materials that are advancing rapidly for use as high-current conductors that can operate not only in strong magnetic fields but also at elevated temperatures (20–77 K). These high-Tc materials have difficulty pinning flux lines if the temperature and field become too high; reversible motion of the flux lines occurs and thus current transport without generating a voltage and losses is no longer possible. This effect occurs above the line that has been named the ‘irreversibility line’ and is denoted by Bi r r e v (T) or Ti r r e v(B), which is lower than the line defined by Bc 2 (T) or Tc (B) (Müller et al 1987, Yeshurun and Malozemoff 1988). Although a very pronounced effect in high-Tc superconductors, the irreversibility line can also be observed in low-Tc materials (Suenaga et al 1991). Much discussion as to the origin of this effect has been published and it can be related to global effects that cause depinning of flux lines such as flux-line lattice melting, 3D to 2D transitions and strong flux creep at high temperatures. In this case the knowledge of Bi r r e v (T) is necessary for device design. The experimental problem is to obtain reliable measurements for Bc 2 (T) and Bi r r e v (T)—different methods give different results and care is needed when making comparisons. A further complication for high-Tc materials is the effect of anisotropy which gives rise to different Bc 2 values and transport currents for different directions of field and current with respect to the superconductor’s crystallographic directions. Some examples of Bc 2 for practical materials are the following: for NbTi Bc 2 (T = 4.2 K) ~ 11 T, for Nb3Sn it is ~23 T and for PbMo6S8 (Chevrel phase) it is ~50 T. At an operating temperature of 77 K the Bc 2 of the high-temperature superconductors YBaCuO and Bi(Pb)SrCaCuO is ~30 T and at 4.2 K this value is over 100 T However, the irreversibility field is much lower than these values: for example for

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YBaCuO this field is displaced by 10–15 K below Bc 2 (T) in moderate fields of 1–10 T (see for example the discussion by Larbalestier and Maley (1993)). For the more anisotropic high-Tc superconductors such as Bi(Pb)SrCaCuO that are being developed as conductors for large-scale applications the irreversibility line is shifted to much lower temperatures (Shi et al 1993). The high values of Bc 2 impose some experimental measurement problems: high DC fields of 20–35 T are possible, but for higher fields (up to ~60 T) pulsed techniques are necessary (Siertsema and Jones 1994, Ryan et al 1996, Foner 1995). For certain applications it is necessary to know the external field for which the magnetic flux profile penetrates to the centre of a material (for example for fields beyond this field the regime in AC loss calculations changes), or through the wall of a hollow cylinder (for example for screening-type fault current limiters). For low-Tc materials this macroscopic penetration field, Bp , is easily described by critical-state models. Again, high-Tc materials are different and often present extended E—J characteristics (due to strong flux creep and other effects). This means that the transition from a pinned flux-line regime to a flux-flow regime is more gradual and can lead to phase shifts between applied and internal magnetic fields as well as a frequency dependence of Bp , even at low frequencies. Recently, the modelling for the electromagnetic behaviour for materials with extended E—J curves has been developed (Rhyner 1993, Gürevich 1995, Brandt 1996). B7.2.2 Critical fields: Bc , Bc 2 , Bi r r e v and B p To illustrate the various critical fields a magnetization curve for a type II superconductor is shown in figure B7.2.1. For applied fields near to the lower critical field, Bc 1 , the flux lines are spaced approximately

Figure B7.2.1. Magnetization curve for a type II superconductor showing the critical fields Bc 1 (= µ0Hc 1 ), Bc 2 (= µ0HC 2 ) and Bc 3 , (= µ0Hc 3 ) as well as the macroscopic penetration field, Bp .

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λ apart (λ is the magnetic penetration depth) and at the upper critical field, Bc 2 , the flux lines overlap each other significantly and are only spaced apart by ξ (where ξ is the coherence length (λ/ξ = κ and is >1/p2 for a type II superconductor)). Bc 1 and Bc 2 are intrinsic material properties describing the superconductivity of the material. The macroscopic penetration field, Bp , however, depends on the sample geometry and the pinning strength. In the simplest case for slab geometry (width 2a) and with a constant critical current density Jc then Bp = µ0 Jc a shown in the inset in figure B7.2.1. The existence of a reversible superconducting region above the irreversibility line where flux pinning is ineffective and thus the transport critical current density, Jc , is zero is shown schematically in figure B7.2.2.

Figure B7.2.2. Different regimes for a type II superconductor—the existence of a large reversible region is particularly apparent for high-temperature materials although low-temperature materials also show this effect to a lesser extent.

B7.2.3 Measurement techniques for Bc 2 As Bc 2 represents a maximum upper limit for useful applications some basic experimental techniques and their limitations are discussed. B7.2.3.1 DC magnetization measurements In a DC magnetization experiment the reversible magnetization contribution approaches Bc 2 linearly with slope proportional to 1/(2/κ 2 – 1). For practical materials with high κ values (~100) then the change of slope at Bc 2 may be difficult to detect; extrapolation of the linear region below Bc 2 to intersect the normal region above Bc 2 can overcome this problem. Also, sweeping the temperature in constant field to measure Tc ( B ) can be used (see the discussion by Campbell (1992)). The equivalence of this approach can be seen from figure B7.1.1. B7.2.3.2 AC susceptibility measurements In AC inductive techniques (see figure B7.1.3) in the presence of a magnetic field the resulting signals for χ′ and χ″ depend strongly on the pinning whilst the effect of the reversible magnetization tends to cancel out. Thus, these measurements are more sensitive to the irreversibility field rather than Bc2- However, a small positive χ′ signal at the fundamental in the reversible region between Tc ( B ) and Ti r r e v ( B, f ) can be detected (Couach and Khoder 1991) for low values of f, the frequency of the AC field. Whilst AC techniques benefit from considerable noise reduction by using lock-in techniques the use of harmonics such as the third harmonic can also increase the sensitivity of the measurements.

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Figure B7.2.3. Adiabatic and non-adiabatic calorimetry.

B7.2.3.3 Resistive measurements For resistive measurements of Bc 2 the techniques and apparatus are very similar to those presented in the previous section (see figure B7.1.2). Here, to measure Bc 2 (T) the applied field is gradually swept up and back to trace out a transition whilst a (usually) small DC or AC measuring current is passed through the sample. Conversely, the temperature can be swept at constant field to obtain Tc (B). A difference in this type of measurement is that the sample is always in the flux-flow state and this can lead to considerable broadening of the transition (Evetts 1992). Also, inhomogeneity will also broaden the transition. In strong-pinning materials a variation in Bc 2 may be responsible for the pinning and thus transition broadening can be expected. With the appropriate measuring current and by extrapolating the linear portion of a measured transition a ‘bulk’ value denoted by B*c2 can be obtained (see figure B7.1.1). An important factor in the resistive measurement of Bc 2 is consistency of experimental technique.

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In interlaboratory comparisons, for example for Nb—Ti wire (VAMAS, Tachikawa et al 1995), with the applied field direction (perpendicular to the wire), a range of different measuring currents (in this case 10 mA–1 A) and a resistive transition characterized by its offset (i.e. fully superconducting), 10%, 50%, 90% and onset temperatures are specified. With this method the scatter between laboratories was only ~0.6%. To increase the range of applied fields pulsed magnetic fields can be used (up to ~60 T, Siertsema and Jones 1994, Ryan et al 1996, Foner 1995). When using a resistive measurement in a pulsed magnetic field the inductive pick-up due to the rapidly changing magnetic field has to be carefully balanced. This can be done by a special symmetric arrangement of the sample and a dummy sample. The resulting voltage trace that occurs in the ~20 ms measurement can be analysed using the Bean model to extract the critical current as a function of applied field and thus Bc 2 can be obtained (Ryan et al 1996). For results to be valid the skin depth (at the effective frequency of the measurement) must be larger than the sample size. An alternative is the use of a balanced RF bridge; this is especially useful for powder samples or where contacts cannot be made (Foner 1995). An alternative to applying high magnetic fields is to employ scaling laws and extrapolation methods. A successful approach for A15 materials (e.g. Nb3Sn) is to use the Kramer plot (Kramer 1973). Firstly, measurements are made of Ic (T, B) over the range of temperatures and fields available from which the critical current density variation, Jc (T, B), can be calculated. The pinning force Fp = Jc B is assumed to follow a b1/2(1–b)2 dependence, where b = B/B*c2. Thus, a plot of ( Jc B 1/2 )1/2 is linear in B with intercept at B*c2(T). This is a useful method but care has to be taken as not all materials follow the same scaling laws. The value of Bc 2 (0) can be estimated from dBc 2/dT and the appropriate theory (Evetts 1983, Suenaga and Welch 1980, Dew-Hughes 1974, Werthamer et al 1966; see also section B7.3). B7.2.3.4 Specific heat measurements A very powerful method of investing Bc 2 and its variation is to measure the specific heat as a function of temperature and field. The advantage is that the response of the sample is volumetric and the characteristic distance is very small, ~ξ. At the transition temperature there is a jump in the specific heat, ∆C associated with the second-order superconducting phase transition. From the BCS theory, ∆C = 1.43γTc (where γ is the Sommerfeld constant). The specific heat jump is directly proportional to the volume of superconducting material just below Tc . When a magnetic field is applied not only can Tc (B) (and thus Bc 2 (T)) be obtained but also information on the volume fraction of material within a specified range of Bc 2 values (Evetts et al 1981, Cors 1990). The specific heat can be measured by adiabatic or non-adiabatic techniques. Early techniques have been enhanced by the use of computers, high-conductivity and low-thermal-mass sample supports (silicon) and refined analysis (Bachman et al 1972, Regelsburger et al 1986, Sullivan and Siedel 1968, Junod 1979, Schwall et al 1975, Forgan and Nedjat 1980). Figure B7.2.3 shows both an adiabatic calorimeter and a non-adiabatic calorimeter. In the first case it must be possible to initially lower the temperature of the sample (exchange gas, liquid helium, radiation, ...) and then isolate it from its surroundings. Heat is applied to the sample/sample holder and temperature changes recorded. It is very important to maintain (through the use of differential thermometry) the shield at the same temperature as the sample to ensure adiabatic conditions. Early measurements of this type were made step by step with careful correction for temperature drift. Subsequently swept measurements have been developed and recently such a method has detected flux-lattice melting in a pure YBaCuO sample (Roulin et al 1996). For the non-adiabatic calorimeter the ideal situation is difficult to obtain and an analysis of the actual

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situation has to be made. In this case, the equations governing the heat flows are

where Q(t) is the heat input, Ca and Cs are the thermal capacities of the addenda (sample holder, thermometer etc) and sample respectively, K0 and Ks are the thermal conductances linking the sample holder to the regulated thermal block and the sample and T0 , Ta and Ts are the temperatures of the thermal block, the sample holder (with integrated thermometer and heater) and the sample respectively. AC techniques offer the possibility of high sensitivity through the use of lock-in amplifiers. In AC calorimetry the temperature variation in relation to the specific heat is given by (see Sullivan and Seidel 1968)

with heat being supplied as (by passing a current at frequency ω through the heater)

where F(ω) is a transfer function that depends on the system time constants; τ1 is the the combined sample holder and sample to thermal block time constant and τ2 is the sample to sample holder time constant (including the internal temperature stabilization time constant)

where ε is a usually small geometrical factor (Sullivan and Seidel 1968). Best results are obtained when ω 2τ12 » 1 and ω 2τ22 « 1 and thus F(ω ) ≈ 1. In a variation of the relaxation experiment (Bachman et al 1972, Cors 1990), the power, P ( T ), flowing through K0 to the thermal block is calibrated over the required temperature range. The temperature of the sample is then raised to a temperature Te above T0, and allowed to relax back to T0. At each instant in time, the thermal capacity is given by

By repeating the measurement at several overlapping segments of Te and T0, the consistency of the results can be checked. As an illustration of the interaction between the various quantities figure B7.2.4 shows two examples: firstly the appearance of a second time constant if the sample is not well bonded to the sample holder (Ks variation) in a relaxation experiment and secondly the effect of too high a measuring frequency in an AC experiment (f variation). Thus, for successful measurements, the specific heat of the addenda and the thermal conductances between sample, sample holder and cryogenic bath need to be well characterized as a function of temperature and the correct frequency chosen in an AC measurement. B7.2.4 Measurement techniques for the irreversibility field, Bi r r e v The irreversibility field is difficult to measure. Physically, it corresponds to the field at which the flux-line pinning becomes ineffective. Different methods, such as resistive and inductive, give different results and

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Figure B7.2.4. Solutions of the differential equations (B7.2.1) for the temperature variation: two examples illustrating the need to control very carefully the experimental parameters in non-adiabatic calorimetry. The parameters that correspond to the diagram in figure B7.2.3 are T0 = 10, Ca = 50, Cs = 200, K0 = 50 and Ks = 2000 unless otherwise indicated.

can depend on sample size and measurement frequency. Furthermore, each measurement comes with its own definition for the irreversibility field. Some commonly employed definitions of the irreversibility field and temperature are shown in figure B7.2.5. As an experimental example to illustrate this problem, figure B7.2.6 shows values obtained for the same material by two different magnetization measurement methods. In the VSM (vibrating-sample magnetometer) case, the field sweep is faster than that in the SQUID magnetometer case (SQUID, superconducting quantum interference device). Note that the SQUID values are much lower than the VSM values. This is because the SQUID takes several minutes per point, allowing significant creep to take place. B7.2.5 Measurements of the macroscopic flux penetration field, Bp † To illustrate the problem of an extended E—J curve in high-Tc superconductors the screening characteristics over a range of low frequencies (1–200 Hz) at liquid-nitrogen temperatures (77 K) for a hollow tube are considered. Typical results for the central field amplitude, Bi , versus the applied AC field amplitude, B0 , are shown in figure B7.2.7 for a stack of Bi-2212 rings inside a brass cylinder (produced by the composite reaction texturing process: Watson et al 1995). The screening threshold field at which significant flux begins to penetrate into the central region of the hollow tube increases with frequency. This is because of the shallow E—J curve in these materials and the low flux-flow resistivity giving rise to skin-depth screening effects. As there are also associated phase shifts between the drive field and the field in the centre of the tube we have employed a whole-waveform

† Based on the discussion by Cave et al (1995)

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Figure B7.2.5. Some commonly used methods of measuring the irreversibility field (and temperature).

Figure B7.2.6. ‘Irreversibility line’ (i.e. the line defined by the field and temperature at which bulk pinning becomes immeasurably small) as a function of temperature determined by two different magnetization measurement techniques: (a) from swept-field magnetization loops using a vibrating-sample magnetometer (VSM) and (b) from zero-field-cooled (ZFC) and field-cooled (FC) swept-temperature magnetization curves measured with a sensitive SQUID magnetometer. The difference in result is because the SQUID magnetometer takes longer to make each measurement, thereby allowing more flux creep to take place. Data from N Adamopoulos, University of Cambridge; the sample is composite reaction textured Bi-2212 (see Watson et al 1995 for details).

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Figure B7.2.7. Screening measurement on a high-Tc hollow tube to illustrate the consequence of extended V—I characteristics on measurements. Also shown (lower graph) is a comparison of the experimental data (points) with theory (solid lines) using an effective field-dependent resistivity, ρ = ρ0(B/Bp ) (data from Cave et al 1995).

technique (with averaging) using a digital oscilloscope to obtain the central field amplitude and its phase rather than a phase sensitive detector. For low-temperature superconductors and for high-Tc materials under certain conditions the critical state analysis of this type of experiment yields good agreement. However, in materials where the E—J curve is shallow this may not be the case. An inset in figure B7.2.7 shows schematic E—J curves described by the often used empirical relationship E ~ Jn For n <~ 8 the superconductor behaves more like a low-resistivity linear material (Rhyner 1993). For a low-resistivity linear material Chen (1991) has developed the appropriate solutions of the magnetic diffusion equation. In particular, the solution for the internal field, Bi , and its phase in the central region of a hollow tube (see figure B7.2.7) is given by (inner radius, rb ; outer radius, ra ; AC drive

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field, B0; skin depth, δ )

where k = (1 + i)/δ with = p -1 and K and I are modified Bessel functions. To take into account the decrease in Jc and increase in resistivity as the applied field, B0 , increases we introduce an approximate analysis using an effective field-dependent resistivity and skin depth ρ = ρ0( B/Bp ) y and δ = ( 2ρ/µ0ω )1/2 and figure B7.2.7 shows good agreement between theory and experiment for brass-reinforced CRT rings (at low applied fields the phase angle tending to 0° is due to a small leakage flux). These results show that at power frequencies some high-Tc materials can be more appropriately described using the magnetic diffusion equation rather than the critical-state model. Gürevich (1995) reviews recent developments in non-linear flux diffusion in high-Tc superconductors and Brandt (1996) develops a theoretical treatment. B7.2.6 Conclusions Measurements of Bc 2 (T) are necessary for comparison to theoretical models and also as a parameter in scaling relations. The commonly used resistive method shows broadening due to inhomogeneity near to Bc 2 and also because the sample is in the flux-flow state for fields greater than Bc 1 . The use of a suitable measuring current and extrapolation plots lead to a ‘bulk’ value B*c2. In high-Tc materials the irreversibility field can be measured in a variety of ways. However, results are sample and frequency dependent. The knowledge of this limiting field is extremely important for applications of these materials. The specific heat method is volumetric and sensitive on the scale of ~ ξ. Although the techniques require care, the use of computers and fast data acquisition promise to make this a very useful method and its use should to be encouraged. The macroscopic penetration field Bp is a measure of the transport current carrying capacity and is useful in developing devices. In high-Tc superconductors, the extended E—J curve must be taken into account when analysing results. Acknowledgments The author would like to acknowledge the support of a NATO collaborative research grant for work with the University of Cambridge on flux dynamics in superconductors. He also thanks Wen Zhu and Phillipe Ouimet for their help in the preparation of the diagrams in sections B7.1 and B7.2. References (See also the references and further reading list for section 7.1.) Bachmann R, Disalvo F J Jr, Geballe T H, Greene R L, Howard R E, King C N, Kirsch H C, Lee K N, Schwall R E, Thomas H U and Zubeck R B 1972 Heat capacity measurement on small samples at low temperatures Rev. Sci. Instrum. 43 205–14 Brandt E H 1996 Superconductors of finite thickness in a perpendicular magnetic field. I. Strips and slabs Phys. Rev. B 54 4246–64 Campbell A M 1991 DC magnetization and flux profile techniques Magnetic Susceptibility of Superconductors and other Spin Systems ed R A Hein (New York: Plenum) pp 129–55

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—1992 Measurements in superconducting materials Concise Encyclopedia of Magnetic and Superconducting Materials ed J Evetts, R W Cahn and M B Bever (Oxford: Pergamon) Cave J R, Watson D R and Evetts J E 1995 Screening properties of Bi-2212 superconducting tubes Applied Superconductivity (Inst. Phys. Conf. Ser. 148) (Bristol: Institute of Physics) pp 539–42 Chen Q Y 1991 AC inductive measurements: application to the studies of high-Tc superconductivity Magnetic Susceptibility of Superconductors and other Spin Systems ed R A Hein (New York: Plenum) pp 81–105 Couach M and Khoder A F 1991 AC susceptibility responses of superconductors: cryogenic aspects, investigation of inhomogeneous systems and of the equilibrium mixed state Magnetic Susceptibility of Superconductors and other Spin Systems ed R A Hein (New York: Plenum) pp 25–48 Cors J 1990 Propriétés supraconductrices sous champ magnétique du composé PbMo6S8 étudiées par chaleur spécifique Thèse 2456 Université de Genève Faculté des Sciences Département de Physique de la Matière Condensée (Geneva: Imprimerie National) Dew-Hughes D 1974 Practical superconducting materials Superconducting Materials and Devices—Large Systems Applications ed S Foner and B B Schwartz (New York: Plenum) Evetts J E 1983 The characterization of superconducting materials—conflicts and correlations IEEE Trans. Magn. MAG-19 1109–19 —1992 Resistive transition and flux flow in superconducting materials Concise Encyclopedia of Magnetic and Superconducting Materials ed J Evetts, R W Cahn and M B Bever (Oxford: Pergamon) Evetts J E, Cave J R, Somekh R E, Stanton J P and Campbell A M 1981 Characterization of Nb3Sn diffusion layer material IEEE Trans. Magn. MAG-17 360–3 Foner S 1995 High-field magnets and high-field superconductors IEEE Trans. Appl. Supercond. AS-5 121–40 Forgan E M and Nedjat S 1980 Heat capacity cryostat and novel methods of analysis for small specimens in the 1.5–10 K range Rev. Sci. Instrum. 51 411–7 Gürevich A 1995 Nonlinear flux diffusion in superconductors Int. J. Mod. Phys. B 9 1045–65 Junod A 1979 An automated calorimeter for the temperature range 80–320 K without the use of a computer J. Phys. E: Sci. Instrum. 12 945–52 Kramer E J 1973 Scaling laws for flux pinning in hard superconductors J. Appl. Phys. 44 1360–70 Larbalestier D C and Maley M P 1993 Conductors from superconductors: conventional low-temperature and new high-temperature superconducting conductors MRS Bull. August 50–6 Regelsberger M, Wernhardt R and Rosenberg M 1986 Fully automated calorimeter for small samples with improved sensitivity J. Phys. E: Sci. Instrum. 19 525–32 Rhyner J 1993 Magnetic properties and AC-losses of superconductors with power law current-voltage characteristics Physica C 212 292–300 Roulin M, Junod A and Walker E 1996 Flux line lattice melting transition in YBa2Cu3O6.94 observed in specific heat experiments Science 273 1210–2 Ryan D T, Hole C J R, van der Burgt M, Jones H, Christopher M D, Christopher R M G, Goringe M J C and DewHughes D 1996 D.C. transport critical current measurements on high temperature superconductors in pulsed fields up to 50 T IEEE Trans. Magn. MAG-32 2803–5 Schwall R E, Howard R E and Stewart G R 1975 Automated small sample calorimeter Rev. Sci. Instrum. 46 1054–9 Shi D, Wang Z, Sengupta S, Smith M, Goodrich L F, Dou S X, Liu H K and Guo Y C 1993 Critical current density, irreversibility line, and flux creep activation energy in silver-sheathed Bi2Sr2Ca2Cu3Ox superconducting tapes IEEE Trans. Appl. Supercond. AS-3 1194–6 Siertsema W J and Jones H 1994 The Oxford pulsed magnetic field facility IEEE Trans. Magn. MAG-30 1809–12 Suenaga M, Ghosh A K, Xu Y and Welch D O 1991 Irreversibility temperatures of Nb3Sn and Nb-Ti Phys. Rev. Lett. 66 1777–80 Suenaga M and Welch D O 1980 Flux pinning in bronze-processed Nb3Sn wires Filamentary A15 Superconductors ed M Suenaga and A F Clark (New York: Plenum) Sullivan P F and Seidel G 1968 Steady-state, AC-temperature calorimetry Phys. Rev. 173 679–85 Tachikawa K, Koyama S, Takahashi S and Itoh K 1995 The VAMAS intercomparison on the upper critical field measurement in Nb-Ti wire IEEE Trans. Appl. Supercond. AS-5 536–9 Watson D R, Chen M and Evetts J E 1995 The fabrication of composite reaction textured Bi2Sr2CaCu2O8+δ superconductors Supercond. Sci. Technol. 8 311

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Werthamer N R, Helfand E and Hohenberg P C 1966 Temperature and purity dependence of superconducting critical field Hc 2 . III Electron spin and spin-orbit effects Phys. Rev. 147 295–302 Yeshurun Y and Malozemoff A P 1988 Phys. Rev. Lett. 60 2202

Further reading Brandt B L and Rubin L G 1982 Low-temperature thermometry in high magnetic fields V. Carbon-glass resistors Rev Sci. Instrum. 53 1129–36 Decroux M, Selvam P, Cors J, Seeber B, Fischer ∅, Chevrel R, Rabiller P and Sergent M 1993 Overview on the recent progress on Chevrel phases and the impact on the development of PbMo6S8 wires IEEE Trans. Appl. Supercond. AS-3 1502–9 Gelinas C, Lambert P, Dubé D, Arsenault B and Cave J R 1993 Texturing of thick films on a metallic substrate Supercond. Sci. Technol. 6 368–72 Green S M, Lobb C J and Greene R L 1991 The irreversibility line of (Bi,Pb)2Sr2Ca2Cu3O10 determined by DC magnetization IEEE Trans. Magn. MAG-27 1061–4 Hettinger J D, Kim D H, Gray K E, Welp U, Kampwirth R T and Eddy M 1992 Effective electric field in dc magnetization measurements: comparing magnetization to transport critical currents Appl. Phys. Lett. 60 2153–5 Junod A, Bonjour E, Calemczuk R, Henry J Y, Muller J, Triscone G and Vallier J C 1993 Specific heat of an YBa2Cu3O7 single crystal in fields up to 20 T Physica C 211 304–18 Junod A, Wang K-Q, Tsukamoto T, Triscone G, Revaz B, Walker E and Muller J 1994 Specific heat up to 14 tesla and magnetization of Bi2Sr2CaCu2O8 single crystal thermodynamics of a 2D superconductor Physica C 229 209–30 Li Q, Wiesmann H J, Suenaga M, Motowidlo L and Haldar P 1994 Observation of vortex-glass to liquid transition in the high-Tc superconductor Bi2Sr2Ca2Cu3O10 Phys. Rev. B 50 4256–9 Malozemoff A P 1993 Superconducting wire gets hotter IEEE Spectrum December 26–30 Müller K A, Takashige M and Bednorz J G 1987 Phys. Rev. Lett. 58 1143 Naughton M J, Dickinson S, Samaratunga R C, Brooks J S and Martin K P 1983 Thermometry in high magnetic fields at low temperatures Rev. Sci. Instrum. 54 1529–33 Otabe E S, Matsushita T, Matsuno T and Yamafuji K 1991 Irreversibility line and flux flow noise in superconducting Nb-Ta IEEE Trans. Magn. MAG-27 1033–6 Ramsbottom H D 1996 Flux profiles of superconductors in high magnetic fields Thesis University of Durham Seeber B, Cheggour N, Perenboom J A A J and Grill R 1994 Critical current limiting factors of hot isostatically pressed (HIPed) PbMo6S8 wires Physica C 234 343–5

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B7.3 Critical current of wires

B Seeber

B7.3.1 Introduction The measurement of the critical current of a superconducting wire is, at first glance, rather simple. The standard method consists of sending a dc current through the sample which is slowly increased until a voltage drop is detected. This is normally done as a function of a perpendicularly applied field and, in particular cases, also as a function of strain. Another parameter of interest may be the temperature. The applied criterion is arbitrary, but there is some convention either to define the critical current as the current where a certain voltage per metre is measured ( electric field criterion, e.g. 10 µV m–1 ) or a defined electrical resistivity appears (e.g. 10–13 Ω m). Depending on the specific application of a superconducting wire, the criterion has to be selected. In reality, a practical Ic measurement is not very easy because high magnetic background fields and high currents must be controlled at the same time as very small voltages have to be measured. In addition, depending on the kind of superconducting wire (Nb—Ti, Nb3Sn, highTc superconductors, etc) particular care is required in the handling. The critical current density is defined either as the engineering critical current density Jc e (or overall Jc ), which is the critical current divided through the total cross-section of the wire, or as the non-copper critical current density (without electrical stabilization), or as the critical current density of the superconducting filaments (critical current divided by the cross-section of the superconducting filaments). Finally one can define a critical current density of the winding package of a superconducting magnet. Here one has to take into account the filling factor of the winding (a round wire has a lower filling factor than to a rectangular wire), the thickness of the electrical insulation (a heat-resistant glass insulation is much thicker than an enamel insulation) and eventually the mechanical reinforcement (in large magnets). In this section the most important practical aspects of an Ic measurement are discussed. This means that Ic is determined as outlined above and other methods, like magnetization measurements or the measurement of the penetration of a magnetic field in a superconductor, are not described. The latter methods require a model for the calculation of Ic and a number of conditions must be fulfilled. Although important for the understanding and the development of new superconducting materials, they are of less interest to the engineer. Further information on alternative methods for the determination of Ic can be found in the work of Campbell and Evetts (1972a, b). B7.3.2 Sample holders The wire to be measured must be fixed on an appropriate sample holder. Different kinds of sample holder have been developed in the past and they are summarized in figure B7.3.1. One can distinguish essentially between four sample geometries: short straight, hairpin, coil and long straight (Goodrich and Fickett 1982). The straight geometries have the advantage that the wire does not have to be bent. This may be

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Figure B7.3.1. Different methods of fixing a wire sample onto a sample holder. The measured part of the wire must be perpendicular to the applied magnetic field. Small deviations from the 90° angle introduce an error which can often be tolerated: (a) short-straight geometry; (b) hairpin geometry; (c) coil geometry; (d) long-straight geometry.

of interest for thick wires or for the development of new superconductors where the influence of bending strain on Ic is not yet known. For multifilamentary superconductors one of the problems to take into account for the selection of an appropriate sample geometry is the current transfer length (Ekin 1978). This is the length of the wire required for the current to go through the matrix into the superconducting filaments and is given by

where L is the current transfer length, d is the diameter of the filament region of the wire, n is the resistive transition index characterizing the abruptness of the superconducting transition, ρm is the resistivity of the matrix and ρc is the equivalent of the applied resistivity criterion. The higher the resistivity of the matrix the longer is the current transfer length. In other words, the distance between the current contact and the first voltage tap has to be longer. In addition, the length between the current contact and the first voltage tap must be increased when the Ic criterion is more sensitive. The situation is shown in figure B7.3.2 where the relative transfer length L/d is plotted versus the resistivity criterion for the determination of Ic (Ekin 1978). Because the resistivity of the Cu—Sn matrix of an Nb3Sn wire is much higher than that of the Cu matrix of an Nb—Ti wire, the current transfer length is substantially increased in the case of Nb3Sn. A typical current transfer length for a 10–13 Ω m resistivity criterion for an Nb3Sn bronze conductor is ~20d. In the case of Nb—Ti the current transfer length is reduced to ~2d. These estimates are conservative and shorter transfer lengths are possible if the length of the current contact itself is increased (Goodrich and Fickett 1982). Because of the above-mentioned current transfer problem and in order to push the sensitivity of the Ic measurement, today’s standard geometry is the coil geometry. In such a geometry the length of the current contact is about the circumference of the coil. Voltage taps can be far away from each other and the definite distance depends on the winding pitch and the homogeneity of the background field. What follows is a more detailed discussion of the coil geometry. The interested reader who wants to know more about the other geometries is referred to the article by Goodrich and Fickett (1982).

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Figure B7.3.2. The relative transfer length L/d versus the resistivity criterion for Nb—Ti and Nb3Sn. L is the current transfer length and d is the diameter of the filamentary zone. The resistivity of the copper matrix of Nb— Ti is 1.4 × 10– 10 Ω m and that of the bronze matrix of Nb3Sn is 2 × 10–8 Ω m. Reproduced from Ekin (1978) by permission of the American Institute of Physics.

An important aspect of the coil sample geometry is the thermal contraction of the sample holder upon cooling. Care must be taken that the sample holder does not contract less than the wire sample otherwise uncontrolled tensile strain may damage it. This is of less relevance for Nb—Ti, but is particularly important for brittle superconductors like Nb3Sn and high-Tc superconductors. Commonly used materials to fix the superconducting wire are glass-fibre reinforced plastics (GRPs), alumina ceramic, stainless steel, brass and copper. In the case of an electrically conducting support there is a parallel shunt path for the metered current. If the wire is homogeneous, this does not introduce an error when stainless steel is used but corrections are required in the case of brass and copper (Warnes and Dai 1992). An electrically conducting mandrel is particularly helpful for the protection of the superconducting wire against burnout. The thermal expansion as a function of temperature for different materials used as sample holders (measurement mandrel) is displayed in figure B7.3.3. Also shown is the range of thermal contraction for Nb3Sn wires at 77 K (Goodrich et al 1990). Because the layout of superconducting multifilamentary wires can vary considerably, the thermal contraction upon cooling can only be given approximately. It is interesting to note that typical Nb3Sn wires behave similarly to stainless steel (AISI 304L, AISI 316) and copper (not shown in figure B7.3.3). In the case of GRPs, like G10CR and G11CR, there is a strong anisotropy of the thermal contraction. The situation can be explained with the help of figure B7.3.4. One has to distinguish three directions in a GRP: the normal to the fabric planes and the warp and fill directions. Whereas the anisotropy of thermal contraction is relatively weak in the fabric plane, it is important between the normal and the fabric plane. For instance, for a cool-down of G10CR from 293 K to 4.2 K the normal contracts by 0.71% and the warp and fill directions by 0.24% and 0.28% respectively (Clark et al 1981). This is the main reason why measurement mandrels should be either a rolled tube or a so-called plate tube. The rolled tube makes it possible to adjust the thermal contraction by the appropriate choice of the ratio between the wall thickness and the outer radius (Goodrich et al 1990). A recent and more detailed discussion of the thermal contraction problem can be found in an article by Goodrich and Srivastava (1995) and references therein. Because during a critical current measurement important Lorentz forces can act on the wire (F ∝ J × B) the latter must be fixed on the sample holder. The simplest way is to machine a Vshaped spiral groove into the mandrel and to ensure that the Lorentz force is directed inwards in the coil.

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Figure B7.3.3. The low-temperature thermal expansion of different materials used as a measurement mandrel. The behaviour of the glass-fibre reinforced plastics G10CR and G11CR is in the direction of the warp. A Nb3Sn wire behaves like stainless steel or copper (not shown because its behaviour is nearly identical to that of stainless steel).

Figure B73.4. (a) The main directions of GRP. As a measurement mandrel one should use either a so-called plate tube (b) or a rolled tube (c). According to the ratio of the tube thickness to the outer tube diameter, the thermal contraction of the rolled tube can be adjusted to the superconductor. Reproduced from Goodrich et al (1990) by permission.

If the thermal contractions of the wire and the mandrel are matched, the wire is pressed onto the sample holder during the measurement and cannot move. This is an ideal situation but often a thin layer of silicon

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vacuum grease or a glass-fibre reinforced tape wound around the coil is required to bond the wire onto the mandrel. Sometimes the wire is glued or soldered to the sample holder. Anyway, the bonding has to be tested from case to case. Any movement of the wire will induce a spike in the measured voltage which may trigger a quench (thermal runaway). If the measurement is repeated the quench occurs at higher currents. This is a clear indication that wire movements are present. The current contacts are normally copper blocks at both ends of the mandrel and the superconducting wire is soldered onto them. A minimum copper cross-section of about 35 mm2 per 1 kA is recommended. The current contacts should be shorter than one complete turn in order to allow a faster decay of eddy currents which are induced by a change of the background field. Care has to be taken that the wire seats over the whole length tightly on the sample holder. Voltage taps may be soldered or attached by silver paint. It is important to twist the leads of a pair of voltage taps and, in a noisy background field (e.g. hybrid magnets), to co-wind them with the superconducting wire. The same technique allows a reduction of the voltage offset due to the finite inductance of the coil geometry and any temporal change of the current in the wire sample (V ∝ LdI /dt ). In the coil geometry, the wire is not exactly perpendicular to the direction of the field due to the pitch of the spiral groove of the sample holder. The angular dependence of the Ic measurement has been studied in detail and the result is summarized in figure B7.3.5 (Goodrich and Fickett 1982). The depicted curves are for multifilamentary Nb-Ti and Nb3Sn and do not depend on the applied field, nor on the selected Ic criterion. The origin of the different behaviours of Nb—Ti and Nb3Sn is not known. It is interesting to note that the wire can be quite misoriented before the measurement error becomes important. For instance a misalignment of 8° gives an increase of Ic of ~2%. In the case of a wire with a rectangular cross-section there is also a dependence of Ic on the angle between the wide face of the conductor and the direction of the field. For angles up to ~20° the error is negligible and changes to ~9% and ~–5% at 9° (wideface is perpendicular to the field direction) for Nb—Ti and Nb3Sn respectively (Goodrich and Fickett 1982).

Figure B7.3.5. The relative error of the Ic measurement as a function of the misorientation angle between the axis of the superconducting wire and the applied magnetic field for Nb—Ti and Nb3Sn multifilamentary conductors. The misorientation angle is defined as the angle of deviation from the perpendicular orientation. Reproduced with modifications from Goodrich and Fickett (1982, p 233) by permission of Elsevier Science Ltd.

If the critical current has to be studied as a function of uniaxial strain, special sample holders have been developed for the short-straight, long-straight or coil geometries. The main difficulty is the application of stress along the wire axis in a restricted space because the wire must be perpendicular to the field. A possible way of achieving this is shown in figure B7.3.6 (Ekin 1980). A superconducting wire of ~40 mm

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Figure B7.3.6. A schematic drawing of a sample holder for the measurement of Ic under uniaxial strain in a transverse field. Reproduced from Ekin J W (1980, p 612) by permission of Elsevier Science Ltd.

length is soldered onto two copper blocks. The left copper block can be moved by a lever arm and the pull-rod, which puts the wire under tensile strain. The strain is measured directly on the sample by a strain gauge extensometer. Longer straight wire samples can be measured if a split-coil magnet is available. Such a magnet has a horizontal field direction and allows vertical access to the superconducting wire. So the length of the wire is not restricted by the bore size of the magnet. A further advantage is that larger forces can be applied and there are no problems with the current transfer length because current contacts are usually far away from the voltage taps. This is also the method which allows the measurement of Ic superconducting cables under strain (Specking et al 1989). The magnetic field of a split-pair magnet is limited to ~15 T (see chapter G1). If higher fields and longer wire lengths are required, the so-called Walters spring is an elegant solution (Walters et al 1986). The superconducting wire is soldered on a highly elastic spiral spring and, by applying a torque, the wire can be put either under tensile or compressive strain. A schematic illustration is shown in figure B7.3.7. A typical diameter of the spring is 40 mm with six turns. This allows one to measure Ic of wires longer than 0.7 m and the situation is very similar to that of the coil geometry. The strain of the wire is measured by strain gauges on the sample holder. One drawback of this concept is that the zero strain of the wire is not defined due to the differential thermal contraction between the wire and the spring. However, in the case of Nb3Sn, this can be overcome by scaling the maximum of Ic versus strain by another independent measurement (Katagiri et al 1995). Another possibility, not limited to Nb3Sn, would be to make a V-shaped groove in the spring and to fix the wire with no bonds or only with a weak bond. During cool-down the torque must be measured and kept near zero. Ic measurements under uniaxial strain do not give a full description of the influence of strain on Ic . There are other strain components, e.g. radial strain, which can also be of importance. A more detailed discussion of the problem and other strain apparatus can be found in (ten Haken 1994).

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Figure B7.3.7. A schematic drawing of the Walters spiral. Reproduced from Walters et al (1986, p 409) by permission of Elsevier Science Ltd.

B7.3.3 Background field For practical applications the critical current density must be known at a specific magnetic field. Today fields of up to ~20 T can be produced by superconducting solenoid magnets. Higher fields are available using hybrid magnets where the outer part is superconducting and the inner part is normal conducting. Fields above 30 T can be achieved by this technique. The advantage of a purely superconducting magnet is that it can work in the persistent mode which gives a very stable and quiet field. As a consequence there are fewer noise problems for the Ic measurement. According to the VAMAS recommendations (Versaille Project on Advanced Materials and Standards) the magnetic field should be as accurate as 1% and the homogeneity should be ±1% over the distance of the voltage taps (VAMAS Technical Working Party for Superconducting Materials 1995). A simple test of whether the field homogeneity influences Ic is to fix more than one pair of voltage taps along the sample. If the wire sample is homogeneous one can estimate the error due to an inhomogeneous field. For this purpose it is recommended to use the fully characterized Nb—Ti standard reference material SRM 1457, available from the National Institute of Standards and Technology (NIST), Boulder, CO (appendix A). For the long-straight sample geometry a split-coil magnet is required which allows radial access to the field. The inconvenience of such a background field is that only a short part of the wire can be measured due to the rather inhomogeneous field. In addition, such a magnet is much more expensive than a solenoid magnet and the field is limited to ~15 T due to the substantially higher field inside the coils and the very high forces between the pairs of a split-coil system.

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Attention should be drawn to the hysteretic behaviour of superconducting magnets near zero field. If no current goes through the magnet, there is a small residual magnetic field due to magnetization currents and trapped flux in the superconducting filaments. Depending on the construction of the magnet and the size of the filaments, the strength of this field varies. However, this is only of concern for Ic measurements of superconducting materials under self-field conditions. B7.3.4 Measurement technique A schematic set-up of the measurement of the critical current is shown in figure B7.3.8. The earlier used x—y recorder, to trace out the current—voltage (I—V) characteristics of a superconducting wire, is today often replaced by a data acquisition system. The computer pilots the current source for the sample and records the current and the voltage. For very high-sensitivity measurements, a ripple noise as small as possible of the sample current source is important. However, for an ordinary Ic criterion of ≥10 µV m–1 the results do not vary by more than 0.5% if a battery (no ripple) and a silicon controlled rectifier (SCR) regulated power supply are compared (Goodrich and Fickett 1982). It is also important to have a precise control of the current and the size of current steps. For instance to have the possibility to control the current to within ~100 mA at a sample current of 1000 A can be very helpful for the Ic measurement, in particular for less stabilized wires. In practice, the current can be brought near to Ic rather fast (a few seconds) and small current steps are only used to trace out the I—V characteristic of the superconducting wire. Modern digital voltmeters can measure voltages below 1 µV quite fast so the result of an Ic measurement does not depend on how fast the sample current is increased. However, the ramp rate of the sample current is limited by the inductive offset voltage, which should be considerably lower than the required voltage for the applied Ic criterion. A further limitation comes from the self-field of the transport current where the measured electrical field depends on the history of the transport current change (Duchateau et al 1976). Finally, the ramp rate may be limited by micro-movements of the wire under the Lorentz force and induced eddy currents in metallic parts of the sample holder and the cryostat/magnet (heating). The recommended ramp time near Ic is of the order of one minute with a typical rate of 1 A s–1. As mentioned above, modern digital voltmeters can measure voltages below 1 µV quite quickly (~50 ms), which makes averaging possible. At a fixed sample current the voltage is measured several times and then averaged. In this mode the noise of the voltage data can be reduced. The programming of the data acquisition system should allow one to adjust the time interval between the moment when the

Figure B7.3.8. A schematic diagram of a set-up for the measurement of the critical current. V—voltmeter, I—sample current source, B—magnet power supply, T—temperature control, PC—(personal) computer.

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sample current is constant (after a current step) and the time when the voltage is measured. This time is in the range of 0.1 s and the right choice can substantially contribute to less noisy data. There is a second time interval which must be adjustable by software. In order to speed up the measurement, it should be possible to increase the sample current from zero to near Ic very quickly (a few seconds). Before the measurement continues with small current steps, one has to wait until the inductive offset voltage is approaching zero. Typical waiting times are in the order of 10 s. Obviously this is only possible if one knows approximately the critical current. Ic can either be taken from the specification of the wire or from a (slow) measurement at the highest field. When the field is reduced, Ic increases and one can quickly ramp up the current to a value corresponding to the Ic of the previous measurement at higher fields. If such a sequence has been selected, i.e. Ic is measured versus decreasing field, the Lorentz force acting on the wire increases up to a maximum which is in the range of 5 T for both Nb-Ti and Nb3Sn. It has been suggested in the literature that at the beginning of an Ic measurement one should apply a field corresponding to the maximum Lorentz force. This allows the wire to settle into the most stable position on the sample holder. However, because in this field range critical currents are high, the probability of damaging the wire in the event of an uncontrolled quench is not negligible. There are also Nb3Sn wire constructions which are less stable at low fields and the danger of the wire burning out is high. For this reason it is recommended to start the Ic measurement at the highest field and to decrease it subsequently. If wire movement occurs at a certain field, it is possible to decrease somewhat the field (higher Lorentz force), to send a current through the sample up to ~Ic , and then go back to the required higher field. To be complete, Ic can also be measured by using a continuously increasing sample current with a simultaneous recording of the voltage. This is a typical configuration where an x—y recorder is employed. Care must be taken to achieve the correct current ramp rate, which is determined by the voltage measurement system. Another technique is to send a current pulse through the sample and to store the I—V response with the help of an oscilloscope. The application of a dc current with a superposed ac component of fixed frequency has also been used for Ic measurements. Here the voltage is detected with a lock-in amplifier. A more detailed discussion of these alternative techniques can be found in the article by Goodrich and Srivastava (1992) and in references therein. Because Ic depends strongly on the temperature it is necessary to have some kind of control. One possibility is to monitor the temperature with one or several sensors (see chapter E4). They have to be selected and positioned so that the error due to the magnetic field is negligible. Another possibility is to measure the vapour pressure above the helium bath. In the case of a liquid nitrogen bath there is a tendency to stratification which causes temperature variations over the depth of the bath. One has also to take into account the hydrostatic pressure which is at the origin of significantly warmer nitrogen at the bottom of the liquid-nitrogen reservoir. A precise temperature control may also be of interest for the cool-down of the sample holder. The magnet power supply can also be controlled by the data acquisition system. It is up to the user to introduce refinements of this control such as persistent mode operation of the magnet with or without current in the leads, a protocol for the virgin run of the magnet, lambda plate operation, control of the level of cryogen etc. B7.3.5 Analysis B7.3.5.1 Standard analysis In figure B7.3.9 definitions of different critical current criteria are shown. The electric field criterion is the most common one. The current generating a voltage of Vc = LEc is defined as the critical current. L is the length of the wire between the voltage taps and Ec is the electric field criterion. Ec is nor mally 10 µ m-1 or 100 µV m-1. The resistivity criterion is another possible way of defining Ic . Here, Ic

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Figure B7.3.9. Definition of different Ic criteria. Note that the resistivity criterion Ic ( ρc ) depends on the sample current.

is equal to the sample current yielding a voltage of Vc = Iρc L/A. I is the sample current, ρc is the resistivity, L is the length of the wire between the voltage taps and A is the total cross-sectional area of the wire. Values for ρc are 10–14 Ω m or 10–13 Ω m. The I—V characteristic of a superconductor near Ic can be described analytically by

where V0 and I0 are a reference voltage and current respectively, n is the resistive transition index which is a measure of the abruptness of the transition. The higher n is the faster the voltage V increases with the current I. n is also known as the n-value and varies typically between 10 and 100. The n-value is also an important parameter for the characterization of superconducting wires. The higher n is the more homogeneous is the wire and the smaller can be the Ic margin for the construction of magnets. As an example, magnets which work in the persistent mode without a drift require wires with a high n-value, typically >30 at the highest field. If n is smaller, the operating current for this application must be further reduced with respect to Ic which increases the size of the magnet and its costs, n can be calculated from the slope of a double logarithmic plot of log V versus log I. Because in such a plot n may be nonlinear, one has to check whether the calculated n-value is in the linear regime. Finally one should specify in what electric field range or electrical resistivity range n has been determined. For a fast and simplified analysis the following equation can be used.

where Ic ( 100 ) and Ic ( 10 ) are the critical currents according to an applied criterion of 100 µV m–1 and 10 µV m–1, respectively. There may be several difficulties in recording data suited for further analysis. If the correct set-up of the whole system is used, voltage noise is not considered as a real problem for commonly used criteria. However, there is always an offset voltage which must be taken into account. The offset voltage should be observed during the measurement and any time or current dependence is an indication of complications. For instance, a drift of the offset voltage, either positive or negative, comes probably from thermally induced voltages They appear when the system is not in thermal equilibrium, e.g. after the cool-down or

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a quench of the wire. After a few minutes of waiting the problem is normally solved. If the offset voltage depends on the current in the sample, a current transfer problem exists. This shows up as a finite slope in the I—V characteristic before Ic has been reached. If the current transfer voltage is small with respect to the applied criterion, good Ic measurements can still be made but the slope should be subtracted from the data. The correction for a slope is particularly important for the calculation of the n-value. Finally, corrections may be necessary where the measurement mandrel is an electrical conductor. There is current sharing between the wire and the metallic sample holder and the metered current does not correspond exactly to the sample current. A study has shown that stainless steel does not require any corrections for current sharing. However, in the case of brass and copper, Ic and n are strongly influenced and must be corrected (Warnes and Dai 1992). B7.3.5.2 Advanced analysis The measured voltage in an I—V curve of a superconductor is a consequence of the onset of the movement of magnetic flux lines. If it is possible to analyse this curve in more detail, vital information on the behaviour of flux lines, and finally of their pinning, can be obtained. This information can be particularly beneficial for the development of new superconducting wires and for the optimization of the critical current. Let us assume an arbitrary distribution of the critical current along a wire f(Ic). When the current I is higher than the local critical current Ic , there will be a detectable voltage (Baixeras and Fournet 1967).

A is the only adjustable parameter and its physical origin, as well as its value, depends on a model. f( Ic ) dIc is the fraction of the wire with a local critical current density between Ic and Ic + dIc. The distribution function f(Ic) must fulfil the condition

Then the distribution of the critical current can be found

According to this equation f (Ic) can be obtained by calculating d2V/dI2 from the I—V curve. As long as equation (B7.3.1) is valid, i.e. A does not depend on the current, the pre-factor 1/A acts as a scaling factor and does not influence f (Ic). d2V/dI2 can be calculated by numerical techniques (see e.g. Savitzky and Golay 1964). Once f (Ic) is known, these are several other useful parameters that can be calculated. For instance, one can define an average critical current 〈Ic 〉

or the fraction of the wire in a dissipative state (flux flow state) at a specific current

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Knowledge of FD(I) is important for the optimization of Ic in superconductors. As an example FD(I) at the critical current goes up to 30% for Nb—Ti and Nb3Sn (Warnes and Larbalestier 1986); a non-optimized heat treatment of a Nb3Sn wire may give less than 1%. For further details the reader is referred to the article by Warnes and Larbalestier (1986). From the practical point of view, an analysis of the I—V curves in the above-mentioned manner requires the complete distribution f (Ic). This is only possible when the linear regime of the I—V curve above Ic is reached (d2V/I 2 = 0), as illustrated in figure B7.3.10. Because the sample current can be substantially above Ic , measures must be taken to prevent thermal runaway, or damage of the wire. The upper limit is given by the amount of stabilizer in a wire and the critical current at a specific field. For instance, superconducting wires are more stable at high fields (low Ic ) and it may happen that they become unstable at low fields. Thus a conducting measurement mandrel with low electrical resistance is necessary (e.g. brass, copper). Although the Ic and the n-value must be corrected to take account of the shunted current in the sample holder, no corrections are required for the determination of the distribution of the critical current f (Ic) (Narang and Warnes 1993). The only consequence of a low-resistivity sample holder is a reduction of the measured voltages, so the method is limited by the sensitivity of the voltmeter.

Figure B7.3.10. The distribution of the critical current f (Ic ) = d2V /dI2 . In order to reach the linear regime of the I—V curve above Ic one has to measure V at higher values than is usually required for the ordinary Ic criteria.

B7.3.5.3 Scaling law Many Ic measurements of different superconductors with substantially varying layout can be described by the empirical equation

where P v is the volume pinning force, Jc is the critical current density, B is the applied field, S is a prefactor determined by the microstructure, κ is the Ginzburg—Landau parameter and b = B/Bc 2 is the reduced field, m, n, p and q are obtained from the experiment and are characteristics of the investigated superconductor. If it is assumed that the same pinning mechanism acts over the considered temperature, strain and field ranges, then S and κ m will not depend strongly on temperature and strain and they can be put together into one constant C. The above equation can be simplified to

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Note that the right-hand term b p( 1—b )q depends only on the reduced field b, whereas Bnc2 is a function of temperature and strain. Pv , when plotted versus the reduced field, goes through a maximum. Then one can plot the reduced Pv / Pvmax versus the reduced field b and one obtains a universal curve. If the temperature, or the strain state, of the Ic measurement changes, Pv can be scaled with respect to Pvmax and it then falls exactly on this universal curve. It is said that the considered superconductor obeys a scaling law. Such behaviour is very useful because with the knowledge of Jc versus field at a given temperature and/or strain and the knowledge of how Bc2 changes with temperature and strain, Ic for different temperatures and strains can be calculated. The scaling law is also helpful for the extrapolation of Ic at fields above the experimental limits imposed by the available magnet. The temperature scaling law was first described by Fietz and Webb (1969) and the strain scaling law by Ekin (1980). How Bc2 depends on temperature and strain for Nb—Ti and Nb3Sn will be discussed further in appendices A and B respectively. It is important to underline that the n exponent in the scaling law is different for temperature scaling and strain scaling. However, the p and q exponents are independent of temperature or strain scaling (see appendices A and B). B7.3.5.4 Bending strain Ic versus uniaxial strain measurements can be used to estimate the behaviour of the superconductor under bending strain. If a wire is bent, there is inside it a neutral axis which does not see any strain. If there is no yielding, the neutral axis is in the centre of the wire with a bending diameter of D. For diameters > D the wire is under tensile strain and for diameters < D under compressive strain. The maximum bending strain is obtained at the wire diameter d which is either compressive or tensile. The degradation of Ic due to bending strain can be calculated by averaging over the uniaxial strain curve (Ic versus strain). One has to distinguish between two cases, wires with a long twist pitch and those with a short twist pitch. The length of the twist pitch, l, of the superconducting filaments has to be compared with the current transfer length L , already discussed at the beginning of this section. circular conductor.

square or aspected condcutor. Ic 0 is the critical current without bending strain, ε0 is the intrinsic strain of the superconducting filament and ε is the induced bending strain. The case l > L applies also to monofilamentary conductors. If the length of the twist pitch is short ( l < L ) the following relation holds circular conductor with ε0 < 0. Note that averaging over the conductor goes now from ε = 0 to ε = εB (and not from ε = −εB to ε = εB ) Further details can be found in the article by Ekin (1981b) and references therein. Appendix A Nb—Ti wires Nb—Ti is today’s most used superconductor. The material allows the generation of fields up to ~11 T and one of the most important advantages is its ductility. So Nb-Ti wires can be used to wind magnets with

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almost no degradation of the critical current due to bending strain. Because Nb—Ti wires do not need any heat treatment after winding one can use a thin layer of enamel for electrical insulation which allows a high current density of the winding package. How the critical current is reduced (reversibly) when a uniaxial strain is applied is shown for a Nb—Ti standard reference material (SRM) in figure B7.3.A1. The applied field is a parameter in this presentation. Note that Ic ( 7 T ) is reduced by less than 4% at a rather high strain of 1%. The SRM has been produced by the NIST, Boulder, CO and can be purchased under the number SRM 1457. A very detailed description of Ic measurements of this Nb—Ti SRM is available which is helpful in finding the impact of different error sources and their importance (Goodrich et al 1984). The SRM is also well suited for checking equipment and helps to locate errors in the experimental set-up or the data acquisition software.

Figure B7.3.A1. Normalized critical current Ic /Ic 0 versus uniaxial strain of the Nb—Ti SRM 1457. Ic 0 is the critical current without strain. Reproduced from Goodrich et al (1984) by permission.

The critical current of Nb—Ti obeys the scaling law

where Pv is the volume pinning force, Jc is the critical current density, B is the applied field, C is a constant and b = B/Bc 2 is the reduced field. If temperature alone is considered, one has to know how Bc 2 depends on the temperature. This is shown in figure B7.3.A2 for Nb—Ti wires from different manufacturers (Lubell 1983). The data can be fitted by the empirical equation

with Bc 2(0) = 14.5 T for B < 10 T and Bc 2 (0) = 14.8 T for B > 10 T. Tc (at zero field) is 9.2 K. Inversely, Tc (B) can be calculated from

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Figure B7.3.A2. Upper critical field as a function of temperature for Nb—Ti wires from different manufacturers. Data were taken for MCA, Supercon-1 and Supercon-2 from Iwasa and Leupold (1982), for IMI-1 from Hampshire et al (1969), for IMI-2 from Hudson et al (1981) and for Airco from Spencer et al (1979). The calculated line was obtained using Bc 2 (T) = Bc 2(0) (1 - T/Tc (0))1.7 with Bc 2(0) = 14.5 T and Tc (0) = 9.2 K. Reproduced from Lubell (1983) by permission of IEEE.

If strain is considered, Bc 2 (ε0 ) data can be fitted according to (Ekin 1981a)

where ε0 is the intrinsic strain of the superconductor which is the strain acting on the superconductor without the presence of the matrix. However, due to the differential thermal contraction upon cooling, matrix materials can introduce a prestrain (tensile or compressive). The exponent n for strain scaling is n =4 (Ekin 1981a). The exponent n for temperature scaling, as well as p and q, can vary substantially and depends on the degree of cold work and on the heat treatment. For instance, heavily cold-worked Nb-Ti has n up to 4, p > 1 and q < 1. In contrast, a thoroughly aged (heat-treated) Nb—Ti has n ~ 2, p ≤ 0.7 and q > 1. With respect to Ic , optimized Nb-Ti wires have n = 2, p _~ 1 and q _~ 1 and the maximum pinning force (Jc B) is at the reduced field b = 0.5 (Wada et al 1985). Appendix B Nb3Sn wires Nb3Sn has a much higher Bc 2 than Nb—Ti, of the order of 23 T at 4.2 K for unalloyed Nb3Sn and up to 29 T at 4.2 K for Nb3Sn with Ta and Ti additions (see also section B8.1). This superconductor is quite brittle and intrinsic strain above ~0.5% can damage the wire. For this reason magnets are mostly built using the ‘wind and react’ technique. The unreacted wire does not contain the Nb3Sn phase and is therefore ductile. After winding, the whole magnet must be annealed at temperatures around 700 °C in order to form the superconducting Nb3Sn. Electrical insulation can be achieved by a heat-resistant and flexible glass insulation. Because of the brittle character of Nb3Sn, particular precautions are necessary for the measurement of critical currents. An extensive discussion of problems related to Ic measurements of Nb3Sn can be found in the article by Wada et al (1995). In this work results from the VAMAS initiative over the last ten years have been compiled. A characteristic example of how the critical current behaves as a function of uniaxial strain in a Nb3Sn multifilamentary wire is depicted in figure B7.3.B1. The strain where the

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Figure B7.3.B1. The critical current Ic as a function of uniaxial strain of a Nb3Sn multifilamentary wire. The strain where the maximum of Ic appears is εm = 0.32%. For ε < 0.32% the Nb3Sn filaments are under compressive strain (intrinsic strain ε0 < 0). For strains ε > 0.8% the Nb3Sn filaments are irreversibly damaged (cracks). Reproduced ~ from Ekin (1980, p 613) by permission of Elsevier Science Ltd.

maximum of Ic appears is εm = 0.32%. For ε < 0.32% the Nb3Sn filaments are under compressive strain (intrinsic strain ε0 < 0). For strains ε > 0.8% the Nb°3Sn filaments are damaged (cracks). ~ Before any Ic measurement can be carried out, the wire must be heat treated. So the wire must be wound on a reaction mandrel which should have the same dimensions as the measurement mandrel. Care must be taken that the wire does not stick by diffusion bonding onto the reaction mandrel. A surface oxidized stainless steel mandrel may help. Another, and very important, precaution is to take into consideration the thermal expansion of the annealing support with respect to the wire. Measures must be taken to prevent any tensile strain of the wire during cool-down. For example, one can fix one end of the wire firmly on the reaction mandrel and the other end can move slightly although still being kept in position. There are techniques where the reaction mandrel is also used as a measurement mandrel with the advantage that the Nb3Sn wire must not be transferred from one mandrel to the other. This mandrel can be of alumina ceramic or a stainless steel with a ceramic coat. The critical current of Nb3Sn obeys the scaling law

where Pv is the volume pinning force, Jc is the critical current density, B is the applied field, C is a constant and b = B/Bc 2 is the reduced field. If temperature alone is considered, one has to know how Bc2 depends on the temperature. Because Bc 2 and Tc of Nb3Sn can vary from wire to wire the best way to obtain this information is as follows. Bc 2 (4.2 K) can be obtained by the so-called Kramer extrapolation (Kramer 1973). Here one plots J1/2 B1-4 of an Ic measurement as a function of field. This gives, c in the case of Nb3Sn, a straight line and the intersection for Jc1/2 B1/4 = 0 at the field axis gives Bc 2 (4.2 K). If one knows Tc , which can be assumed constant as a first approximation (Tc = 18 K),Bc 2 (t) can be calculated

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where t = T/Tc is the reduced temperature and ρAG(t) is the Abrikosov—Gorkov function which is tabulated in table B7.3.B1. If Tc is 18 K , then t = 0.23 at 4.2 K and ρAG(0.23) = 0.254 15. Then the complete Bc 2 (t) curve can be calculated. For temperature scaling the exponent n is n = 2.5. If strain is considered, Bc 2 (ε0 ) data for commercial Nb3Sn conductors can be fitted according to (Ekin 1981a)

where ε0 is the intrinsic strain of the superconductor which is the strain acting on the superconductor without the presence of the matrix. However, due to the differential thermal contraction upon cooling, the matrix introduces a compressive pre-strain, so the total strain acting on the superconductor will be ε = εm + ε0 , where εm is the strain coming from the matrix. The exponent n for strain scaling is n = 1 ± 0.3 (Ekin 1980, 1981). Finally, the p and q exponents of the scaling law for Nb3Sn do not change as a function of metallurgical treatment, unlike the case of Nb—Ti, and are p = 0.5 and q = 2. The maximum pinning force is at b = 0.2, which is characteristic for grain-boundary pinning. Table B7.3.1. The Abrikosov-Gorkov function ρAG (t).

References Baixeras J and Fournet G 1967 Pertes par déplacement de vortex dans un supraconducteur de type II non idéal J. Phys. Chem. Solids 28 1541–7 Campbell A M and Evetts J E 1972a Flux vortices and transport currents in type-II superconductors Adv. Phys. 90 199–28 —1972b Critical Currents in Superconductors (London: Taylor and Francis) Clark A F, Fujii G and Ranney M A 1981 The thermal expansion of several materials for superconducting magnets IEEE Trans. Magn. MAG-17 2316–9 Duchateau J L, Turck B, Krempasky L and Polak M 1976 The self-field effect in twisted superconducting composites Cryogenics 16 97–102 Ekin J W 1978 Current transfer in multifilamentary superconductors, I. Theory J. Appl. Phys. 49 3406–9 — 1980 Strain scaling law for flux pinning in practical superconductors. Part 1: basic relationship and application to Nb3Sn conductors Cryogenics 20 611-24 — 1981a Strain scaling law for flux pinning in NbTi, Nb3Sn, Nb-Hf/Cu-Sn-Ga, V3Ga and Nb3Ge IEEE Trans. Magn. MAG-17 658–61 — 1981b Mechanical properties and strain effects in superconductors Superconductor Material Science: Metallurgy, Fabrication and Applications ed S Foner and B B Schwartz (New York: Plenum) pp 455–510

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Fietz W A and Webb W W 1969 Hysteresis in superconducting alloys-temperature and field dependence of dislocation pinning in niobium alloys Phys. Rev. 178 657–67 Goodrich L F, Bray S L and Stauffer T C 1990 Thermal contraction of fibreglass—epoxy sample holders used for Nb3Sn critical current measurements Adv. Cryogen. Eng. 36 117–24 Goodrich L F and Fickett F R 1982 Critical current measurements: a compendium of experimental results Cryogenics 22 225–41 Goodrich L F and Srivastava A N 1992 Comparison of transport critical current measurement methods Adv. Cryogen. Eng. Mater. B 38 559–66 — 1995 Thermal contraction of materials used in Nb3Sn critical current measurements Cryogenics 35 S29–32 Goodrich L F, Srivastava A N, Yuyama M and Wada H 1993 n-value and second derivative of the superconductor voltage-current characteristic IEEE Trans. Appl. Supercond. AS-3 1265–8 Goodrich L F, Vecchia D F, Pittman E S, Ekin J W and Clark A F 1984 Critical current measurements on a NbTi superconducting wire standard reference material NBS Special Publication 260-91 1–52 Hampshire R, Sutton J and Taylor M T 1969 Effect of temperature on the critical current density of Nb—44 wt% Ti alloy Low Temperature and Electric Power, Annexe M69-I (London: International Institute of Refrigeration Commission I) pp 251–7 Hudson P A, Yin F C and Jones H 1981 Evaluation of the temperature and magnetic field dependence of critical current densities of multifilamentary superconducting composites IEEE Trans. Magn. MAG-17 1649–52 Iwasa Y and Leupold M J 1982 Critical current data of NbTi conductors at sub-4.2 K temperatures and high magnetic fields Cryogenics 22 477–9 Katagiri K, Okada T, Walters C R and Ekin J W 1995 V-2: effects of stress/strain Cryogenics 35 S85–8 Kramer E J 1973 Scaling laws for flux pinning in hard superconductors J. Appl. Phys. 44 1360–70 Lubell M S 1983 Empirical scaling formulas for critical current and critical field for commercial NbTi IEEE Trans. Magn. MAG-19 754–7 Narang G and Warnes W H 1993 Extended measurements of the resistive critical current transition IEEE Trans. Appl. Supercond. AS-3 1269-72 Savitzky A and Golay M J E 1964 Smoothing and differentiation of data by simplified least squares procedures Anal. Chem. 36 1627–39 Specking W, Nyilas A, Klemm M, Kling A and Flükiger R 1989 The effect of axial stresses on Ic of subsize NET Nb3Sn conductors Proc. MT-11 (Tsukuba, 1989) vol 2, ed T Sekiguchi and S Shimamoto pp 1009–14 Spencer C R, Sanger P A and Young M 1979 The temperature and magnetic field dependence of superconducting critical current densities of multifilamentary Nb3Sn and Nb—Ti composite wires IEEE Trans. Magn. MAG-15 76–9 ten Haken B 1994 Strain effects on the critical properties of high-field superconductors PhD Thesis Technical University of Twente, The Netherlands ten Haken B, Godeke A and ten Kate H H J 1993 New devices for measuring the critical current in a tape as a function of the axial and the transverse strain, the magnetic field and temperature IEEE Trans. Appl. Supercond. AS-3 1273–6 VAMAS Technical Working Party for Superconducting Materials 1995 VI-1: recommended standard method for determination of d.c. critical current of Nb3Sn multifilamentary composite superconductors Cryogenics 35 VAMAS Suppl. S105–12 Wada H, Goodrich L F, Walters C and Tachikawa K (eds) 1995 Critical current measurement method for Nb3Sn multifilamentary composite superconductors Cryogenics 35 VAMAS Suppl. S1–126 Wada H, Itoh K, Tachikawa K, Yamada Y and Murase S 1985 Enhanced high-field current carrying capacities and pinning behaviour of NbTi-based superconducting alloys J. Appl. Phys. 57 4415–20 Walters C R, Davidson I M and Tuck G E 1986 Long sample high sensitivity critical current measurements under strain Cryogenics 26 406–12 Warnes W H and Dai W 1992 Shape measurements of the resistive transition in SSC strands Adv. Cryogen. Eng. 38 709–13 Warnes W H and Larbalestier D 1986 Critical current distributions in superconducting composites Cryogenics 26 643–53

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B7.4 Critical current measurements of superconducting cables by the transformer method

P Fabbricatore and R Musenich

B7.4.1 Introduction Several large-scale projects, ranging from particle accelerators for high-energy physics to nuclear fusion and energy storage, are being developed at the present time on the basis of the use of superconducting magnets generating fields up to 13 T in large volumes. These magnets are wound with conductors carrying currents as high as 20–40 kA. Remarkable examples are the huge magnets for the Large Hadron Collider experiments at CERN (CMS and ATLAS) (Desportes 1994) and the toroidal coils for the ITER nuclear fusion project (Thome 1994). The conductors used for such applications are complex structures composed of subconductors. The simplest component generally is a multifilamentary NbTi or Nb3Sn strand of diameter 1 mm. One of the most important pieces of information that needs to be known about the conductor is the dependence of the critical current on the magnetic field at the operating temperature. A very simple way to determine the critical current of the complete conductor is to measure the critical current of all the strands, which in most cases does not exceed 1000–2000 A at the magnetic fields of interest. Nevertheless this process takes a lot of time and requires a comparison with the critical current of the complete conductor. Furthermore the measurements on multistrand conductors are of interest, because the stability margin against thermal disturbances or the electrical resistance of the joints can be only obtained by feeding the current to the complete conductor. Generally speaking, the measurement of the critical current of a high-current cable cannot be avoided in order to gain information about the real performance of the conductor. When performing such measurements, the classical method supplying the samples under test with a d.c. high-current—low-ripple power supply has several disadvantages: (i) high cost of the power supply and current leads (ii) high liquid helium consumption (about 3 1 h–1 each kA) (iii) the necessity to control several devices such as the ancillary equipment of the power supply (cooling water, power leads, electric power control) (iv) large dimensions of the devices. Due to these drawbacks, only a big laboratory with expensive electrical and cryogenic facilities could perform critical current measurements on cables. To overcome these basic problems related to the complexity of the experimental apparatus, alternative methods of feeding the current were developed in different laboratories. These methods use superconducting transformers to charge the samples, avoiding the use of high-current power supplies. In the following sections we will describe the basic principles of the transformer methods in relation to the equations controlling the current in the sample and the voltage along the sample. Then typical experimental set-ups will be described with emphasis on the sample holders

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and the measurement techniques. As a last item the problem of the critical field will be discussed. In fact due to the set-up and to the high current, the self-field can be an appreciable fraction of the applied field, so that the field at the cable is not homogeneous, generating ambiguities in the definition of critical field. B7.4.2 The superconducting-to-normal transition of a cable As for wires, the transition from the superconducting to the normal state of a cable is a smooth process ruled by the semi-empirical law

ρs being the electrical resistivity of the sample, I the current crossing through the sample and n an integer giving information about the quality of the superconducting cable. The critical current is defined on the basis of the resistive criterion (Clark and Ekin 1977), which states that the current flowing through a sample has attained the critical value when an electrical resistivity ρc = 10–14 Ω m is measured. It is important to stress that the resistive criterion corresponds to a well determined regime of the fluxon dynamics. In fact the resistivity range 10–16–10–13 Ω m is related to the thermally activated flux creep (Kim et al 1962, 1963), even for a low-Tc superconductor. Other limiting values for the electrical transport properties, such as the quench current, do not have the same general physical meaning, being strongly influenced by the thermal heat exchange of the sample with the coolant (Fabbricatore et al 1990, Kovàc et at 1991), by the level of thermal disturbances and by the peculiarities of the sample holder (Fabbricatore et al 1990). These effects are particularly critical for high-current cables and could lead to a sample quench below the critical current. The critical current is not the only information required because the knowledge of the n value is of great importance for the determination of the electrical transport properties of a cable as demonstrated by Warnes and Larbalestier (1986). The complete information is obtained through the measurement of the current-voltage (I—V) characteristic, which can be approximated by

where l is the distance between the voltage taps and Ss c is the superconducting cross-section. For a multistrand cable the determination of Ss c and l should be made considering the twisting of the strands. The simplest way to obtain Ss c is to multiply the superconducting cross-section by the number of strands. These parameters should be known as accurately as possible in order not to introduce systematic errors in the critical current measurement. A simple calculation can be made to check this assertion. The error on the measured critical current due to the errors on voltage, current and geometry is given by

For a typical value of the index n = 20 and considering a possible 10% error in l and Ss c , we have 1% error on Ic only due to our poor knowledge of the sample geometry. B7.4.3 The transformer methods After defining the quantities of interest we can analyse the methods of inducing a current in a circuit including the sample to be tested. From the point of view of the electrical set-up, there are two main transformer methods, as shown in figures B7.4.1(a) and B7.4.1(b).

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Figure B7.4.1. Schematic diagrams of the electrical circuits used in the transformer methods: (a) indirect method, (b) direct method.

The method of figure B7.4.1(a) is the most common one and is used in several laboratories. The principle of operation is very simple: the sample to be tested, immersed in a fixed d.c. magnetic field, is connected to the secondary turns of an air core superconducting transformer. The second method (figure B7.4.1(b)) is even simpler, because the primary coil is the magnet giving the background magnetic field, whilst the sample itself is the secondary of the transformer. These two methods will be named respectively, and arbitrarily, the indirect method and the direct method. B7.4.4 The indirect transformer method For the indirect transformer method we refer to figure B7.4.1(a), where L p = self-inductance of the primary R p = resistance of the circuit including the primary I p = current in the primary L s = self-inductance of the secondary L s p = self-inductance of the sample

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Measurement techniques for the characterization of SC wires and cables R s = resistance of the sample (when going to normal) R j = resistance of the joints in the secondary circuit I s = current in the secondary M = mutual inductance between primary and secondary. The equations in the primary and secondary are

For simplicity, in these equations, we have assumed that the primary is only coupled with the secondary and not also with the sample. We are mainly interested in finding the expression for the current in the secondary, which can be easily determined if the current rate in the primary is constant, i.e. assuming − dIp /dt = C = constant and that the resistance of the sample is zero (the sample is in the superconducting state). We obtain, with the starting condition Is , (t = 0) = 0

where τ is the time constant given by τ = ( L s + Ls p )/R j . Equation (B7.4.4) can be written in a more convenient way since the constant C is given by C = -(Ip( t ) - I0 )/t

Equations (B7.4.4) and (B7.4.5) allow us to obtain the two important pieces of information related to the transformer: the current transformer ratio and the maximum current that can be induced in the secondary. The ideal current transformer ratio, also called the current amplification factor, i.e. the ratio between the current in the secondary and the current variation in the primary, can be obtained from equation (B7.4.5) in the limit t /τ → 0

where k is the coupling constant between primary and secondary. The maximum current that can be induced in the secondary circuit, obtainable from equation (B7.4.4) in the limit of t/τ → ∞, is

Several considerations can be made on the basis of these simple expressions. (i) The operation of inducing a current in the secondary circuit takes a certain amount of time so that the effective current transformer ratio is lower than the ideal one. From equations (B7.4.5) and (B7.4.6), we have that

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(ii) When the current is approaching the critical current the resistance of the sample Rs could be significant with respect to the joint resistance Rj . As shown in equation (B7.4.1) this resistance is a power function of the current, so that its presence in equations (B7.4.3) leads to nonlinear effects. In this case equation (B7.4.5) is only a rough solution of equation (B7.4.3). In order to know whether the used device can feed the critical current to the sample, we can replace the joint resistance in equation (B7.4.7) with the total secondary resistance at the critical current

where Rc is the critical resistance of the sample, i.e. the resistance corresponding to the critical resistivity. (iii) The current amplification factor as given by equation (B7.4.6) is determined by the inductances of both the primary and the sample. For a given sample with inductance Ls p , the maximum value of the amplification factor, depending on the primary inductance, is reached if Ls = Ls p . This is shown in figure B7.4.2, for a sample inductance Ls p = 10– 6 H, a primary inductance ranging from 0.1 to 1 and 10 H and a coupling constant k = 0.8. The curves also show that it is convenient to have the secondary inductance slightly higher than the sample inductance, so that the amplification factor is a smooth function of the secondary inductance. It is interesting to note that, for a given typical value of the sample inductance, the amplification factor changes by one order of magnitude (102 to 103) on changing the primary inductance by two orders of magnitude (0.1 to 10 H).

Figure B7.4.2. The current amplification factor as a function of secondary inductance for three values of the primary inductance (sample inductance Ls p = 10– 6 H).

B7.4.5 The direct transformer method The direct method, for which a schematic diagram is shown in figure B7.4.1(b), is a simplification of the indirect method. The secondary is the sample itself; the primary is the magnet generating the background field. The equation ruling the secondary circuit is the equation (B7.4.3b) modified by replacing Ls + Ls p with Ls p . The current amplification factor is simply given by α0 = k (Lp/Lsp)-. As the background magnet is the primary, its inductance is generally high (10 H) so that an obvious advantage of this method is related 1 2

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to the possibility of having high amplification factors. Since the sample is also the secondary circuit, it is required that its inductance is low but not too low, in order to have a good coupling with the primary. The best geometry in this case is the round loop. The primary magnet should be a solenoid with a bore large enough to accommodate the sample. The sample to be tested is wound in a single loop, with the two ends connected by a single low-resistance joint. In the indirect method, there are no special problems related to the voltage signals to be measured along the sample for the critical current measurement. In the direct method the measured signals are more complex because the electromotive force due to the magnetic flux variations is generated in the sample itself. Consequently the connection of the voltage taps on the superconducting loop and the position of the measuring wires must be carefully studied. We will consider three different signals as shown in figure B7.4.3: (i)

VA− B is the voltage drop across half a loop. It can be written by considering half the loop containing the joint as

or half the loop not containing the joint

In both cases using equation (B7.4.3b), equations (B7.4.10) become

Consequently the voltage VA−B can be used to measure the electrical resistance of the joint, if the current is known and if the loop is in the superconducting state.

Figure B7.4.3. A schematic diagram of the voltage taps and wiring needed to detect the sample resistance in the direct method.

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(ii) VC− D is the voltage signal of half the loop enclosing the joint. The wires of the voltage taps have to be placed very close to the sample so that by considering the loop given by the half of the sample not containing the joint and the measuring wires, we have

where the parameter a1 is introduced to take into account the inductive uncoupling between the cable and the voltage-tap wires. If the voltage-tap wires are well coupled with the sample a1 = 1, and we find using equation (B7.4.3b) again that

If the sample is in the superconducting state we again obtain a signal depending on the joint resistance. (iii) VE− F is the voltage signal of half the loop not enclosing the joint, obtained by looking at the voltage drops along the circuit composed by half the sample containing the joint and the voltage-tap wires.

where a2 has the same function as a1 in equation (B7.4.12). If the voltage-tap wires are well coupled, also the constant a2 = 1, so that

This is the voltage employed to measure the critical current as will be better shown in the following sections. B7.4.6 Overview of the experimental set-ups After the pioneering work of Gillani and Britton (1969) in the late 1960s and early 1970s, the first remarkable device based on a superconducting transformer was developed by Purcell and DesPortes (1973) to measure the critical current of the conductor for the 15 ft (~4.6 m) magnet built at Argonne. Figure B7.4.4 shows the short-sample test device. The primary was a superconducting coil made by 400 turns of 0.76 mm monofilamentary wire. The secondary, made by the same material as the conductor to be tested, was wound directly on the outside of the primary. A part of the secondary was used to make a small loop which included a pickup coil. When the current was induced in the secondary, the pickup coil sensed the field variation. The integration over time of the pickup-coil signal was proportional to the current in the sample. The sample to be tested was soft soldered to the secondary and put in a dipole magnet of 5 T in a useful bore of diameter 25.4 mm. Using such a device, induced currents in the secondary were obtained up to 16 000 A with a current amplification factor of 100. Several other current transformer devices were developed in the 1980s. The set-up changes with respect to the first device were related to the number of joints (one to three), the magnet generating the field applied to the sample (dipole, solenoid or quadrupole), the sample geometry (hairpin, round sample

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Figure B7.4.4. A schematic diagram of the indirect transformer device developed at Argonne in 1972.

Figure B7.4.5. A schematic diagram of a typical set-up for the direct transformer method.

or coil) and to the methods for measuring the current (Leung et al 1988, Shirshov et al 1985, ten Kate et al 1988). The direct method, which uses the sample itself as the secondary, was developed at KfK Karlsruhe (Schmidt 1983, 1984, 1988) for critical current and stability measurement of the conductor for the fusion project ‘Large Coil Task’. The same method was also used at the University of Twente (Mulder et al 1988), and at INFN Genoa (Fabbricatore et al 1991). Figure B7.4.5 shows a schematic diagram of the set-up. The sample is placed in a sample holder. A heater is in thermal contact with the sample to reset the current to zero by quenching the sample. Hall probes are used to measure the self-field generated by the current flowing through the sample. Table B7.4.1 summarizes the main characteristics of some transformer devices. B7.4.7 Methods for measuring the current A very important feature of the transformer devices is related to the methods for measuring the current. One of the most accurate methods was developed by ten Kate and coworkers (ten Kate et al 1986) and is schematically shown in figure B7.4.6. Two superconducting toroidal coils (Rogowski type) are coupled to a conductor carrying a current in the presence of a high external magnetic field. The toroid Lt p senses the field variation, when the current in the conductor is changed. If Lt p is connected to a superconducting circuit including a coil Lc s far from the magnetic field, we have a field signal at the Hall probe (or other field-sensing device) placed inside the coil Lc s . This signal can be used to drive a control unit supplying a current to the toroid Lt p . This current is proportional to the current in the conductor. The circuit is self-balancing allowing high precision (less than 0.5%) in the current measurement.

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Table B7.4.1. Examples of devices for critical currrent measurement by the transformer method.

Figure B7.4.6. The set-up for measuring the current in the indirect method.

Due to the geometrical and magnetic configuration, the direct method cannot use Rogowski coils to measure the current by sensing the magnetic flux variations. However, Hall probes can be arranged so that accurate measurements can be performed anyway. Figure B7.4.7 shows a possible set-up for the field sensors. The probe HP1 senses only the self-field generated by the current in the sample whilst the second probe HP2 is placed at the centre of the sample so that it senses both the field and the self-field. A third probe HP3, placed far from the sample, senses only the applied field. The probe HP2 is used for calibration, which is an important feature of the measurement. The current in the sample can be determined if the relation between the measured self-field and the current is known. This relation can be

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Figure B7.4.7. Positioning of the field sensors in the direct method set-up.

determined with high accuracy using a numerical method to compute the field, known as the HP2 position. In order to minimize the error due to mispositioning of HP2, it is convenient to put it in the centre of the sample, where the self-field is a smooth function of the position (order of 10– 4–10– 3 T cm–1). Close to the conductor the self-field gradient is so high (10– 3–10– 2 T cm–1 ) that large errors (10%) on the current measurement are possible. The calibration is carried out at low external field due to the orientation of HP2, which maximizes the sensitivity to the external field. The magnetic field is lowered from a preset value to zero while monitoring the HP1 and HP2 signals. This makes it possible to determine the amplification factor and to relate the current in the sample to the HP1 signal. At high field (B > 1 T) only the HP1 signal, which is mainly determined by the self-field, can be used to measure the current. Nevertheless it is not possible to neglect the background magnetic field at HP1, so that the current measurements are affected by a bias signal depending on the background magnetic field strength. A different way of performing the measurement after the calibration is to use the difference between the signals at HP2 and HP3, which is proportional to the current in the sample. B7.4.8 Mechanical supports Unlike the indirect method, which uses noninductive samples, the direct method needs inductive samples. This fact implies high stresses in the samples due to the magnetic load so that the sample must be supported by a mechanical structure, which provides the hoop strength. Analysis of a typical situation can better show the mechanical problems related to these kinds of measurement. If a conductor is bent into a loop of radius R = 0.2 m, immersed in a magnetic field Bz = 5 T, whilst carrying a current I = 50 kA, we have a total radial force of

The longitudinal hoop force is Ft = Fr /2π = 50 kN. Using a mechanical structure made of copper, which has a yield strength s0.2 = 80 MPa, we need a total cross-section of minimum value Ft /s0.2 = 625 mm2. Furthermore it is important that the magnetic loads are applied radially outward, because if the forces were inward a dangerous torque could take place, as shown in figure B7.4.8. This can be simply obtained by ramping down the background magnetic field, so that the current in the sample has the same direction as the current in the background magnet.

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Figure B7.4.8. Effect of the Lorentz force on the mechanical stability of the sample holder.

As already stressed, the indirect method does not usually suffer from this problem, because the sample is arranged to be noninductive (figure B7.4.4) and the total force is zero, though attention must be paid to the torque. On arranging a sample holder with the direct method, a reliable support for the conductor can be obtained by soft soldering it into a seat made on a copper ring. This solution is recommended for Rutherford cables, made by twisting several strands in flat conductors. This choice has the following advantages. (i) (ii)

(iii) (iv) (v)

The Lorentz force is borne by the copper ring, without applying any pressure, so that the sample holder structure is largely simplified. One of the main problems with sample holders for high-current cables is given by the mechanical disturbances, producing noise on the measured voltage (up to the same signal level at the critical current) and premature quenching of the samples. Soldering prevents all movement, so that the noise is reduced. The heat produced by the Joule effect at the joint is removed more efficiently due to high thermal conductance of the copper and to the large exchange surface with the liquid helium bath. The thermal capacity and the thermal conductivity of the copper ring stabilize the temperature of the sample at the bath temperature. Last but not least, the copper ring has the function of protecting the background magnet. When the sample goes to the normal state, the carried current decays fast. Current and voltage are thus induced at the primary winding. Since only a part of the primary is coupled with the sample, high internal voltage can be generated inside the background magnet. To estimate the order of magnitude of the maximum induced voltage, we can look at the term

where Is 0 is the initial current in the sample and R is the resistance of the sample in the normal state. For a Rutherford conductor composed of 40 NbTi/Cu strands of 1.3 mm wound on a 1.5 m loop, R = 8.5 µΩ. With Is 0 = 50 kA and α0 = 500, we have Vi n d = 200 V. In general the use of high conductivity support rings avoids dangerous voltages in the primary. As discussed before, the copper cross-section has a minimum value to fulfil the electrical, thermal and mechanical requirements. On the other hand there is a maximum allowed cross-sectional area, to be determined on the basis of the maximum current that can flow through the copper without affecting the critical current measurement. The current flowing through the copper can be calculated using

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where λ is the copper-to-cable ratio and Ic , is the critical current of the cable. Since ρc = 10–14 Ω m and ρc o p p e r = 1.710–10 Ω m (with no magneto-resistivity effect), setting λ = 100 will mean that at the critical current the ratio Ic o p p e r /Ic a b l e = 0.6%, which can be considered too low to influence the superconducting-to-normal transition of the cable. A further effect to be analysed is that every variation of the magnetic field induces a current in the copper ring. This current decays with a time constant of some seconds, with respect to the time constant of 102 or 103 s of the secondary loop, so that this extra current does not affect the measurements. B7.4.9 Heater and electrical joints Both direct and indirect methods need a system to quench the sample. This can be made by using thermal heaters or a.c. coils inducing dissipation in the secondary circuit. The power needed to cause a quench and reset the current to zero depends on the set-up configuration. For the direct method, where the sample/secondary is immersed in a liquid helium (LHe) bath, a power as high as 100 W could be necessary. Both transformer methods need excellent electrical joints to be able to carry out the measurements. Usually the inductance of the secondary is of the order of 1 µH, so that a suitable time constant of 500–1000 s for the current decay implies a joint resistance of maximum 2 nΩ. For a copper-stabilized conductor the best way to make the joints is to use a soft soldering technique. When using tin-lead alloy solders, it is better not to use superconducting alloys, because part of the current can flow in the solder causing disturbances, which can lead to quenches of the joint. This drawback is mainly relevant for the indirect method because for the direct method the joint is placed in the applied field, so that the solder is in the normal state. Nevertheless in this case the joint resistance grows when the magnetic field is increased as shown in figure B7.4.9 for an aluminium-stabilized conductor. In order to obtain a good joint the two ends of the conductor should be overlapped for almost a twist pitch in the clasping hands configuration, which allows good current transfer between the two cables. As regards the soldering temperature, it must be kept between 200°C and 250°C to obtain a good joint quality and to avoid thermal degradation of the critical current. Good soldering of the whole cable on the copper ring is needed to reduce the voltage noise due to mechanical movements and to prevent quenches that could impede the critical current measurement.

Figure B7.4.9 Resistance versus applied field for an indium joint of an Al-stabilize conductor.

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B7.4.10 Typical measurement by the direct transformer method In this section we will describe the practical procedure adopted at the INFN Genoa laboratory to carry out critical current measurements on large superconducting cables. Figure B7.4.10 shows the conductors under study for the CMS and ATLAS magnets (Desportes 1994). Both the cables are of the aluminium-stabilized Rutherford type; the conductor for CMS has a mechanical reinforcement in the Al alloy. Measurements have been performed on both the pure-aluminium-stabilized conductors and the Rutherford cable using the direct transformer method.

Figure B7.4.10. The conductors under study for the Large Hadron Collider detector magnets at CERN: (a) ATLAS conductor; (b) CMS conductor.

The measurement device is composed of a superconducting solenoid and a cryostat with a doublewall insert to separate the magnet helium bath and the experimental zone. The magnet is supplied by a 1000 A–10 V power supply and generates 6.0 T at 1000 A in 50 cm free bore, or 8 T at 920 A in a 38 cm bore, with the addition of an insert solenoid. As a result of the solenoidal geometry the magnetic field at the conductor is slightly higher. The sample holders are hosted into the insert and suspended from the upper flange by means of six stainless steel tie rods about 1.5 m long. Even though the measurement method was the same in both cases, it is interesting to describe the sample holders and the sample preparation as these were different for different cables. The sample holders are shown in figure B7.4.11. The Rutherford cable, bent in a single turn, is hosted in the spline of a copper ring with the two ends superimposed for about one twist pitch length. Then the cable is soft soldered using either pure indium or a tin—lead eutectic alloy. To have good bonding, the copper ring is precoated with the same metal as used for the soft soldering, and a soldering flux is added when the whole ring has reached the metal melting temperature. During this operation, two voltage taps are soldered to the cable diametrically opposite one another and equidistant from the joint. Generally, the distance between the voltage taps and the joint should be higher than a twist pitch length of the cable in order to allow good sharing of the current by all the strands. This is an empirical rule that can be easily experimentally verified. In fact, whether or not current transfer occurs in the region between the voltage taps, a nonlinear resistive voltage is measured well below the critical current. The copper ring (5 cm wide and 1 cm thick) has been chosen on the basis of the considerations described in section B7.4.8, but it is not strong enough to support the Lorentz forces which are of the order of 3 X 105 N. Thus it is connected, via 18 bolts, to a stainless steel ring having the same dimensions.

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Figure B7.4.11. Sample holders for high-current conductors: (a) Rutherford cable with the magnetic field normal to the wide face, (b) Rutherford cable with the magnetic field parallel to the wide face, (c) aluminium stabilized conductor with the magnetic field parallel to the wide face.

A heater able to supply 60 W is glued on the cable near the joint and covered by plastic foam in order to thermally insulate it from the helium bath. The sample holder used for the aluminium-stabilized cables is completely different: due to the difficulty in soldering aluminium, the sample is supported in an aluminium alloy cylinder and clamped by a conical ring (divided into four parts), which in turn is pushed by another conical ring (see figure B7.4.11(c)). The two ends of the cable, machined on one side to remove aluminium and coated with indium, are superimposed. The two voltage taps are inserted between the sample and the conical ring sectors and the heater is supported between the cable and the external cylinder. Finally, the whole sample holder is heated in an oven at about 140°C and the tie rods pushing the conical rings are further tightened. The measurement procedure is then the same for the two kinds of cable. The sample holder is connected to the upper flange and the measurement wires are soldered to the voltage taps. The wires must be coupled to the cable to allow the resistive voltage detection, as described in section B7.4.5. As regards the Rutherford cable, the wires are positioned directly over the sample and locked by plastic foam slabs tied to the sample holder. In the aluminium-alloy sample holder the wires are supported in a spline in the internal wall of the outer cylinder, in order to be close to the conductor. It must be pointed out that the voltage signals are only a few µV, so that particular care must be taken in positioning, soldering and twisting the wires. Two Hall probes are used to detect the magnetic field (figure B7.4.7): one (HP1) is positioned over the cable to detect the self-field in such a way that it is insensitive to the external magnet field. The distance between the probe and the sample is a few centimetres (3–5 cm). The other probe (HP2) is positioned at the centre of the sample holder to measure the magnet field and for calibration (see section B7.4.7). It must be pointed out that it is not convenient to lock the first probe to the sample holder, because it could be slightly deformed by the Lorentz forces affecting the current measurement. The position of the two Hall probes, in particular HP2, must be determined with accuracy by measuring the field in order to calculate the current flowing through the sample. The relation between the sample current and the self-field measured by the Hall probe can be calculated once the position of the probe and the cable geometry are known. The sample holder is finally introduced into the measurement cryostat and cooled down. During the cool down, when the sample temperature is still above the transition temperature, the magnet current is increased up to the operation condition. It is important to avoid the transition of the sample to the superconducting state because the current induced in the sample during the magnet charge up generates inward Lorentz forces, which in turn generate mechanical instability (see figure B7.4.8), so, if necessary, the heater is used. Once the operating field is reached, the sample is covered with liquid

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helium and it is charged by lowering the magnet current at a constant rate (between 0.15 and 0.75 A s–1, i.e. about 100–500 A s–1 in the cable). The Hall probe signal, sample voltage and magnet current are recorded. Figure B7.4.12 shows a typical I—V characteristic: the initial increase of voltage is due to the voltage induced by the constant current rate. After a plateau, the voltage further increases according to equation (B7.4.1b) (curve (a) in figure B7.4.12) and when the critical current has been reached, the ramp is stopped. Then, at constant magnet current, the sample current begins to decay because of the resistance of the sample and the joint (curve (b) in figure B7.4.12). The critical current can be obtained from both curves, but the use of curve (b) is preferred because it is obtained at constant external field. The determination of the critical current is done by the intersection of the I—V curve with the critical voltage line on a logarithmic graph. The measurement shown in figure B7.4.12 has been carried out by lowering the magnet field from 4.54 T to 4.16 T.

Figure B7.4.12. The I—V characteristics of a cable measured by the transformer method.

A problem arises in the current measurement via the Hall probe, namely the HP1 probe misalignment with respect to the magnetic axis. It generates a signal dependent on the external field when no current is flowing though the sample (see section B7.4.7)

where θ is the angle between the probe and the magnetic axis. θ generally being small, cos θ ≈ 1 while the second term cannot be neglected due to the high value of Be x t with respect to Bs f . The zero-current signal can easily be measured during the magnet charge up and it does not represent a problem if it is lower than or comparable to the signal at the transition. In the described measurement, the zero signal is about 1 µV A–1 i.e. about 1 mV at the maximum field, while the current signal at the transition (43.5 kA) is about 1.7 mV. B7.4.11 Measurement error analysis The critical current values obtained by the described measurements are affected by an error of about ±2.5% which is mainly caused by the errors on the current value and on the field value. Let us analyse each contribution, i.e. temperature, voltage, current, external field and the sample self-field.

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(ii)

(iii) (iv)

(v)

Measurement techniques for the characterization of SC wires and cables The cable temperature error is due to the variation of the atmospheric pressure and the ohmic heating. The former leads to a variation of about 0.01 K per 10 mbar. The latter depends on the critical current value and on the joint resistance: if Ic = 50 kA and R j = 10– 9 Ω , considering the voltage drop Vc on the whole cable to be about 30 µV, the ohmic dissipation is about 4 W. As the exchange surface is very large ( about 1000 cm2 ) the ohmic dissipation leads to a temperature increase lower than 0.01 K. Using Lubell’s formulae (Lubell 1983) for NbTi wires, an inaccuracy in the sample temperature of ±0.02 K leads to an error on the critical current of ±0.3%. The errors on the voltage measurement, on the voltage tap positions and on the ratio between the NbTi cross-sectional area and the matrix area, do not seriously affect the critical current measurement because, according to equation (B7.4.2), they must be divided by the value of n. The total error on Ic due to these effects is less than ±0.3%. The error on the current measurement, depending on the position and linearity of the Hall probe, is ±1%. The error on the field determination can be evaluated separately or included in the critical current error via Lubell’s formulae. The error is due to the error on the magnet current (±0.5%) or, if the Hall probe is used, is due to its linearity (±0.5%). The error on the sample self-field is mainly related to the error in the current measurement, i.e. ±1%, and must be added to the error in the applied field, taking into account the relative weights. If we refer to the previously described measurement and apply the peak field correction (see the next section), the self-field value at the transition is 1.39 T, compared with an external field of 4.16 T, so that the error on the magnetic field due to the self-field determination is about ±0.3%. The error is lower at higher external field, i.e. for lower values of the critical current.

As a final remark, we stress that the weights of most of the errors do not depend on the choice of measurement method (transformer or classical method). The direct transformer method is responsible for the relatively high errors in the current measurements (±1.3%, taking into account the errors described in (iii) and (v)). The other errors are given by the transformer methods as well as the classical methods. B7.4.12 The self-field correction When performing critical current measurements on cables carrying high current, the problem arises of how to define the critical magnetic field, i.e. the field experienced by the sample at the transition. The magnetic field generated by the current flowing through the sample can be of the same order of magnitude as the external field, and the resultant applied field at the cable (vectorial sum of the external field and the self-field) is strongly inhomogeneous (see figure B7.4.13). As an example, we can refer to the measurement described in the previous section: in that case the maximum value of self-field at the cable is about 0.32 T kA–1, which means 1.39 T at the critical current (assumed to be 43.5 kA). Adding the external field ( 4.16 T ), which has the same value and direction all over the cable, the resulting field value at the cable ranges between 2.77 T and 5.55 T. As a result of the nonohmic behaviour of the superconducting cable (see equation ( B7.4.1)) the self-field cannot be neglected even if its average value is zero. The effect of the self-field can be reduced, but never cancelled, by the noninductive coupling of two or more samples connected in series. Of course, coupling is never possible when using the direct transformer method. The definition of the critical field is not only an academic problem, but it is fundamental when comparing the results obtained by different laboratories, which use experimental set-ups with different geometries. In fact the value of the self-field depends on several factors, such as the measured critical current, the distance between the coupled samples and the dimension and the shape of the cable, so that different measurement set-ups can give different results. A typical example is the comparison between the critical current of a Rutherford-type cable, composed of n strands, and the sum of the critical current of

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Figure B7.4.13. The magnetic field distribution due to the superposition of an external field and a self-field in a round cable.

each strand. If the measurements are referred to a fixed external field, the critical current of the cable will be lower than the sum of the strand critical currents, because, due to its higher transport current, the cable experiences a magnetic field higher than the strands. The problem of the definition of the critical field is to calculate the correct value which must be added to the external field. It is generally defined as the applied field at the transition of the peak field, i.e. the maximum value of the field experienced by the sample (external field plus maximum value of the self-field at the sample). Though this assumption is not supported by a satisfactory theory, it is an empirical criterion confirmed by a large set of experimental data as reported by Garber and coworkers (Garber et al 1989). Figure B7.4.14 shows the critical current versus the field of a Rutherford-type cable for both uncorrected and corrected data. Though the peak field criterion gives good results in comparing measurements performed by different experimental set-ups, the problem of the critical field is still open. Another criterion, which takes into account the inhomogeneity of the magnetic field at the sample, has been proposed by Fabbricatore and coworkers (Fabbricatore et al 1989, 1990): an effective critical field is calculated on the basis of the critical current—field and I—V characteristics and of the sample geometry. The effective critical field criterion gives good agreement between different measurements but is not simple to apply, so that generally the application of the easier peak field criterion is suggested. A further problem related to the self-field is the effect of the field inhomogeneity on cables showing critical current degradation in some parts of them. That is the case with Rutherford-type cables, the strands of which are sharply bent at the cable edges: the critical current density of the misshapen zones is lower than that of the other parts. By measuring the whole cable, an average value of critical current is obtained but the results are generally affected by the orientation of the cable with respect to the direction of the

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Figure B7.4.14. Critical current versus magnetic field for a Rutherford cable.

external magnetic field. As shown in figure B7.4.15, because of the self-field, the magnetic field is a maximum at the edge of the cable if the external field is perpendicular to the wide face and is a maximum at the wide face if the external field is parallel to it. The former configuration gives a lower value of the measured critical current than the latter. Differences of the order of 5–10% are generally measured. As the maximum field of a magnet wound with a Rutherford-type cable is generally at the edge of the cable, the short-sample measurements carried out with the field perpendicular to the wide face are more reliable than the others.

Figure B7.4.15. Dependence of the peak field position on the orientation of a flat conductor.

B7.4.13 Summary The transformer method allows critical current measurements well above the current range generally available with the classical measurement set-up (based on a power supply and current leads). In particular, the direct tranformer method is simple and feasible. The main drawback of the method is due to the high value of the self-field which could cause ambiguity in the interpretation of the results. This is generally true for all the methods (classical, direct transformer and indirect transformer methods) but is emphasized

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in the direct transformer method where a noninductive configuration is not allowed to compensate for the self-field. In order to compare critical current measurements performed in laboratories with different experimental set-ups we suggest, as a general rule, to give the results (critical current, applied field, temperature) together with information about the adopted self-field criterion, the self-field value and the sample configuration. References Clark A F and Ekin J W 1977 Defining critical current IEEE Trans. Magn. MAG-13 38–40 Desportes H 1994 Advanced features of very large superconducting magnets for SSC and LHC detectors IEEE Trans. Magn. MAG-30 1525–32 Fabbricatore P, Musenich R and Parodi R 1991 Inductive method for critical current measurement of superconducting cables for high energy physics applications Nucl. Instrum. Methods A 302 27–35 Fabbricatore P, Musenich R, Parodi R, Pepe S and Vaccarone R 1989 Self field effect in the critical current measurement of superconducting cables and wires Cryogenics 29 920–5 Fabbricatore P, Musenich R, Parodi R and Vaccarone R 1990 Effect of the n-value and the field inhomogeneity on the quench current of superconducting cables IEEE Trans. Magn. MAG-26 3046–55 Garber M, Ghosh A K and Sampson W B 1989 The effect of the self field on the critical current determination of multifilamentary superconductors IEEE Trans. Magn. MAG-25 1940–4 Gillani N V and Britton R B 1969 Critical current of superconductors in low fields Rev. Sci. Instrum. 40 949–51 Kim Y B, Hempstead C F and Strnad A R 1962 Critical persistent current in hard superconductors Phys. Rev. Lett. 9 306–9 Kim Y B, Hempstead C F and Strnad A R 1963 Flux creep in hard superconductors Phys. Rev. 131 2486–54 Kovàc P, Gömöry F and Cesnak L 1991 Influence of conductor temperature on the real voltage—current characteristic of composite superconductors Supercond. Sci. Technol. 4 172–8 Leung E M W, Arrendale H G, Bailey R E and Michels P H 1988 Short sample critical current measurements using a superconducting transformer Adv. Cryogen. Eng. 33 219–26 Lubell M S 1983 Empirical scale formulas for critical current and critical field for commercial NbTi IEEE Trans. Magn. MAG-19 754–7 Mulder G B J, ten Kate H H J, Krooshoop H J G and van de Klundert L J M 1988 On the inductive method for maximum current testing of superconducting cables Proc. MT-11 (Tsukuba, 1988) pp 479–84 Purcell J R and DesPortes H 1973 Short sample testing of very high current superconductors Rev. Sci. Instrum. 44 295–7 Schmidt C 1983 Critical current, stability and AC-loss measurement on the Euratom LCT conductor IEEE Trans. Magn. MAG-19 707–10 Schmidt C 1984 Stability tests on the Euratom LCT conductor Cryogenics 24 653–6 Schmidt C 1988 Stability of poloidal field coil conductors: test facility and subcable results Proc. ICEC 12 (London: Butterworths) pp 794–7 Shirshov L S and Enderlin G 1985 Apparatus for critical current measurement of high current superconductors Cryogenics 25 527–9 ten Kate H H J, Nederpelt W, Juffermans P, van Overbeke F and van de Klundert L J M 1986 A new type of superconducting direct current meter for 25 kA Adv. Cryogen. Eng. 31 1309–13 ten Kate H H J, Uytterwaal W, ten Haken B and van de Klundert L J M 1988 The Twente high-current conductor test facility, first results on critical current and propagation in two cables Adv. Cryogen. Eng. 33 211–8 Thome R J 1994 Design and development of the ITER magnet system Cryogenics 34 ICEC Supplement 39–46 Warnes W H and Larbalestier D C 1986 Critical current distribution in superconducting composites Cryogenics 26 643–53

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B7.5 A.c. losses in superconducting wires and cables I Hlasnik,M Majoros and L Jansak

B7.5.1 Introduction A.c. losses in superconducting multifilamentary wires and cables are of two types—electromagnetic and mechanical. Electromagnetic a.c. losses are divided into three components:

(i) hysteresis losses arising due to currents flowing only in superconducting filaments; (ii) coupling current losses arising in normal-metal matrix and filaments due to currents between superconducting filaments in a composite or between composites or subcables in a cable; (iii) eddy current losses due to Foucault’s currents in the normal metals. As described in preceding chapters of this book (see, e.g., B4.1 and B4.2) in the frame of the critical state model (CSM) and at transport current smaller than the critical current Ic the electromagnetic losses arise only due to the time-varying external magnetic field Ba and/or the time-varying transport current I. Their amplitude depends much more strongly on the size, form and structure of superconductors as well as on the amplitude, frequency and orientation of B and I with respect to each other and to the characteristic directions of the sample than in normal metals. Nevertheless it is to be noted that in the frame of the CSM electromagnetic losses arise in a superconductor also in the d.c. regime if I > Ic i.e. when the electric field E0 ≠ 0. The material characteristics giving the relation between the electric field E and the current density J in superconductors, called the E–J characteristics, are strongly nonlinear. Usually they are expressed as E = E0( J/J0 )n where E0 is the electric field at current density J0 and n is the characteristic of the superconductor. The behaviour of superconductors with n smaller than about ten deviates noticeably from that predicted by the CSM and with decreasing n it approaches that of a normal metal. Actually for a normal metal n = 1. The exponent n < 10 occurs in some high-Tc superconductors as well as in some multifilamentary composites with irregularly deformed filaments (sausaging) of low-Tc superconductors. Sometimes this should be taken into account in the study of a.c. losses in superconducting wires and cables. There are two types of mechanical loss: (i)

mechanical losses due to the external friction on the surface of the conductor caused by its movement against the frictional force; (ii) mechanical losses due to the internal friction inside the conductor material caused by its deformation. Both the electromagnetic and mechanical a.c. losses have played and are still playing very important roles in the design, development and application of superconductors. This is due to the fact that they strongly influence the behaviour of superconducting devices and also represent a crucial parameter in the economical feasibility of superconductor a.c. applications especially at industrial frequencies of 50–60 Hz.

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They depend on many internal as well as external parameters of the wires and cables used as well as on the winding structure. Therefore the measurements of losses are very important from the point of view of research and development work, as well as the manufacturing and application of superconductors. Accordingly they can be classified into two groups: (i) loss measurements with the aim of acquiring the basic characteristics of superconducting materials; (ii) loss measurements in specific forms of models or products of superconducting devices to characterize their technical and/or economical parameters. This section is divided into three subsections describing calorimetric, electromagnetic and mechanical methods for measuring losses according to the type of measured quantity used for their determination. Each section is subdivided according to the principle and measurement techniques used as well as depending on whether the origin of a.c. losses is a.c. external magnetic field Ha , transport current I or both. Circuit diagrams and the accuracy and sensitivity of realized apparatus are also presented. In practice their choice depends on the type and level of a.c. losses, the sample form and size, the use of measurement results and at last but not least on the disposable measurement equipment. B7.5.2 Calorimetric methods and techniques Calorimetric methods are the most direct methods noted for measuring total a.c. losses. They offer from medium to very high sensitivity and good precision for both small samples and superconducting apparatus and devices. They can be classified on the basis of an energy balance equation which relates loss energy W and power P dissipated in the sample with thermal quantities and time

where ∆t is the time interval during which the losses are dissipated, V the sample volume, p the volumetric density of a.c. loss power, C the volumetric specific heat, T the temperature, S the sample surface and k the sample thermal conductivity near its surface. The last term in equation (B7.5.1) is generally described as the heat transferred to the coolant. When the sample is immersed in a cryogenic fluid at constant pressure and perfectly thermally isolated from the exterior, this term can be expressed by the amount of evaporation heat

where h is the heat transfer coefficient from the sample surface to the cryogenic medium, δT = TS — Tm is the difference between the sample surface temperature TS and that of the cooling cryogenic medium Tm . Vl (Tl , pl ), L l (Tl , pl ) and Vg(Tg , pg ), Lg(Tg , pg ) are the volume and latent heat of evaporation of the liquid or that of the gas at temperatures Tl , Tg and/or pressure pl , pg respectively. There are three types of calorimetric method depending on the value of terms on the right-hand side of equation (B7.5.1), namely (i) isothermal, (ii) adiabatic and (iii) semi-adiabatic calorimetry. B7.5.2.1 Method and techniques of isothermal calorimetry This method corresponds to the regime when ∂T/∂t = 0 i.e. when W is given by equation (B7.5.1a). Then the loss power is given as P = (∂Vl /∂t )Ll or as P = (∂Vg /∂t )Lg where Ll and Lg are the latent heats of evaporation per unit volume of the cryogenic liquid or per unit volume of the evaporated gas at

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given temperature and pressure respectively. Accordingly the loss power P can be measured either by measuring the evaporation rate of the cryogenic liquid ∂Vl /∂t or by measuring the gas flow rate ∂Vg /∂t of the evaporated cryogenic gas respectively. The evaporation has to occur at a constant temperature, i.e. at constant pressure in a cryostat or in a thermally isolated calorimeter. Also the gas flow rate has to be measured at constant temperature as well as at constant pressure. The stationary regime defined by these conditions is established only after a time interval of the sample excitation ∆t > τt h where τt h , is the thermal equilibrium time constant of the sample in the cryostat or of the gas flow through the flowmeter, whichever is longer. A schematic illustration of a calorimeter for isothermal calorimetry is shown in figure B7.5.1. Liquid helium in the cryostat is separated into two regions by the sample housing. The amount of liquid in the housing is replenished through the bottom of the container. A sample exposed to a time-varying external magnetic field and/or fed by an a.c. transport current is positioned in the sample housing which is immersed in the liquid helium. As a result of the different evaporation rates and different hydraulic resistances in the cryostat and in the housing a pressure difference can occur in them resulting in a difference in the He levels in them. This should be avoided as the same helium vapour pressure in the cryostat and in the housing reduces the heat flux between the housing and the cryostat through its walls or through the thermal conductivity of current leads and the heat dissipated in them.

Figure B7.5.1. A schematic illustration of an isothermal calorimeter.

The evaporation rate ∂Vl /∂t is determined from the measurement of the position of the cryogenic liquid level in the cryostat or the calorimeter while the gas flow rate ∂Vg /∂t is measured by different types of flowmeter. Table B7.5.1 gives the values of ∂Vg /∂t for different cryogenic fluids at normal atmospheric pressure and T = 0 °C for P = 1 mW. There are several methods of measuring the evaporation rate of cryogenic liquids using continuousreading level indicators or the cryogenic gas flow rate by either mass flowmeters or semiconducting temperature-sensitive elements as gas flowmeters (see also chapter E2). For 4He as cryogenic fluid the sensitivity of the method using commercial continuous-reading level meters or mass flowmeters is within 0.1 to 1 mW with measurement errors of about ±10% at a level of 5 mW or 5% at a level of 0.5 W (Kovachev 1991). The sensitivity can be further increased by using semiconducting temperature-sensitive resistive elements as gas flowmeters. Their principle is based on the fact the heat transfer coefficient from the surface of the temperature-sensitive resistive element depends on the pressure and velocity of the gas

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Table B7.5.1. Evaporation rates at normal boiling point of cryogenic fluids corresponding to a dissipation of 1mW.

in which it is placed. If such an element is transporting a constant current I its temperature T, i.e. also its resistance R or the voltage drop U across it at constant pressure, will be a function of the gas flow rate V• g. Having a calibrated curve V• g—U and measuring U one can determine V• g . With such gas flowmeters it is possible to achieve higher sensitivity by about one order of magnitude than with metallic sensors of He gas flow from tungsten or platinum. The main source of error and the factor limiting the sensitivity of this method is the background boil-off which is due to the parasitic heat influx through the cryostat walls, current leads and sample suspension as well as to its variation in time. To reduce this factor to a minimum, elements of as low as possible heat conductivity and cross-section (thin-walled stainless steel tubes and capillaries, thin nylon fibres, etc) are used for calorimeter mountings. To combat this error the loss power corresponding to measured V• l or V• g is calibrated after each measuring point by measuring the electric power dissipated in the electric heater (see figure B7.5.1) and producing the same V• l or V• g . Obviously, the method of isothermal calorimetry is very simple and reliable. Nevertheless it has some disadvantages compared with electrical methods. Some of them are its low sensitivity and precision as well as its discrete variable temperature intervals corresponding to the boiling temperature intervals of different cryogenic liquids like helium, hydrogen, neon and nitrogen. As seen from table B7.5.1, the evaporation rates of other cryogenic liquids are much smaller than that of 4He. This means that the sensitivity of the method of isothermal calorimetry utilizing liquids with boiling points higher than about 5.2 K will be much lower. The long thermal equilibrium time constant τt h and long pulse train needed to measure one experimental point represents another drawback of this method. B7.5.2.2 Method and techniques of adiabatic calorimetry This method corresponds to the regime in which the heat evacuated through the sample surface is negligible compared with that absorbed by the heat capacity of the sample, i.e. when the second term on the right-hand side of equation (B7.5.1) is much smaller than the first one. It is based on measuring the temperature increase ∆T of a sample perfectly thermally isolated from the surrounding medium after being exposed to an a.c. magnetic field and/or to a current pulse of duration ∆tp . According to equation (B7.5.1) the a.c. loss energy dissipated in the sample is given as W = CV∆T and a.c. loss power as P = W/∆tp , where CV is the heat capacity of the sample. The measurement of the temperature increase ∆T should be performed at time tm when the temperature distribution in the sample after the field and/or the current pulse is already uniform. Nevertheless the time interval ∆tm between the pulse end and the temperature measurement time tm should be small enough to eliminate any heat evacuation from the sample to the surrounding medium. It means the condition τi n << ∆tm << τe x should be fulfilled where τi n is the time constant of the thermal equilibrium process in the sample and τe x is that of the thermal equilibrium process between the surrounding medium and the sample. The maximum temperature rise ∆T during the measurement should be sufficiently low to limit the error from the change of sample thermal and electric parameters due to

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their temperature dependence. A schematic illustration of the adiabatic calorimeter described by Buchhold and Molenda (1962) is shown in figure B7.5.2. The sample glued to a resistor thermometer holder is fixed in a nylon container which isolates it from the helium bath and fixes its position in the superconducting solenoid. The calibration of the apparatus can be made in situ by measuring the power P = W/∆tp of an energy pulse W delivered to the electric heater tightly wound on the sample, the pulse being of the same duration ∆tp and leading to the same temperature increase ∆T as those occurring during the measurement. Therefore neither the thermal capacity of the sample nor the thermometer need be measured or calibrated separately. The sensitivity of this method is quite high (of the order of 10– 4 W m– 2 ). The main disadvantages of this method are the changing temperature of the sample during the measurement, the long thermal equilibrium time constant after the measurement of each point and the long time required for sample installation which makes the method rather slow.

Figure B7.5.2. A schematic illustration of an adiabatic calorimeter.

B7.5.2.3 Method and techniques of semi-adiabatic calorimetry This method is very similar to the methods used by isothermal as well as adiabatic calorimetry because it works in a stationary thermal regime characterized by ∂T/∂t = 0 but differs from them in measuring the slight temperature increase ∆T of the sample due to a.c. losses and a finite interchangeable thermal resistance Rt >> 1/hS through which the sample is coupled to the cryogenic liquid bath. The calorimeter and the sample fixation techniques are very similar to those of adiabatic calorimetry. A schematic illustration of the semi-adiabatic calorimeter described by Schmidt (1985) is shown in figure B7.5.3. The a.c. loss power is determined as P = ∆T/Rt . The thermal resistance Rt given in kelvin per watt is determined by calibration using an electric heater in a similar way to the description in section B7.5.2.2. The interchangeable thermal resistance Rt allows us to change the range of a.c. loss measurement according to the experimental needs. This method allows us to increase the sensitivity compared with that of the method of isothermal calorimetry and to reduce the thermal time constant between two measuring points compared with that occurring in adiabatic calorimetry. The maximum temperature rise ∆T during the measurement should be sufficiently low to limit the error from the change of sample thermal and electric parameters due to their temperature dependence.

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Figure B7.5.3. The principle of the method of semi-adiabatic calorimetry.

Using an arrangement described by Schmidt and Specht (1990) a resolution down to 10– 8 W has been attained. The main drawback of this method is a rather lengthy process of sample preparatory work. This can be eliminated but at the cost of lowered sensitivity with a resolution of about 5 µW being given for a new modified technique described by Schmidt (1994) which replaced the vacuum vessel by a Teflon cylinder. Quite good accuracy of better than 2.5% has been achieved. B7.5.3 Electromagnetic methods and techniques Electromagnetic methods and techniques can be classified into two groups: (i) a.c. loss measurements with the aim of acquiring the basic a.c. loss characteristics of superconducting materials and (ii) a.c. loss measurements in specific forms of conductors or models of superconducting devices to characterize their technical and/or economical parameters. In the first group of a.c. loss measurements the question is to determine the relation between the local loss power or loss energy density at as accurately defined as possible electric and magnetic field distributions and material characteristics. Electromagnetic methods consist in measuring electromagnetic quantities such as the local macroscopic field strength and induction of the electric and magnetic fields E , H and B , respectively, the macroscopic current density J , current I , voltage U , total magnetic moment m , magnetization M and magnetic flux Φ and in processing measured data according to the theoretical relations in order to obtain either a.c. loss power P and loss energy W or local loss power density p and loss energy density w. According to macroscopic electromagnetic theory the energy flux density per m2 and second in the electromagnetic field is given by the Poynting vector Π = E × H. Its dimension is [Π] = W m– 2. Accordingly the a.c. electromagnetic loss energy W dissipated in the volume V confined by the surface S and containing a superconducting sample during one period T of a periodic external a.c. magnetic field Ha and/or the current I can be determined as the time integral through the period T of the influx of the Poynting vector Π through the surface S. Using some formulae of vector analysis and Maxwell’s equations (see e.g. Stratton 1941) W can be expressed as

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with n the unit vector in the direction of the normal to and oriented outside the surface S. The last term in equation (B7.5.2) results from the fact that in superconductors and a normal-metal matrix at low frequencies ( f < 1 MHz) J » ∂D/∂t. Equation (B7.5.2) indicates that for determining the total loss energy W dissipated in the sample per cycle the time integral through one period T either of the surface integral of E × H through the surface S containing the sample or of the volume integral of E ⋅ J + H ⋅ ∂B/∂t through the sample volume has to be measured or calculated. It is to be noted that at low frequencies E can be locally measured practically only on the sample surface while the measurement of H can be performed only outside the sample. In the general case the measurement of the vector Es on the sample surface requires the measurement of its two noncollinear, e.g. perpendicular, components. This needs two pairs of contacts as shown in figure B7.5.4(a). The lead from one voltage tap of each pair, A or C for example, is carried along the shortest path to the other voltage tap of the pair, B or D, respectively, and is situated in close contact on the sample surface. After that the two potential leads of each pair have to be twisted.

Figure B7.5.4. (a) A schematic illustration of the position of voltage taps for surface electric field measurement. (b) A schematic illustration of Hall probes and of pickup coils for surface magnetic field measurement.

If the potential leads A—A′ and C—C′ are infinitely thin then the induced voltages Ui A′ B′ = −∂ΦA′ABB′ /∂t and Ui C′ D′ = —∂ΦC′ C D D′ /∂t are practically zero. Therefore ES would be given as ES = ES a + ES c = a 0 lima → 0(a–1UA′ B′ ) + c 0 limc → 0(c–1UC′ D′ ), a 0 and c 0 being the unit vectors in the directions of vectors a and c , respectively. The corresponding components HS a and HS c of the magnetic field on the surface S parallel to ES a and ES c respectively, can be measured for example either by Hall probes or by small pickup coils the centres of which are near the cross-point G in figure B7.5.4(b) and the dimensions of which in the direction perpendicular to the surface are sufficiently small. Sometimes the magnetic constant km of the magnet can be used to express the surface field as BS = Ba = km I. Local values of H , B , E and J in the volume of the sample cannot be measured directly. Nevertheless they can be calculated in the frame of some models such as the critical state model, flux creep model, etc. In these models necessary material characteristics relating J with E , B and T as well as B with H allow us to solve Maxwell’s equations in the sample for given boundary conditions. The determination of a.c. losses, i.e. of the surface or volume integrals mentioned in equation (B7.5.2), is greatly simplified if at least one quantity entering their integrands, e.g. E , H or J , is constant on the surface S or in the sample volume V. In such cases the constant quantity can be put in front of the integral and only the integral of the other quantity is to be determined. Owing to Maxwell’s equations the latter

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integrals sometimes can be measured directly. Using Faraday’s or Ampère’s law for example, the path integrals of E or H along a closed curve s are equal to the induced voltage Ui = −∂φ/∂t or to the current I, respectively, both of which can be measured directly, Φ and I being the magnetic flux or the current coupled by the curve s. As an example of the use of such integral quantities for the determination of the total loss energy per cycle W it can be mentioned that the measurement of losses dissipated in a coil or in a winding transporting the current I supplied by a power source with output voltage Ua gives

where E is the electric field on the curve s in the sample between two points A and B near the power supply terminals, ds is the elementary length of this curve and Φ is the magnetic flux coupled by this curve s and by a straight line joining the points A and B outside the sample, Ua r is the resistive and L dI/dt the inductive voltage across the sample, L being the self-inductance of the sample. The voltage Ua and the current I can be measured directly by standard methods. Another such example is the measurement of the total loss energy per cycle, W, dissipated in a sample with total magnetic moment m in a homogeneous external magnetic field Ha when

In most practical cases the superconductors are working in the regime of a special skin effect where neither the magnetic field nor the current penetrate into the whole volume of the sample. According to Bean’s critical state model, which postulates that current density in the superconductor can be zero or equal to the critical current density Jc , this occurs, for example, when Ha is parallel to the sample surface and smaller than the penetration field or when I is smaller than the critical current. Such an uncompleted penetration of the external magnetic field Ha and/or of the transport current I depends strongly on the form, size and structure of the sample as well as on the orientation and amplitude of the external magnetic field and current. Often it results in an inhomogeneous distribution of H , E and J which complicates the measurement of a.c. losses. Only a few special configurations of the field, sample form and of the choice of surface S enclosing the sample exist, allowing easy and sufficiently precise measurement of a.c. losses. To such configurations belong: (i)

samples of ellipsoidal form and long straight samples or samples of small dimensions placed in homogeneous magnetic field Ha ; (ii) long straight circular and elliptical cylinders as well as coils or a winding fed by a.c. current I from a power supply with output voltage Ua ; (iii) some sample forms defined in (i) or (ii) placed simultaneously in homogeneous external magnetic field Ha and transporting a.c. current I .

In the following electromagnetic methods and techniques using such samples and field configurations are described first of all. For some sample forms and field configurations differing from the above mentioned ones the methods of calculating or measuring the calibration factor kc between a.c. losses and the measured output quantity are presented too. Generally in the processing of measured quantities two mathematical operations, namely the multiplication and the integration in space and/or in time, have to be performed. In principle this can be done using analogue or digital processing techniques. As the level of a.c. losses in superconductors is very low, signals corresponding to measured quantities are extremely low. Moreover, the signal-to-noise ratio is often much smaller than unity.

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All this requires special measuring circuits and very sensitive techniques for measuring a.c. losses in superconducting wires and cables. A typical problem encountered during electromagnetic loss measurements in superconductors is the presence of spurious voltage signals induced by the electromagnetic induction due to external as well as internal magnetic fields. Actually in superconductors the loss energy per cycle W is in general much smaller than the maximum total energy Wm accumulated in the conducting electrons and in the magnetic field during the cycle. Therefore the signals proportional to the loss power which has to be measured are often very much smaller, even by several orders of magnitude, than those proportional to the reversible part of the electromagnetic energy H∂B/∂t (see equation (B7.5.2)) or L dI/dT (see equation (B7.5.3)). These large interfering terms have two negative effects: (i) they overload the input amplifier, the subsequent multiplier or integrator so that a small loss signal cannot be measured correctly and (ii) even if this overload does not occur, any small amplitude and/or phase errors in these operational units could cause a big error in measuring the loss. Therefore for the sake of measurement precision and sensitivity the spurious signals should be compensated before or during the processing of measured data. This can be done by using compensation coils placed in the external magnetic field or by putting a linear mutual inductance into the current leads of the magnet or of the sample transporting current and by subtracting the signals proportional to dB/dt or dI/dt so obtained from the corresponding measured U or E, respectively. This is of primary importance especially in the measurement of a.c. losses due to strong external magnetic field (Ha >> Hp , Hp being the field of full penetration of the sample) or in superconducting coils and short samples with L dI/dt >> Ua r (Ua r being the loss component of the measured voltage). Other ways of fighting against the spurious signals in a.c. loss measurement at harmonic external field or transport current is the use of the phase-sensitive detector with lock-in amplifier and tracking bandpass filter and/or of the isolating transformer reducing the common mode rejection error. In the following the electromagnetic methods and techniques for measuring a.c. losses in superconducting wires and cables are classified into three main groups depending on the type of the source of measured a.c. losses which are: (i) a.c. external magnetic field Ha (ii) a.c. transport current I or (iii) both a.c. Ha and I. Further subdivision of these groups is made following the form of the sample, measurement configuration and technique used. B7.5.3.1 Methods and techniques for measuring magnetization a.c. losses due to a periodic external magnetic field Ha As mentioned above the sample geometry and structure have an important influence on the absolute value of a.c. losses in superconductors as well as on the choice of appropriate measurement technique. As for the sample geometry it is well known that samples of ellipsoidal geometry made from homogeneous and isotropic magnetic material placed in an external homogeneous magnetic field Ha give rise to a constant internal magnetic field Hi , magnetic induction B i and magnetization M which are given as

where M = χ H i is the magnetization in the sample corresponding to the resulting internal field Hi , Hd = −N ⋅ M is the demagnetizing field, I is the identity tensor and N the demagnetization tensor (see e.g. Stratton 1941). The demagnetizing field Hd changes the field in the sample from Ha into Hi , due to the constraints resulting from the boundary conditions on the sample surface. The latter are Hi t = He x t and B i n = Be x n , where Hi t and He x t or Bi n and Be x n represent the surface tangential or normal components of H or B

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inside or outside the sample respectively. They can be rewritten as Hd t = HM t and Mn + Hd n = HM n , where HM is the magnetic field outside the sample due to M . If the three principal ellipsoid axes coincide with the x, y and z axes of the coordinate system, respectively, N has the form

with Nx + Ny + Nz = 1 and Nn being the demagnetization factor corresponding to the nth principal axis. Nn is always smaller than unity. For an infinitely long circular cylinder the demagnetization factor corresponding to the cylinder axis z is Nz = 0 and those corresponding to x and y axes are Nx = Ny = 0.5. For a sphere Nn = 13 for all three axes. If Ha is along the z principal axis of the ellipsoid, internal magnetic field Hi , internal induction B i and magnetization M are constant and parallel to Ha . They are given as

It is to be pointed out that M is 1/(1 – Nz ) times higher than (B i — Ba )/µ0. This is due to the fact that H i is different from Ha by the demagnetizing field Hd . This is very important from the point of view of the measurement of magnetization loss density as will be shown later. As a result of the symmetry of the ellipsoid, the sample electric field on the sample surface in the midplane is parallel to both the sample surface and the midplane and perpendicular to Hi . From the above expressions according to equation (B7.5.2) it follows that the a.c. loss energy density per cycle wH a in the x y midplane can be expressed as

where all quantities correspond to the cross-section of the sample through the midplane and where Sx y is the sample cross-section, dS the elementary area on the surface S enclosing the elementary volume Sx y dz, E × H and n the Poynting’s vector and the unit vector on the surface S enclosing the elementary volume Sx y dz, Hi = He x = Ha — Nz M the tangential components of H on the sample surface inside or outside the sample, in the x y midplane, respectively, ds the elementary length of the perimeter of the sample cross-section, U1 = – ∂φ/∂t = —Sx y ∂Bi /∂t is the voltage induced per turn in a pickup coil tightly wound on the sample surface and φ = Sx y Bi the magnetic flux coupled by the sample cross-section area. In derivating the different expressions in equation (B7.5.5e) integration by parts and the fact that the integral over one period of any quantity X by dX vanishes, were used. Internal magnetic induction Bi can be experimentally determined by integrating in time the induced voltage U1 as Bi = - S-1xy ∫ T0 U1 dt

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As Bi and Hi are constant in the ellipsoid sample, expression (B7.5.5e) gives the energy loss density per cycle in all the sample. The longer is the major axis az , the smaller is Nz (see e.g. Stratton 1941). For an ellipsoid with an infinitely long major axis az the demagnetizing factor Nz = 0, i.e. Ha = Hi = He x and Bi = (H i + M ). This is also true for an infinitely long circular cylinder or a long sample with constant cross-section Ss in planes perpendicular to its long axis and placed in a longitudinal Ha . We shall call such samples long straight samples. In this case ωHa, can be obtained from equation (B7.5.5e) substituting Nz = 0

Equation (B7.5.5e) serves as the basis for wH a measurement in ellipsoidal samples while equation (B7.5.5f) does the same for long straight samples placed in a homogeneous longitudinal a.c. magnetic field Ha . It is said that wH a can be measured either as the time integral through the period T of the product of U1 and Hi or Ha divided by the sample cross-section area Sx y , respectively, or as the area of the magnetization hysteresis loop –µ0M versus Ha , with all quantities related to the x y midplane. The methods using this principle are called fluxmetric methods as they are related to the magnetic flux coupled by the sample in its midplane. They are used also for the measurement of wH a in the midplanes of samples of forms other than ellipsoids or long straight samples, e.g. circular cylinders with a finite length-to-diameter ratio γ or different open-ended coils in which both Hd and M are space dependent. For such loss measurements Hd and M are characterized by their average values 〈Hd〉 and 〈M〉 across the midplane cross-section of the sample by calculating their so-called fluxmetric demagnetizing factor Nf = −∫s Hd dS / ∫s M dS = −〈Hd〉 / 〈M〉. For Nf values of finite length cylinders, see e.g. Chen et al (1991) and for different open-ended coils and coupling current losses see e.g. Bruzzone and Kwasnitza (1987). Actually in this case according to equation (B7.5.5e) the measured mean value of 〈M〉 can be determined from

and hence the loss wH a in the midplane can be calculated too. The values of Nf for cylinders with the ratio of length to diameter γ ≥ 10 and for susceptibilities −1 ≤ χ ≤ 1 with χ =dM/dH, Nf ≤ 0.005 (see Chen et al 1991), which is true for the majority of practical samples made of superconducting wires and cables. Therefore in such cases the influence of Nf on a.c. loss measurement can be neglected. However, when γ ≈ 1 and −1 ≤ χ ≤ 1 ≈ 0.23 and the effect of the demagnetizing field Hd should be considered when calculating the calibration constant. There are still other groups of measurement methods for determining WH a or wH a . Usually they are classified in the following groups: magnetometric methods, force methods and measurement techniques using transient phenomena. Magnetometric methods are based on measuring quantities related to the magnetic field HM outside the sample which is due to the magnetization M in the sample volume. For a given distribution of M in the sample volume, HM can be calculated as the volume integral through the sample volume V of the field due to elementary magnetic dipoles M dV, i.e. as (see e.g. Stratton 1941).

where r0 is the unit vector of the radius vector r of the point at which the field is calculated with regard to the centre of the elementary volume dV with magnetization M .

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Force methods are based on measuring quantities related to the mechanical force Fm or torque Tq acting on the sample with magnetic moment m and are given as

Measurement techniques using transient phenomena are based on determining characteristic time constants θs of coupling currents in multifilamentary composites and cables, or decay rates of the proper oscillations in electric LC resonance circuits or in mechanical torsion systems containing a sample in an a.c. magnetic field. There exists another method to calculate or model the magnetic field HM of a known distribution of the sample magnetization. It consists in replacing the sample magnetization M either by an equivalent current distribution within the sample volume and/or on its surface with volume current density JM V and superficial current density JM S or by an equivalent distribution of the ‘magnetic charges’ with volume charge density qMV and/or surface charge density qMS which are given as (see e.g. Stratton 1941)

or

respectively, where n is the normal unit vector to the sample surface oriented outside of the sample. Using equations (B7.5.8) or (B7.5.9) Hd can be calculated numerically using the Biot—Savart law or Coulomb’s law, respectively, if an approximation of the distribution of M is assumed

or

where r0 and r are the unit vector and modulus of the radius vector r of the point at which Hd due to M is calculated with regard to the point at which the elementary volume dV or surface area dS are considered respectively. If the distribution of M is known it is possible to use this model to determine Hd experimentally as the field of a coil has the same form as the sample and the current density given by equations (B7.5.8). Also the flux due to HM coupled by a turn can be determined from the mutual inductance measured between the model coil and the turn. Samples of superconducting wires and cables differ in two points from ellipsoidal or long straight magnetic samples. (i)

Losses in them are due to two types of macroscopic currents instead of microsopic ones in magnetic materials. Macroscopic currents induced in superconducting filaments during the viscous flow of fluxoids give rise to hysteresis losses while the coupling currents induced by changing magnetic field and flowing in the normal-metal matrix between the filaments in composites or between composites or subcables in cables lead to coupling current losses. In a low field, the hysteresis losses in filaments are influenced by the field due to magnetization in other filaments, i.e. by the demagnetization field Hd f

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of the filament array. This problem has been solved by Zenkevitch and Romayuk (1979, 1980a, b). However, at Ha >> Hp (Hp is the field of full penetration, see section B4.1), Hd f due to M in filaments is negligible. (ii) As a rule these samples have the form of open-ended coils or stacks of several pieces of individual conductors or cables (see figure B7.5.5) in which the resulting field is not homogeneous and can rarely be calculated analytically. Coupling currents in a matrix partially screen the interior of the wire itself but influence the field in other composites. This problem of collective interaction was treated by Bruzzone and Kwasnitza (1987), Campbell (1982), Sumiyoshi et al (1980) and Zenkevitch and Romanyuk (1979, 1980a, b). The latter authors have calculated fluxmetric demagnetizing factors for different forms of open-ended coils. The field in the interior of composites can be strongly influenced by Mc c , the magnetization due to coupling currents.

Figure B7.5.5. Standard forms of samples made of superconducting wires and cables: (a) open-ended coil; (b) stack.

Both these losses depend on many factors such as type of conductor (monofilamentary or multifilamentary wire, cable), filament diameter, matrix-to-superconductor ratio, matrix and superconductor material characteristics, pitch length and direction of the twist of strands, subcables and cables and the size, form and structure of samples as well as on the amplitude, frequency and direction of Ha with regard to significant sample directions. Detailed analysis of their influence on both types of loss was performed, for example, by Bruzzone and Kwasnitza (1987), Campbell (1982), Carr (1983), Ries (1977) and Zenkevitch and Romayuk (1979, 1980a, b). The characteristic parameter for hysteresis losses is the penetration field Bp = µ0d Jc /π where d is the filament diameter and Jc the critical current density while that for coupling currents losses is the ωθ0 product where ω is the angular frequency of Ba and θ0 = (lp / 2π)2 µ0/ρc c the characteristic time constant of the sample when the filaments are fully penetrated, i.e. B > Bp which occurs when µ ≈ µ0 , and lp and ρc c are the twist pitch length and effective transverse resistivity of the composite respectively. From the point of view of a.c. loss measurement, cylindrical open-ended coils with their axes parallel to Ha and the axis z of the coordinate system seem to be more advantageous than stacks. This is because the effects due to the ends of cut wires or cables in stacks are eliminated and the rotational symmetry of coils makes the electric lines of force outside the sample almost concentric circles with E practically constant on them. Moreover, the magnetic field in the vicinity of the midplane has mainly an Hz component. Therefore, here the Poynting’s vector almost has only a radial component. The energy

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loss density per sample unit volume in the central part of the sample during a cycle can be expressed as

where V1 is the sample volume per unit sample length in m2, Hs is the mean value of the Hz component in the volume of the pickup coil wound around the sample in the midplane and U1 is the voltage per turn of this pickup coil. Hs can be determined by integrating over time the differential voltage induced in two concentric coils connected in series opposition and wound with the same number of turns and coil length on mandrels with outer diameters near to the inner and outer diameters of the pickup coils respectively. In some cylindrical open-ended coils made of single composite or of cable with one cable stage, Nf can be calculated numerically as a fluxmetric demagnetization factor of the cylinder circumscribed around the sample, see e.g. Bruzzone and Kwasnitza (1987) or Chen et al (1991). Nf can also be determined experimentally by replacing the sample by two model coils with mandrel diameters equal to the inner and outer diameters of the sample, respectively, and with ampere turns per metre equal to JMS according to equation (B7.5.8b). Special attention is to be paid to the cooling conditions of the sample winding when a.c. losses are too high. In this case cooling channels are usually used in the sample as well as in the sample holder, in order to ameliorate the heat transfer from the sample to the cooling medium and to keep the sample temperature constant. The following four subsections describe the principles, circuit diagrams and techniques belonging to the above-mentioned four groups of fluxmetric, magnetometric and force methods as well as measurement techniques using transient phenomena for measuring a.c. losses in superconductors due to an a.c. homogeneous external magnetic field. (a) Fluxmetric methods to measure the Poynting’s vector influx In the following, three very sensitive a.c. loss measurement systems based on equations (B7.5.5e), (B7.5.5f) and (B7.5.5g ) will be described. According to figure B7.5.6 to increase the sensitivity and precision of the measurement the inductive component Up i of the voltage induced in the pickup coil (1) and proportional to dBa /dt is compensated by the voltage Uc from the compensation coil (2) which is connected in series opposition with the coil (1). Coil (2) is placed at a point where the magnetic field due to magnetization M of the sample S is negligible. These systems differ in the circuits for adjusting the

Figure B7.5.6. A block diagram of the magnetization hysteresis loop measurement technique using an electronic integrator: SM—solenoid; PS—power supply; Rs h —shunt resistor; I—solenoid current; S—sample; 1—pick-up coil; 2—compensation coil; R, R1 , R2—compensation circuit potentiometers; INT—integrator; XY—x—y recorder.

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compensation of the voltage Up i by Uc but also by the main instruments processing the differential voltage Ud = Up — Uc of the two coils after the compensation is adjusted. These instruments are an electronic integrator and a x—y plotter, a phase-sensitive detector or an electronic wattmeter. (i) Magnetization hysteresis loop measurement using an electronic integrator In ellipsoidal as well as in long straight samples with the z principal axis parallel to Ha according to equations (B7.5.5e), (B7.5.5f) and (B7.5.5g), wH a can be measured as the surface area of the magnetization hysteresis loop wH a = ∫ c y c l e − µ0M dHa = ∫ c y c l e −[(Bi−Ba )/(1−Nz )]dHa where Nz is the demagnetizing factor corresponding to the direction z . For a long straight sample Nz = 0. The principal scheme of the measuring circuit as described by Fietz (1965) is represented in figure B7.5.6. It consists of the pickup coil (1) and of the compensation coil (2) which are located in the working space of a solenoid (SM) with their axes parallel to Ha . The coil (1) embraces tightly the central part of the sample placed in a homogeneous magnetic field. The coil (2) is far enough from the sample so that the magnetic field due to its magnetization is negligible there. The coils are connected in series opposition and designed so that their differential voltage Ud in the absence of the sample is zero during the cycle, i.e.

where Up , Uc , N1 , N2 , S1 , S2 , and Ba , Ba 2 are the voltages, number of turns, cross-sectional areas and magnetic fields in coils 1 and 2 respectively. The resistances R, R1 and R2 serve to adjust precisely the circuit if the compensation condition (B7.5.11) is not fulfilled exactly. Their values have to fulfil the condition R1 + R2 ≈ R assuring the same phase shift of currents in both pickup and compensating coils. Moreover, they should be sufficiently high to reduce this phase shift to a negligible level (R >> ωL1 and R1 + R2 >> ωL2, L1, L1, being the self-inductances of the pickup and compensation coil respectively). Then in the presence of the sample the differential voltage Ud = N1S1 ∂( Bi — Ba )/∂t = N1S1(1 – Nz )µ0 ∂M/∂t. After its integration by analogue electronic integrator INT the output voltage U0 is

where τ is the time constant of the integrator. Applying the voltage Us h , = Rs h I, = Rs h Ba /km across the shunt resistor Rs h connected into the power supply circuit of the solenoid and the output voltage U0 to the input of the x and y channels of the x–y recorder XY, respectively, one obtains the magnetization loop M–Ba of the sample. According to equations (B7.5.5e) and (B7.5.5g) a.c. loss energy wH a per unit volume of the sample and per cycle is proportional to the hysteresis loop area AM − B , i.e.

where Km = Ba /I is the magnetic constant of the solenoid, AM− B is the area of the sample magnetization loop in m2 and kx and ky are the sensitivities of the x and y channels of the analogue recorder XY in mV–1. If a transient recorder is used instead of the analogue recorder kx = ky = 1 and the hysteresis loop area is in V 2. It is to be noted that for a long straight sample, i.e. if the z semi-axis of the sample is longer than approximately five times those in the x y plane, Nz ≈ 0. If the sample is not ellipsoidal and/or not long enough or the pickup coil is not sufficiently thin and tightly wound on the sample surface the voltage Ud and consequently also the measured wH a will be smaller than the theoretical values given by equation (B7.5.13). Therefore the latter must be multiplied by a correction factor kc which must be determined experimentally. This can be done using a physical model based on equations (B7.5.8a) and (B7.5.8b) replacing the sample magnetization by the corresponding current density distribution. Assuming

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M to be constant in the sample and parallel to its axis the model consists of two coils wound of thin Cu wire on two coaxial mandrels the outer surfaces of which are identical with the inner and the outer surface of the modelled sample. The coils are connected in series opposition and supplied by a current I to simulate a surface current density JM S = ±M = ±n1I1 where n1 is the number of turns per unit length of each model coil and I1 is the current in them. The factor kc can then be determined as

where Uo c is the voltage measured at the output of the integrator INT when the model coils substituting M by ampere turns n1I1 are present in the pickup coil and fed by the current I1 and Uo m is that voltage measured when the sample is present on the pickup coil. In order to measure a.c. losses of samples with large diameters, comparable with the diameter of the inner bore of the solenoid, the empty compensation coil is positioned under or above the pickup coil containing the sample which is usually placed in the centre of the solenoid. The distance between the coils should be as large as possible to reduce the coupling by the magnetic field due to the sample magnetization. The influence of the nonideal capacitance in the feedback of the integrator on its work is treated by Fietz (1965). Because of the drift of the integrator due to its nonzero offset the static magnetization characteristic cannot be measured by this method. The lower frequency range is given by the condition that the time integral of the offset voltage Uo f f at the input of the integrator during the period T is always very much lower than the integral of the differential voltage Ud , i.e.

The fulfillment of this condition is checked before recording the hysteresis loop by drift compensation using a variable d.c. voltage at the input of the integrator until the hysteresis loop is closed. The upper frequency range is determined mainly by the frequency which can be sustained by the superconducting solenoid and the x—y recorder. Actually produced low-loss multifilamentary composites allow us to work at frequencies around the power frequency 50–60 Hz and at field amplitudes Bm > 2 T (Hlasnik 1984). To lower the power supply voltage for superconducting solenoids a series resonance capacitance bank circuit is put into the magnet current leads. Using a special configuration of two a.c. and one d.c. superconducting solenoids it is possible to measure a.c. losses at different bias d.c. magnetic field levels (de Reuver et al 1985, Polak et al 1995). It is to be noted that at low external magnetic field amplitudes (Ha m << Hp ) M is relatively high (M ~ _ —Ha ) but the width of the hysteresis loop, i.e. the irreversible part of the magnetization Mi r is still relatively very small (Mi r « M) as the losses are still small. In this case the method of compensation characterized by equation (B7.5.11) is not suitable. In fact as M >> Mi r , the amplification factor during the recording of the hysteresis loop is limited by Mm a x rather than by Mi r m a x because the signal proportional to M is much larger than that due to Mi r . Actually the voltage U2 from the compensation coil (2) would be much higher than U1 from the pickup coil (1) so that the differential voltage Ud would also be much higher than U1 Therefore in this case it is more convenient to use equation (B7.5.5e) in the form

i.e. to integrate the smaller voltage U1 after a tiny compensation for the leakage flux between the pickup coil (1) and the sample only in order to obtain the hysteresis curve in a horizontal position. This allows us to use smaller τ, i.e. higher amplification during the integration. This means that it is more convenient

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to measure the hysteresis curve Bi /(1 – Nz )–Ha instead of the µ0M–Ha curve. This can be achieved by proper adjustment of resistances R1 and R2, see figure B7.5.6. The possibility of measuring at temperatures higher than the liquid-helium temperature depends mainly on the level of disturbing effects of eddy currents induced in the metallic parts of the variable temperature insert used. Instead of electronic analogue integration of the compensated voltage and successive plotting as well as measuring the area of the magnetization hysteresis loop, the signals can be digitally processed. When the sample losses are high either method will serve; however, analogue integration is better for low-loss samples. (ii) Magnetization a.c. loss measurement using a phase-sensitive detector From expression (B7.5.5 f) it is seen that wH a can be determined by measuring the root mean square (rms) values of Ha and of that component U1r of the voltage U1 induced in one turn of the pickup coil which is in phase with Ha . Figure B7.5.7 shows a block diagram similar to that published by Kovachev (1991) of the compensation and processing circuits for measuring U1r using a phase-sensitive lock-in detector. The partially compensated signal from the pickup coil (1) and compensation coil (2) as well as the signal from an additional compensation coil (3) are amplified separately by preamplifiers A2 and A3 respectively. The final compensation of the inductive component of the pickup coil voltage is observed visually on the oscilloscope OSC. It is carried out by selecting the amplification gains G2 and G3, of these preamplifiers, respectively, so as to prevent the saturation of the amplifier A2 and simultaneously to minimize the sum of their output signals. These output voltages enter two independent channels of the differential input of the phase-sensitive detector of a lock-in amplifier. The amplification factor of the latter is GL. The signal Us h from the shunt resistor Rs h proportional to the magnet current I and amplified in the preamplifier Al with gain G1 is used as the reference signal of the phase-sensitive detector. The output voltage of the lockin amplifier U0 is then proportional to the loss component Up r of the pickup coil voltage which is in phase with Ha . According to equation (B7.5.5f) the energy loss per sample unit volume and cycle is then given as

where T is the period of Ha , ϕ is the phase shift between Ha and U1, Ss is the cross-sectional area of the

Figure B7.5.7. A block diagram of the a.c. magnetization loss measurement technique using a phase-sensitive detector: SM—solenoid; PS—power supply; Rs h —shunt resistor; I—solenoid current; S—sample; 1—pickup coil; 2— compensation coil; 3—additional compensation coil; A1, A2, A3—amplifiers; OSC—oscilloscope; lock-in—phasesensitive detector of a lock-in amplifier; REF—reference voltage; Ud —amplified differential voltage; U0 —output voltage.

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sample in the plane perpendicular to Ba and N1 the turn number of the pick-up coil 1. Values of Ha and U1 as well as U0 and Us h are rms values and km = Ba /I is the magnetic constant of the magnet. From equation (B7.5.16) it follows that in order to measure wH a multiplication of U0 and Us h , still has to be performed. This method for a.c. loss measurement requires accurate phase adjustment. Therefore the selfinductance Ls h , of the shunt resistor Rs h has to fulfil the condition Ls h << Rs h T/2π. Instead of Us h a signal from a Rogowski coil embracing the current lead to the solenoid can be used for 90° phase adjustment of the reference signal. By shifting the phase of the reference voltage of the lock-in amplifier by–90° with regard to the current I of the magnet, i.e. to Us h , the output voltage of the lock-in amplifier allows us to. determine the inductive component U1 sin ϕ by this measuring system. Using these two various working modes of the lock-in amplifier, the real and imaginary parts of the a.c. susceptibility can be measured by this technique (Gomory and Cesnak 1991). In the case of symmetric magnetization characteristics M (H ) loops can be measured too (Gömöry 1991). These two working modes can be realized simultaneously by the use of a two-channel lock-in amplifier. This second variant of this measurement system is often used for a.c. loss studies in high-temperature superconductor samples. (iii) Magnetization a.c. loss measurement using an electronic wattmeter The electronic wattmeter is a complex instrument which accomplishes all operations described in the preceding set-up, i.e. it amplifies, compensates, integrates, multiplies and averages the received signals for the determination of the average loss power. The block diagram of the measurement system given by Kovachev (1991) is illustrated in figure B7.5.8. It is more or less a combination of those in figures B7.5.6 and B7.5.7. Voltages Up and Uc of the pickup and compensation coils (1 and 2) are amplified by the amplifiers A1 and A2 with gains G1 and G2 , respectively, both being about ten. The dividing ratio of the potentiometer RD is automatically controlled by the automatic control circuit ACC. The balanced differential voltage U1r enters the amplification cascade A3 with the amplification gain G3. The voltage Uc from the compensation coil (2) amplified in the amplifier A2 enters the input of the integrator INT. There it is integrated and amplified by a gain Gi = 1/τ , where τ is the time constant of the integrator feedback circuit, to give a signal proportional to Ha . Its output voltage UI N T together with the output voltage UA3 of the cascade amplifier A3 are multiplied in the four-quadrant

Figure B7.5.8. A block diagram of the a.c. magnetization loss measurement technique using an electronic wattmeter: SM—solenoid; PS—power supply; Rs h —shunt resistor; DVM—digital voltmeter; I—solenoid current; S—sample; 1— pickup coil; 2—compensation coil; A1, A2, A3—amplifiers; ACC—automatic control circuit; RD—potentiometer; OSC—oscilloscope; INT—integrator; MLT—four-quadrant multiplier; DSP—digital display.

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multiplier MLT. Its output voltage UM L T is displayed on the digital display DSP as an rms value. Using this value the loss power density per cycle can be expressed as

where K14 = UA3UI N T /UM L T is the constant of the multiplier MLT in Vr m s , N1S1 and N2S2 are the products of the number of turns and the cross-sectional areas of the coils 1 and 2, respectively, and G1, G2, G4 are the gain factors of the amplifiers A1, A2 and A3 respectively. A very important part of the electronic wattmeter is the four-quadrant multiplier which gives the correct amplitude as well as sign of the output voltage according to the product of the input signals. In this way, when the adjustment of the wattmeter is correct, the quality of the compensation does not significantly affect the performance of the device. Nevertheless as high as possible complete compensation monitored visually on the oscilloscope (OSC) is necessary to secure the amplifier performance in the linear region and to avoid saturation of the wattmeter electronics by parasitic inductive signals. The error of the electronic wattmeter in loss measurements is due to the spurious signals from the instrument leads, eddy currents in the pickup coils, slight position dependence of the magnetic field phase inside the working space of the solenoid and relative phase shifts in the elements of the wattmeter. The errors associated with the phase shifts in the amplifier cascade and in the multiplier of the wattmeter are negligible (less than 1%). All the elements of the wattmeter are matched and adjusted in such a way that the total sum of the instrument phase shift represents a very small value (of the order of 10-5 rad). The total error of the wattmeter described by Kovachev (1991) was experimentally estimated to be smaller than 5%. To reduce possible sources of error in analogue circuitry a numerical processing of the loss signals is possible (Reilly and Morgan 1992). (b) Magnetometric methods for hysteresis loop measurements In this section four magnetometric methods of recording magnetization hysteresis loops based on the measurement of the magnetic field BM outside the sample due to magnetization M are presented. The magnetometers used are vibration magnetometers, Hall probes, superconducting quantum interference device (SQUID) magnetometers and extraction magnetometers. In all four methods the relation between measured quantities and the magnetization in the sample or its total magnetic moment have to be calibrated experimentally or calculated following the procedures described in section B7.5.3.1. Finally the a.c. losses are determined from the magnetization hysteresis loop area. (i) The vibrating-sample magnetometer technique The principal scheme of the vibrating-sample magnetometer is shown in figure B7.5.9(a). The principle of this method consists in measuring the magnetic moment m of the sample (S) positioned in a homogeneous harmonic a.c. external magnetic field Ba of frequency fa and according to equation (B7.5.4) to plot the hysteresis curve –µ0m versus Ha for the determination of W as ∫c y c l e −m ⋅ dBa . To measure m sample S oscillates harmonically with small deflection ∆z(t) parallel to Ba at frequency f >> fa . The mechanical oscillations are generated by the transducer Tr, usually a loudspeaker cone or a mechanical can, fed by audio generator AG and coupled with the sample S and coil 2 using a stainless steel tube. In two equal stationary pickup coils (1) with axes parallel to the z axis connected in series opposition, the magnetic field BM is proportional to the sample magnetic moment m and will have an average oscillating z component 〈Bz (t )〉 = ±[∂(〈B M z (t )〉)/∂z ]∆z(t ) = ±kc mz (t )∆z(t ) where 〈BM z 〉 is the average z component of BM through the volume of coils 1 at ∆z = 0 and kc is the proportionality factor between ∂(〈BM Z (t )〉)/∂z and mz(t) which is the z component of m (t). Voltage U1 induced in coils 1 is

where Np and Sp are the number of turns and effective area of each pickup coil (1), respectively, and Kc = 2NpSpkc . From (B7.5.18) it is seen that U1 has two components. One of them is proportional to

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Figure B7.5.9. A block diagram of the a.c. magnetization hysteresis loop measurement technique using a vibratingsample magnetometer: (a) SM—solenoid; PS—power supply; Rs h —shunt resistor; I—solenoid current; S—sample; 1—pickup coil; 2—compensation coil; 3—coils generating inhomogeneous magnetic field; Tr—loudspeaker cone; AG—audio generator; R—potentiometer; AMP—amplifier; PSD—phase-sensitive detector; XY—x-y recorder, (b) P—pole pieces of the magnet; S—sample; C1–sample pick-up coils; Ref—reference sample; C2—reference sample pickup coils; Rp—potentiometer; M—measuring apparatus.

[∂mz(t )/∂t ]∆z(t ) and the other one to mz(t)[∂∆z(t )/∂t ]. As the aim of the measurement is to determine mz , the second voltage component is to be detected by a phase-sensitive detector (PSD) controlled by a voltage UAG proportional to ∂∆z(t )/∂t, i.e. to the current IT r in the transducer Tr delivered by audiogenerator AG. However, to increase the sensitivity of the measurement especially at low fields (Ha ≤ Hp ) it is desirable to compensate a part of the voltage U1 induced in the pickup coil (1) which is proportional to the reversible part of M or mz , i.e. to —Ha because it does not contribute to the loss signal but it could saturate the amplifier AMP. This can be done by subtracting a fraction r of the voltage U2 induced in coil 2, oscillating in phase with the sample, and placed in the inhomogeneous magnetic field generated by coils 3 and proportional to Ba . These coils are mutually connected in series opposition and supplied by the same current I as the magnet SM. The potentiometer R serves for adjusting the necessary fraction r of the voltage U2. The condition 2π f L2 << R and 2π f L1 << (1 - r)R + Ra i , where Ra i is the input resistance of the amplifier AMP, has to be fulfilled to limit the phase shift error in the compensating circuit for the measurement error. L1 and L2 are the self-inductances of coils 1 and 2 respectively. After the amplification of the differential voltage Ud by amplifier AMP with a gain GA , its output voltage UA together with a reference voltage UA G are connected to the inputs of the phase-sensitive detector PSD with the amplification gain KP S D . The detected signal Um = KCGAKP S D mz(t) = Kmmz(t) together with the voltage Us h from the shunt resistor Rs h are recorded by the x-y recorder XY to obtain the mz versus Ba hysteresis loop. The total

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loss energy per cycle

can be measured either as the ratio of U1/[ ∂∆z(t )/∂t ] at constant mz(t) of a standard sample or of a model coil made according to equation (B7.5.8), or by calculation or measurement of U1/[ ∂mz(t)/∂t ] at constant ∆z(t) ≠ 0 and mz(t ) modelled either by a standard sample with known mz[Ba(t)] or by a coil according to equation (B7.5.8). The latter case is equivalent to the measurement of a mutual inductance between coils 1 and the model coil because U1 = –M dI/dt, where I is the current of the model coil according to equation (B7.5.8). The standard samples as well as the model coils must have the same form and size as well as the same position with regard to the coils 1 as the measured sample. It is obvious that for cylindrical samples Km will depend on the length-to-diameter ratio γ of the sample as well as on the distance d of the central planes of coils 1 from the end planes of the sample at ∆z = 0. According to Chen et al (1991) the magnetometric demagnetizing factor Nm of a cylindrical sample decreases more slowly with increasing γ than the fluxmetric demagnetizing factor Nf does. Therefore in these measurements γ should be higher than 50 in order to reduce the error in Km below 1% by neglecting the influence of Nm . The frequency and amplitude of mechanical oscillations are about 100 Hz and 0.1 mm when a loudspeaker cone or about 20 Hz and 2 mm when a mechanical can are used as generators respectively. If the loudspeaker cone is used the sample mass should be less than 100 mg. An alternative version of a vibrating-sample magnetometer is shown in figure B7.5.9(b) (Graham 1992). The signal from the pickup coils C1 around the sample S is balanced against a signal from pickup coils C2 around reference sample Ref of fixed magnetic moment. If the samples and pickup coils have the same geometry, then the magnetic moment of the measured sample ms = r mR e f where r is the dividing ratio of the potentiometer Rp when the current flowing through indicator M is zero. Here also the resistance of the potentiometer Rp should be much higher than the impedance 2π f L2 of the reference pickup coil C2, L2 being its self-inductance. Vibrating-sample magnetometers are commercially available with accessories to permit measurements in the temperature range from 2 to 1000 K, and are fully automated. Their limiting sensitivity is usually quoted as 10-8 A m2 (10-5 emu). (ii) Extraction magnetometer The basic principle of the extraction magnetometer is that a sample with magnetic moment m is moved through a pair of pickup coils, wound as a gradient pair, from below one coil position P0 , to just above the other one, position Pf (see figure B7.5.10). The waveform of the voltage U induced in the pickup coils is captured on a fast digital voltmeter V. This is then downloaded to the computer where a curve is fitted to the data and then integrated to give the sample moment. A drive unit is used to move the sample rapidly through both coils. This procedure is repeated for different values of Ba to obtain the hysteresis curve m versus Ba . According to equation (B7.5.4) the total loss per cycle is numerically calculated as W = ∫c y c l e −m ⋅ dBa . Extraction magnetometers are commercially produced as fully automated devices with the use of variable temperature inserts. When used with high-field superconducting magnets they allow the measurement of hysteresis M (H) curves and of a.c. losses too, in magnetic fields of 10-20 T. The estimated sensitivity is typically 10-7 A m2 (10- 4 emu). (iii) Measurement techniques using Hall probes The principle of this method proposed by Krempasky (1976) and Krempasky et al (1979) consists in measuring directly the magnetic field BM due to the sample magnetization M at a point A outside the sample by two Hall probes instead of integrating the differential voltage induced in two searching coils due to ∂φ/∂t. The schematic configuration of the measurement assembly is shown in figure B7.5.11.

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Figure B7.5.10. A block diagram of a static magnetization hysteresis loop measurement using an extraction magnetometer: Ba —external magnetic field; S—sample; GPPC—gradient pair of pickup coils; P0 , Pf —initial and final sample position respectively; V—fast digital voltmeter; data acquisition system.

Figure B7.5.11. A block diagram of the magnetization hysteresis loop measurement technique using Hall probes: SM—solenoid; PS1—power supply of the solenoid; Rs h —shunt resistor; I—solenoid current; S—sample in the form of open-ended coils; HPA—Hall probe at position A; HPB—Hall probe of compensation at position B; PS2 and PS3—power supplies for Hall probes HPA and HPB respectively; ∆UH M —the difference between the output voltages of both Hall probes proportional to BM due to the magnetization in the sample.

The sample S, in the form of an open-ended coil, is placed in the working space of the superconducting solenoid SM with the homogeneous field Ba parallel to the z axis. The measurement of BM is performed using two linear Hall probes HPA, HPB placed at points A and B, respectively, with planes perpendicular to Ba . The point A is chosen in the centre of the sample so that it has BM parallel to Ba . However, point B is at some distance away from the sample where B is almost equal to Ba . In general the Hall voltage is given as

where SH and IH are the sensitivity and the current of the Hall probe and Bn is the magnetic flux density component perpendicular to the Hall probe plane respectively. The Hall probes are supplied from

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separate current sources and their Hall voltages are connected in series opposition. Therefore the currents in Hall probes can be set up so that in the absence of the sample the Hall voltages totally compensate each other. Then from the differential Hall voltage ∆UH M measured in the presence of the sample one obtains BM = ∆UH M /SH IH . Krempasky (1976) and Krempasky et al (1979) analytically calculated the proportionality factor K0 = BM/M using equation (B7.5.6). Putting ∆UH M to the input of the y channel and the voltage drop UR s h = RS h I = Rs h Ba /km to the x channel of the x-y recorder XY and using K0 as well as the magnetic constant km = Ba /I of the solenoid the plot of the hysteresis loop −µ0M−Ha can be obtained. According to equation (B7.5.5e) wH a is then given as

where AM⎯B is the area of the plotted hysteresis loop and kx , ky are the sensitivities of the x and y channels expressed in metres squared and metres per volt respectively. When M is supposed to be constant in all the volume of the sample and parallel with Ba then for the open-ended coil with inner diameter 2a1 outer diameter 2a2 and length 2b shown in figure B7.5.11. K0 = µ0β [(α 2 + β 2 )−1/2 − (1 + β 2 )−1/2 ] where α = α2 /α1 , β = b/a1(see Krempasky 1976). The constant K0 for the magnetic field in the air gap between two cubic stack samples is given by Krempasky et al (1979). However, the assumption of a space-independent M parallel to Ha is true only if the internal magnetic field Hi = Ha - Hd = Ha - NM is constant and parallel to Ha . From these expressions for Hi it is seen that the assumption concerning M can be fulfilled if either Hd = - NM is constant and parallel with Ha , which is the case for isotropic ellipsoidal samples, or when the demagnetizing field Hd = - NM << Ha . This condition can be fulfilled either by a small demagnetizing factor N << 1 which is the case for long samples in a longitudinal field or when magnetization fulfils the condition M << Ha . The latter condition is often fulfilled in superconducting wires and cables at slowly varying Ha when M is due mainly to the magnetization of superconducting filaments, i.e. when M ≤ 2Hp << Ha (Hp being the field of full penetration of the filaments) or at moderate Ha (when the main contribution to the magnetization is due to coupling currents which, however, do not saturate the filaments) when Mc c = θs ∂Ha /∂t << Ha (θs being the time constant of the coupling currents; see section B7.5.3.1(c)(i). Conditions N << 1 and M << Ha also enlarge the applicability region of this method for nonellipsoidal samples. The advantage of this method is that the application of an integrator which decreases the measurement accuracy and sensitivity because of its drift is not necessary. So this method enables us to measure static magnetization characteristics too and a.c. losses for very low frequencies up to power frequencies. This measuring technique can also be used for measuring the magnetization of anisotropic material. (iv) Magnetic dipole moment measurement with a SQUID magnetometer There exists another configuration which allows us to measure a.c. losses due to Ha by magnetic dipole moment hysteresis loop measurements using a SQUID magnetometer (see, e.g. Libbrecht et al 1994). This method is appropriate for small samples when the maximum sample dimension d << 2RH a « 2LH a with 2RH a and 2LH a being the diameter and the length of the cylindrical working volume V with highly homogeneous external magnetic field Ha parallel to the axis z. For measuring the magnetic flux coupled by one turn on the surface of the working volume due to m of the sample as a function of z, a SQUID coupled to an appropriate input flux transformer offers unsurpassed sensitivity (see figure B7.5.12 and chapter I2 of this book). The commercially available SQUID magnetometers utilize superconducting detection loops composed as a highly balanced secondderivative coil set, designed to reject the flux of the uniform field from the superconducting magnet. These coils are wound in such a way that the planes of two outer detection loops are placed at a distance ±A from the centre of the magnet where two oppositely wound loops are situated (see figure B7.5.12).

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Figure B7.5.12. Measuring set-up using a SQUID magnetometer to measure the total magnetic moment m of the sample.

During the measurement at a given value of Ha when the sample is moved through this set of pickup coils the magnetic moment m of the sample induces currents in the (superconducting) detection loops. The current passes through an inductance L, generating a magnetic flux which is fed to the SQUID, resulting in an output voltage V in the electronic tank circuit, which depends on the position z of the sample (figure B7.5.12). This response V(z) is used to fit the response of an ideal magnetic point dipole, described by the so-called response function. The latter allows one to calculate the magnetic moment of the sample. The response function is given by

where z is the distance of the dipole with respect to the centre of the magnet, where the central pickup loops are located, and R H a is the radius of the coils of the pickup loop. During the measurement a constant background together with a drift, linear in z , are superimposed on this response. The actual signal is therefore described by

where a and bz denote the background and drift, respectively, and c the amplitude of the response function. The parameter d was introduced to correct for eventual mispositioning of the sample with respect to the centre of the magnet. By fitting fA(z) to the measured V(z) the software of the magnetometer generates values for a,b,c and d. The parameter c is directly proportional to the magnetic moment of the sample. The commercially available SQUID magnetometers enable us to detect magnetic moments as small as 10-12 A m2 (10-9 emu) in very stable d.c. fields as high as 9 T (the superconducting magnet works in persistent mode) over the temperature range from 2 K up to about 1000 K. The total loss per cycle is numerically calculated as W = ∫c y c l e −m ⋅ dBa . Two mutually perpendicular second-derivative pickup-coil sets enable the measurement of both magnetic moment components mz and m⊥ which are parallel or perpendicular to the applied magnetic field Ba respectively. Together with the possibility of rotation of the sample it enables the measurement of m for anisotropic materials too. Mounting three mutually perpendicular second-derivative pickup coils enables the measurement of all three components mx , my and mz of the magnetic moment. The disadvantage of the method from the point of view of magnetization measurements of wires and cables is the possibility of measuring only hysteresis a.c. losses in samples of quite small volume (about

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≈100 mm3 ). The advantage of the method is its high sensitivity which allows the measurement of a.c. losses in individual filaments of superconducting wires and cables. (c) Measurement techniques using transient phenomena In this section three a.c. loss measurement methods using transient regimes of magnetization are described. They are the pickup coil technique, the LC resonance circuit method and the torsion resonance oscillating system method. The first one concerns the measurement of coupling current losses in composites and, with some limitation, in transposed cylindrical cables placed in an external field Ha . It consists in measuring the time constant of coupling currents induced between superconducting filaments in the normal-metal matrix of a single multifilamentary composite and between strands or subcables of transposed cylindrical cables with a low number of cabling stages made from such composites. The method is used mainly in research and development work on multifilamentary composites and cables for applications with periodically changing magnetic fields, e.g. for particle accelerator magnets and a.c. 50-60 Hz superconducting machines and devices or for those with a single strong magnetic field pulse such as for a tokamak’s poloidal and toroidal windings. The second method concerns the measurement of total a.c. losses in superconducting samples of arbitrary geometry placed in an a.c. harmonic magnetic field. It consists in measuring the difference of the decay in time of electric oscillations arising in an electric resonance LC circuit with a superconducting coil in the regimes when a superconducting sample is placed in the superconducting coil or when the coil is without the sample (see Ishigohka et al 1994). This method allows us to study the total a.c. loss power or energy in samples of irregular form placed in a harmonic external a.c. magnetic field with exponentially time decreasing amplitude. The third, very sensitive method is suitable for a.c. loss measurement in small superconducting samples exposed to a rotating or oscillating magnetic field. It is based on measuring the torque acting on a superconducting sample placed in a rotating magnetic field or on measuring the difference between the decay in time of the deflection angle amplitude of a mechanical resonance oscillation system consisting of a thin elastic thread on one end of which a superconducting sample is suspended and the other end of which is fixed to a shaft through which a torque can be transmitted to the thread. (i) Pickup coil technique for the measurement of coupling-current time constants Analytical calculations of coupling-current losses in cylindrical as well as rectangular composites and cables exposed to a time-varying transverse homogeneous external magnetic field Ba have been performed, for example, by Bruzzone and Kwasnitza (1987), Campbell (1982), Krempasky and Schmidt (1994a,b), Ries (1977), Zenkevitch et al (1979, 1980a,b) and Turck et al (1982). Their results are collected in section B4.3 of this book. The main conclusion of the calculation constituting the basis for this method is the fact that the coupling-current loss energy density per cycle wc c is a function of the time constant of coupling currents θs and of the amplitude and time dependence of the external magnetic field. In some cases measurement of θs can be used for determining coupling-current losses. In general the measurement of this time constant allows us to determine wc c much more easily than using other direct methods for a.c. loss measurement. As θs is influenced by the structure of the composite and/or of the cable as well as by the form of the sample (single composite, stack of short pieces of composites or cables, open-ended coils, etc), it is often used for comparing the influence of different conductor parameters on the level of coupling-current losses in them as well as to separate these losses from other components of total a.c. losses. The calculation of wc c is often made under two simplifying assumptions. The first one supposes stationary conditions, i.e. the time derivative of the internal magnetic field ∂Bi /∂t inside the composite is space independent. This assumption is justified if the period T of Ba is much longer than θs . The second assumption supposes that coupling currents do not saturate the superconducting filaments. In this case the

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following differential equation for the internal magnetic field Bi in the composite is obtained

According to a known formula from the theory of differential equations its general solution is

where the coupling-current time constant

with N the fluxmetric demagnetizing factor of the sample and θ0 the basic time constant of the composite

where lp is the twist pitch length, µ and ρt the effective transverse permeability and resistivity of the filamentary zone located in the central part of the composite up to the radius Rf , ρn the resistivity of the outer normal-metal shell, ρc c = 1/ρt + e/ρnRf the effective resistivity of the total composite with apparent radius Rf and e the thickness of the outer shell. Coil sides a and b perpendicular and parallel to the coil axis as well as distances dr and dz between the axes of two neighbouring composites in the radial or axial directions, respectively, and the filamentary zone radius Rf are indicated in figure B7.5.5. Expressions of N for different rectangular open-ended coils with their axes parallel to Ba are given, for example, by Bruzzone and Kwasnitza (1987) as

for a round composite or for a square coil

for a long single-layered coil

for a wide pancake coil

for a coil with a/b < 1 and

for a coil with a/b > 1. N for the infinite single-layered coil was calculated for the limit a/b → 0 and that for the wide pancake coil for the limit a/b → ∞ respectively.

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For stacks in the first approximation N can be calculated from expressions for coils supposing that field Ba is perpendicular to the side a and parallel to the side b , see figure B7.5.5(b). For other sample forms N should be determined by numerical calculation or experimentally using standard samples or models according to equation (B7.5.8) or equation (B7.5.9). According to section B7.5.3.1, equation (B7.5.5c) and equation (B7.5.23) the reaction field of the coupling currents ∆Bi can also be expressed as

Then according to equation (B7.5.5e) the coupling-current energy loss per cycle and filametary zone unit volume is

The corresponding expressions for wc c of a single round or rectangular composite or cylindrical cable are given, for example, in section B4.3. As an example the expressions for Bi and wc c for a harmonic transverse external magnetic field Ba = B0 sin(ω t) and no saturation of filaments are given here as

and

with ϕ = tan−1 (ωθs ). From equation (B7.5.31) it follows that in this case the frequency dependence of wc c is a function of two nondimensional variables ωθ0 and ωθs with demagnetizing factor N as a parameter. The coupling-current time constant θs = θ0(1 – N) is a characteristic which reflects both the basic factors of the sample θ0 and N. This function displays a linear section for all types of coil until ωθs = ωθ0(1 – N ) is smaller than about 0.2, then at ωθs = ωθ0(1 – N) = 1 it attains a maximum and then decreases monotonically with ωθs . Figure B7.5.13 shows two sets of the functions wc c /πB02/µ for three coils tightly wound from the same composite but with different demagnetizing factors N and plotted as functions of ωθs or ωθ0 . The curves with N equal to 0.1, 0.5 and 0.8 correspond to a tightly wound single-layered coil, to a square coil and to a pancake coil, respectively, axes of all of them being parallel with Ba . From this figure it is seen that the coupling-current losses wc c in samples made from the same composite, i.e. with the same time constant θ0 depend very strongly on the sample form through the demagnetization factor N. For a given frequency they increase with increasing N. From figure B7.5.13 it is seen also how the choice of the independent variables ωθs or ωθ0 influences the form of the dependence of wc c on frequency ω and N. The simplest experimental technique for measuring θs consists in recording by an oscilloscope or a transient data recorder the voltage Ui induced in a pickup coil tightly wound around the sample after a pulse of external magnetic field Ba , see figure B7.5.14. The sample should have a form with a defined

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Figure B7.5.13. The dependence of coupling-current losses in relative units wc c /πB02/µ for three different sample forms: an infinite single-layered coil with demagnetizing factor N = 0.1; a square coil or single cylindrical composite with N = 0.5; a wide pancake coil with N = 0.8 with different dimensionless parameters: (a) ωθs , (b) ωθ0 .

Figure B7.5.14. The experimental set-up for the measurement of coupling-current time constants.

demagnetizing factor, usually an open-ended coil or stack of short pieces of composites or cables. If at t > t0 Ba = 0 then according to equation (B7.5.24) Bi = Bi (t0)exp[–(t — t0 )/θs ] independently of the form of the field pulse and the voltage induced in the pickup coil at t > t0 is

where Ss and n are the cross-sectional area of the sample and the number of turns of the pickup coil respectively. This means that θs can be simply measured as the slope of the 1n Ui (t) curve. To increase the sensitivity a compensation coil with the same product of cross-section and turn number as that of the pickup coil can be used. If it is connected in series opposition with the pickup coil and placed in the same external magnetic field as the sample but far from the sample to exclude the magnetic field due to

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coupling currents it will reduce parasitic signals induced during the field pulse and by spurious magnetic fields. Moreover, it is to be noted that the differential signal for t > t0, i.e. also the sensitivity of the measurement, will be maximum at a given amplitude of Ba if Ti n /θs >> 1 and Td /θs << 1, where Ti n and Td are the time duration of the increasing and decreasing branches of the field pulse respectively. The higher Ti n /θs is the higher will be the flux penetration into the sample during the increase of the field; the smaller Td /θs is the higher will be the induced coupling currents, i.e. the higher will be Bi (t0 ). Recently a theoretical as well as experimental study of the diffusion of the magnetic field in a slab of homogeneous normal metal of thickness 2w and resistivity ρ was performed as described by Krempasky and Schmidt (1994a, b). The theoretical solution was found on the basis of Maxwell’s equations and took the boundary conditions strictly into account. The voltage Ui s induced in the pickup coil after the pulse of the magnetic field was found in the form

where v = 1, 3, 5,…, ∞, Bv(t0 ) are the coefficients of the Fourier series of Bi (t0, x) = –ΣBv (t0)sin(vπx/2w) and τv = (2w/v π)2µ0/ρ. Comparing this expression for Ui s with that obtained for a round composite under the assumption of space independent B• i in all the volume of the sample given by equation (B7.5.32) one sees that the form of their dependence in time differs mainly for time near t0. Actually for t > t0 , Ui given by equation (B7.5.32) for a composite sample is represented only by one exponential term characterized by the time constant θs = (1 – N)θ0 = (1 – N )(lp /2π)2µ/ρc c , while Ui s from equation (B7.5.33) is represented by a series of exponential terms with time constants τv = ( 2w/v π)2µ0/ρ where v = 1, 3, 5,…, ∞. However, for t - t0 > τ1, Ui s also approaches one term with the time constant

In the article by Krempasky and Schmidt (1994a) the experimental records of Ui s for a Cu slab as well as for a stack of Nb-Ti/Cu strands exhibit a form corresponding qualitatively to the theory. In the case of the copper slab a very good quantitative agreement between the theory and experiment has also been found. However, in the case of the stack of multifilamentary composite the calculated values of Ui s were by a factor of about 1.8 higher than the measured ones. Since for the slab the demagnetization factor N = 0, the time constant τ1 = (2w/π)2µ0/ρ in Ui s should correspond to θ0 = (lp /2π)2µ0/ρc c rather than to θs in Ui . If θ0 is compared to τ1 one can conclude that the coupling-current time constant θ0 of a twisted multifilamentary composite is the same as τ1 for a slab of thickness 2w = lp /2, permeability µ and resistivity ρc c the values of which equal lp /2, µ and ρc c respectively. Therefore the following conclusions can be drawn from this discussion concerning the pickup coil technique for measuring coupling current losses: In general to apply this method either both time constants θ0 and θs = (1 – N)θ0 or one of them and the demagnetization factor N of the sample are to be known. (ii) θs can be determined from the measured signal of the compensated voltage ∆Ui but at t — t0 > θs

(i)

because ∆Ui for t — t0 < θs is characterized by a steep and successively decreasing slope in time and only at t — t0 ≥ θs does it acquire the form of a single exponential the logarithmic decrement of which corresponds to θs . (iii) θ0 = θs /( 1 – N ), which is a characteristic of the composite structure, can be determined from θs using N obtained either by calculation for the given sample form or by measuring θs on a sample with known N, e.g. an open-ended coil with a square cross-section, when N = 0.5. (iv) If the time Ti of the increase of the applied external magnetic field fulfils the condition Ti » θs i.e. 1 << ωaθs (ωa = 2π/4Ti is an apparent angular frequency), when really the internal magnetic field Bi

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can be considered as sufficiently homogeneous in all the volume of the sample and B• i ≈ B• a during the majority of the cycle the expression for coupling-current loss power density per unit volume of the filamentary zone and one cycle has a very simple form wc c = (θ0/µ0 ) ∫c y c l e (dBa /dt )2dt. For a harmonic external magnetic field this is true when ωθs < 0.2. This regime is the most favourable for the application of the time constant measurement to coupling-current a.c. loss determination in cylindrical cables. (v) As the coupling currents are strongly related to the periodic structure of filaments in the composite each disturbance of that periodicity, e.g. by an additional twisting at the edges of a flat cable (Hlasnik et al 1980), or due to the fact that the ratio of the sample length ls to the twist pitch length lp is not an integer (Ries and Takacs 1981) or due to the inhomogeneous distribution of the magnetic field along the sample (Takacs 1982) can substantially influence the magnitude of coupling-current losses in composites and cables. Therefore these effects should be taken into consideration at the sample preparation as well as at the interpretation of experimental results obtained for coupling-current time constant measurements by the pickup coil technique. (ii) Method using an LC resonance circuit with a superconducting coil The principle of this method consists in measuring the difference of the decay rate of the proper oscillations in a high-quality LC resonance circuit composed of a superconducting coil SM with selfinductance L and a capacitor with capacitance C when a superconducting sample S is present or absent in the coil, respectively, see Ishigohka et al (1994) and figure B7.5.15. Actually the total mean loss dissipated in the circuit during one period of oscillations, ∆Wl can be measured either as the change of the energy accumulated in the self-inductance after k consecutive current maxima In up to In +k , or as an analogous change of the energy accumulated in the capacitor at k consecutive voltage maxima Un and Un +k divided by k. As, in general, the self-inductance of a superconducting coil is a function of the current amplitude, it is more appropriate to use the mean loss per cycle by the change of the energy accumulated in the capacitor rather than that in the coil. So the sample mean loss per cycle can be expressed as

where the subscripts s or 0 denote the measured values when the sample was or was not present in the magnet respectively. To assure the same mean value of Ba during both measurements, the conditions Usn ~ _ U0n and Usn +ks ~ _ Uon +k0, i.e. the numbers ks and k0 should be different.

Figure B7.5.15. A block diagram of a.c. loss measurement using an LC resonance circuit with a superconducting solenoid: d.c. power supply—for charging capacitor C or the superconducting solenoid SM with the self-inductance L; S—sample; search coil—to measure the current I ; S1, S2—switches.

To initiate resonance oscillations in an LC resonance circuit, two different methods can be used. Either after having Charged the capacitor C with switch S2 open and switch S1 closed, switch S1 is opened and switch S2 is closed or after having closed both switches and established the current I through the magnet L , switch S1 is opened. The oscillations decay characteristics of the peak current are obtained by recording the induced voltage of the search coil or the voltage across the capacitor C. Changing the

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value of capacitance C or using the superconducting coils with different self-inductances L the resonance frequency of the resonance circuit ωr = (LC)−1/2(1−R2C/L)1/2 can be changed. This enables the measurement of the frequency dependence of a.c. losses in the sample. The smaller ∆W10 is the higher will be the sensitivity of this method. This is the reason for the requirement of high quality for the LC resonance circuit. (iii) Method using a torsion resonance oscillating system This is a very sensitive method for measurement of a.c. losses due to the action of a d.c. external magnetic field Ba on a cylindrical superconducting sample which oscillates around the sample axis perpendicular to the vector Ba . It is based on the measurement of the change of decay of oscillation amplitude in a mechanical resonance system containing a superconducting sample when the latter is or is not exposed to the external magnetic field. It is suitable for measuring a.c. losses in superconductors exposed to a transverse oscillating or rotating magnetic field. The measurement set-up used, e.g. by Andronikashvili et al (1969) and Chigvinadze (1973), is represented in figure B7.5.16. The sample in a cylindrical form is attached to a long glass rod which is suspended on the elastic thread in the dewar vessel. The other end of the thread is attached to the shaft which passes through the gasket in the top of the cryostat. The two ends of the thread can be fixed from the outside of the cryostat. Two mirrors 1 and 2 are attached to the sample and shaft ends of the thread respectively. The torsion angle of the thread ϕ can be measured from the light spot reading obtained from two mirrors by electronic photomultipliers The cryostat filled with liquid helium is placed between the pole pieces of an electromagnet. The sample can be driven into periodic oscillations by turning the shaft manually from its equilibrium position by an angle ϕ0 at a fixed position from the sample end of the thread. Then after fixing the new position of the shaft end of the thread its sample end is released to

Figure B7.5.16. A block diagram for a.c. loss measurement in an external oscillating magnetic field using a mechanical torsion resonance oscillating system consisting of a sample attached to a glass rod suspended on an elastic thread the upper end of which is attached to the shaft; mirror 1 and mirror 2 allow one to measure the deflection angle ϕ between the thread ends. From Andronikashvili et al (1969).

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start the torsion oscillations. Actually the total mean loss dissipated in the system during one period of oscillations ∆Wl= 0.5kt(ϕ 2n+k - ϕ2n)/k, where kt is the proportionality factor between the reaction torque of the elastic thread and its torsion angle ϕ and ϕn +k , ϕn , are the deflection maxima measured at the nth and (n + k)th oscillation period. So the sample mean loss per cycle can be expressed as ∆Wl s = ∆Wl B –∆W10 where ∆Wl B and ∆W10 are the total loss per cycle measured in the presence or in the absence of the magnetic field respectively. The number k of cycles should be selected in such a way as to obtain an easily measurable change in the deflection amplitudes ϕn −ϕn +k during the measurement. ∆Wl s can be measured as a function of Ba as well as of the frequency of oscillations. The latter can be varied by the change of the tension in the elastic thread, i.e. of the factor kt , or by the change of the inertial moment I of the system, respectively. Chigvinadze (1973) quotes a very high sensitivity of the apparatus of 10-17 W. (d) Mechanical force methods This section deals with techniques of a.c. loss measurement based on determining the sample magnetic moment m by measuring the mechanical force F acting on the sample due to the magnetic field Ba with F given by equation (B7.5.7a). According to equation (B7.5.4) using the dependence of m on Ba the total a.c. loss per cycle W = ∫c y c l e −m ⋅ dBa can then be determined using the magnetization hysteresis loop area Am−B . Three such methods called the balance method, the alternating gradient method and the cantilever magnetometer method are described here. (i) Balance method The principal scheme of the experimental set-up of this measurement technique is represented in figure B7.5.17. The sample S is suspended between tapered pole pieces P of an electromagnet where it is subjected to an average field H inducing a parallel magnetic moment m in the sample and having a known field gradient ∇Ba. The force F acting on the sample and given by equation (B7.5.7a) can be measured by a type of balance B. The magnetic moment m can be determined from measured force F if the field gradient is known. This method is variously known as the Faraday or the Curie method and the apparatus is often called a magnetic balance. It is capable of measuring small magnetic moments and, since the only connection to the sample is a support wire, the method is well suited to measurements at low temperatures. It is often used for a.c. loss measurement in high-temperature superconductors.

Figure B7.5.17. A block diagram of the a.c. magnetization loss measurement technique using mechanical force measurement with a field gradient produced by tapered pole pieces: P—tapered pole pieces of an electromagnet; S—sample; B—balance; Fm and Fg —magnetic and gravitational forces respectively.

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Using a balance with a sensitivity of 0.1 mg and a field gradient of 400 kA m-2 (typical for tapered pole pieces), the limiting sensitivity is about 10-6 A m2 (10-3 emu). Accuracy is limited by the fact that it is difficult to make the field gradient uniform over a large volume and constant over a range of fields and it is also difficult to measure the field gradient accurately. Therefore the apparatus and the sample form have to be calibrated. The system is usually calibrated by measuring a sample of definite form and known magnetic properties, frequently pure nickel. The problems are avoided if the sample is made in the form of a long cylinder or a thin strip extending from the point with maximum field to a region of essentially zero field. In this case (called the Gouy arrangement) the force on the sample does not depend on the shape of the field gradient. Actually if the sample and field are rotationally symmetric or the sample is thin the total force FM on the sample is given as FM = ∫S (MBS /2)dS where S is the surface of the sample at the end which is in the strongest field Bs assumed to be perpendicular to the sample surface and M is the magnetization corresponding to Bs . This means that the Gouy arrangement allows us to measure the M—B hysteresis loop. The field gradient in the arrangement of figure B7.5.17 vanishes as the field approaches zero, so this method is not useful for low-field measurements. If water-cooled or superconducting field gradient coils are used instead of tapered pole pieces, the field gradient becomes independent of the average field and it can be varied or reversed and the volume of uniform gradient can be made larger. Furthermore, the gradient and therefore the sensitivity can be increased by about a factor of ten. However, the gradient coils require an increase in the magnet gap and, consequently, reduce the maximum average field (Graham 1992). (ii) Alternating-gradient magnetometer If the gradient is produced by coils C in the magnet gap and is made a.c. instead of d.c., the sample is subjected to an alternating force F (see figure B7.5.18). The latter can be made either parallel or perpendicular to the d.c. field of the magnet depending on the orientation of the coil generating the field gradient grad Ba . The displacement caused by this force can be detected by mounting the sample S on a thin support arm r which is attached to a piezoelectric crystal Pz . The a.c. output voltage of the piezoelectric crystal is proportional to the torque r × F where r is the radius vector of the sample centre with regard to the basic plate V. It is detected and amplified by a lock-in amplifier. If the frequency of the gradient field is tuned to the mechanical resonance frequency of the support arm and sample, the amplitude of the vibrations is increased by the quality Q of the resonance system, which is of the order of 100 (Graham 1992). This device is called an alternating-gradient magnetometer and has a sensitivity

Figure B7.5.18. A block diagram of the a.c. magnetization loss measurement technique using mechanical force measurement with an alternating field gradient produced by an appropriate coil configuration: P—electromagnet pole pieces; C—alternating field-gradient coils; S—sample; r—support arm; Pz —piezoelectric crystal motion detector; V—vibration-damping support table; lock-in—lock-in amplifier.

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of up to 10-10 A m2 (Flanders 1988, Frey et al 1988). (iii) Cantilever magnetometer This is a modern miniaturized version of the balance technique, similar to the alternating-gradient magnetometer, which allows us to measure the static magnetization characteristic of the magnetic moment m versus external magnetic field Ha of very small samples. Its principal layout is represented in figure B7.5.19. It consists of a cantilever represented by a beam or plate with an arm a, to the free end of which a small superconducting sample S is attached at a distance ra from the fixed end. When placed in an external magnetic field Ba = jBa and the field gradient grad H = kj∂Hy /∂z the torque ra × Fz due to the z component of the force F is given as

leads to a deflection ∆z of the cantilever arm in the ∆z direction. ∆z can be determined by measuring the change ∆C of the capacitance C between the arm plate a and the fixed plate P.

Figure B7.5.19. A schematic illustration of a cantilever magnetometer for magnetization hysteresis loop measurements: a—cantilever arm; S—sample; Fz —magnetic force acting on the sample; ra —distance of the sample centre from the fixed arm end; d—capacitor gap width between the arm and the fixed plate P; H—the resulting magnetic field; ∆z—displacement of the arm a.

Such a situation can be realized by putting the cantilever magnetometer in a homogeneous magnetic field Ha = jHa parallel with the y axis and simultaneously the sample is situated in a quadrupole, the axis of which is parallel to the x axis of the magnetometer and passes through the sample centre. After having obtained by calibration the function Fc(Ba , grad H) giving Fz as Fz = Fc(Ba , grad H)∆C the measured changes ∆C of the capacitance C allow us, using equation (B7.5.36), to obtain the hysteresis curve my versus Ha by measuring ∆C as a function of Ha . Then according to equation (B7.5.4) the total loss energy per cycle W = ∫c y c l e −m y ⋅ dBa can then be determined using the magnetization hysteresis loop area Am− B . Single-crystal silicon cantilever chips are commercially available. The unique feature of the device is the incorporation of null deflection circuitry on the cantilever surface. This can be used both for calibration and in the null deflection mode for force and torque measurements. The devices are micromachined to a thickness of approximately 4.5 µm and a few millimetres long. Sensitivities of 10-10 A m2 can be achieved. At applied fields above 1 T, the sensitivity of this method exceeds that of conventional commercial SQUID magnetometers. In combination with superconducting magnets and temperature controllers the cantilever

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magnetometers can operate in a typical temperature range of 1.5-300 K in magnetic fields up to about 20 T. The method is suited for magnetization and a.c. loss measurement on small single-crystal samples of high-temperature superconductors. B7.5.3.2 Electrical measurements of a.c. losses caused by transport current According to equation (B7.5.3) the basic principle of these methods is to measure external voltage Ua and current I applied by the power supply to the coil or a short sample of superconducting wire or cable and to process them so as to obtain a quantity directly proportional to the net energy fed to the coil or short sample. However, in the case of a.c. loss measurement on a part of the short sample sometimes it is difficult to determine the external voltage applied to this part of the sample when the magnetic field on its surface is not constant. In this case it is necessary to determine the appropriate form of potential leads or to resort to the use of equation (B7.5.2), i.e. to measure E and H on the surface enclosing this part of the sample and to perform the corresponding surface integral of E × H according to equation (B7.5.2) numerically. Both analogue as well as digital data processing techniques are used. According to the measured objects these measurements can be categorized into two groups: (i)

measurements on superconducting magnet systems and windings;

(ii) measurements on superconducting short samples. (a) Electrical measurement of a.c. losses in superconducting magnets and windings One basic problem which arises in this case is due to the fact that the induced voltage Ui is in general by several orders of magnitude higher than the loss voltage Ur . Moreover, at industrial frequencies of 50-60 Hz the amplitude of the former attains kilovolt levels even for small laboratory magnets. Another problem is the fact that the absolute value of the a.c. loss power is small especially with regard to the amplitude of the current involved. So conventional wattmeters with an electromagnetic system are not sufficiently sensitive. Therefore untraditional measuring techniques must be used. In this subsection electrical measuring techniques for a.c. loss measurement in superconducting magnets and windings are presented and mutually compared. All are based on measuring the voltage Ua and current I applied to the magnet or winding but differ in the processing of the measured data to obtain the total a.c. losses. First of all we will describe four systems used for a.c. loss measurements in pulsed magnets for elementary particle accelerators where the working frequency f < 1 Hz. The simplest described approach performs the direct multiplication of Ua by I using a Hall probe multiplier to obtain a signal proportional to the instantaneous power input to the magnet. The output voltage UH from the multiplier is then integrated over a complete magnet cycle to obtain a signal proportional to the net loss per cycle. Furthermore, it is shown how the compensation of the inductive component Ua i of the terminal voltage allows us to increase the sensitivity and precision of such a measurement. Then another method, also using the Ua i compensation but based on measuring the hysteresis loop of the magnet instead of the multiplication of Ua by I, is presented. Beside the amplitude of a.c. losses the form of the hysteresis loop also conveys more information about the internal behaviour of the magnet. Finally in the last described technique the graphical integration of the hysteresis loop of a magnet is replaced by an electronic integration which allows us to obtain the energy loss per cycle in digital form. The latter method is suitable also for a.c. loss measurement at higher frequencies when harmonic currents are used. We then show that some of the techniques described in section B7.5.3.1, namely those using a phase-sensitive detector as well as the described electronic wattmeter, can also be used for measuring a.c. losses due to a harmonic transport current.

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To close this section an apparatus with a selective nanovoltmeter and a special compensating coil is described. (i) A wattmeter using a Hall probe The method using a Hall probe multiplier for the multiplication of Ua by I and an electronic integrator for integration of the output voltage of the Hall probe with respect to the time during one period yields an output voltage directly proportional to the energy loss per cycle (see Gilbert et al 1968). The circuit diagram of the apparatus is shown in figure B7.5.20. The power supply PS delivers current I to the magnet S with self-inductance Ls . The magnet terminal voltage Ua is connected via an interchangeable resistor R into the current input of the Hall probe HS. The condition I >> IH is supposed to be fulfilled. The Hall probe is placed in a copper wire coil K connected in series with the measured magnet and produces a magnetic field B = kkI perpendicular to the Hall probe plane. The output voltage of the Hall probe is then UH = SH kk IUa /(R + RH ) = PSH kk /R′ where SH [VT−1A−1 ] and RH are the sensitivity and the input resistance of the Hall probe, respectively, R′ = R + RH , and P = UaI is the instantaneous power delivered to the coil. The output voltage of the Hall probe is integrated in time during the period T by the electronic integrator INT. Its output voltage U0 is

where τ is the time constant of the integrator and W is the energy loss per cycle. Accordingly

The sensitivity of the wattmeter SW = SHkk /τ R′ is directly proportional to SH and kk , which are the characteristics of the Hall probe and of the coil K, respectively, but is inversely proportional to τ and R′ which can be optimized.

Figure B7.5.20. A block diagram of a wattmeter for a.c. loss measurement in pulsed superconducting magnets using a Hall probe: S—sample in the form of a solenoid; PS—power supply of the solenoid; Ls —self-inductance of the solenoid; I—solenoid current; K—copper coil generating magnetic field B proportional to I and perpendicular to the plane of the Hall probe HS; Ua —voltage applied to the sample; R —interchangeable resistor limiting the current IH of the Hall probe; UH —output voltage of the Hall probe; INT—electronic integrator with output voltage U0.

The value of R′ = R + RH has to fulfil two conditions. Firstly R′ has to limit the input current IH of the Hall probe to its maximum admissible value IH m at the maximum value of the applied voltage Ua m . Secondly R has to reduce the influence of the change ∆R of RH due to the magnetoresistance on IH during the cycle. These conditions can be written as Ua m /IH m ≤ R + RH » ∆RH whichever is greater. In the majority of practical cases Ua m /IH m >> ∆RH . Then from the point of view of maximum sensitivity

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the optimum value of R′ is R′ = Ua m /IH m and the maximum sensitivity SW = SH kk IH m /Ua mτ. The latter is inversely proportional to τ and Ua m . The lowest value of τ is limited by the maximum admissible integrator output voltage Uo m above which the error due to the nonlinearity of the integration increases very rapidly. It can be shown that this condition leads also to the requirement of as small as possible Ua m . As in this method the a.c. losses are measured by processing the total voltage Ua , the error in measuring W will be proportional to Ua while W is proportional to Ua r only (Ua r is the resistive component of the terminal voltage). Therefore the error in measuring W will be approximately a factor of Ua /Ua r higher than it would be if Ua r were directly processed. Because of the drift of the integrator due to its nonzero offset voltage Uo f f a relative error of the measurement occurs (see equations (B7.5.15) and (B7.5.37))

which increases with Uo f f the period of the cycle T and with R′ = Ua m /IH m and decreases with increasing loss per cycle W, Hall probe sensitivity SH and magnetic constant kk of the coil K. The apparatus described by Gilbert et al (1968) had a moderate sensitivity and a relative error of about 5% for superconducting magnets with the ratio of the stored magnetic energy to the loss energy per cycle smaller than 100. (ii) Wattmeter using a Hall probe and the compensation of the inductive component of the terminal voltage inductive component The discussion about the sensitivity as well as the accuracy of the preceding type of wattmeter has shown that they are both inversely proportional to the maximum value Ua m of the terminal voltage of the magnet applied to the Hall probe. According to equation (B7.5.3) the terminal voltage of the magnet is the sum of the loss component Ua r and the inductive component Ua i . As in superconducting magnets Ua i is often several orders of magnitude higher than Ua r , it has been proposed by Hlasnik et al (1969) and Bronca et al (1970) to increase both the sensitivity and accuracy of this type of wattmeter by compensating the inductive component Ua i see figure B7.5.21. This compensation is done by putting a linear mutual inductance M1 into the magnet current leads and by subtracting the M dI/dt signal from a part r Ua of the magnet terminal voltage reduced by the resistance divider with dividing ratio r = R2 /(R1 + R2 ) = M1/LS , Ls being the self-inductance of the magnet. Here again to limit the phase error of the compensating circuit

Figure B7.5.21. A block diagram of a wattmeter for a.c. loss measurement in pulsed superconducting magnets using a Hall probe and compensation of the inductive voltage Uai′,differing from that of figure B7.5.20 by the insertion of a linear mutual inductance M1 into the current leads of the sample, of the voltage divider from resistors R1 , R2 and of the voltage-current converter A.

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R1 + R2 >> ω Ls . The differential voltage Ud = r Ua —M1 dI/dt is amplified by the voltage—current amplifier A with a gain 1/R″ (ω–1 ) before being converted to the Hall probe current IH . As IH has to be smaller or at least equal to the maximum allowed value of IH m , R″ ≥ r Ua r m /IH m where Ua r m is the maximum value of the loss component of the voltage applied to the magnet. Then the Hall probe output voltage UH = SH kk Ir Ua r /R″ = PSH kk r/R″ and the loss per cycle is

This means that the maximum sensitivity of the wattmeter with the Ua i compensation Sw c = r SH kk /R″τ ≤ SHKkIHm/U arm τ would be Ua m /Ua r m times higher than that of the preceding case at the same value of τ. For the parameters chosen by Hlasnik et al (1969) this modification allowed the authors to increase the sensitivity of the apparatus by a factor of 20 compared with that without Ua i compensation. (iii) A wattmeter using an electronic integrator (Wilson’s method) A simple technique for measuring a.c. losses in superconducting magnets has been proposed by Wilson (1973). The principle of this method is that instead of using the multiplication of Ua by I and the integration of this product with respect to time during one period the a.c. loss energy per cycle is obtained by integrating Ua with respect to time and using consecutive graphical registration of this integral as a function of I in the form of a hysteresis loop of a magnet. The energy loss per cycle is proportional to the area A of this hysteresis loop which can be readily measured, e.g. by a planimeter. Using integration by parts equation (B7.5.3) can be rewritten as

If the magnet is cycled from −Im to +Im and to −Im or from 0 to Im to 0, then the first term on the righthand side of (B7.5.41) is zero, i.e. W = -∫ 0T(Uadt)dI which is proportional to the area of the hysteresis loop of the integral ∫ Uadt versus current I. A circuit diagram of the apparatus executing such a.c. loss measurement of a magnet is reproduced in figure B7.5.22. It represents essentially the circuit diagram of figure B7.5.21 in which the amplifier A and the Hall probe HS are omitted and the coil K is replaced by the shunt resistor Rs h . The differential voltage Ud = r Ua − M dI/dt is connected directly to the input of the integrator INT. The voltage Us from the shunt Rs h and the integrator output voltage U0 enter the x and y channels of the x—y plotter XY respectively. The curve obtained from the x—y plotter is a sort of averaged hysteresis loop Ud dt — I of the magnet. W is therefore proportional to its area A as

Figure B7.5.22. A wattmeter for a.c. loss measurement in pulsed superconducting magnets using an electronic integrator (Wilson’s method): PS—power supply; Ls —measured magnet; M—compensation linear mutual inductance; R1 , R2 —resistors; Rs h —shunt resistor; INT—integrator; XY—x—y recorder.

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where τ is the time constant of the integrator, and Sx and Sy are the sensitivities of the x and y channels, respectively, expressed in metres per volt. The sensitivity of the apparatus is Sw = A/W = SxSyRsr/τ. The degree of compensation does not change the area of the loop but influences the maximum value of the input voltage to the integrator INT and by this the lowest admissible τ , i.e. the maximum sensitivity. Moreover, by the form of the hysteresis loop beside the total energy loss this method conveys information about the internal processes which take place in the magnet during the cycle, such as the winding deformation under the influence of the electromagnetic forces or the effect of iron saturation in magnets with an iron core, eddy current losses, etc (see Wilson 1973). For the analysis of the possible error sources one can refer to Fietz (1965). At a higher rate of change of the magnet current an additional error can occur due to the phase shifts arising in the compensation circuit and in the voltage divider. It can be shown that this error is of the order of ∆W given as ∆W = 2Ls(dI/dt )Im a x (τc −τs ), where τc = M/(RM + R2 + Ri n ) and τs = Ls /(R1 + R2 ) are the time constants of the compensating circuit and of the magnet respectively. RM is the resistance of the secondary winding of the compensating mutual inductance M1 and Ri n is the input resistance of the integrator INT. The influence of the capacitance of large magnets and compensation coils on measured hysteresis loops of superconducting magnets and windings was analysed by Gömöry (1986). At long periods T the integrator drift is the main source of error. However, it is easily detectable because it means the hysteresis loop is not closed. At short periods T the measurement of the loop area and the inertia of the x—y recorder at a high rate of field change are the main sources of error. To eliminate them Jansak and Chovanec (1980) have proposed the electronic wattmeter described below. (iv) A wattmeter using double integration In principle this wattmeter replaces the graphical registration and integration of the hysteresis loop of a magnet by the multiplication of the integral ∫ Ud dt by M2dI/dt , a signal proportional to dI/dt (see equation (B7.5.41), using an electronic multiplier M and integrating the output voltage of the multiplier with respect to time by the integrator INT2 (see figure B7.5.23). The energy loss per cycle is directly proportional to the output voltage Uo u t , of this integrator and is displayed by the digital voltmeter DVM via joulemeter constant kw . The latter can be determined in a similar way to that described in the preceding case and

where r is the divider ratio, τ1 and τ2 are the time constants of the integrators INT1 and INT2, respectively, M2 is the auxiliary mutual inductance and cm is the multiplier constant given as the ratio of the multiplier output voltage U12 to the product U1 U2 of the input voltages of the multiplier. When the parameters r, τ1, τ2, M2 and cm are properly chosen, one can obtain the joulemeter constant in decade form, e.g. 1, 10, 100 J V-1 and thus direct digital reading of the loss per cycle is possible.

Figure B7.5.23. A wattmeter for a.c. loss measurement in superconducting magnets using double integration: PS— power supply; LS —measured magnet; M1—compensation linear mutual inductance; M2 —Rogowski coil; R1, R2—resistors; Ud = Ua — M1 dI/dt the differential voltage; INT1, INT2—electronic integrators; M—electronic multiplier; DVM—digital voltmeter.

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(v) A wattmeter using a phase-sensitive detector If the a.c. current is harmonic the a.c. loss power may be determined as

where current I and the in-phase first harmonic of the input voltage Ua 1 cos ϕ are rms values and ϕ is the shift angle between I and Ua 1. The latter can be measured using a phase-sensitive detector of a lockin amplifier (see section B7.5.3.1(a)(ii)). As the terminal voltage Ua has a very high inductive component Ua i , to eliminate the saturation of the amplifier and to diminish the error due to the amplitude and phase distortion of the signal the component Ua i must be compensated. The circuit diagram of such an apparatus is similar to that of figure B7.5.23 in which the integrator INT1, the multiplier M and the integrator INT2 are replaced by a lock-in amplifier. The latter also preamplifier the differential signal Ud , then multiplies the amplified signal by the reference signal M2 dI/dt taken from a linear mutual inductance and shifted by 90° and finally passes through a low-pass filter to the digital voltmeter. For M2 a Rogowski coil, for instance, can be used. This is a coil wound on a hollow cylinder or on a flexible ribbon which can be deformed into a hollow cylinder with a constant number of turns per unit length on its perimeter. The magnetic flux coupled by a Rogowski coil is proportional to the current passing through its hole independently of the position of the conductor inside the hole. As has been already mentioned, the terminal voltage of a superconducting magnet can be very high, of the order of kilovolts. Therefore it is very important to protect the input circuits of the lock-in amplifier against overvoltage and common mode rejection problems. Both these problems can be reduced or eliminated by either choosing the measured parts of the circuit to be as near as possible to the grounded point or to separate them by an isolating transformer and form a virtual ground when using differential inputs (see the following section and figure B7.5.26 below). It is to be noted that this type of wattmeter is generally used for a.c. loss measurement on small experimental superconducting samples rather than on large magnets. (vi) A wattmeter using digital data processing This principle consists in transforming the analogue measured data into digital data using analogue—digital converters (ADCs) with a resolution of at least 12 bits or high speed digital voltmeters (Gömöry and Cesnak 1985). The digitized data are then either stored in a transient digital data memory and latter processed by a personal computer or directly processed by an on-line computer to obtain the a.c. loss energy. The circuit diagram of one such apparatus used by Fukui et al (1994) is shown in figure B7.5.24. It uses the analogue compensation of Ua i by a voltage from a linear smoothly controllable mutual inductance M as described by Kokavec et al (1993) and works with only two analogue amplifiers and two ADCs. The compensated differential signal Ud is amplified and converted to the digital data using one channel of the transient digital data recorder A1 which consists of an analogue amplifier and an ADC. The voltage proportional to the sample current UI = Rs h I , with Rs h being the resistance of the shunt resistor put into the current leads of the sample, is amplified and converted to digital data using another channel of the transient digital data recorder A2 via the amplifier and ADC. The digitized signals are processed using the computer (PC) providing the a.c. loss value according to equation (B7.5.3). The sensitivity of this apparatus is proportional to r G1G2Rs h , where G1 and G2 are the gain factors of amplifiers A1 and A2 of the transient digital data recorder respectively. In the article by Fukui et al (1994) the lowest measured power was of the order of 50 µW. Moreover, this apparatus allows us to correct the analogue compensation of Ua i by a complementary digital compensation which adds a correcting term ∆M dI/dt generated by PC using the experimental data on I and the chosen value of ∆M.

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Figure B7.5.24. A block diagram of a.c. loss measurement in superconducting magnets or samples transporting a.c. current and using digital data processing: S—superconducting magnet or short sample; Ua —voltage applied to the magnet or short sample; M—linear compensation mutual inductance; Ud —compensated differential voltage; Rs h —shunt resistor; A1, A2—channels of the transient digital data recorder; PC—personal computer.

(b) Electrical measurements of a.c. losses due to the transport current in short samples Because of the short sample length (a few centimetres to a few metres) and consequently of the extremely low signal (down to nanovolt level), the instrumentation for a.c. loss measurement on short samples with transport current needs to have a much higher sensitivity than that for measurements on superconducting magnets and windings. Therefore the apparatus should be carefully designed and realized. Actually the significance of different error sources is strongly increased. Apart from the error sources present for a.c. loss measurements in magnets and windings some complementary error sources also occur. They are due mainly to inhomogeneities of magnetic and electric fields on the surface as well as in the interior especially of tape samples due to their noncircular cross-section, very short length and material inhomogeneities. Electric field due to the transfer of the current from the normal metal into the superconductor and vice versa near the current contacts and near material inhomogeneities makes it difficult to determine the intrinsic properties of the conductor. The same is true with regard to the influence of the magnetic field due to the current in the current leads. Further problems are related to the inhomogeneous current and field distribution in superconducting cables. To reduce possible sources of error due to the analogue circuitry, it is recommended to avoid any preprocessing of the loss signals or, if necessary, to use high-quality amplifiers that introduce minimal amplitude, frequency and phase distortion. Another way to do this is to use the digital technique in the acquisition as well as in the processing of the experimental data. In this section a description of special constructions of coils for Ui compensation as well as discussion about the use of a Rogowski coil as a sensor for the signal proportional to the current amplitude and phase are first presented. A scheme for the creation of a virtual ground when using differential inputs to minimize the common mode rejection problems is described too. Then a discussion of the influence of the position of potential taps and of the form of potential wires and current leads on the value of sample voltage and its inductive and resistive components measured on a tape sample is given. The rules for the choice of the right form of potential as well as of current leads for a.c. loss measurements on this type of sample are also given. In the second part of this section some typical measuring set-ups using phase-sensitive detectors, selective nanovoltmeters or digital data processing are described too. (i) Elements and schemes for spurious signal compensation As in superconducting magnets and windings, the sample voltage U between the voltage taps is composed

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of the voltage Ur , which is in phase with the transport current and proportional to the a.c. losses, and of the inductive voltage Ui which is 90° out of phase with the current. The latter is usually higher than the former by up to several orders of magnitude. As has been shown in section B7.5.3.2(a)(ii) the simplest way to cancel the inductive component from the sample voltage is to subtract a pure inductive voltage Ui c , from it. Uic must be exactly proportional to the sample current derivative and, if possible, galvanically isolated from the sample circuitry. This can be done by putting a linear mutual inductance M into the sample current leads and by subtracting the MdI/dt signal using an analogue or digital procedure. Unlike in magnets and windings the amplitude of Ui in short samples is very low; so relatively small compensation coils placed in a low magnetic field proportional to I are usable. However, it is necessary to use fluently variable mutual inductances to allow complete compensation of the inductive voltage Ui . At the same time the divider ratio r should be as near as possible to unity so as not to decrease the useful differential signal Ud . Moreover, as will be shown later, the potential leads must not be led close to the surface of the sample if the sample has a noncircular cross-section. There are three basic requirements on the compensation signal from the mutual inductance: (i) it should be free of in-phase components due to, e.g., eddy currents in the mutual inductance windings; (ii) spurious signals due to the external a.c. magnetic field should be suppressed as much as possible; (iii) it should be adjustable to allow as good as possible compensation of Ui without the use of a potentiometric divider with sliding contacts. Several such types of mutual inductance have been proposed. One of them (see figure B7.5.25(a) and Boggs et al 1992) consists of a toroidal primary coil C1 and of two small secondary flat circular coils C2a and C2b which are situated within the toroid. Secondary coil C2a has about ten times the turn area of the coil C2b. The secondary coils can be turned around their axes which are perpendicular to that of the primary coil. Secondary coils C2a and C2b are connected to a combination 6:1, 36:1 reducer so that the orientation of the secondary coils to the axis of the toroid can be adjusted with very fine resolution.

Figure B7.5.25. Variable linear mutual inductances for transport current a.c. loss measurement in short samples: (a) C1—toroidal primary coil; C2a, C2b—flat circular secondary coils turning around their axes perpendicular to the solenoid axis; (b) the base plate bears a circular rod as the primary winding connected to the sample current lead and two flat compensation coils as a secondary winding connected in series opposition to each other and fixed to swivel joint strips allowing their displacement with regard to the circular rod.

The voltages of the secondary coils and the sample are connected in series. The compensation procedure begins by turning the coil C2a with higher turn area to obtain a rough minimum of the total differential voltage. Then by turning the coil C2b the full compensation is established.

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Three other modifications of variable mutual inductance without sliding contacts and with suppressed spurious signals due to eddy currents in the primary winding and to external a.c. magnetic field have been proposed by Jansak and Kokavec (1994) and Kokavec et al (1993). To suppress the influence of eddy currents in the primary winding the latter is made either in the form of a solenoid from a cable with sufficiently thin insulated Cu wires or in the form of a straight round Cu cylinder put into the sample current leads. To suppress the influence of the external a.c. magnetic field the secondary winding consists of two identical coils with parallel axes connected in series opposition. They are placed at different positions in the inhomogeneous magnetic field of the primary coil. The voltages induced in the secondary coils by the external a.c. magnetic field cancel each other while those induced by the field of the primary coil add. The compensation procedure begins again by moving one secondary coil into the position of stronger coupling with the field of the primary coil so as to obtain a rough minimum of the total voltage, the other secondary coil being in the region of low coupling. Then by moving the latter full compensation is established. Figure B7.5.25(b) shows such a mutual inductance as described by Jansak and Kokavec (1994). Two identical parallel compensation coils are fixed on the swivel joint strips placed on both sides of the central conductor rod. The configuration is held on the support base plate. Compensation coils are connected in series opposition therefore the voltages induced by the external magnetic field cancel each other. By positioning each coil independently, the mutual inductance can be set from zero to its maximum value and the polarity can be changed too. At the beginning of the balancing both compensation coils are placed in the region of the lowest field. Then one of them is moved to compensate roughly Ui and after that complete compensation is achieved by moving the other coil. A Rogowski coil is also a suitable configuration which eliminates both the external spurious electromagnetic field effects and the influence of eddy currents in the primary winding if the rotational symmetry of the central current rod is secured. The disadvantage of the classical Rogowski coil or another constant mutual inductance is that the compensating voltage adjustment must be made using a resistance divider with sliding contact. Moreover, the Rogowski coil is suitable for obtaining a voltage which can be used as a reference signal shifted 90° with regard to the current. The phase shift ϕ of the compensating or reference voltage taken from the secondary winding of the mutual inductance with respect to the current I or the sample voltage U and due to the parasitic impedance of the coaxial cable connecting the secondary winding with the reference input of the lock-in amplifier or other measuring instrument with sufficiently high input impedance can be calculated from the expression tan ϕ = ω C(Rc − R2 )/[ 1 − ω 2C(Lc + L2 ) ] where C , Rc and Lc are the connecting cable capacitance, resistance and inductance respectively, and R2 , L2 are, respectively, the resistance and inductance of the secondary coil of the compensating mutual inductance. Due to the fact that the capacitance per metre of a standard coaxial cable C1 is of the order of 10−10 F m−1 and the resistances Rc ≈ R22 ≤ 1Ω as well as the impedance ω (L2 + Lc ) << 1Ω this tan ϕ is of the order of 10− 8 , i.e. negligible. (ii) A wattmeter using the phase-sensitive detector of a lock-in amplifier This instrument is often used for a.c. loss measurement in short samples because of its high voltage sensitivity in the range down to a few nanovolts even when the signal-to-noise ratio is much smaller than unity. The circuit diagram of the apparatus using the lock-in amplifier technique is shown in figure B7.5.26. The sample current is delivered by the secondary winding of the isolating transformer CT minimizing the common mode rejection error. Its primary winding is connected to the output of the power amplifier AMP which is piloted by the internal generator of the lock-in amplifier with frequency f. A load resistor RL connected in series with the sample converts the voltage source into the current source and a shunt resistor Rs h provides information about the amplitude and waveform of the current. To minimize the common mode rejection error differential inputs can be used and a virtual ground can be created using the common point of two equal resistors connected in parallel to the sample. This virtual ground is connected to the

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Figure B7.5.26. A block diagram of a.c. loss measurement in superconducting samples transporting a.c. current and using a phase-sensitive detector: AMP—a power amplifiser piloted by a voltage of frequency f for an internal generator of the lock-in amplifier; CT—isolating transformer; Rs h —shunt resistor; TR—electronic transient recorder; RL— load resistor; S—sample; 1 k, 1 k—resistors forming a divider to provide a virtual ground; RC—Rogowski coil giving a voltage for accurate phase setting of the lock-in amplifier; HP-IB—bus interface between the lock-in amplifier and the computer PC.

shield of the cables which transfer the sample voltage to the differential input of the lock-in amplifier. The voltage of the Rogowski coil RC is used for an accurate phase setting of the reference signal with respect to the sample current. Because the Rogowski coil voltage is shifted by 90° with regard to the current, the ‘in-phase’ sample voltage component is that which is shifted by –90° with regard to the reference voltage. In some modifications of this method, the Rogowski coil is used also for the determination of the sample current. As the lock-in amplifier output voltage U0 is usually given as an rms value the corresponding current should be given as an rms value too to obtain the loss power according to equation (B7.5.4) as P = U0 I. (iii) Apparatus with a selective nanovoltmeter and compensating coil Figure B7.5.27 shows the circuit diagram of the apparatus described by Kokavec et al (1993). The sample current is provided by the voltage-controlled current supply VCCS piloted by a harmonic voltage of the function generator FG. The shunt resistance Rs h is put into the sample current leads and the voltage Us h across it is measured by transient recorder TR. The inductive component of the sample voltage Ui is compensated by the voltage M dI/dt induced in the secondary winding of a linear smoothly variable mutual inductance M the primary winding of which is connected in series with the sample. The compensated sample signal Ud is connected to the input transformer IT of the selective nanovoltmeter SnV. The latter is designed for the selective measurement of extremely weak a.c. voltages in the frequency range from units of hertz to hundreds of kilohertz. The signal-to-noise ratio is effectively improved by the bandwidth limitation with an octave selectivity of at least 54 dB. The measurement starts with a frequency adjustment of the selective nanovoltmeter to the frequency of the sample current using an uncompensated signal. From the phasor diagram it can be shown that the compensated differential voltage Ud is minimum when Ui = −M dI/dt. The minimal reading of the

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Figure B7.5.27. A block diagram of the set-up for a.c. transport current loss measurement on a small superconducting sample using a selective nanovoltmeter and a variable linear mutual inductance: FG—function generator; VCCS— voltage controlled current supply; Rs h —shunt resistor; TR—transient recorder; S—sample; M—variable linear mutual inductance; comp on/off—switch; IT—input transformer; Ud = U — M dI/dt—differential voltage; SnV—selective nanovoltmeter.

compensated signal Ud m is then found by smoothly varying the mutual inductance M of the compensating coil. The loss power P is then given as P = Ud m I; both Ud m and I are given as rms values. (iv) A.c. loss measurement in tape samples—the role of the voltage tap position and of the form of the potential leads As pointed out by Fukunaga et al (1993) and confirmed by Ciszek et al (1994), in a superconducting tape sample transporting harmonic current I, the standard method for measuring the voltage U on the sample surface, as well as its component Ur which is in phase with current, using potential leads positioned tightly on the sample surface and in parallel with the sample axis as shown in figure B7.5.4(a) gives different values of U as well as Ur depending on the position of the voltage contacts on the sample perimeter if I < Ic . This means that both the electric field E and its component Er , as well as the magnetic field H, are space dependent on the sample surface. Therefore the loss energy W per cycle on the length ∆l (∆l being the distance between the potential taps) cannot be expressed as Ur IT because

where Ur and I are rms values and T is the period of the current. In fact for a long straight tape sample the magnetic field H is parallel with planes perpendicular to the sample axis. Contribution dH due to the current element J dSc c with J the current density and dSc c the elementary surface in the sample cross-section perpendicular to the sample axis is given by the Biot—Savart law (see e.g. Stratton 1941) as

where d and r ′ are the the radius vectors of the point at which H is considered or of the point at which the current element J dS is situated with regard to the sample axis, respectively, and r = d — r ′, k, d 0 and r ′0 are unit vectors of the sample axis and of vectors d and r ′, respectively (see figure B7.5.28(b)). As for thin tapes r ′/d is smaller than or approximately equal to w/d, from equation (B7.5.46) it is clear that when d/w >> 1 then H = (k × d 0 )I/2πd. In this case H is constant on the cylinder of radius d >> w

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Figure B7.5.28. A schematic illustration of the situation for voltage measurement and calculation of the magnetic field due to transport current in a tape sample: (a) different potential tap positions and different forms of potential wires for transport current a.c. loss measurement on a superconducting tape sample; (b) an illustration of the vectors occurring in equation (B7.5.46).

and the magnetic flux lines are almost circles. This means that in this case the loss per cycle on the length ∆l can be expressed as

where Ur is measured using potential leads in the form shown in figure B7.5.28(a) with d >> w. Ur and I are rms values and T is the period of the current. Ciszek et al (1995) have shown experimentally on a AgIBSCCO-2223 tape that for d/w > 2.5 the apparent loss power Ur I is practically independent of the ratio d/w and is in very good agreement with the theoretical model for an elliptical or round wire given, for example, by Norris (1970) as

where i = Im /Ic with Im the amplitude of the current and Ic the sample critical current. It is to be noted that for elliptical or round wires in which the central part of the conductor with the cross-sectional area Sn does not contain any superconductor, for Ic in equation (B7.5.47) it is necessary to substitute the value Ic* = Ic m (1 + Sn /Sf z ) with Ic m the measured critical current and Sf z the real cross-sectional area of the filamentary zone (see Hlasnik et al 1994b). To suppress the influence of the magnetic field Hc l due to the current in the volume of current leads on measured a.c. losses in short samples, it is necessary to form the current leads in such a way as to fulfil the condition Hc l << Hs , with H s being the magnetic field on the sample surface between potential contacts due to the current in the volume of the sample. This can be achieved by current leads in a form close to that of a coaxial cylinder to the sample. To reduce the error due to losses caused by current transfer from the current leads through the contact resistance and the normal matrix into the superconductor the length of the current contact Lc should be sufficient, Lc ≥ 2( r/R )1/2. Here r is the contact resistance across the solder between the current lead and the sample surface as well as across the normal-metal shell

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and the contact resistance between it and the superconducting part of the sample per unit length and R is the longitudinal resistance per unit length of the normal-metal shell (see Wilson 1983). To eliminate the possible influence of the transfer resistance on the a.c. loss measurement it is necessary to find the region where the electric field E measured on the sample surface is constant. This can be done by measuring the voltage drop across several pairs of potential taps along the sample. B7.5.3.3 Measurement of a.c. losses due to an a.c. field Ha and current I In a.c. windings the multifilamentary composites are transporting a.c. current I under the simultaneous influence of the external magnetic field Ba created by all turns of the winding. In this case Ba is proportional to the current, and the total loss energy per cycle in such windings can be measured using one of the set-ups described in the section B7.5.3.2(a). Nevertheless there exist special cases in which the external magnetic field present in the winding is created by another winding. This is, for example, the case for a toroidal winding of a tokamak which is in the magnetic field of the poloidal coil and in that created by the current in the plasma. Another such case is the experimental set-up in which the influence of different external parameters such as the amplitude, orientation, frequency and the phase of the external magnetic field Ba on total a.c. losses in composites or cables transporting a.c. current should be studied. For this purpose it is necessary to dispose of a winding or magnet supplied by the current Im from a power supply, other than that supplying the current I to the sample. Then the total a.c. loss energy per cycle W in the sample consists of two parts WI and WH a which according to Gurevich et al (1987) and equations (B7.5.3) and (B7.5.4) can be expressed as

Ua and m being the applied voltage and the total magnetic moment of the sample respectively. WI and WH a are the losses covered by the power supply delivering the current I to the sample and by that delivering the current Im to the magnet respectively. The component WI can be measured by one of the sets-ups described in section B7.5.3.2(b) as ∫0T Ua Idt where T is the period of the current I. The total sample magnetic moment m can be measured by some of the magnetometric methods described in section B7.5.3.1(b), e.g. using a vibrating-sample magnetometer or Hall probes, or by some of the fluxmetric methods described in section B7.5.3.1(a) measuring M and multiplying it by the sample volume Vs . One such configuration of the magnetic field and sample form was described by de Reuver et al (1985), see figure B7.5.29. It was used to measure a.c. losses in a bifilar coil transporting a.c. current and exposed to an external magnetic field Ha parallel with its axis. Ha has a d.c. as well as an a.c. component Hd c and Ha c respectively. Hd c is generated by the solenoid DC SM, coaxial with the nonmetallic central tube CT as well as with a set of two coaxial a.c. solenoids AC SMi and AC SMe connected in series opposition and generating a.c. field Ha c in the space between them. By appropriate design of dimensions and number of turns of the a.c. and d.c. coils coupling between the d.c. coil and the a.c. field was prevented. (Another configuration of windings, when the sample can be placed on the axis of two a.c. coils, is described by Polak et al (1995)) In the working space with Ha c two coaxial pickup coils PUC1 and PUC2 with the same number of turns and connected in series opposition surround a part of the bifilarly wound test wire S. Two compensation coils PCC1 and PCC2 identical to the pickup coils are placed far enough below the sample where the field BM due to the magnetization M in the sample is negligible. They are connected in series opposition with the pickup coils for measuring the magnetization of the sample. Full compensation of the inductive component Up i of the pickup coils, i.e. the minimization of the differential voltage Ud = Up — Uc , is adjusted by secondary voltages of variable linear mutual inductances Ma c and MI , the primary windings of which are put into the current leads of the a.c. magnet and of the sample

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Figure B7.5.29. A block diagram of the set-up for a.c. loss measurement in a superconducting sample transporting a.c. current and exposed to a.c. as well as d.c. external magnetic fields: CT—nonmetallic central tube; DC SM—coil generating d.c. magnetic field; AC SMi , AC SMe —internal and external coils, respectively, generating a.c. magnetic field in the working volume between external and internal pickup and compensation coils PUC1, PUC2 and PCC1, PCC2, respectively, where the sample is also situated.

respectively. The coils Ma c and MI are situated outside the working volume of the a.c. coils and they are not shown in figure B7.5.29. Trapezoidal ramp currents and fields were applied. A computer generated the ramp, recorded the relevant responses and performed the integration required to obtain the loss values. Such a set-up allows us to simulate the situations corresponding to different values of relevant parameters. For example, different values of Jc can be simulated by the change of the amplitude of the d.c. magnetic field Bd c . Different forms in time of the current or field (triangular or sinusoidal), their frequency and amplitude as well as different ratios Ba c /I and I/Ic can be secured by the appropriate power supplies. By changing the slope of turns in the sample winding with regard to the sample bobbin axis the ratio of the longitudinal and perpendicular component of Ba c can be adjusted too. If, however, the simultaneous action of the transport current I and of the external magnetic field Ba as well as the geometry of the sample give rise to space and time multiharmonic forms of currents, voltages and fields then the best way of measuring the total losses is the use of an appropriate calorimetric method. B7.5.4 Measurement of a.c. losses due to mechanical effects The study of the influence of a.c. losses due to mechanical effects on the performance of superconducting magnets and devices is important. It is necessary for understanding, diminishing or eliminating such significant problems of superconducting devices as the training and the degradation of their quenching currents as well as the complementary a.c. losses in many superconductor applications with time-variable magnetic fields. To give some idea about the importance of different types of mechanical factor we shall mention here some quantitative data on them. For example, the energy associated with the movement of a conductor transporting current at Jc = 109 A m− 2 and B = 5 T under the influence of a Lorentz force against the external friction on a distance of 1 µm is sufficient to increase the temperature of the superconductor at 4.2 K by about 1 K (see e.g. Hlasnik 1978). On the other hand the losses due to the internal friction under the cycled loading of the superconductor in pulsed superconducting magnet systems could be of the

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same order of magnitude as hysteresis or eddy current losses and should be considered in the design of the magnet (see e.g. Akin and Moazed 1977). The sources of mechanical losses in SMS could be categorized into the two following groups: (i) external friction on the conductor surface; (ii) internal friction in the conductor, insulation and construction materials. In the following two measurement methods of losses due to the external and internal friction are described. B7.5.4.1 Method for the measurement of losses due to external friction In this section one method will be described for the measurement of losses due to the movement of the conductor in relation to a sudden transient decrease of the static friction coefficient (see Iwasa 1981). The mechanical losses due to the external friction during the time T0 of such an event occur on the conductor surface at its movement under the influence of the Lorentz force against the dynamic frictional force between two conductors or between a conductor and cooling channel spacers or a bobbin. Let the Lorentz force acting on an element ds of the conductor be dFL = I ds × B and let the elementary displacement of this element be dr = υ dt where υ is its velocity, then the total energy supplied to the winding during T0 by the Lorentz force is given as

Ut f can be measured as the change of the differential voltage Ud = r Ua −M dI/dt where r Ua is a part of the magnet terminal voltage reduced by the resistance divider with dividing ratio r = R2/(R1 + R2 ) = M/LS , where M dI/dt is the compensating voltage across the secondary winding of a linear mutual inductance put into the magnet current leads (see e.g. figure B7.5.22) and LS the self-inductance of the magnet. Usually to detect microscopic slip events, an acoustic emission (AE) transducer attached to the magnet is also used. In the experiment individual AE events rather than counting rate are emphasized. Figure B7.5.30 from the article by Iwasa (1981) shows voltage and acoustic emission signals from conductor motion events recorded in two different large superconducting magnets, a high-energy physics beam-handling dipole (figure B7.5.30(a)) and a large ( 170 ton, 170 MJ ) MHD dipole magnet (figure B7.5.30(b)). Both magnets have been wound from Nb—Ti/Cu monolithic conductors; however, the copper-to-superconductor ratio in the second one was many times higher than that in the first one.

Figure B7.5.30. Voltage and acoustic emission signal traces from conductor motion events recorded in two large superconducting magnets: (a) a high-energy physics beam-handling dipole; (b) a large (170 ton, 170 MJ) magnetohydrodynamics generator dipole magnet (Iwasa 1981).

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The voltage was balanced across the magnet terminal. In the first case the energy density of 1.6 mJ g-1 released adiabatically during about 400 µs can raise the conductor temperature from 4.2 K to 9 K. In the second case the damped oscillatory nature of the voltage signal suggests that a conductor segment is vibrating over a time period of about 10 ms. Together with a higher copper-to-superconductor ratio, in this case the corresponding temperature increase was more than one order of magnitude lower and could not lead to a magnet quench. B7.5.4.2 Measurement of mechanical losses from internal friction The mechanical losses due to the internal friction occur in the conductor volume at the deformation of the material. Their energy per unit volume and cycle can be expressed as

where σ is the stress and ε the strain in the material. This means that Wi f can be measured as the area of the hysteresis loop of the strain versus stress characteristic of the conductor during a cycling mechanical load. There are two components of the internal friction. One in due to microplastic effects and the other to macroplastic effects. The microplastic effects demonstrated in Nb—Ti by Schmidt and Pasztor (1977) occur even at relatively low strain levels (ε << 0.5%) in the superconducting material itself, while the macroscopic effects such as serrated yielding and plastic deformation appear at higher strain of the conductor (ε > 0.5%). The latter effects are due to a large extent to the deformation of the matrix material. The microplastic effects are irreversible in the sense that they appear only at the first loading cycle. In following cycles they start only at strain values exceeding the previous maximum strain. In contrast, macroplastic effects in a composite material consisting of Nb—Ti filaments in a normalmetal matrix occur at both the first and the following loading cycles as is seen from figure B7.5.31. They can be measured by standard σ–ε measurements appropriately adapted to work at cryogenic temperatures.

Figure B7.5.31. The macroplastic stress—strain characteristic measured on a Nb—Ti multifilamentary composite with a copper matrix.

References Akin J E and Moazed A 1977 Finite element calculation of stress induced heating of superconducters IEEE Trans. Magn. MAG-13 124

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Andronikashvili E L, Chigvinadze J G, Ker R M, Lowell J, Mendelson K and Tsakadze J S 1969 Flux pinning in thermodynamically reversible type II superconductors Cryogenics 9 119–21 Boggs S A, Collings E W and Parish M V 1992 AC losses in HTSC conductor elements IEEE Trans. Appl. Supercond. AS-2 117–21 Bronca G, Hlasnik I, Lefrancois C and Pouillange J P 1970 Problems and recent developments concerning superconducting magnets for synchrotrons Proc. 3rd Int. Conf on Magnet Technology (Hamburg, 1970) pp 701–67 Bruzzone P and Kwasnitza K 1987 Influence of magnet winding geometry on coupling losses of multifilament superconductors Cryogenics 27 53—44 Buchhold T A and Molenda P J 1962 Surface electrical losses of superconductors in low frequency field Cryogenics 2 344–7 Campbell A M 1982 A general treatment of losses in multifilametary superconductors Cryogenics 22 3–16 Carr W J Jr 1983 AC Loss and Macroscopic Theory of Superconductors (New York: Gordon and Breach) Chen D X, Brug J A and Goldfarb R B 1991 Demagnetizing factors for cylinders IEEE Trans. Magn. MAG-27 3601–19 Chigvinadze J G 1973 Influence of surface and volume defects on dissipative processes in type II superconductors Sov. J. Exp. Theor. Phys. 65 1923–7 Ciszek M, Campbell A M, Ashworth S P and Glowacki B A 1995 Energy dissipation in high temperature superconductors Appl. Supercond. 3 509–20 Ciszek M, Campbell A M and Glowacki B A 1994 The effect of potential contact position on AC loss measurements in supercoducting BSCCO tape Physica C 233 203–8 de Reuver J L, Mulder G B J, Rem P C and van de Klundert L J M 1985 A.C. loss contribution of the transport current and transverse field caused by combined action in a multifilamentary wire IEEE Trans. Magn. MAG-13 173–6 Fietz W A 1965 Electronic integration technique for measuring magnetization of hysteretic superconducting materials Rev. Sci. Instrum. 36 1621–6 Flanders P J 1988 An alternating gradient magnetometer J. Appl. Phys. 63 3940–6 Frey Th, Jantz W and Stibal R 1988 Compensated vibrating reed magnetometer Appl. Phys. 64 6002–8 Fukui S, Hlasnik I, Tsukamoto 0, Amemiya N, Polak M and Kottman P 1994 Electric field and losses at AC self field mode in MF composites IEEE Trans. Magn. MAG-30 2411–4 Fukunaga T, Maruyama S and Oota A 1993 AC loss in Ag-sheathed Bi-based (2223) superconductors Advances in Superconductivity VI, Proc. 6th Int. Symp. on Superconductivity (Hiroshima, 1993) Gilbert W S, Hintz R E and Voelker F 1968 AC loss measurements in superconducting magnets E D Lawrence Laboratory Report UCRL 18176 Gömöry F 1991 Use of a phase sensitive detector for measuring magnetic hysteresis loops Rev. Sci. Instrum. 62 2019–21 —1986 Measurement of the magnetization curves and losses in superconducting magnets at pulse duration of ≈1 s Cryogenics 26 273–80 Gömöry F and Cesnak L 1985 Loss and magnetization measurement of superconducting magnets pulsed at very low ramp rates Cryogenics 25 375 —1991 Temperature dependence of ac losses in multifilamentary superconductors measured by ac susceptibility technique Proc. Int. Symp. on AC Superconductors (Smolenice, 1991) pp 53–7 Graham C D Jr 1992 Measurements in magnetic materials Concise Encyclopedia of Magnetic and Superconducting Materials ed J Evetts (Oxford: Pergamon) pp 297–300 Gurevich A V, Mints R G and Rakhmanov A L 1987 Physics of Composite Superconductors (Moscow: Nauka) (in Russian) Hlasnik I 1978 AC losses in superconducting magnets Proc. 6th Int. Conf on Magnet Technology ed M Polak et al (Bratislava: Alfa) pp 575–95 —1984 Prospects of multifilamentary superconductors AC 50 Hz applications J. Physique Coll. 45 C1459–66 Hlasnik I, Fukui S, Amemiya N, Tsukamoto 0, Ikeda N, Kumano T, Suzuki E, Polak M, Kokavec J and Kottman P 1994a Peculiarities in electromagnetic behavior of MF NbTi composites for AC use IEEE Trans. Magn. MAG-30 1681

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Hlasnik I, Fukui S, Ito M, Tsukamoto 0, Polak M, Kokavec J and Kottman P 1994b Electromagnetic phenomena due to transport current in superconducting NbTi composites for ac use Applied Electromagnetics in Materials (JSAEM Studies in Applied Electromagnetics 3) (Kyoto: Japan Society of Applied Electromagnetics) Hlasnik I, Lefrançois C and Pouilange J P 1969 Appareil de mesure des penes de bobines supraconductrices alimentées en courant pulsé Report SEDAP 69-308 SUP/84 (Saclay: CEN) Hlasnik I, Ries G, Polak M and Krempasky L 1980 An additional loss source in cabled multifilamentary superconductors Cryogenics 20 491–8 Ishigohka T, Maruhashi T and Inomiya A 1994 AC loss measurement using LC resonance circuit with superconducting coil Cryogenics 34 (ICEC Suppl.) 575–8 Iwasa Y 1981 Conductor motion in the superconducting magnet—a review Proc. Workshop ‘Stability of Superconductors in Helium I and Helium II’ (Paris: International Institute of Refrigeration) pp 125–37 Jansak L and Chovanec F 1980 An analogue digital double integration joulemeter for ac loss measurement in superconducting magnets Cryogenics 20 125–6 Jansak L and Kokavec J 1994 The inductive component compensation in the AC loss measurements Biennial Report 1993-1994 (Bratislava: Institute of Electrical Engineering SAS) pp 73–4 Kokavec J, Hlasnik I and Fukui S 1993 Very sensitive electric method for AC loss measurement in SC coils IEEE Trans. Appl. Supercond. AS-3 153–5 Kovachev V 1991 Measurement techniques Energy Dissipation in Superconducting Materials ed D Dew-Hughes (Oxford: Clarendon) Krempasky L 1976 A simple method for measuring magnetization of superconducting wires Cryogenics 16 178–9 Krempasky L, Polak M, Chovanec F and Stofanik F 1979 Measurement method of magnetization anisotropy of superconducting cables Sov. J. Technol. Phys. 49 623–7 Krempasky L and Schmidt C 1994a Theoretical analysis of time constant measurements of technical superconductors IEEE Trans. Magn. MAG-30 2654–7 —1994b Time constant measurement in technical superconductors: a theoretical solution of the problem J. Appl. Phys. 75 4264–6 Libbrecht S, Osquiguil E and Bruynsraede Y 1994 Influence of field inhomogeneity on magnetization of YBCO films Physica C 225 337–45 Norris W T 1970 Calculation of hysteresis losses in hard superconductors carrying a.c.: isolated conductors and edges of thin sheets J. Phys. D: Appl. Phys. 3 489–507 Polak M, Pitel J, Majoros M, Kokavec J, Suchon D, Kedrova M, Kvitkovic J, Fikis H and Kirchmayr H 1995 Superconducting DC/AC magnetic system for loss and magnetization experiments operating up to 50/60 Hz IEEE Trans. Appl. Supercond. AS-5 717 Reilly K M and Morgan G M 1992 A digital technique for the measurement of power losses in high temperature superconductors IEEE Trans. Appl. Supercond. AS-2 181–3 Ries G 1977 AC loss in multifilamentary superconductors at technical frequencies IEEE Trans. Magn. MAG-13 524 Ries G and Takacs S 1981 Coupling losses in finite length of superconducting cables and in long cables partially in magnetic field IEEE Trans. Magn. MAG-17 2281–4 Schmidt C 1994 Calorimetric ac. loss measurement in the microwatt range: a new simplified measuring technique Cryogenics 34 3–8 —1985 Measuring a.c. losses of superconductors Cryogenics 25 492–5 Schmidt C and Pasztor G 1977 Superconductors under dynamic mechanical stress IEEE Trans. Magn. MAG-13 116–9 Schmidt C and Specht E 1990 AC loss measurements on superconductors in the microwatt range Rev. Sci. Instrum. 61 988–92 Stratton J A 1941 Electromagnetic Theory (New York: McGraw-Hill) Sumiyoshi F, Irie F and Yoshida K 1978 Magnetic field dependence of ac losses in multifilamentary superconducting wires Cryogenics 18 209–13 —1980 The effect of demagnetization on the eddy-current loss in a single-layered multifilamentary superconducting coil J. Appl. Phys. 51 3807–11 Takacs S 1982 Coupling losses in cables in spatially changing ac fields Cryogenics 22 661–5 Turck B, Lefevre F, Polak M and Krempasky L 1982 Coupling losses in a rectangular multifilamentary superconducting composite Cryogenics 22 441–50

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Wilson M N 1973 An improved technique for measuring hysteresis loss in superconducting magnets Cryogenics 13 361–3 —1983 Superconducting Magnets (Oxford: Clarendon) Zenkevitch V B and Romanyuk A S 1979 The effect of magnetic properties of a composite superconductor on the losses in variable magnetic field. Part 1: theory Cryogenics 19 795 —1980a The effect of magnetic properties of a composite superconductor on the losses in variable magnetic field. Part 2: experiment Cryogenics 20 11–7 —1980b The effect of magnetic properties of a composite superconductor on the losses in variable magnetic field. Part 3: collective interaction of turns Cryogenics 20 79-85

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B8.1 Conductors for d.c. applications

H Krauth

B8.1.1 Introduction Practical superconductors for applications in energy and magnet technology are composite wires consisting of the superconducting material embedded as filaments in a normal-conducting matrix usually with a high longitudinal conductivity. Although, of course, the basic superconducting properties such as critical temperature Tc and upper critical field Bc 2 are key factors in the selection of a superconducting material, other parameters are of similar importance for successful wire fabrication and performance. First of all, superconductor and matrix material must be compatible such that coprocessing as a composite becomes possible. This processing must not deteriorate the Tc and Bc 2 values too much and must allow optimization of the critical current density Jc by producing a pinning-active microstructure and by maintaining homogeneous properties of the superconducting filaments along wire lengths of many kilometres. The most important characteristics are chemical compatibility, with respect to reaction and diffusion processes during heat treatments, and mechanical compatibility, i.e. the hardness values of the components must not be too different in order to allow hot and/or cold working without introducing defects in one component or destroying the integrity of the composite. Last but not least the cost of the prematerials and of the manufacturing processes must allow economic production of the superconducting composite. As a consequence of these requirements only two superconducting materials have so far reached the commercial stage. The most successful technical superconductor is based on the NbTi solid solution alloy at a Ti content between 46 and 52 weight per cent, with NbTi47 exhibiting the most favourable combination of properties. The main reason for the success of NbTi is its excellent ‘workability’ within a Cu matrix although its superconducting properties are moderate (Tc ~ 10 K, Bc 2(4.2 K) ~ 10.5 T). Nearly all magnets with fields below 9 to 10 T are therefore built with NbTi conductors. Magnetic fields higher than 10 T can be achieved with NbTi at lower temperature (e.g. 1.8 K) or by using Nb3Sn in the highfield sections. As Nb3Sn is a brittle intermetallic compound it cannot be processed by mechanical working. Several techniques have been developed all using ductile precursor materials for composite production and processing. Essentially Nb filaments are embedded in a matrix containing Cu and Sn in elemental or alloy form and the Nb3Sn compound is formed by a diffusion and reaction heat treatment only at the final diameter of th e wire. As a result of the higher Tc ~ 18 K and Bc 2(4.2 K) ~ 23–29 T (depending on exact chemical composition) Nb3Sn conductors can be used to generate fields up to 20 T or can be used at temperatures above 4.2 K to generate fields of several tesla. The emergence of the Cu-oxide-based high-Tc superconductors with their superior superconductor properties of Tc ~ 100 K and Bc 2 of several hundred tesla at low temperature has opened new perspectives with respect to the generation of fields well above 20 T or to the operation of superconducting systems at higher temperatures up to the temperature of liquid nitrogen (77 K). Due to the ceramic nature and the high

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chemical reactivity of these materials the production of composite wires with high current density turned out to be very difficult. Only silver in elemental or a certain alloy form (or even more exotic materials) seems to be usable as a matrix material because of its transparency to oxygen and its low reactivity with the ceramics. In addition, it turned out to be very difficult to produce a microstructure in the high-Tc material that would allow a high critical current density. The most promising approach to wire fabrication seems to be the powder-in-tube method based on Bi2Sr2CaCu2O8 or Bi2Sr2Ca2Cu3O10 in a silver matrix. A survey of these conductors is given in section B9.2. It has already been mentioned that in technical wires the superconducting material has to be made into filaments. There exists a hierarchy of arguments leading to this requirement. A very basic reason for this is magnetic stability. It is most important for low-Tc superconductors because of the small temperature margin between the operational temperature To p and Tc , and because of the small heat capacity at low temperature. Nevertheless, Nb3Sn conductors usually have smaller filaments than NbTi conductors. The reason for this is the fact that the superconducting Nb3Sn phase is formed by a solid-state diffusion process which produces only thin layers. Another reason for filaments being smaller than required by stability arguments is the a.c. losses in the time-dependent external fields or with a.c. currents. Depending on frequency and amplitude very small filament diameters in the sub-micrometre range may be needed. A.c. losses, phase formation arguments and, in addition, mechanical performance are other reasons why high-Tc superconducting wires will consist in most cases of composites with multifilamentary superconducting material. As a summary of these introductory remarks it can be stated that the field of technical superconductors represents a multidisciplinary area combining the basic physics of superconductors, the technology of composite materials including their mechanical processing, the electrodynamics of composite superconductors and the cryogenic and cooling aspects. Conductor design and fabrication have to take into account all these different aspects. B8.1.2 Design criteria The design of a technical superconductor for a given application is a relatively complex task in which system requirements have to be reconciled with conductor technology aspects like fabricability, performance characteristics and cost. The most important design criteria are given in the following sections. B8.1.2.1 Superconductor material selection The boundary between superconducting and normal-conducting behaviour is characterized by three critical parameters, the critical temperature Tc , the upper critical field Bc 2 and the critical current density Jc . The values of Tc and Bc 2 are mainly determined by the electronic and phononic properties of the material and depend on the chemical composition and the crystal structure. They can be manipulated only marginally by the wire fabrication process. The critical current density Jc is strongly dependent on temperature and magnetic field. It is determined by the microstructure and can therefore be manipulated during the wire fabrication process. Optimization of Jc is achieved by introducing finely dispersed microstructural features acting as pinning centres, while maintaining good connectivity and homogeneity along the wire length. It is important to note that all critical parameters are not exactly defined by their absolute value especially in technical superconductors like wires. Remaining inhomogeneities tend to smear out the transitions such that measured Tc , Bc 2 and Jc values depend on the applied criterion. All values of Tc and Bc 2 given in this section are therefore only approximate. The importance of a clear definition will be pointed out in the example of a resistive Jc measurement. Usually the transport Jc is determined by measuring the current—voltage (I—V) characteristic of a wire specimen placed in a transverse magnetic field with a four-point technique. By definition the critical current

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Ic is reached when the voltage per unit length of the wire (electric field) or its resistivity has reached a certain level. The usual criteria are 0.1 µV cm–1 or 1 µV cm–1 for the electric field method and 10–13 Ω m or 10–14 Ω m for the resistivity method. Clearly the resulting Ic value depends on the criteria and the differences are biggest for smooth transitions between normal conductivity and superconductivity. The critical current density is calculated from Ic by dividing through an appropriate area, which for technical purposes is not necessarily the area of the superconductor material itself but includes, for example, the bronze area in the case of Nb3Sn. Another method of determining Jc consists in magnetization measurements. Usually these are measurements with slow variations of an external magnetic field, leading to low electrical fields and therefore to higher sensitivity in terms of an electric field criterion. In many cases the I—V curve can be approximated by a power law U ∝ In. The exponent n describes the steepness of the transition and can be taken as a measure of the quality of a technical superconductor as inhomogeneities such as variations of the filament area tend to decrease the steepness of the transition. The higher the n values the less dependent are the measured Ic values on the applied criterion. The n value, although not very well defined and depending relatively strongly on the measuring and evaluation method, is often used in specifications of technical superconductors as a quality index. It is also a function of the magnetic field when n → 1 for B → Bc 2 . For high-quality conductors n increases up to values above 60 in low fields. For conductors exhibiting more inhomogeneities n levels off at smaller values in low fields. For magnet applications in persistent mode values of n ≥ 30 are desirable. As already mentioned, only NbTi and Nb3Sn are so far of industrial and commercial relevance. The solid solution alloy NbTi47 exhibits Tc ~ 10 K with Bc 2 (4.2 K) ~ 10.5 T and Bc 2 (2 K) ~ 13.5 T. The intermetallic compound Nb3Sn exhibits Tc ~ 18 K. Bc 2 for the binary compound is about 23 T at 4.2 K. By further alloying with about 7.5% Ta and/or about 0.2% Ti, Bc 2 can be enhanced up to 26–29 T, with the highest value being achieved by simultaneous alloying with Ta and Ti. The Jc values of commercial NbTi, Nb3Sn, (NbTa)3Sn and (NbTaTi)3Sn wires are shown in figure B8.1.1 as a function of magnetic field and for two temperatures (4.2 K and 2 K). In the case of Nb3Sn the conductors were produced by the bronze route (see below) and the bronze area is included in the calculation of Jc . The Jc value in the Nb3Sn material can be estimated by multiplying the given values by about three. B8.1.2.2 Magnetization and stabilization Any superconductor exposed to a magnetic field tends to screen itself from this field. Other than in type I and ideal type II superconductors the shielding in high-Jc ‘hard’ superconductors is dominated by flux pinning effects. The key to understanding most phenomena is the critical state model, which essentially states that the current density prevailing in the material is either 0 or Jc . The result is a current and flux profile in the superconductor depending on the external field, the transport current and their history. This leads to a certain magnetization of the superconductor. A typical major magnetization curve of a round filamentary superconductor in a transverse magnetic field is shown in figure B8.1.2(a). The width of the magnetization loop ∆M is related to Jc m by

where f is a geometry factor equal to -34 π for a superconducting cylinder in a field perpendicular to its axis. It has to be pointed out that in contrast to magnetic materials ∆M cannot be interpreted as a materials property of a microscopic volume but is given by macroscopic currents as reflected by the product of Jc m and the dimension d over which the currents are flowing. In the case where the filaments of a wire have isotropic and homogeneous Jc values and are completely decoupled, d is equal to the filament diameter

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Figure B8.1.1. Non-Cu Jc versus B characteristics of NbTi, binary Nb3Sn and doped (NbTa)3Sn and (NbTaTi)3Sn multifilamentary superconductors.

and Jc m is nearly equal to Jc as measured by the transport current. A difference between Jc m , and Jc may occur because of the different sensitivities of resistive and magnetization-type measurements, as discussed in section B8.1.2.1. This interpretation breaks down when the filaments are interrupted or are very inhomogeneous longitudinally, or are coupled transversely either by the formation of bridges through the matrix or by a proximity effect. A magnetization curve above the inception of coupling due to small filament spacing s is shown in figure B8.1.2(b) and in figure B8.1.2(c) a strongly coupled case is shown, together with the decoupled case where the Cu matrix is replaced by a CuNi matrix. In the case of coupling, sometimes the concept of an effective filament diameter de f f is used; de f f is then calculated from ∆M by dividing through by Jc as measured by the resistive transport current method. However, this concept must be treated carefully e.g. because coupling usually is strongly field dependent such that de f f is field dependent. The occurrence of superconductor magnetization has important consequences for conductor performance. Obviously the critical state tends to be unstable because of the magnetic energy stored in the shielding currents. In the case of sufficiently large external disturbances an instability can occur leading to a flux jump with large energy dissipation which finally can lead to a quench of the superconductor. The disturbance initiating such a flux jump can be either of magnetic, thermal or mechanical nature. The existence of a positive feedback loop can be prevented by two measures: (i)

reduction of the magnetic energy available by fine subdivision of the superconductor volume such that the heat capacity of the material is high enough to limit heating up: ‘adiabatic stabilization’ (ii) damping of the flux change rate by a highly conducting normal conductor (Cu, Al) to reduce heat generation and allow heat removal to the coolant (e.g. helium at 4.2 K or 1.8 K): ‘dynamic stabilization’.

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Figure B8.1.2. Magnetization curves of NbTi multifilamentary wires: (a) 5 µm filaments in a Cu matrix without coupling (s = 0.5 µm); (b) 2.5 µm filaments in a Cu matrix above inception of coupling (s = 0.25 µm); (c) 0.54 µm filaments in a Cu matrix showing strong coupling and in a CuNi matrix without coupling (s = 0.12 µm).

The required subdivision can be calculated by energy and power balance calculations, respectively. In the case of a magnetic field perpendicular to the wire axis the adiabatic stabilization requires a diameter d of the superconducting material such that

with C the volumetric specific heat of the conductor and

Usually Jc decreases linearly with temperature such that ∆T = Tc ( Bo p ) — To p where Bo p is the operational field and To p the operational temperature. For typical parameters of NbTi at 4.2 K this yields d < 100 µm. The current-carrying capacity of such small filaments is limited to quite low values. Therefore a large number of such filaments is bundled together. This is usually done by embedding the filaments

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in a normal-conducting matrix. If this matrix is electrically and thermally highly conductive one can achieve high ‘dynamic stabilization’. To avoid an unacceptable temperature difference inside a filament, the following criterion must be fulfilled to prevent flux jumping

with ks c the thermal conductivity of the superconducting material, ρs t the resistivity of the stabilizer and α the stabilizer to superconductor area ratio. For NbTi in a highly conductive Cu matrix this requires d < 80 µm. It should be pointed out that the coincidence with the adiabatic stability criterion is purely incidental as completely different material properties enter equations (B8.1.2) and (B8.1.3). In practical conductors and magnet designs both types of stabilization occur simultaneously, such that the given criteria tend to be conservative. As can be seen from equation (B8.1.3), the resistivity of the stabilizer ρs t is an important parameter for the stability of a superconducting composite. In technical specifications this parameter is usually specified in terms of the residual resistivity ratio (RRR) which ranges typically from 70 to 150 in the case of Cu and 200 to 1000 in the case of Al. RRR is defined as the ratio of the resistance of the stabilizer at room temperature to the resistance at operating temperature and in zero magnetic field. In an external field the magnetoresistance has to be taken into account. The given stability criteria for the filament diameter do not represent absolute limits. Instability and quenching may also occur below the given limits due to large disturbances, e.g. by conductor movement. On the other hand successful operation of a conductor with larger filaments may be possible even above the limit (e.g. at 200 µm to 300 µm for NbTi at 4.2 K) in the case of high mechanical integrity (no conductor movement), high stabilizer content and good cooling conditions of the device. For some applications finer filaments are needed than required by the stability criteria. First of all, any magnetization current produces macroscopic external fields with a topology different from that of the transport current e.g. in a dipole magnet a sextupole field is generated. In high-quality magnets such as those required for particle accelerators (effective) filament diameters as small as 5 µm are needed to reduce the field disturbance to an acceptable level. Secondly, in the case of a pulsed field the magnetization leads to hysteresis losses which are given by the area of the magnetization loop Qh , as shown in figure B8.1.2, or any minor loop inside the major loop. For major loops these losses are proportional to Jc d o r Jc de f f . In order to calculate the losses from Jc without measuring the magnetization itself the field dependence Jc (B) must be known and filament bridging and/or proximity coupling must be excluded.

If we assume Jc = constant, i.e. a relatively small sweep amplitude ∆Bm but one large enough for full reversal of the magnetization, the hysteresis losses per cycle Qh in a perpendicular magnetic field can be estimated by

Equation (B8.1.5) is only valid for full penetration of the filaments by the alternating field, i.e. for Bm /Bp » 1 with the penetration field Bp = (4/π)µ0 Jc d. Well below penetration, i.e. for minor magnetization loops not leading to full reversal of the magnetization the losses are proportional to 1/( Jc d ). It has to be mentioned also that the presence of a transport current I increases the losses by a factor (1+i ) with i = I/Ic . This factor is also valid only for the full-penetration case.

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B8.1.2.3 Effects of filament coupling Filamentization and embedding of the superconducting filaments in a (highly conductive) matrix inevitably lead to coupling effects between the filaments. A rather trivial effect is the formation of direct contacts between the filaments by irregularities of the filamentary array introduced by an improper conductor assembly technique, by inhomogeneous deformation, i.e. filament sausaging and texture formation, and, for Nb3Sn conductors, by the growth of the filament volume during the reaction heat treatment. Filament bridging can be avoided by maintaining a highly homogeneous filamentary array and by a sufficient separation of the filament through conductor design (see section B8.1.3). Superconducting coupling can also occur, especially at low fields, through proximity effects when the spacing s between the filaments becomes comparable to the effective superconducting coherence length in the normal-conducting matrix material. At 4 K this length is of the order of 500 µm for pure copper but much smaller for Cu alloys such as CuNi and CuSn and especially in alloys with Mn additions because of the effective decoupling through the breaking up of the superconducting Cooper pairs by spin-flip scattering of electrons at the magnetic Mn atoms. Values of the coherence length down to 15 nm can be achieved (see section B8.2). Filament bridging and proximity coupling can both lead to a significant increase of magnetization effects, i.e. broadening of the peak at low external fields and widening of the magnetization curve (see figure B8.1.2). As a consequence, field distortions by magnetization currents and hysteresis losses may be much larger than expected from the nominal filament diameter. This has to be taken into account in finefilament NbTi conductors (d ≤ 6 µm, s < 1 µm) for pulse field or accelerator magnets and especially in a.c.-type superconductors (d ≤ 0.5 µm, s ≤ 100 nm) for which a Cu-alloy matrix has to be used. For NbTi-based a.c. conductors a pure CuNi matrix is used in the filamentary area. As already mentioned, Nb3Sn conductors tend to filament bridging and the related effects (field distortions and increase of hysteresis losses) may be very important if no appropriate measures are taken to avoid bridging. In some Nb3Sn conductors with very high current density even (partial) flux jumps have been observed due to the large effective filament diameter de f f . Even when superconducting coupling effects can be avoided, the matrix leads to normal-conducting coupling of filaments when the external field or the transport current is changed. This is most evident for a changing field transverse to the wire. In the case where all filaments run parallel to the wire axis a large voltage is introduced in the loop formed by the outer filaments driving currents in these filaments, with the current loops being closed by currents flowing partly through the normal-conducting matrix at the conductor ends. This coupling can effectively be reduced by reducing the area of the current loops. This is done by twisting the wire around its axis. As a result the filaments are running like a screw with a pitch lp . Nevertheless, a residual coupling remains leading to so-called coupling losses Qc when the external field is being cycled. For linear field sweeps with a rise time of Tm and an amplitude Bm these losses are given by

with the conductor time constant τ being

In (B8.1.7) ρ⊥ is the transverse resistivity of the composite wire which usually is close to the transverse resistivity of the matrix and (B8.1.6) is valid if τ « Tm . As a consequence of equations (B8.1.6) and (B8.1.7) coupling losses can be reduced by using a matrix with high transverse resistivity and applying a tight twist pitch. For mechanical reasons the twist pitch is usually limited to lp ≤ 10D with D equal to the

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wire diameter. For cost reasons such tight twist pitches are only applied when required by the operational conditions for a specific application. The second parameter in equation (B8.1.7) which can be used to reduce transverse coupling effects is the transverse resistivity ρ⊥. By using a Cu/CuNi mixed matrix instead of a pure Cu matrix (low transverse resistivity) or a pure CuNi matrix (bad stability and quench protection properties), a high transverse resistivity and a low longitudinal resistivity can be achieved simultaneously. At a wire diameter of 1 mm and a tight twist pitch Cu matrix conductors typically exhibit time constants of several milliseconds; with a mixed matrix time constants ≤ 100 µs can be achieved. The expected coupling losses can be estimated from equation (B8.1.6) for a given field cycle with characteristic time Tm . Unfortunately, twisting only decouples the filaments for transverse field components. When longitudinal field components occur in magnet systems with complicated winding and magnetic field geometry twisting in fact introduces a coupling which is not present in untwisted wires. In such cases losses in changing longitudinal field have to be considered and the application of a moderate twist may be appropriate. Finally, a current in a wire produces an azimuthal self-field and twisting has no influence on self-field effects. As a consequence large-diameter wires exhibit high losses with changing currents, and stability criteria with respect to flux jumps under self-field conditions have to be observed. As a rule monolithic wires should not have diameters larger than a few millimetres, especially under pulsed operation. Larger conductors are produced as fully transposed cables where all wires of the cables are changing position inside the cable cross-section within a transposition length Lp . Equations (B8.1.6) and (B8.1.7) can be used to estimate losses in transposed cables when replacing lp by Lp . However, it should be mentioned that the transverse resistivity ρ⊥ of a cable is difficult to predict because of the contact resistances involved. B8.1.3 Conductor fabrication As described in the previous sections, technical superconductors are mono- or multifilamentary composite wires with continuous superconducting filaments embedded in a normal-conducting matrix. The usual process for producing such a composite wire is to stack together the individual components at larger dimensions and to work down the composite by hot and/or cold working while maintaining a regular geometry of the filamentary array and continuity of the filaments. At least one hot-working step is desirable in order to achieve a good metallurgical bond between the individual components of the composite to allow its homogeneous deformation. This hot-working step usually consists of an extrusion process. In addition to bonding, extrusion allows easy assembly of the components in the form of billets so that large quantities of conductors can conform to the length requirements under economic conditions. Typical dimensions of extrusion billets are 150 to 250 mm diameter and 600 to 800 mm active length, resulting in typical units of 100 kg to 250 kg. This corresponds to lengths of about 5 to 100 km of wire, depending on wire diameter. Any heat treatment or hot-working step performed during composite fabrication for Jc optimization, annealing or metallurgical bonding is accompanied by diffusion and reaction processes at the interfaces which may lead to the formation of hard reaction products. This in turn makes it more difficult to work the composite and reduces the filament quality by producing sausaging effects and finally leads to filament and wire breakages. Not all combinations of matrix and filament materials are suitable for the fabrication of technical superconductors. For example, pure A1 is too soft in comparison with Nb or NbTi and a reliable working process is not possible. On the other hand Cu and Nb or NbTi are relatively well matched and coworking processes are possible. In fact Cu/NbTi composites so far represent the most successful technical superconductors. Nevertheless conductor design and fabrication processes have to be appropriately chosen to result in high-quality conductors.

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For the production of Cu/NbTi composites several methods of billet assembly may be used. For low filament numbers (up to a few hundred) NbTi rods and Cu tubes with an outer hexagonal geometry are assembled in an outer billet tube. Appropriate pieces of Cu are used to fill the voids between the filamentary array and the circular billet tube. For larger numbers of prefabricated filaments, hexagonalshaped monocore conductors are bundled inside the billet tube. Above several thousand filaments the hexagons become very small and their handling and regular bundling becomes difficult and expensive. Sometimes, therefore, round elements are used but the array tends to exhibit defects typical for twodimensional lattices in addition to cross-overs of elements along their lengths. A reliable fabrication is difficult to achieve. The typical approach for large filament numbers is therefore a two-stage bundling process whereby hexagonal stacking is applied in both stages. CuNi and Cu/CuNi matrix composites are produced in a very similar way by using CuNi tubes instead of or in addition to Cu tubes, thus providing high transverse resistivity while, in the case of a Cu/CuNi matrix, maintaining high longitudinal conductivity. At final dimension the wire must exhibit a high critical current density. The most active pinning centres in NbTi are elongated normal-conducting α − Ti precipitates. Accordingly the wire fabrication process includes intermittent heat treatments at typical temperatures of 380–400°C with a total heat treatment time of typically 100 h. In order to achieve the critical current densities given in figure B8.1.1 large area reductions by cold working are required, resulting in the optimum microstructure needed for high critical current densities. As a consequence, for NbTi the large billet sizes mentioned above are required not only to achieve long lengths but also to result in high Jc . Above wire diameters of about 1 mm gradual degradation of Jc compared to the values given in figure B8.1.1 has to be taken into account. Hot extrusion and also the heat treatments for Jc optimization can also have an adverse effect on Jc values through the formation of hard Cu—Ti intermetallics at the filament surface (figure B8.1.3(a)). Especially at filament diameters smaller than 10 µm the intermetallic particles formed lead to filament sausaging (variation of the area along their lengths) and a related degradation of Jc . To reduce and almost avoid this effect, fine-filament NbTi conductors are produced by applying an Nb diffusion barrier around each filament and thus using mono-core NbTi/Nb/Cu elements for the stacking process. The Nb thickness is designed to be thick enough to prevent intermetallic formation down to the diameter of the last heat

Figure B8.1.3. Filament quality: (a) NbTi filaments (20 µm diameter) with large amounts of NbTi/Cu intermetallics; (b) NbTi filaments (about 5 µm diameter) with Nb barriers preventing intermetallic formation.

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treatment of the multifilamentary wire (figure B8.1.3(b)). The final steps to make a NbTi composite wire fit for magnet application are twisting, insulation by applying a varnish and optional profiling into a rectangular shape. In the case of high-current conductors additional fabrication steps, such as cabling and the application of additional stabilizing and reinforcing components, have to be applied. Obviously Nb3Sn conductors cannot be produced in the same way as NbTi conductors due to the brittleness of the Nb3Sn phase. Nevertheless several fabrication methods have been developed for the production of multifilamentary Nb3Sn conductors. Some of them will be described here. In the ‘externalSn’ process Nb filaments are assembled in a Cu matrix and the Nb/Cu composite is worked in the same way as NbTi/Cu composites. At final diameter the wire is coated with Sn and, in a sequence of heat treatments with increasing temperature, Sn and Cu react to form CuSn alloys and Sn diffuses to the Nb filaments to form Nb3Sn. Unfortunately, the process is limited to relatively small diameters below 0.2 mm because of the restrictions related to stable layer thickness and the need to supply enough Sn. In the ‘internal-Sn’ process Sn sources are provided inside the composite as localized Sn reservoirs, either in the form of pure Sn or an Sn alloy containing small amounts of Cu or Mg in order to harden the tin and to optimize reaction heat treatment characteristics. Again the Nb3Sn phase is formed at final wire diameter by a sequence of heat treatment steps. This process in principle allows high critical current densitites if the Sn content in the composites is kept high. Disadvantages of the process are related to the softness and the low melting point of the Sn (alloy). Hot extrusion of composites containing Sn is not possible due to melting of Sn. As a consequence, metallurgical bonding of the elements in the composite is not optimum. This, in addition to the softness of the Sn (alloy) in comparison with the other elements of the composites, increases the tendency to mechanical instabilities during wire drawing and subsequent wire breakages. Research is still going on to resolve these issues. The discussed problems can be avoided in the ‘bronze’, process, which uses Nb filaments in a CuSn solid solution matrix. As a result of the solubility limit of Sn in Cu the content of Sn in the alloy is usually limited to about 13 to 15 wt%. In order to provide enough Sn a relatively high CuSn matrix to Nb filament area ratio of about three is required. This implies that the overall critical current densities of bronze-process conductors tend to be smaller than those of internal Sn conductors. On the other hand the hardness of the CuSn alloy allows an excellent filament quality with high n values, as required for persistent-mode magnets. Also, by redistributing the CuSn in such a way that the filament separation is sufficiently high, bridging of the filaments in the reacted stage can be avoided, allowing low hysteresis losses without sacrificing critical current density and workability of the composite. Figure B8.1.4 shows examples of filamentary areas with and without bridging. Usually for both the bronze process and the internal-Sn process the rod/filament design is used. Another alternative is to use the ‘jelly roll’ process in which alternate Nb and Cu sheets are formed into a roll and act as a filament bundle. A modification of this approach is the ‘modified jelly roll’ in which an extended Nb mesh instead of a solid Nb sheet is used. The jelly roll approach can be used in the bronze process as well as in the internal-Sn process. Unfortunately, it seems to be difficult to scale the process up for large quantities and large wire cross-sections. The Nb3Sn wires described so far are all based on the same basic principle of an intermediate formation of a CuSn bronze and a final reaction to form Nb3Sn at about 650–700°C for typically 50–200 h. As a consequence the composite would contain no stabilizing Cu. Any Cu must be added as a separate element and be protected from Sn poisoning by a diffusion barrier. Candidate barrier materials are Nb, Ta and V. Vanadium would be the cheapest choice but tends to lead to a reduction of the RRR value of the Cu due to nonoptimum barrier performance, at least at temperatures above 650°C. In addition, the formation of a nonsuperconducting Sn intermetallic phase tends to reduce the critical current density by Sn depletion. Niobium is often used as a barrier, especially in internal-Sn-type conductors because the mismatch of the hardness with respect to Sn and Cu is smaller than with Ta. It has to be taken into account that an Nb3Sn

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Figure B8.1.4. Bridging of filaments: (a) partly reacted Nb3Sn superconductor showing bridging of the filaments within a filament bundle; (b) fully reacted Nb3Sn with large filament separation without filament bridging.

layer is formed at the interfaces to the filamentary area. This tends to increase the critical current density but leads also to larger magnetization and hysteresis losses and increases the susceptibility to flux jumping. Tantalum as a barrier material avoids all the above-mentioned complications and is therefore the preferred choice, at least for bronze-route conductors, because of its compatibility with the hardness of the bronze matrix. An interesting approach avoiding the need for an extra barrier element can be found in the ‘Nb-tube’ process. Here Nb tubes embedded in a Cu matrix are filled with Sn sources, either in the form of Sn (alloy) or of an NbSn compound powder (e.g. NbSn2 ). During reaction heat treatment, an Nb3Sn layer is formed from the inside of the Nb tube and the outer part of the tube acts as a diffusion barrier. This approach leads to wires with very high critical current densities, but exhibits large filament diameters and is relatively expensive and difficult to scale for large quantities and for industrial production. Examples of cross-sections of internal-Sn, bronze-route and Nb-tube conductors are shown in figure B8.1.5. B8.1.4 Mechanical properties and strain sensistivity A superconductor in a magnet has to accommodate different mechanical loads, e.g. during coil winding, cool-down and operation. Usually the stresses during operation due to the magnetic forces are dominant. The typical stress level σ is given by

where J is the overall current density, B the magnetic field and R the coil radius. The conductor must withstand these stresses without mechanical damage and without degradation of the critical current density. NbTi conductors exhibit an Rp 0.2 value (stress at which the irreversible strain εi r r = 0.2%) well above 100 MPa, depending, of course, on the Cu to NbTi ratio because of the high strength of the NbTi alloy (Rp 0.2 ~ 500 MPa). The strength of the Cu matrix depends on the degree of

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Figure B8.1.5. Cross-sections of Nb3Sn conductors produced by different routes: (a) internal Sn (0.8 mm diameter); (b) bronze route (1.5 mm diameter); (c) Nb tube (0.5 mm diameter).

final cold working as does the RRR value. An optimum relation between Rp 0.2 and RRR, as defined by the magnet design, can be achieved by an adequate combination of final cold work and/or heat treatment. For Nb3Sn the strength of the composite is mainly given by the filamentary area and by the barrier material, because after reaction treatment the Cu stabilizer and also the bronze are in a soft condition. Typical Rp 0.2 values are between 100 MPa and 250 MPa depending on the Cu content and the barrier material and area ratio. The critical current density of a superconductor depends not only on temperature and magnetic field but also on the strain state. This dependence is small in NbTi but very pronounced in Nb3Sn, especially in high fields close to Bc 2 . Figure B8.1.6 shows Jc (ε ) for the case of binary Nb3Sn under uniaxial longitudinal strain. As the strain dependence scales with Bc 2 it is, at a given field value of B, less pronounced in Ta-and/or Ti-doped materials. It has to be noted that with no externally applied strain the Nb3Sn material is in a condition of intrinsic compressive strain εi ≈ 0.2–0.3%, depending on the conductor design such that applying an external tension first leads to an increase of Jc . It is also important to mention that the

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Figure B8.1.6. Strain sensistivity Jc (ε ) of Nb3Sn conductors. From Ekin (1983).

degradation under tensile strain becomes irreversible at a few tenths of one per cent because of crack initiation in the brittle Nb3Sn material. B8.1.5 Conductor examples B8.1.5.1 NbTi conductors Standard NbTi conductors for laboratory or nuclear magnetic resonance (NMR) applications typically have filament diameters between 40 µm and 120 µm with filament numbers between 40 and a few hundred and a Cu to NbTi ratio α between 1.3 and about 4. Examples of such wires are shown in figure B8.1.7. Smaller α values (down to about α = 0.9) are feasible, but the magnet must be designed carefully to avoid problems with quenching, training and degradation. In larger systems with larger stored energy, for example, for whole-body magnetic resonance imaging (MRI) scanners, higher α values are necessary mainly for protection purposes. Typical values of α are 4 to 15. A conductor cross-section of a 24-filament conductor with α = 6.5 is shown in figure B8.1.8(a). In this case the total amount of Cu is extruded and worked down as a part of the composite. Another possibility is to use a standard low-α wire and to solder the completed wire into a Cu profile. Such a ‘wire-in-channel’ conductor is shown in figure B8.1.8(b). This method allows flexible and cheap production of conductors with very high Cu to NbTi ratios. A certain disadavantage lies in the fact that this type of conductor cannot be insulated in a straightforward way with a varnish. Usually a braiding with textile yarn is used. For beam line magnets of particle accelerators with high field quality requirements fine-filament conductors are needed to reduce field distortions due to magnetization currents. Conductors with filament diameters down to 2.5 µm have been developed successfully. Typical examples are shown in figure B8.1.9(a) and figure B8.1.9(b) for single-stage bundling with 6000 filaments and double-stage bundling with 8900 filaments, respectively. For pulsed magnets, as needed in Tokamak-type nuclear fusion devices, low-a.c.-loss wires with a Cu/CuNi matrix are needed. Figure B8.1.10(a) shows an example of such a conductor with CuNi barriers reducing coupling losses (τ of the order of 200 µs at a wire diameter of 1.2 mm), fine filaments (10 µm)

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Figure B8.1.7. NbTi/Cu wires with low (a) and medium (b) Cu to NbTi ratios (diameter/height 1 to 2 mm, typically).

Figure B8.1.8. NbTi/Cu wires with a high Cu to NbTi ratio (diameter/height 1 to 2 mm, typically), (a) Monolith, (b) wire-in-channel.

to reduce hysteresis losses and large amounts of Cu to maintain high stability and good quench protection characteristics. This wire is suitable for cables of very large pulsed field coils such as the poloidal field coils of Tokamaks. A wire with less Cu stabilizer and therefore larger overall Jc suitable for superconducting machines (turbo generators) is shown in figure B8.1.10(b). B8.1.5.2 Nb3Sn conductors A description of different Nb3Sn conductor designs has already been given in section B8.1.3 and shown in figure B8.1.5. In this section some more details are given on bronze-route-type conductors. The conductor shown in figure B8.1.5(b) is based on so-called external stabilization i.e. the Cu is located at the periphery of the wire. This design allows Cu to non-Cu ratios of 0.3 to about 1.5. Lower values are impracticable for geometrical reasons and much higher values by hot- and cold-working limits. The regime of Cu to non-Cu ratios below 0.3 can be achieved by the design with internal stabilization, i.e. location of the Cu

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Figure B8.1.9. Fine-filament NbTi/Cu wires: (a) single-stage bundling of 6000 filaments (0.65 mm wire diameter); (b) double-stage bundling of 8900 filaments (1.1 mm wire diameter).

Figure B8.1.10. Mixed-matrix NbTi/Cu/CuNi wires (1.2 to 1.7 mm diameter): (a) for Tokamak poloidal field coils (large amount of Cu); (b) for turbo generators (small amount of Cu).

at the wire centre (figure B8.1.11(a)). The regime well above 1.5 is accessible by applying external and internal stabilization simultaneously. The conductors shown in figures B8.1.5(b) and B8.1.11(a), are designed to achieve high filament quality, i.e. high n values. They are therefore excellently suited for use in magnets operated in the persistent mode, e.g. in high-resolution NMR magnets. At present 18.8 T NMR magnets using these types of conductor are becoming standard and NMR systems with fields above 20 T are under development. Because of the small spacing of the filaments within a bundle these standard conductors exhibit substantial bridging (figure B8.1.3(a)) and are therefore not suited for applications requiring low hysteresis

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Figure B8.1.11. Nb3Sn conductors after using the bronze route: (a) internally stabilized wire (standard conductor for NMR, 0.8 mm diameter); (b) externally stabilized wire with low hysteresis losses and a large amount of Cu (for the ITER Tokamak reactor, 0.8 mm diameter).

losses. By redistribution of the bronze to achieve larger filament spacings bridging can be significantly reduced (figure B8.1.4(b)). Figure B8.1.11(b) shows the Nb3Sn strand developed for the ITER Tokamak fusion reactor. The effective filament diameter of this wire is only slightly higher than the calculated one such that the hysteresis losses are smaller than 100 mJ cm-3 per non-Cu volume for a full +3 T/– 3 T cycle. B8.1.5.3 Cabled conductors For magnets with large dimensions and/or large stored energy, high-current conductors (several kiloamperes to several tens of kiloamperes) are needed for protection purposes. These conductors are produced by cabling of a number of wires (‘strands’) to form a fully tranposed cable. The simplest form of a fully transposed cable is the flat (or ‘Rutherford’-type) cable. A cable produced for the Large Hadron Collider (LHC) at CERN is shown in figure B8.1.12. In this case the cable is slightly keystoned, i.e. made trapezoidal in order to better accommodate the geometry of the winding pack of a dipole magnet.

Figure B8.1.12. Flat (slightly keystoned) cable for LHC dipole magnets consisting of 26 strands with 1.29 mm diameter and 27 600 filaments each (5 µm diameter).

For some applications additional stabilizer material may be necessary to improve protection characteristics. An effective method to add stabilizing material is the co-extrusion of high-conductivity Al onto the completed cable. An example of such co-extruded conductors for detector magnets is shown in figure B8.1.13. Another method is to solder a superconducting cable into a Cu profile as described in section B8.1.5.1 for a wire-in-channel conductor.

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Figure B8.1.13. Example of an Al-stabilized conductor for large particle detector magnets (11 x 4 mm2).

For very large magnet systems such as those used in nuclear fusion applications mechanical considerations dominate the conductor design. This includes addition of mechanical reinforcing members into the conductor design and forced internal cooling to allow impregnation of the winding pack to guarantee high mechanical integrity. An example of such a cabled conductor is shown in figure B8.1.14. It was used in the POLO project for the demonstration of the feasibility of fast pulsed coils suitable for the poloidal field coils of Tokamak reactors and was therefore designed also to exhibit low a.c. losses. A strand similar to that shown in figure B8.1.10(a) was used and, in addition, highly resistive barriers were applied within the cable to increase ρ⊥ and thereby to decrease coupling losses in the cable. The cabled conductors for the international fusion project ITER are of modified but similar design and will use Nb3Sn low-loss strands as shown for example in figure B8.1.11(b).

Figure B8.1.14. Internally cooled cabled conductor with a stainless steel conduit as used in the POLO demonstration experiment for poloidal field coils in Tokamaks (25 x 25 mm2 ).

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B8.1.6 Conclusion and outlook Multifilamentary wires based on NbTi and Nb3Sn have reached a high degree of technical perfection. Standard conductors are available for industrial applications such as MRI and spectroscopy with NMR. In the latter case the frontier is still being pushed to higher resolution, i.e. fields higher than 20 T. For applications in large-scale basic research like beam line and detector magnets in high-energy physics and energy technology (nuclear fusion devices, superconducting magnetic energy storage (SMES), turbo generators) specially tailored strands and cables are available. In energy technology, except for nuclear fusion where no solution other than superconducting magnets exists, the introduction of superconductivity is still slow because of overall technical (related to the real or perceived complexity of the related cryogenic systems) and economical considerations. Whether the emergence of high-Tc superconductors will change the situation and will open ever new fields of applications in energy technologies such as current limiters and power cables has yet to be shown. Further reading Ekin J 1983 Materials at Low Temperature ed P R Reed and F C Clark (Metals Park, OH: American Society for Metals) Evetts J (ed) 1992 Concise Encyclopedia of Magnetic and Superconducting Materials (Oxford: Pergamon) Foner S and Schwartz B B (eds) 1981 Superconductor Materials Science (New York: Plenum) Krauth H 1988 Recent developments in NbTi superconductors IEEE Trans. Magn. MAG-24 1023 Krauth H, Szulczyk A and Thöner M 1994 Critical current density and magnetization of NbTi and Nb3Sn fine filament superconductors Critical Currents of Superconductors ed H W Weber (Singapore: World Scientific) Osamura K (ed) 1994 Composite Superconductors (New York: Dekker) Reed P R and Clark F C (eds) 1983 Materials at Low Temperature (Metals Park, OH: American Society for Metals) Suenaga M and Clark A F 1980 (eds) Filamentary A15 Superconductors (New York: Plenum) Thöner M, Krauth H, Szulczyk A, Heine K and Kemper M 1991 Nb3Sn multifilamentary superconductors: an updated comparison of different manufacturing routes IEEE Trans. Magn. MAG-27 2027 Wilson M N 1983 Superconducting Magnets (Oxford: Clarendon)

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B8.2 Low-Tc superconductors for 50–60 Hz applications T Verhaege, Y Laumond and A Lacaze

B8.2.1 Introduction The development in the 1960s and 1970s of new superconductors featuring properties such as high field, high current density and operation at liquid-helium temperatures made it possible to envisage many new applications in power equipment (Foner and Schwartz 1974). Initial applications such as in d.c. generator field windings were successful. Nevertheless, due to the magnitude of the a.c. losses, a.c. applications could not be developed at the same rate, type II superconductors being lossy in time-varying magnetic fields. To cope with this type of problem, filamentary superconducting wires started to be developed in the 1970s: the solution selected involved using smaller and smaller diameter filaments. When, finally, at the beginning of the 1980s wire fabrication technology for fine filamentary superconductors was sufficiently developed to make use of superconductors in a.c. power equipment attractive, the diameter of an individual wire filament could be brought down to below 1 µm (Dubots et al 1984, Hlasnik 1984). The research and development work that followed in the 1980s allowed problems such as proximity coupling between closely spaced filaments to be overcome. The production of long lengths of very low-loss ultra-fine multifilamentary wire for use in power equipment is now possible commercially. In these wires, the filament size is approximately 100 nm and the a.c. losses are lower than those of copper in a similar situation (Cave 1987, 1992). Some of the applications that can be envisaged are fault current limiters, lightweight 50–60 Hz power transformers and power generators with stator and rotor made from superconducting wire (Fevrier 1987, Verhaege et al 1993b). Another possible application consists of quick-response power regulators which are capable of injecting or absorbing both active and reactive power by way of an energy storage magnet. B8.2.2 Wire design As a result of high current densities (several 109 A m-2 ) and low a.c. losses, the use of fine filamentary wires in superconducting electrical machines is advantageous. The above properties result in lighter and more efficient machines. In addition, a high electrical resistance beyond the conductor critical current generates a natural current limiting effect during fault conditions. This only results in a small increase in magnetic forces and surge currents whilst allowing a simplification of the mechanical design. Thermo-electromagnetic stability is another property which requires the use of multifilamentary wire with a.c. currents and fields (when the wire must not quench suddenly to the normal state (Gueraud et al 1989). It is especially important to have stability in low applied fields with high critical current density. The stability is influenced by the filament diameter and twist pitch, matrix material, positioning of the filamentary region and also the diameter of the wire (the wire is more stable if it has a smaller diameter)

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therefore different wire configurations have been designed for different applications, depending on how dense the magnetic field is (Lacaze et al 1991). When we consider that 1 W dissipated at helium temperatures requires 500–1000 W of refrigerator power at room temperature, we are reminded of the fact that a.c. losses in these materials should be minimized. It is extremely difficult to obtain an accurate prediction of a.c. losses, because both hysteretic and eddy current contributions are needed (Fevrier 1983). It is, however, possible to apply approximate formulae in some cases. When the changing magnetic field has completely penetrated the filaments (usually relevant to ultra-fine filaments subjected to moderate field variations) the overall hysteretic losses in W m-3 are given approximately by

where α is a wire-dependent constant (of order unity), df is the effective filament diameter (where the filament diameter is the lower limit, increasing gradually as successive layers of filaments become saturated with current), Jc o v is the overall composite critical current density (dependent upon both temperature and field), Jo v is the circulating overall current density and |dBt /dt| is the instantaneous variation of the transverse magnetic induction. We can estimate the eddy current contribution ( also in W m-3 ) from

where β is a wire-dependent constant (of order unity), σ is the matrix effective conductivity (in the filamentary region) and p is the filament twist pitch. So we can deduce from equation (B8.2.1a) that by reducing the filament diameter we achieve a lowering of hysteretic losses, while keeping the overall critical current density high. Likewise, equation (B8.2.1b) shows that by shortening the filament twist pitch and decreasing the interfilament matrix conductivity eddy current losses can be diminished. The ductility of niobium—titanium means that it is used a great deal in industrial applications. Figure B8.2.1 shows a micrograph of the filamentary structure near the centre of the high-performance ultrafine multifilamentary wire made by AISA (GEC Alsthom Intermagnetics SA). The wire contains 956 340 Nb—46.5%Ti filaments embedded in a Cu–30%Ni matrix. The process used to produce such a wire is one of multiple hot extrusion, drawing and compaction. The filament sizes can be reduced to about 100 nm after these three stages. The wire in figure B8.2.1 at a final diameter of 0.15 mm has filaments of 131 nm in diameter and a twist pitch of 1.2 mm separated by 126 nm of a resistive copper-nickel matrix (with a resistivity of about 4 x 10-7 Ω m at liquid-helium temperatures, which is several thousand times more resistive than copper at the same temperature). Typical a.c. losses are shown in figure B8.2.2. Nb3Sn fine multifilamentary wires are more fragile but have a higher critical temperature (18 K instead of 9 K) and upper critical field (25–30 T instead of 11 T) and are currently being developed (Kubota et al 1986). B8.2.3 Critical currents While the filament diameter is being reduced, the critical current rises to a maximum at a filament diameter of about 100 nm and then it decreases as the diameter is reduced further (see figure B8.2.3). Although a complete model does not exist, magnetization measurements show that both reversible and irreversible surface currents and the possibilty of pinning matching effects are significant to explain the initial increase and the high values attained in Jc . If there are very high surface currents ≈ Hc l /λ , giving several times 1010 A m- 2, or depairing currents » 0.5 Bc /λ , giving several times 1011 A m-2, flowing at the filament surface, it is to be expected that the low-field critical current density will rise in line with the ratio of

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Figure B8.2.1. View of an ultra-fine multifilamentary wire, near the centre of the wire; each filamentary packet is about 8 µm in diameter and contains 1038 Nb—Ti filaments separated from each other by a highly resistive Cu—Ni matrix (AISA). Courtesy of GEC Alsthom.

Figure B8.2.2. Schematic a.c. losses for ultrafine filamentary wires measured in W A−1 m−1; the comparison with copper carrying 5 x 106 A m−2 has been normalized to 4.2 K using a refrigerator efficiency of 500 (the true copper losses at 300 K are about 4 x 10−2 W A−1 m−1).

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Figure B8.2.3. Critical current densities (at 1 µV cm−1) of the superconducting fraction for nontwisted short samples in a transverse field at 4.2 K. The jump in the curves for 0 T and 0.5 T, shown by arrows, is due to the self-field contribution of the current which is more evident for the larger-diameter wire.

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surface to volume, i.e. 1/df , up to the point where there is a significant overlap between the contributions from opposite surfaces. Proximity effects which reduce both Tc and Bc 2 can be held responsible for the subsequent depression of Jc for filaments less than ≈ 100 nm. The highest twist pitch that can possibly be used (down to just four times the wire diameter) is employed to reduce eddy current losses in a.c. uses. The problem caused here, inevitably, is that as the outer filaments increase in length with respect to the inner ones, filament distortions and a distribution in diameters occur. In figure B8.2.4(a), we can see the voltage—current characteristics for twisted and insulated wires with different filament diameters. The ‘n’ dependence (see figure B8.2.4(b)) shows the systematic flattening of the characteristics: n is the exponent in the empirical relation V = kIn , k = constant. This is commonly applied when measuring filament quality. There is a correlation between both the filament Jc distribution and the current transfer from filament to filament and the n values (for example, this can be caused by surface inhomogeneities) (Takacs 1988, Warnes 1986). Where the filaments are ultra-fine, these effects should be more noticeable as the ratio of surface to bulk increases. This could also partly explain the observation here that the n values are low and they depend on df , but the critical currents, for a criterion of 1 µV cm−1, remain very high, and the a.c. loss performance is apparently unaffected.

Figure B8.2.4. (a) Voltage—current characteristics at 1.5 T normalized to Ic for twisted and insulated wires showing the dependence on filament diameter, (b) The n index at 1.5 T (between 0.1 µV cm−1 and 1 µV cm−1 ) of the same wire as shown in (a).

B8.2.4 A.c. losses Proximity effects and flux-line effects are the two changes in behaviour which are observed as the filament diameter is reduced below about 1 µm. B8.2.4.1 Proximity effects Strong proximity coupling occurs if the spacing between filaments falls below a critical value which is characteristic of the matrix material. This proximity coupling increases the effective diameter of the

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Figure B8.2.5. Evidence of proximity effects (df is the filament diameter; the filament spacing is 0.26df ).

filaments and thus the losses. Figure B8.2.5 demonstrates this type of behaviour because the ratio of measured losses to calculated losses increases for spacing less than about 100 nm in a Cu–30% Ni matrix. Supercurrents are able to cross the matrix by means of proximity coupling; so filament packets mainly operate as bulk, rendering the low filament dimensions inefficient. There has been a great deal of research into proximity effects in low fields and a general feature is an exponential dependence on both normal layer thickness and field. The coupling critical current Jc between filaments can be written in the form

where Jc t 0 is a constant, dn the inter-filament distance and ξn e f f is dependent on the nature of the matrix and is the effective depth of penetration into the normal metal. It is possible to estimate the variation of ξn e f f with alloy content from its dependence on resistivity ρn and the mean free path ln from ξn e f f ≈ ln− 1/2 . So the estimate of the value of ξn e f f for Cu–30% Ni is thus ≈15 nm at 4.2 K; since the filament spacing is several times this value strong coupling is expected. The proximity effects can be reduced by increasing the filament spacing or by decreasing the value of ξn e f f , the latter either by increasing the matrix resistivity or by the addition of magnetic impurities such as manganese and nickel. For specific matrix materials, the important factors are the level of spacing at which the onset of coupling occurs and the optimum spacing for a given field application. Small coupling effects at low inductions are tolerable because the field dependence will strongly reduce them. B8.2.4.2 Flux-line effects in fine filaments A second type of behaviour is found for submicrometre filaments that are separated above the critical proximity coupling distance, showing losses that are much lower than those predicted by the Bean model. This behaviour is illustrated in figure B8.2.6, showing that this effect is amplitude dependent, the reduction being more marked for the finer filaments at low inductions (Cave et al 1992). This effect is interpreted as being a consequence of the beneficial role of the low-field properties of small superconductors such as the increase in effective Bc 1 , a reduction of flux entry and exit hysteresis, the low dimensionality of the flux lattice and reversible movement of the flux lines.

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Figure B8.2.6. Magnetization loops (only half of each loop is shown) for a transverse field excursion of +0.34 T symmetric about zero field for various filament diameters and spacings: a—df = 324 nm, dn =312 nm; b—df = 210 nm, dn = 202 nm; c—df = 131 nm, dn = 126 nm; d—df = 105 nm, dn = 102 nm.

This fact is illustrated by magnetization loops (figure B8.2.6). Hysteretic losses are defined as the loop area. A strong decrease occurs when the filament diameter decreases from 324 nm to 131 nm; on the other hand, proximity effects become apparent at low fields, for finer filaments and spacings. Optimized wires are thus a compromise between fine filaments, sufficient spacings and current transport capacities. B8.2.4.3 Losses of 50–60 Hz windings The calculation of a.c. losses in a 50 Hz winding is a hard task: one has to combine Maxwell equations with the Bean model, using proximity and flux-line effect models, in a complex geometry of assembled, multifilamentary and twisted wires. The self-field and the external transverse and longitudinal varying magnetic fields have different effects on the current distribution and resulting losses. (i)

The self-field pushes the current into the outer filament layers, as in the case of the ‘skin effect’ for a normal conductor; external filaments become saturated, with enhanced losses and compromised stability. The self-field thus makes necessary fine wire diameters. (ii) The transverse field component creates current loops in the filaments, which produce hysteretic losses, but also tend to moderate the self-field effects; both effects vanish with decreasing filament diameter. (iii) The transverse field also creates ‘coupling currents’, which use longitudinal superconducting pathways (filaments) and transverse resistive pathways (matrix). The matrix conductivity and the twist pitch must be reduced, in order to reduce the coupling current losses. (iv) The longitudinal component of the field tends to create current loops in twisted wires; these loops use the external filaments for one pathway, and the internal filaments for the other pathway. The phenomenon leads to saturation, and so to enhanced hysteretic losses and problematic stability. The twist pitch must be kept at a sufficient value to make this phenomenon acceptable in specified conditions of longitudinal field. A fairly good agreement is obtained between calculations and experiments for moderately complex conductors and windings. A typical result is given in figure B8.2.7, obtained for a solenoid made of a 0.3 mm wire, containing ≈ 106 Nb—Ti filaments in a Cu—30% Ni matrix (observe that the transport current varies proportionally

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Figure B8.2.7. Calculated and measured losses of a typical 50 Hz winding.

with the magnetic field). The ratio of the losses to the reactive power is generally less than 10−4, thus leading to an excellent efficiency, in spite of the high cryogenic cost of losses. B8.2.5 Stability The thermo-electromagnetic stability of the individual filaments at industrial frequencies is an important problem because of the low values of magnetic induction generally used in applications. One can typically define three values from 0.1 to 1 T. The first value, 0.1 T or self-induction, corresponds to fault current limiters (Fevrier 1987); the second one, 0.5 T, is the typical induction of a transformer; the last one, about 1 T, is the value relative to the stator of an alternator. These low values give rise to very high critical current densities, Jc , and thus to large self-field effects so that the current distribution within the wire becomes very inhomogeneous. The consequence of this is that the external layers of filaments will tend to reach their critical current before the inner layers transport a significant proportion of the current. This effect is more marked when k = B/I, i.e. the coupling coefficient between the magnetic induction and the current is low. The detrimental aspect of this effect, from an electromagnetic point of view, is that several saturated layers behave as a massive layer of the same thickness (Fevrier 1983). One of the main characteristics of Cu—Ni is a magnetic diffusivity much higher than the thermal diffusivity. This leads to an adiabatic stability criterion. Referring to the adiabatic stability model of a semi-infinite sheet, the criterion for the stability of a multifilamentary wire can be written

where Ns is the number of saturated layers, Ns M the maximal number of saturated layers before instabilities can occur, δ F the filament diameter, C the specific heat of the superconducting material, Jc the critical current density and dJc /dT the variation of Jc with respect to temperature. This expression shows that the electromagnetic stability is improved when the magnetic induction and the specific heat increase and when the temperature gradient of the critical current decreases. Moreover, a wire will be completely stable if Ns M is higher than the number of layers of filaments in the wire. To calculate the number of saturated layers (Ns ) we need to know the variation of the specific heat and of the critical temperature with the magnetic induction.

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The critical current density is specific to the wire processing, which is not the case for the specific heat. Figures B8.2.8 and B8.2.9 show the variation of the latter parameter and of the critical temperature. Figure B8.2.10 compares critical currents with d.c. and a.c. quench currents for a typical 50 Hz wire of 0.3 mm diameter. Experimental a.c. quench currents correspond roughly to theoretical values, with an important dispersion due to mechanical disturbances. One problem to solve is the lack of mechanical stability. Impregnation can be a solution. As a result of the bad coefficient of heat transfer of the epoxy, we must have good cooling conditions of the winding (channels, etc). This type of wire is devoted to applications in magnetic fields of 1 T and more, where its stability can be considered as sufficient. For applications in lower magnetic fields, finer wires are required.

Figure B8.2.8. Specific heat of Nb—Ti.

Figure B8.2.9. Variation of the critical temperature with the induction for filaments having a diameter of: broken curve—5 µm; full curve—0.13 µm.

Figure B8.2.10. Limits of stability for a 50 Hz wire of 0.3 mm diameter: full curve—d.c. critical current; long-dashed curve—a.c. theoretical quench current; I—a.c. experimental quench current (short sample); ♦ and • —d.c. and a.c. experimental quench current (coil).

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B8.2.6 Protection Protection against the consequences of an unexpected quench is one of the most worrying problems for superconducting-windings applications. The intensive and localized Joule heating can produce catastrophic damage: (i) (ii) (iii)

conductor breaking or deformation, due to expansion stresses, breakdown, due to over-voltage, structural mechanical damage, due to helium pressure or unbalanced electromagnetic forces.

In the most usual mode of protection, a large part of the conductor is made of copper (or aluminium), with a very low resistivity at low temperature (≈ 10-10 Ω m). Active protection detects the quench, and transfers part of the stored energy out of the cryostatic zone, for example into a resistor or into the network. However, the speed of current decay is limited by the maximum acceptable inductive voltage of the winding. An essential parameter for protection is the normal-zone propagation velocity: high velocities are favourable, insofar as they aid the current decay, and reduce the levels of localized overheating and stress. The classical mechanism of propagation combines Joule heating in the normal zone, and thermal diffusion along the wire, across the propagation front (see figure B8.2.11(a)). The velocity in that case is less than the adiabatic limit

which gives practical values of 30–100 m s−1. Here V (m s−1 ) is the velocity of axial propagation, J (A m−2 ) the overall current density, γ c (J m−3 K−1 ) the volumic heat capacity at Ts , k (W m−1 K−1 ) the thermal conductivity at Ts , r (Ω m) the electrical resistivity at Ts , T0 (K) the temperature of

Figure B8.2.11. Different quench propagation modes: (a) the usual propagation mode; (b) the transfer propagation mode; (c) the mass transition mode.

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the superconducting zone and Ts (K) the temperature on the propagation front, currently calculated as -12(Tc (B) + Tg ), where Tg is the ‘current-sharing temperature’ for which J = Jc (Tg , B). The axial normal-zone propagation is accompanied by transverse propagation, from turn to turn and from layer to layer, possibly assisted by heat drains. As copper creates a.c. losses, the copper content of a.c. wires has to be minimized. 50 Hz wires have such a high Jc and such a high resistivity in the normal state that they cannot withstand this normal state without excessive thermal excursion, unless the current decays extremely rapidly. This is possible, using fast propagation modes (Iwakuma et al 1990, Verhaege et al 1992, Vysotsky et al 1992); some of them are based on the presence of a resistive wire in the assembled conductor. In the usual mode (figure B8.2.11(a)), propagation is governed by the thermal diffusion from the normal (quenched) zone to the superconducting zone. In the case of the ‘transfer propagation mode’ (figure B8.2.11(b)), heat is produced far away in the superconducting zone by current transferring from the superconducting wires to a resistive wire. In the case of the ‘mass transition mode’ (figure B8.2.11(c)), an insulated resistive wire carries part of the current and operates as a heater on the whole winding. The transfer propagation mode (figure B8.2.11(b)) is a useful arrangement. The conductor contains a central resistive wire, which holds much more copper than the superconducting wires; consequently, the former has a much lower resistivity than the latter in the normal state, especially at cryogenic temperatures. Important contact resistances are assumed to exist between the central and the superconducting wires. Far back into the normal zone, a large part of the current is carried by the central wire, whereas far into the superconducting zone, the current is totally carried by the superconducting wires; a transfer zone exists and, the larger the value of the contact resistance, the longer this zone. As a consequence, heat is produced in the superconducting zone, downstream from the propagation front. This creates the rise in temperature necessary for ultra-rapid propagation. In a simplified approach, we can suppose that the transfer is completed on the propagation front, where a current I0 (A) = Ig(rs /(rs + r )) flows in the central wire. The front advances at a speed V, which has to be determined. The current I (A) in the central wire can be calculated as

where Ig (A) is the total current in the conductor, t (s) the time, x (m) the distance downstream from the propagation front at t = 0, s (Ω−1 m−1) the contact conductance between the superconducting and central wires, α (m-1) = pr σ, r (Ωm-1) the resistance of the central wire at low temperature and rs ((Ω m−1 ) the resistance of the superconducting wires in the normal state. The power P1 (W m−1 ) = r I 2 dissipated in the central wire is identical to the power P2 = (1/σ ) (dI/dx)2 dissipated in the contact resistance. The sum and integration of these two powers determine the total energy dissipated

in particular on the propagation front:

If we assume, as an approximation, an adiabatic evolution of the strand, Q(0) corresponds to the enthalpy increase, between T0 and Ts , then Q(0) ≈ ∆H , with S (m2 ) the section of the strand and ∆H (J m− 3 ) the enthalpy increase equal to H (Ts ) — H (T0 ).

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The speed V of the front is therefore

This expression bears no similarity to (B8.2.3) and gives typical values of more than 1 km s-1. Such values have effectively been measured. Figure B8.2.12 illustrates the mass quench obtained on an impregnated coil (four layers, L = 2.3 mH , conductor length = 70 m). The conductor was made of six strands around an insulated Cu—Ni wire, used for quench detection. Each strand was made of six superconducting copper-free wires (diameter 0.2 mm), around one insulated Cu/Cu—Ni wire, used for mass transition initiation.

Figure B8.2.12. A combination of active and passive protection.

The current I1 in the central wire is near zero under normal conditions, and rapidly rises after the partial quench initiation, observed near the peak value of the main current I2. The detection system sets an alarm, when I1 overruns a threshold value. In response to this alarm, the circuit-breaker operates after 25 ms. The mass transition appears 7 ms after the partial quench initiation, making the residual current I2 very low, and thus the heat dissipation acceptable, for several tens of milliseconds. B8.2.7 High-current conductors Numerous fine wires must be assembled in order to obtain high 50 Hz current capacities (Lacaze et al 1992, Amemiya et al 1993). This leads to many difficulties, which have not yet been completely solved: (i) (ii) (iii) (iv) (v)

technology and cost of the fabrication processes, mechanical stability of elementary wires inside the assembled conductor, increasing conductor self-field, transfer and eddy current losses in the junctions with the leads liable to initiate quenches, heating induced by losses, which reduces Jc ,

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(vi) unbalanced currents in the different wires, due to unbalanced magnetic fluxes, which are liable to initiate the quench of the most loaded wire. As a first step towards industrial multi-kiloamp conductors, a particular wire type has been defined, called the ‘C-type wire’ (Verhaege et al 1993a). It is composed of six superconducting subwires, around one nonsuperconducting central subwire (a conductor made of six C wires is reproduced in figure B8.2.13). It, however, looks like a monolithic wire of regular circular section. Its coherence is sufficient to allow various handling methods like insulation winding or assembling, without particular problems. It also presents an enlarged flexibility, compared with an actual monolithic wire of the same diameter. The typical wire diameter is 0.5 mm. Each of its six superconducting subwires contains 186 252 Nb—Ti 0.16 µm diameter filaments in a Cu–30 wt% Ni matrix, and is totally copper-free. The central nonsuperconducting wire is made of Cu filaments in a Cu—Ni matrix, designed to optimize the quench behaviour.

Figure B8.2.13. A 500 A root mean square conductor assembly of six C wires.

The fabrication process is made possible by the excellent metallurgical behaviour of the Nb—Ti/Cu— Ni composites. The subwires are twisted and assembled at a chosen diameter in the form of a classical strand. This strand is then compacted and reduced by cold drawing, and possibly completed by twisting. Furthermore, the drastic deformation of subwires does not seriously reduce their critical current densities or their consistency. The lengths and costs involved in the processing are largely reduced when the assembly is made with a relatively high subwire diameter, because the final reduction of the subwire is performed by cold drawing of the relatively large wire. Insulation of the subwires is excluded because of the drastic deformation this would impose. Stability and losses of this type of wire are comparable with those of its elementary subwires, when the current transport is about six times higher. It benefits from the transfer propagation mode, with measure propagation velocities of ~7.5 km s-1. Larger conductors are now being assembled, using ultra-fine C-type wires as basic elements. Figure B8.2.13 shows a typical assembly of six C wires, around a Cu–Ni central wire. Difficulties still appear for currents over 1 kA, which therefore require a high level of carb

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B8.2.8 Conclusion Low-Tc superconducting conductors for 50–60 Hz use are now industrially available, even if difficulties remain in multi-kiloamp applications. Rated current densities are about 100 times higher than in conventional conductors, therefore very unusual specific powers can be obtained. A.c. losses are so reduced that the cryogenic constraints become acceptable. This makes applications, such as in current limiters, transformers or, in the future, in a.c. motors and generators possible. Acknowledgments We are very grateful to J R Cave and A Gueraud for their active contribution to the preparation of this article. References Amemiya N, Ohizumi T, Ikeda N, Hlasnik I and Tsukamoto O 1993 Instabilities in multistrands a.c. superconducting cables caused by longitudinal magnetic field with transverse magnetic field IEEE Trans. Appl.Supercond. AS-3 160 Cave J R 1987 Electromagnetic properties of ultra-fine filamentary superconductors Cryogenics 29 304-11 Cave J R 1992 Ac applications of superconducting materials Concise Encyclopedia of Magnetic and Superconducting Materials ed J Evetts (Oxford: Pergamon) Dubots P, Fevrier A, Renard J-C, Goyer J and Hoang Gia Ky 1984 Behavior of multifilamentary Nb—Ti conductors with very fine filaments under a.c. magnetic fields J. Physique Coll. 45 467–70 Fevrier A 1983 Losses in a twisted multifilamentary superconducting composite submitted to any space and time variations of the electromagnetic surrounding Cryogenics 23 185–200 Fevrier A 1987 Latest news about superconducting a.c. machines Proc. 10th Int. Conf. on Magnet Technology (Boston, MA, 1987) Foner S and Schwartz B (ed) 1974 Superconducting Machines and Devices: Large Systems Application (New York: Plenum) Gueraud A, Tavernier J-P, Fevrier A, Laumond Y, Lacaze A, Dalle B and Ansart A 1989 Thermo-electromagnetic stability of ultra-fine multifilamentary superconducting cables for industrial frequency use Proc. 11th Int. Conf. on Magnet Technology (Tsukuba, 1989) Hlasnik I 1984 Prospects of multifilamentary superconductor ac 50 Hz applications J. Physique Coll. 45 459–66 Iwakuma M, Kanetaka H, Tasak K, Funadi K, Takeo M and Yamafugi K 1990 Abnormal quench process with very fast elongation of normal zone in multistrand superconducting cables Cryogenics 30 Kubota Y, Okon H and Ogasawara T 1986 Development of a Nb3Sn multifilamentary composite conductor for ac use Cryogenics 26 654–9 Lacaze A, Laumond Y, Bonnet P, Fevrier A, Verhaege T and Ansart T 1992 Coil performances of superconducting cables for ac applications IEEE Trans. Magn. MAG-28 767 Lacaze A, Laumond Y, Tavergnier J-P, Fevrier A, Verhaege T, Dalle B and Ansart A 1991 Coils performances of superconducting cables for 50–60 Hz applications IEEE Trans. Magn. MAG–27 2178–81 Takacs S 1988 Resistive state of inhomogeneous superconducting composites Cryogenics 28 Verhaege T, Agnoux C, Tavergnier J-P, Lacaze A and Collet M 1992 Protection of superconducting ac windings IEEE Trans. Magn. MAG-28 751 Verhaege T, Estop P, Weber W, Lacaze A, Laumond Y, Bonnet P and Ansart A 1993a A new class of ac superconducting conductors IEEE Trans. Appl. Supercond. AS–3 164 Verhaege T, Tavergnier J-P, Agnoux C, Cottevielle C, Laumond Y, Bekhaled M, Bonnet P, Collet M and Pham V D 1993b Experimental 7.2 kV rms/1 kA rms/3 kA peak current limiter system IEEE Trans. Appl. Supercond. AS-3 574 Vysotsky V S, Tsikhom V N and Mulder G B J 1992 Quench development in superconducting cable having insulated strands with high resistive matrix IEEE Trans. Magn. MAG-28 735 Varnes W H 1986 The resistive critical current transition in composite superconductors PhD Thesis University of Wisconsin, Madison

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B9.1 Chevrel phases

B Seeber

B9.1.1 Introduction Today’s commercial superconducting ultra-high-field magnets can generate about 21 T at 2.2 K (λ -point of liquid helium) (see also chapter G1). The present upper limit for commercial magnets designed for nuclear magnetic resonance (NMR) spectroscopy is 18.8 T (see section G2.1). In these magnets the used superconductors are multifilamentary Nb—Ti for the outer low-field section of the magnet and Nb3Sn for the inner high-field section. These materials must operate at their limit due to the material-specific upper critical field Bc 2 . Near Bc 2 the critical current density Jc falls rapidly to zero which gives a limit for the economic use of the superconductor. The Jc performance of Nb—Ti is considered to be near the optimum of what can be achieved. In the case of Nb3Sn, work is still needed to ameliorate the Jc at highest fields but, as already said, the limitation caused by Bc 2 is imposed by nature. Depending on the chemical composition, Bc 2 (4.2 K) of Nb3Sn may vary between about 23 T and 29 T (see section B8.1). There are only a few superconductors with Bc 2 higher than for Nb3Sn and most of them belong to the class of high-Tc superconductors (see sections B9.2 and B9.3). The best known superconductor from the Chevrel phase family is PbMo6S8 (PMS) which has an upper critical field Bc 2 of ~51 T at 4.2 K. This very high Bc 2 is obtained although the critical temperature of PMS is ~15 K, similar to that of Nb3Sn (~18 K). Measurements in pulsed magnetic fields in the range of 50 T and specific heat studies in d.c. fields up to 25 T confirm this extremely high Bc 2 (Cors 1990, Foner et al 1974, Odermatt et al 1974, Van der Meulen 1995). Another remarkable property of PMS is the increase of Bc 2 to ~58 T by reducing the temperature from 4.2 K to 1.8 K, leading to an important improvement in the current density. Obviously, this material is suited for the generation of ultra-high magnetic fields above 20 T. Industrial wire manufacturing of PbMo6S8 has been under development since about 1980 (Seeber et al 1981) but slowed down soon after the discovery of high temperature superconductivity. The most promising concept uses molybdenum as an electrical stabilizer around the PMS filaments embedded in a stainless steel matrix. This concept allows a low matrix-to-superconductor ratio α (α = 1.5 has been demonstrated) and outstanding mechanical performance due to the presence of stainless steel. The mechanical strength of PMS wires is much better than that of Nb—Ti, Nb3Sn and high-Tc superconductors. It is important to mention that the increasing field and size of a magnet requires mechanically strong superconductors (the Lorentz force F ∝ B J R with B the magnetic induction, J the current density and R the radius of the considered winding of the magnet). If the superconductor is mechanically weak, an appropriate mechanical inforcement is necessary which decreases the overall current density of the conductor, resulting in an additional increase in the size and the cost of the magnet.

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B9.1.2 Synthesis of PbMo6S8 Due to the volatile character of constituents such as lead (Pb) and sulphur (S) (low melting temperature and high vapour pressure) PMS is normally synthesized by a solid-state diffusion process. The simplest method is to mix Pb with Mo powder and S powder. The concentration corresponds to the stoichiometric composition according to the formula PbMo6S8. The elements are sealed under vacuum in a quartz ampoule and then heated up slowly to about 1000°C. The maximum of the temperature is limited by the onset of softening of the quartz ampoule. There are several variants to synthesize PMS; most of them start either with Pb, Mo and S, or with Pb, Mo and MoS2. As crucibles one can use materials such as quartz, molybdenum or boron nitride. It is interesting to note that the temperature for the formation of the superconducting PMS phase can vary between 450°C and 1650°C (Decroux et al 1993). This indicates an extraordinary phase stability of PMS which is a very important advantage over high-Tc superconductors. Further details of the new low temperature synthesis of PMS are given by Rabiller et al (1994). In this case the PMS particle size can be adjusted down to 200 nm whereas the high-temperature synthesis (1650°C) produces PMS particles of up to 100 µm. In both extreme cases it is possible to achieve a homogeneous PMS powder with an optimal critical temperature, namely ~15 K (Decroux et al 1993). The diversity of high-quality PMS powder is a crucial point for the deformation process required for the fabrication of PMS wires. For instance, it is known from powder metallurgy that the deformation temperature is strongly influenced by the particle size and its distribution. Down to a certain limit a smaller particle size decreases the optimum deformation temperature. Consequently there exists an important margin to adjust the PMS powder to the deformation of the used matrix materials. B9.1.3 Fabrication of wires PbMo6S8 wires cannot be manufactured like Nb—Ti and Nb3Sn. Powder metallurgy must be used and the research and development effort, up to now, has been concentrated on monofilamentary wires. However, the feasibility of obtaining multifilamentary wires by the powder route has been demonstrated (Grill et al 1989, Willis et al 1995). When considering the starting powder, one has to distinguish between two different approaches. The first uses a precursor powder mixture containing the constituents of PMS (in situ method) and the second takes PMS powder (pre-reacted method). An alternative is the mixture of both types of powder. The powders are normally cold isostatically pressed, machined and inserted into a metallic tube. Unfortunately copper cannot be used because it diffuses easily into the PMS and substantially reduces the critical temperature. Useful materials for this tube, which acts as a diffusion barrier for the later heat treatment or the hot drawing process, are niobium, tantalum or molybdenum. After sealing under vacuum, the billet is inserted in a second tube for thermal and mechanical stabilization. Due to the low thermal expansion of the above mentioned barrier materials with respect to PMS, the matrix must also act as a thermal stress compensator (Seeber et al 1987). Without thermal stress compensation, the PMS filament would come under tensile strain by cooling to helium temperatures. Above a certain tensile strain the superconductor will be damaged in an irreversible manner. As stainless steel has a greater thermal contraction than Nb, Ta or Mo, it is put around the latter to adjust the thermally induced pre-strain of PMS at low temperatures which should be preferentially compressive. This also makes it possible to control the strain tolerance of PMS wires for the construction of coils, which will be shown later. Thermal stabilization of wires with an Nb or Ta barrier is achieved by copper (between the barrier and the stainless steel). In the case of an Mo barrier, the latter is a rather good electrical conductor and no additional stabilization seems to be required. Wire manufacturing at room temperature needs preferably a precursor powder or at least 10–20% of this powder mixed to a pre-reacted PMS powder. In the case of a molybdenum barrier of hot deformation is performed and pre-reacted PMS powder is of advantage.

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The next step in wire manufacturing is extrusion and drawing. At present two technologies are available. The first uses an Nb or Ta sheath around the superconductor and wire drawing at ambient temperature is possible under the condition that the density of the powder core is below 80%. If the density of the powder core becomes too high, there is a tendency for the barrier to leak or even for the wire to break. Wires can be manufactured on a laboratory scale up to about 200 m (wire diameter 0.4 mm). After wire drawing, the precursor powder core (also a mixture of precursor and pre-reacted powder) must be annealed in order to synthesize the superconducting PMS phase. Furthermore, the heat treatment is also necessary to sinter the powder particles together in order to achieve high critical currents. The interface between Nb and PMS after different heat treatments was investigated in detail by Rabiller and coworkers (Rabiller 1991, Rabiller et al 1992). There is a tendency for sulphur diffusion towards the niobium barrier, forming a lamella-like interdiffusion layer. With increasing time, the layer grows in the direction of the PMS and in extreme cases destroys the superconductor. Typical annealing parameters to achieve convenient Jc values are between 900°C and 1000°C for 1–2 h, but the interdiffusion layer can be a few micrometres thick, depending on the conditions. Although lower temperatures can be used to form the PMS compound (<500°C) they are probably too low for a good connection between grains and consequently for a good critical current. The kinetics of the inter-layer growth can be influenced by various parameters such as temperature, annealing time, constituents of the precursor powder and the density of the powder core. Some improvement is possible by taking a pre-reacted powder but, up to now, no way has been found to prevent the interdiffusion layer. This problem will obviously become worse in the case of small PMS filament diameters, necessary for thermal stabilization. The second available wire-drawing technology uses molybdenum as a barrier. The main advantage of this technology is not only the ideal barrier material (no chemical interaction between Mo and PMS) but also that standard hot deformation techniques of powder metallurgy can be used. At a first glance it seems that the hot deformation process, required for the deformation of Mo, is a disadvantage. However, once optimized, this can be very beneficial for the quality of grain boundaries which are at present one of the limiting factors for the critical current (see later in this section). Finally it should be mentioned that this type of PMS wire with up to 1 km length (diameter 0.4 mm) has been developed in Europe and Japan with standard industrial production machinery (Seeber et al 1981, Yamasaki and Kimura 1988, Yamasaki et al 1990, 1991). Preferably pre-reacted PMS powder is used, eliminating the difficult procedure of mixing different kinds of precursor powder and segregation problems during wire drawing such as encountered in the Nb/Ta route. Up to now, an annealing was necessary after drawing in order to sinter the powder core. However, if in the future an optimized hot deformation process is available, it is an open question whether this annealing would still be required. Furthermore, with an optimized manufacturing process of PMS wires, one can expect an increase in the acceptable bending strain which would eventually allow the construction of magnets using the react and wind technique. Monofilamentary wires, up to 1 km in length (wire diameter of 0.4 mm), have been reported with the present dimension of the extrusion billet shown in figure B9.1.1. It is worth mentioning that wires up to 40% of PMS (α = 1.5, OD = 0.4 mm) have already been manufactured (figure B9.1.2(b)). The main problem that can occur by decreasing α is a too strong reduction in the thickness of the Mo barrier which increases the probability of leakage. Micrographs of cross-sections with increasing content of PMS are shown in figure B9.1.2. B9.1.4 Intrinsic physical properties B9.1.4.1 Critical temperature Any preparation with no contamination by other elements (e.g. O2 , Fe) and which assures a precise and homogeneous stoichiometric composition of PbMo6S8 gives a Tc between 14 K and 15 K (Decroux et al 1993). It is interesting to note that this is true for synthesis temperatures between 450°C and 1650°C.

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Figure B9.1.1. The construction of the extrusion billet of a PMS wire. The outer can is stainless steel, followed by the diffusion barrier of molybdenum and the cold isostatically pressed and machined PMS powder core. Courtesy of Plansee AG.

PbxMo6Sy remains a single-phase material for a lead concentration in the range ~0.92 < x < 1 and for a sulphur concentration in the range ~7.4 < y < 8. Details of this phase field depend on the temperature and on the appropriate cool-down rate. Depending on the deviation from the exact stoichiometry, Tc may be reduced to about 12.5 K (Hauck 1977, Krabbes and Oppermann 1981, Yamasaki and Kimura 1986). If the critical temperature is further reduced, other effects, or a combination of them, are at the origin. As an example, data are available for the substitution of sulphur by oxygen which show a reduction of Tc down to 10 K for PbMo6S8–y with y = 0.25 (Hinks et al 1983, Selvam et al 1991). This may be the reason why PMS synthesized in quartz ampoules often has a reduced Tc because of the release of oxygen from the quartz (dissociated H2O). Another mechanism is the compressive stress dependence of Tc , which is of the order of -0.2 K kbar–1 (Shelton et al 1975). No data are available on Tc under tensile stress. The stress dependence of Tc may become important in wires where the PMS is normally under compressive pre-stress. Because Bc 2 is related to Tc (see below) the field dependence of the critical current at high fields is influenced. A few words should be said regarding the measurement of the critical temperature. The above-mentioned Tc values are obtained inductively, i.e. by a measurement of the a.c. susceptibility versus temperature and, where indicated, Tc is defined as the mid-point of the transition from the normal state to the superconducting state. Experience with granular superconductors, like sintered high-Tc superconductors and also PMS, shows that there are two transitions: the first at higher temperatures is due to the intrinsic diamagnetic shielding of grains (if the size of grains is bigger then the London penetration depth λL ) and the second, at lower temperatures, is the response from coupling diamagnetic shielding (between grains). Consequently, although this second transition is not directly material dependent, it is a measure of the quality of the grain boundaries (connection of grains) with its own effective Tc , Jc and Bc 2 . For further details see section B7.1 and the article by Goldfarb et al (1992). In the case of PMS one often observes

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Figure B9.1.2. (a) Cross-section of a PMS monofilamentary wire with an outer diameter of 0.4 mm. The fraction of cross-sections is: 31% PMS, 18% Mo and 51% stainless steel. Courtesy of Plansee AG. (b) Cross-section of a PMS monofilamentary wire with an outer diameter of 0.4 mm. The fraction of cross-sections is: 40% PMS, 9% Mo and 51% stainless steel. Courtesy of Plansee AG.

a granular behaviour (Cattani et al 1991, Decroux et al 1990). For this reason Tc values given in the literature should specify the a.c. field amplitude as well as whether the onset or the mid-point of the intrinsic or coupling transition has been considered. B9.1.4.2 Upper critical field Today there is no doubt about the upper critical field of PbMo6S8. Bc 2 ≥ 51 T at 4.2 K and goes up to ≥58 T at 1.8 K. Because this range of field is not accessible for d.c. fields one normally measures the slope ∂Bc 2 / ∂T|Tc up to fields as high as possible. Bc 2 in the dirty limit can then be calculated for other temperatures according to

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Figure B9.1.3. The Bc 2 (orbital critical field in the dirty-limit approximation) of four different PbMo6S8 bulk samples as a function of the critical temperature (isoentropic transition) obtained by measurements of the specific heat in d.c. fields up to 25 T (Cors 1990, Cors et al 1990, Van der Meulen 1995). Experimental data are fitted by a quadratic polynomial.

and

with t = T/Tc the reduced temperature. The pre-factor 0.693 increases to 0.726 for the orbital critical field in the clean-limit approximation (Werthamer et al 1966). ρAG is the Abrikosov-Gorkov function (Helfand and Werthamer 1966, section B7.3). In figure B9.1.3 calorimetric The Bc 2 measurements of four different PMS bulk samples (Cors 1990, Cors et al 1990) are plotted versus the critical temperature and fitted by a polynomial of degree two. The critical temperature in this figure is measured calorimetrically (isoentropic transition). Because the width of the calorimetric transitions are in the order of 0.5 K (independent of the applied field up to 25 T) the samples can be considered to be very homogeneous. Note that there is a maximum of Bc 2 at a Tc of about 13 K. This maximum can be explained by a subtle interplay between the dirty-limit and cleanlimit behaviour of PMS (Cors et al 1990, Decroux et al 1993). As has been shown in the Chevrel phase model system for Mo6Se8, with a Bc 2 (0) of about 10 T and therefore easily accessible by the field of an ordinary superconducting magnet, the measured critical field is above the calculated values of the orbital critical field in the dirty-limit approximation (Decroux 1980). Consequently the Bc 2 in figure B9.1.3 must be considered as a lower bound estimate. Finally it should be emphasized that Bc 2 studies of PMS single crystals allowed an estimate of the anisotropy of Bc 2 which is in the order of 10–20%, parallel and perpendicular to the crystallographic ternary axis (Decroux et al 1978). This is much less than that in high-Tc superconductors and PMS can be considered as quasi-isotropic. In the literature different Bc 2 values of PMS have often been reported. One reason for this is that, under certain preparation conditions, PMS may show a granular behaviour. In such a case one has to distinguish between the Bc 2 of grains and the effective Bc 2 of grain boundaries. The latter is detected by Bc 2 experiments where a critical (transport) current versus field is measured for different temperatures. The situation is similar in experiments where screening currents are used for the determination of Bc 2 such as in an a.c. susceptibility measurement versus temperature at different d.c. fields or in experiments where the penetration of a magnetic field in the superconductor is studied. Screening currents inside the superconductor have, according to the critical state model, a distinct value of Ic . To study the Bc 2 inside grains one has to choose a method where neither critical transport nor screening currents are used. The most suited method is the measurement of the specific heat versus temperature for different fields, as

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described above. Although in such an experiment screening currents are present, they are not used for the determination of Bc 2. For a further discussion see section B7.2. The difference between the Bc 2 of grains and grain boundaries has been nicely demonstrated by Cattani et al (1991). Their results are recalled in figure B9.1.4 where the critical current of PMS bulk samples are plotted versus the effective Bc 2 at grain boundaries. These samples have been intentionally prepared to observe the granular behaviour and consequently the Jc is moderate. In addition, the Bc 2 of grains of four of these samples, measured by a specific heat experiment in magnetic fields, are indicated. Note the important differences of the effective Bc 2 at grain boundaries and the Bc 2 of grains. There is a clear correlation between Jc and the effective Bc 2 at grain boundaries shown by a fit to the data with Jc ∝ B2.4 This fit is based on the scaling law (equation (B9.1.1)) which will be discussed below. c2

Figure B9.1.4. The critical current density of various PMS bulk samples versus the effective Bc 2 (4.2 K) at grain boundaries obtained from inductive measurements. For four of these samples (∆, , ∇, ◊) the Bc 2 of grains has also been measured by specific heat (S, „, T, ). The fit to the data has been calculated according to equation (B9.1.1) with Jc ∝ B2.4 without any correction for the b 0.5( 1 – b )2 term (the error is less than 15%). Reproduced from Cattani c2 et al (1991) by permission of IEEE.

B9.1.5 Extrinsic physical properties B9.1.5.1 Critical current density The critical current density is the most important parameter regarding applications. More precisely it is the engineering critical current density Jc e which is of particular interest. Jc e is the current density over the whole wire cross-section, including the normal-conducting matrix around the superconducting filaments and the electrical insulation. In technical specifications one should quote the fraction between the cross-section of the matrix and the superconductor, α . The lower α , the more superconductor is in the wire and the higher is Jc e . Although extrinsic effects may influence Tc . and Bc 2 , they markedly dominate Jc . All kinds of defect inside the superconductor contribute in a more or less efficient way to prevent magnetic flux flow which finally determines the current transport capability or Jc . Because defects are mainly controlled by the metallurgical treatment of the superconductor, Jc can be strongly influenced. At higher magnetic fields, Jc is also determined by the interaction between flux lines or bundles (elasticity of the flux line lattice) which is again an intrinsic property of the superconductor.

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where Pυ is the volume pinning force, C is a pre-factor characterizing the acting pinning mechanism and b = B/Bc 2 is the reduced field. In a granular superconductor Jc and Bc 2 are the effective critical current density and the effective critical field at the grain boundary respectively. The exponents m, p and q are determined by experiment. It has been shown that m = 2.4 ± 0.2 in high-quality PMS samples and m increases when the samples are of low quality (Cattani 1990). Because the Bmc2 term describes the temperature dependence of Jc , it can be demonstrated that the increase of Jc on changing the operating temperature from 4.2 K to 1.8 K is of the order of 40% at the same reduced field. The term b p(1 – b )q describes the field dependence of Jc and in almost all reported cases for PMS p = 0.5 and q = 2. These exponents yield a maximum of Pv at b = 0.2 as is also observed in Nb3Sn. A maximum at b = 0.2 is an indication that grain-boundary pinning is dominant. However, by improving the quality of grain boundaries of PMS, it is possible to shift the maximum to b = 0.33 corresponding to a field dependence of b(1 – b)2 (Seeber et al 1994). This has been achieved by a hot isostatic pressure (HIP) heat treatment of PMS wires with an Mo barrier. Because the HIP treatment was not optimized, a further shift to higher reduced fields seems to be possible. A shift from b = 0.2 to higher values means that pinning comes from inside the grains. This has nicely been shown in SnMo6S8 bulk samples where the maximum of the pinning force could be shifted up to b = 0.58 (Bonney et al 1995). PbMo6S8 can be considered as a model system for PbMo6S8, although Bc 2 is only about half that of PbMo6S8. Finally it should be emphasized that reducing the temperature from 4.2 K to 1.8 K is very beneficial for the critical current density. The main reason for this is the change of Bc 2 from 51 T to ~58 T. This gives by the Bmc2 term, as already said, an improvement of about 40% and by a shift of the maximum of Pυ to higher fields the b p (1 – b) figure B9.1.5). With respect to Nb—Ti and Nb3Sn this is an important advantage and should be exploited in high-field magnets. Assuming high-quality PMS (optimized Bc 2 and m = 2.4) and optimal grain boundaries (maximum of Pυ at b > 0.2 ) the C pre-factor, determined by the microstructure, is the key factor for further improvements of the critical current density. With the exception that Jc is inversely proportional to the grain size, not

Figure B9.1.5. Critical current density of a Pb0.6Sn0.4Mo6S8 monofilamentary wire versus transverse field at 4.2 K and 1.9 K. The matrix is stainless steel and the barrier is niobium (no thermal stabilization). As a voltage criterion 1 µV cm−1 has been chosen and Jc has been calculated by taking the superconductor cross-section. The obtained effective Bc 2 values are also indicated. The broken curve is the predicted current density at 1.9 K with an optimized Bc 2 of 50 T. Reproduced from Decroux (1997) by permission of IEEE.

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very much is known about the pre-factor (Karasik et al 1984). Because up to now Jc was mainly limited by grain boundaries it was not possible to study the influence of the microstructure inside grains, although an important potential may exist. To the best knowledge of the author, the highest Jc values in a PMS wire have been achieved by a partial substitution of lead by tin and a wind and react technique under HIP conditions. Although not fully understood, it has been shown earlier in bulk samples that the substitution of Pb or the addition of tin to PMS is advantageous for the critical current density (Selvam et al 1991, 1993 and references therein). In figure B9.1.5 the critical current density of such a wire with the composition Pb0.6Sn0.4Mo6S8 is plotted versus field up to 25 T at 4.2 K and 1.9 K (Cheggour et al 1997, Decroux et al 1997). Note the important improvement of Jc by reducing the temperature. The wire has an Nb barrier and a stainless steel matrix (no thermal stabilization). The Jc measurement has been carried out in coil geometry with about 2 m of wire. If we assume 40% superconductor in a PMS wire this gives an engineering critical current density at 20 T of Jc e = 90 A mm– 2 and 230 A mm– 2 at 4.2 K and 1.9 K respectively. Because Pb is partially substituted by Sn, the upper critical field is reduced but, depending on the composition of the Pb0.6Sn0.4Mo6S8, is still of the order of ~50 T at 1.9 K (Cheggour et al 1997, Decroux et al 1997). However, an estimate of the effective Bc 2 from a Jc measurement gives 38 T at 1.9 K, indicating granular behaviour of the superconductor. The extrapolated Jc obtained by the above-mentioned scaling law (B9.1.1) and assuming a Bc 2 of 50 T is also indicated in figure B9.1.5. By 1990 Rimikis had already substituted Pb by Sn and a wire with the composition of Pb0.96Sn0.24Mo6S8 had been manufactured (Rimikis 1990). Because the effective upper critical field was 38.2 T at 4.2 K, he achieved record critical current densities above 21.5 T (e.g. 1 x 108 A m– 2 at 25 T and 4.2 K). By comparing these results with the published work of other groups one generally observes smaller Jc values at lower fields but similar Jc values in the field range near 20 T (Hamasaki and Watanabe 1992, Kubo et al 1993, Seeber et al 1994, Yamasaki et al 1992). The situation seems to indicate a material that has reached its limits of Jc . This is not really the case and will be discussed further in the following paragraphs. To study critical current limiting effects at very high fields, PMS wires with a molybdenum barrier are particularly well suited. Because there is no chemical reaction between the Mo barrier and the PMS, a wide range of temperatures and annealing times can be applied without degradation of the superconductor. In figure B9.1.6 the Jc versus field of such a PMS wire, without any tin additions, after a heat treatment at 1225°C for 4 h and at 110 MPa argon pressure is shown (Seeber et al 1994, 1995). Although samples 1 and 2 are from the same wire, the high-field behaviour is different, allowing a more detailed study of its physical origin. For instance the second derivative of the current—voltage ( I—V ) curve, d2V/dI 2, allows the calculation of the distribution of the local critical current between the voltage taps (see also section B7.3). In figure B9.1.7(a) such a distribution of the critical current is plotted for sample 1. The critical current obtained by the 1 µV cm–1 criterion as well as the mean critical current are also indicated. The knowledge of the distribution of the critical current density allows an estimate of the part of the wire which is in a dissipative state (normal state) at a certain current (see section B7.3). Taking the critical current generating 1 µ Vcm–1 one finds that only 0.2–0.3% of the conductor is in the dissipative state at fields of 20 T and above. In comparison with Nb—Ti and Nb3Sn, values up to 30% have been reported (Warnes 1986) indicating that the heat treatment of the PMS wire has not been optimized. A further indication in favour for this statement is the ratio Jc /〈Jc〉 which is of the order of 0.7 instead of 0.9 as in Nb—Ti and Nb3Sn (Warnes 1986). It is important to note that a relatively high HIP temperature, in the range of 1200°C, was applied for a prototype wire. For an optimized hot-wire manufacturing process this temperature should be substantially lower. In figure B9.1.7(b) the Jc distribution of sample 2 is shown. There are two peaks in the distribution suggesting two different critical current densities. Note that the right-hand peak has a small field dependence whereas the left-hand peak is much more field sensitive. In figure B9.1.8 the field dependence of both peaks has been used to produce a Kramer plot allowing an estimate of the upper critical field.

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Figure B9.1.6. Critical current density of a PbMo6S8 monofilamentary wire versus transverse field at 4.2 K. The matrix is stainless steel and the barrier is molybdenum. As a voltage criterion 1 µV cm−1 has been chosen and Jc has been calculated by taking the superconductor cross-section. In the inset the n value versus field at 4.2 K for sample 1 is shown. The n value at 1.9 K is about twice as high. Reproduced with modifications from Seeber et al (1994) by permission of Elsevier Science.

Figure B9.1.7. (a) The distribution of the critical current density of a PbMo6S8 wire (sample 1 of figure B9.1.6) at 4.2 K. The critical current densities for 1 µV cm– 1 and the mean critical current densities are indicated. Only 0.2–0.3% of the superconductor produces 1 µV cm–1 instead of ~30% for an optimized wire. Reproduced from Seeber et al (1994) by permission of Elsevier Science, (b) The distribution of the critical current density of a PbMo6S8 wire (sample 2 of figure B9.1.6) at 4.2 K. The critical current densities for 1 µV cm–1 are indicated. Note the splitting of the distribution into two peaks with different field dependence. The left-hand peak is attributed to the Jc of grain boundaries and the right-hand peak to the Jc of grains. Reproduced from Seeber et al (1994) by permission of Elsevier Science.

Taking the left-hand peak (Jc 1 ) one obtains a Bc 2 of 31.4 T. By using the Jc obtained from the 1 m V cm–1 criterion one obtains a field value of 30 T (not shown in figure B9.1.8). The same estimate has been undertaken with the data of sample 1 yielding a Bc 2 of 33.3 T. These different Bc 2 values have a quite dramatic effect on the Jc at higher fields (> 10 T) as can be seen in figure B9.1.6. The Kramer plot for

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Figure B9.1.8. Kramer extrapolation of the critical current density of a PbMo6S8 wire (sample 2 of figure B9.1.6) taking the peaks of the Jc distribution of figure B9.1.7(b). The extrapolated Bc 2 values are indicated. Reproduced with modifications from Seeber et al (1994) by permission of Elsevier Science.

the right-hand peak in figure B9.1.7(b) gives a Bc 2 of 46.4 T. Inspecting figure B9.1.4, this value can be attributed to the Bc 2 of PMS grains. In consequence the Bc 2 obtained for the left-hand peak of 31.4 T must be the effective Bc 2 of grain boundaries. There are additional data supporting this interpretation. The Tc of samples 1 and 2, measured by the a.c. susceptibility (375 Hz, 0.1 mTr m s ), shows a low-temperature tail down to 10 K and 8 K respectively (Seeber et al 1995). Because this part of the transition is related to the coupling diamagnetic shielding between grains, the observed Tc and Bc 2 must be correlated to grain boundaries. In contrast, the onset of the superconducting transition must be related to grains if their size is large enough with respect to the London penetration depth. Depending on the position over the length of the wire, the onset of Tc is typically between 13 K and 14 K. Inspecting figure B9.1.3 one observes a qualitative agreement between the Tc and bc 2 of grains and grain boundaries. The agreement has only qualitative character because the superconducting properties vary over the length of the investigated wire. There is one technique that gives an estimate for the Jc inside grains (Cattani et al 1991). In coldpressed PMS bulk samples the Jc was determined by an inductive method where the penetration of magnetic flux into the superconductor is measured (Rollins et al 1974). In a cold-pressed sample the grains are badly connected so the obtained response comes mainly from the grains (and not from the grain boundaries). The Jc inside grains was found to be ~5 x 109 A m– 2 at 4.2 K and 10 T. This is about four times higher than Jc in figure B9.1.5, measured under the same conditions, and even higher than the extrapolated Jc at 1.9 K with a Bc 2 of 50 T. Therefore, once the grain-boundary problem can be solved, a substantially higher Jc than shown in figure B9.1.5 is expected. For applications where the superconducting magnet must work in persistent mode, the abruptness of the transition from the superconducting state to the normal state is of particular importance. Because this transition can be described by I ∝ V n , the exponent n, also named the n value, is a measure of the abruptness of the transition. I is the current through the superconducting wire and V is the voltage drop. The higher the n value the higher can be the operating current of a magnet with respect to the short sample Jc of the used wire. This is of economic interest because less superconducting wire is required for the magnet. A high n value is imperative for a drift-free persistent-mode operation. As shown in the inset of figure B9.1.6 the n value of sample 1 is about 20 at 4.2 K and at 20 T. At reduced temperature,

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e.g. 1.8 K, the n value increases by a factor of about two (Seeber et al 1994) and in other PMS wires a factor of three has been reported (Gupta et al 1995). Because the n value is linked to the distribution of Jc (high n value = small Jc distribution) much better values of n can be expected. Finally, due to the chemically very stable PMS phase, the matrix materials can be etched away easily without attacking the superconductor, which may be of advantage for the fabrication of very low-resistance joints between superconducting wires. B9.1.5.2 Critical current density under mechanical stress A superconducting wire, which has been wound to a coil, is subject to stress due to electromagnetic forces (Lorentz forces). Additional stress comes from the coil winding and from the differential thermal contraction of the materials present in the wire and in the coil. The mechanical stability can be evaluated by Jc measurements under uniaxial strain. Ekin et al (1985) were the first to study Jc of a PMS wire and also of a tape under uniaxial strain in fields up to 24 T. In the wire and tape geometry, there was a different thermal pre-strain on the PMS layer induced by differential thermal contraction between the Mo substrate and the PMS. Then the fracture strain varies from εf r a c t u r e = 0.2% to 0.3%. In the latter situation the PMS is n early free of thermal pre-strain at ε = 0 and Ic /Ic m at ε = 0.2% is of the order of 0.87. It was also shown that there is hardly any field dependence in the range from 8 T to 24 T which is a significant advantage over Nb3Sn. New data have been obtained by Goldacker et al (1989) who measured the influence of a transverse compressive and an axial tensile stress on Jc of PMS and SnMo6S8 wires. Choosing the right composition of matrix materials, it was possible to observe a maximum in the critical current density as a function of uniaxial strain. This behaviour can be explained by the presence of a thermally induced compressive pre-strain on the PMS at ε = 0. By increasing the axial strain, the PMS core approaches a state of minimum stress which corresponds to the maximum of Ic . Above ε = 0.2% the PMS comes under tensile stress and the first cracks appear at ε = 0.85%. Up to this value the curve is reversible and the intrinsic elasticity limit of PMS can be determined to be ε = 0.65%. In comparison, this value is about 30% higher than that of Nb3Sn (εi r = 0.5%). At the intrinsic strain ε = 0.2%, calculated from the maximum of Ic /Ic m , the critical current is reduced to Ic /Ic m = 0.93. The situation is shown in figure B9.1.9 where Ic /Ic m versus uniaxial strain is plotted. In the same figure two wires with a stainless steel (ss) matrix and Mo barrier are also plotted (Seeber et al 1991). Depending on the ss/Mo fraction, which varies between 2.8 and 3.8, reversibility has been observed up to ε = 0.15% and 0.3% respectively. At higher values the PMS core is damaged. It is interesting to note that Ic /Ic m = 0.98 at ε = 0.2% for one of these wires (ss/Mo = 3.8). By choosing a higher fraction of stainless steel, the strain tolerance of the wire can be shifted to higher strain. By introducing HIP, the thermal pre-strain at low temperatures can probably also be adjusted with the temperature, so that an additional parameter for optimization is available. Since a maximum of Ic /Ic m can be seen in one of the PMS wires of figure B9.1.9, one is tempted to consider this strain behaviour of PMS as universal. This is not the case. For instance in the PMS wire with ss/Mo = 3.8 of figure B9.1.9 the Ic /Ic m is reduced to only 0.98 at ε = 0.2% strain. If one takes into account that the intrinsic strain must be higher due to the unknown thermal pre-strain and following the behaviour for the wire with ss/(Cu + Ta) = 3.7, the Ic /Ic m should be much less. Consequently, the curve of a PMS wire with an ss/(Cu + Ta) matrix does not represent a universal strain behaviour of PMS. It seems that the detailed mechanical behaviour of PMS depends very much on the matrix materials used, the matrix/superconductor ratio α , how deformation has been achieved, the annealing temperature (pressure in the case of HIP) and, finally, on the critical current density at ε = 0.

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Figure B9.1.9. Normalized critical current versus uniaxial strain for various PMS wires, α is the matrix-tosuperconductor fraction. There is no field dependence up to the highest measured field so far of 24 T (Ekin et al 1985). The wire with α = 13 (~7% PMS) has a stainless steel-copper matrix and a niobium barrier. Wires with α = 2.3 (30.5% PMS) have a stainless steel matrix with a molybdenum barrier. εi r r is the irreversible strain limit above which the Jc does not recover after removing the strain. Reproduced from Seeber (1991) by permission of IEEE.

B9.1.6 Stress-strain behaviour Few data are available on the stress-strain behaviour of PMS wires with an Mo—ss matrix. At room temperature a typical value for the tensile strength is of the order of 1.2 to 1.5 GPa at a strain of 0.5% and 0.7%, respectively. The former values for the tensile strength are for a wire without heat treatment (as drawn) with 30% and 20% of PMS, respectively, but with constant ss/Mo fraction. This indicates that the mechanical behaviour is essentially determined by the Mo—ss matrix. After heat treatment the tensile strength is reduced (Grill et al 1989). Typical stress-strain curves at 4.2 K for the wires with α = 2.3 of figure B9.1.9 are shown in figure B9.1.10. Note that the PMS part of both wires is constant (30.5% or α = 2.3), but the fraction of stainless steel to molybdenum has been changed. With increasing Mo content the stress-strain curve is steeper, indicating a stronger material with a higher E modulus. If one assumes a linear stress—strain curve up to 0.3% strain, the E modulus is 140 GPa and 154 GPa, respectively, and the corresponding yield strengths σ0.2 is between ~700 MPa and 860 MPa. In comparison typical yield strengths of Nb3Sn vary between 100 MPa and 250 MPa (see also section B8.1). B9.1.7 Thermal stabilization If one has a PMS wire with a niobium barrier one has to introduce an electrically well-conducting layer (normally copper) between the barrier and the stainless steel matrix. Because there is a high-resistivity interdiffusion layer between the PMS and the Nb, known from long current transfer lengths seen in critical current measurements, and copper may easily be contaminated by the constituents of the stainless steel, it is not clear whether such a concept will work in practice. In the case of a molybdenum barrier, the barrier itself can be used as a stabilizer. The room-temperature resistivity of molybdenum is 5 × 10–8 Ω m (copper 1.55 × 10–8 Ω m) (Fickett 1982) and the observed RRR of these wires is in the range of 30 (Herrmann 1990). The RRRs of PMS wires have not been studied systematically but, if required, better values can

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Figure B9.1.10. Stress versus strain of PMS wires at 4.2 K with α = 2.3 (30.5% PMS) and varying ss/Mo fraction. These wires are the same as in figure B9.1.9. Note the very high yield strength between ~700 MPa and 860 MPa with respect to Nb3Sn (100 MPa to 250 MPa).

be achieved by using a purer Mo quality. Attention must be given to the fact that Mo has a higher magnetoresistivity with respect to copper. The magnetoresistance of Mo can be described in a so-called Kohler plot where the transverse magnetoresistance is plotted versus field times RRR (figure B9.1.11). For comparison, the behaviour of copper is also shown in figure B9.1.11. In a practical design of a PMS wire with an Mo barrier, a balance between the RRR and the magnetoresistivity, as well as the cross-section of Mo, must be found. On the other hand, there is a substantial benefit from the high specific heat of the stainless steel matrix (4.7 × 10− 4 cal g−1 K−1 at 4.2 K) which is nearly 20 times that of copper (2.5 × 10−5 cal g−1 K−1). This will reduce the temperature increase of the wire at a quench.

Figure B9.1.11. A Kohler plot of the transverse magnetoresistance ∆R/R0 versus B RRR of molybdenum and copper. ∆R is the magnetoresistance and R0 the electrical resistance without field B. Q — molybdenum data from Fawcett (1962) and O—unpublished results from the author. Copper data have been taken from Fickett (1982).

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B9.1.8 Conclusions After Nb—Ti and Nb3Sn, PbMo6S8 wires are a good candidate for the third generation of superconducting wires for ultra-high-field applications. The PMS phase is extremely stable and can be synthesized in a wide temperature range (450 °C and 1650°C) with a variety of particle sizes. Because PMS wires are manufactured by powder metallurgical techniques, hot deformation processes are preferentially used and the availability of different particle sizes is very helpful. For instance a smaller particle size reduces the optimum deformation temperature. This opens up the possibility, through an optimized hot drawing process, to overcome the granular behaviour which is at present the main limiting factor for the critical current density at high fields. In a hot drawing process no, or very few, intermediate annealings are required which keeps manufacturing costs low. As a consequence of the nearly isotropic physical properties, round or rectangular cross-sections can be produced. Because of its important development potential the concept of a molybdenum barrier and a stainless steel matrix looks promising. During wire manufacturing the temperature of the deformation process is not limited by the molybdenum barrier (no chemical interaction with the superconductor). The inertness of the interface between PMS and Mo allows small PMS filaments. Because Mo has a low electrical resistivity and high RRRs can be achieved by using Mo of higher purity, it is possible to use it as a thermal stabilizer. In such a case quite low matrix-to-superconductor fractions α are possible (α = 1.5 has already been demonstrated). In combination with the stainless steel matrix, the temperature increase after a quench is considerably reduced (the stainless steel matrix has a specific heat about 20 times higher than that of copper). In addition, the electrical resistivity of the stainless steel matrix is high, allowing an efficient decoupling between the filaments of a multifilamentary wire. This is of particular interest in time-varying fields (low a.c. losses). At present the engineering current density at 20 T is sufficient for the construction of high-field inserts. Under the assumption of a PMS wire with α = 1.5, Jc e = 90 A mm– 2 at 20 T, 4.2 K, and increases to Jc e = 230 A mm– 2 by reducing the temperature to 1.9 K. The n value in this field range is about 20 at 4.2 K and about 40 at 1.9 K. Due to the granular behaviour with a reduced effective Bc 2 at grain boundaries, Jc depends unusually strongly on the applied field. However, because PMS has a quasi-isotropic coherence length of 260 nm at 4.2 K, which is between that of Nb3Sn and high-Tc superconductors, and because PMS is thermodynamically extremely stable, it should be possible to overcome granularity by an appropriate hot deformation process and/or heat treatment. Then substantially higher Jc values and a better field dependence would be expected. A few words should be said on the performance of long PMS wires. In our laboratory test coils with up to 90 m of wire have been constructed. Generally the expected performance is a factor of two or three behind that estimated from Jc measurements on short wire samples (1–2 m). The main reasons have been identified as chemical inhomogeneities in wires with an Nb barrier and leakage of the Mo barrier. The latter can be solved by an optimization of the deformation process. A better long-length performance is also expected if a multifilamentary arrangement is used. Probably one of the most important properties of PMS wires is the mechanical strength due to the presence of the stainless steel matrix. The yield strength is of the order of 750 MPa at 4.2 K which is about three times higher than that of Nb3Sn. In addition, an intrinsic strain up to 0.65% has been reported without permanent damage of the PMS superconductor. References Bonney L A, Willis T C and Larbalestier D C 1995 Dependence of critical current density on microstructure in the SnMo6S8 Chevrel phase superconductor J. Appl. Phys. 77 6377–87 Cattani D 1990 Etude des densité de courant critique dans le composé PbMo6S8 PhD Thesis No 2422 University of Geneva

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Cattani D, Cors J, Decroux M and Fischer ∅ 1991 Intra- and intergrain critical current in PbMo6S8 sintered samples IEEE Trans. Magn. MAG-27 950–3 Cheggour N, Decroux M, Gupta A, Fischer ∅, Perenboom J A A J, Bouquet V, Sergent M and Chevrel R 1997 Enhancement of the critical current density in Chevrel phase superconducting wires J. Appl. Phys. 81 6277–84 Cors J 1990 Propriétés supraconductrices sous champ magnétique du composé PbMo6S8 étudiées par chaleur spécifique PhD Thesis No 2456, University of Geneva Cors J, Cattani D, Decroux M, Stettler A and Fischer ∅ 1990 The critical field of PbMo6S8 measured by specific heat up to 14 T Physica B 165&166 1521–2 Decroux M 1980 Quelques aspects des champs critiques dans le phases de Chevrel PhD Thesis No 1987 University of Geneva Decroux M, Cattani D, Cors J, Ritter S and Fischer ∅ 1990 Granular behavior of the PbMo6S8 Chevrel phase Physica B 165&166 1395–6 Decroux M, Cheggour N, Gupta A, Fischer ∅, Bouquet V, Chevrel R, Sergent M and Perenboom J 1997 Overall critical current density of Chevrel wires at high magnetic field IEEE Trans. Appl. Supercond. AS-7 1759–62 Decroux M, Fischer ∅, Flükiger R, Seeber B, Delesclefs R and Sergent M 1978 Anisotropy of Hc 2 in the Chevrel phases Solid State Commun. 25 393–6 Decroux M, Selvam P, Cors J, Seeber B, Fischer ∅, Chevrel R, Rabiller P and Sergent M 1993 Overview on the recent progress on Chevrel phases and the impact on the development of PbMo6S8 wires IEEE Trans. Appl. Supercond. AS-3 1502–9 Ekin J W, Yamashita T and Hamasaki K 1985 Effect of uniaxial strain on the critical current and critical field of Chevrel phase PbMo6S8 superconductors IEEE Trans. Mag. MAG-21 474–7 Fawcett E 1962 Magnetoresistance of molybdenum and tungsten Phys. Rev. 128 154–60 Fickett F R 1982 Electrical properties of materials and their measurement at low temperatures NBS Technical Note 1053 Foner S, McNiff E J Jr and Alexander E J 1974 600 kG superconductors Phys. Lett. 49A 269–70 Goldacker W, Specking W, Weiss F, Rimikis G and Flükiger R 1989 Influence of transverse compressive and axial tensile stress on the superconductivity of PbMo6S8 and SnMo6S8 wires Cryogenics 29 955–60 Goldfarb R B, Lelental M and Thompson C A 1992 Alternating-field susceptometry and magnetic susceptibility of superconductors Magnetic Susceptibility of Superconductors and Other Spin Systems ed R A Hein, T L Francavilla and D H Liebenberg (New York: Plenum) pp 49–80 Grill R, Kny E and Seeber B 1989 Anwendung von Refraktarmetallen in Keramik-Supraleiter Proc. 12th Plansee Seminar, (Reutte) ed H Bildstein and R Eck pp 989–1006 Gupta A, Cheggour N, Decroux M, Perenboom J, Bouquet V, Langlois P, Massat H, Flükiger R and Fischer ∅ 1995 Dependence of critical current densities in Chevrel phase superconducting wires on magnetic fields up to 25 T Physica B 211 272–4 Hamasaki K and Watanabe K 1992 (Pb, Sn)Mo6S8 monofilamentary wires produced by HIP technique Sci. Rep. Res. Inst. Tôhoku Univ. (RITU) A 37 51–8 Hauck J 1977 Phase relation stoichiometry of superconducting PbxMo6S8 —y Mater. Res. Bull. 12 1015–9 Helfand E and Werthamer N R 1966 Temperature and impurity dependence of the superconductivity critical field Hc 2 Phys. Rev. 147 288–94 Herrmann P F 1990 Transition supraconductrice et courant critique dans des fils de PbMo6S8 PhD Thesis No 2453, University of Geneva Hinks D G, Jorgensen J D and Li H L 1983 Structure of the oxygen point defect in SnMo6S8 and PbMo6S8 Phys. Rev. Lett. 51 1911–4 Karasik V R, Karyaev E V, Zakosarenko V M, Rikel M O and Tsebro V I 1984 Vortex-lattice pinning in bulky single phase PbMo6S8 and SnMo6S8 samples with various grain sizes Sov. Phys.-JETP 60 1221–8 Krabbes G and Oppermann H 1981 The phase diagram of the Pb-Mo-S system at 1250 K and some properties of the superconducting PbMo6S8 Cryst. Res. Technol. 16 777–84 Kubo Y, Uchikawa F, Utsunomiya S, Noto K, Katagiri K and Kobayashi N 1993 Fabrication and evaluation of small coils using PbMo6S8 wires Cryogenics 33 883–8 Odermatt R, Fischer ∅, Jones H and Bongi G 1974 Upper critical fields of some ternary molybdenum sulphides J. Phys. C: Solid State Phys. 7 LI3–5

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Rabiller P 1991 Étude et optimisation des propriétés supraconductrices de filaments a base de phase de Chevrel au plomb PhD Thesis No 655 University of Rennes Rabiller P, Chevrel R, Sergent M, Ansel D and Bohn M 1992 Niobium antidiffusion barrier reactivity in tin-doped, in situ PbMo6S8-based wires J. Alloys Compounds 178 447–54 Rabiller P, Rabiller-Baudry M, Even-Boudjada S, Burel L, Chevrel R, Sergent M, Decroux M and Maufras J L 1994 Recent progress in Chevrel phase synthesis: a new low temperature synthesis of the superconducting PbMo6S8 compound Mater. Res. Bull. 29 567–74 Rimikis G 1990 Einflussgrössen und Methoden zur Optimierung der supraleitenden und mechanischen Eigenschaften von Chevrelphasendrähten PhD Thesis University of Karlsruhe Rollins R W, Küpfer H and Gey W 1974 Magnetic field profiles in type-II superconductors with pinning using a new ac technique J Appl. Phys. 45 5392–8 Seeber B, Cheggour N, Perenboom J and Grill R 1994 Critical current distribution of hot isostatically pressed PbMo6S8 wires Physica C 234 343–54 Seeber B, Erbüke L, Schröter V, Perenboom J and Grill R 1995 Critical current limiting factors of hot isostatically pressed (HIPed) PbMo6S8 wires IEEE Trans. Appl. Supercond. AS-5 1205–8 Seeber B, Glätzle W, Cattani D, Baillif R and Fischer Æ 1987 Thermally induced pre-stress and critical current density of PbMo6S8 wires IEEE Trans. Magn. MAG-23 1740–3 Seeber B, Herrmann P, Schellenberg L and Zuccone J 1991 Considerations for practical conductor design of Chevrel phase wires IEEE Trans. Magn. MAG-27 1108–11 Seeber B, Rossel C and Fischer Æ 1981 PbMo6S8: a new generation of superconducting wires? Ternary Superconductors ed G K Shenoy, B D Dunlap and F Y Fradin (Amsterdam: North-Holland) pp 119–24 Selvam P, Cattani D, Cors J, Decroux M, Niedermann P, Fischer Æ, Chevrel R and Pech T 1993 The role of Sn addition on the improvement of Jc in PbMo6S8 IEEE Trans. Applied Supercond. AS-3 1575–8 Selvam P, Cattani D, Cors J, Decroux M, Niedermann P, Ritter S, Fischer Æ, Rabiller P, Chevrel R, Burel L and Sergent M 1991 Tc variation in PbMo6S8: a critical analysis and comparison with pure phases Mater. Res. Bull. 26 1151–65 Shelton R N, Lawson A C and Johnston D C 1975 Pressure dependence of the superconducting transition temperature for ternary molybdenum sulfides Mater. Res. Bull. 10 297–302 Van der Meulen H P, Perenboom J A A J, Berendschot T T J M, Cors J, Decroux M and Fischer Æ 1995 Specific heat of PbMo6S8 in high magnetic fields Physica B 211 269–71 Willis T C, Jablonski P D and Larbalestier D 1995 Hot isostatic pressing of Chevrel phase bulk and hydrostatically extruded wire samples IEEE Trans. Appl. Supercond. AS-5 1209–13 Yamasaki H and Kimura Y 1986 The phase field of the Chevrel phase PbMo6S8 at 900 °C and some superconducting and structural properties Mater. Res. Bull. 21 125–35 —1988 Investigation of the fabrication process of hot-worked stainless-steel and Mo sheathed PbMo6S8 wires J. Appl. Phys. 64 766–71 Yamasaki H, Umeda M, Kimura M and Kosaka S 1991 Current carrying properties of the HIP treated Mo-sheath PbMo6S8 wire IEEE Trans. Magn. MAG-27 1112–5 Yamasaki H, Umeda M and Kosaka S 1992 High critical current densities reproducibly observed for hot-isostaticpressed PbMo6S8 wires with Mo barriers J. Appl. Phys. 72 1–3 Yamasaki H, Willis T C, Larbalestier D and Kimura Y 1990 Microstructure and critical current densities of PbMo6S8 in hot-worked Mo-sheathed wires Adv. Cryogen. Eng. 36 343–51 Warnes W H and Larbalestier D 1986 Critical current distribution in superconducting composites Cryogenics 26 643–53 Werthamer N R, Helfand E and Hohenberg P C 1966 Temperature and purity dependence of the superconducting critical field Hc 2 . III Electron spin and spin orbit effects Phys. Rev. 147 295–302

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B9.2 General aspects of high-temperature superconductor wires and tapes Johannes Tenbrink

B9.2.1 Introduction Soon after the discovery of high-Tc superconductivity in oxides in 1986 (Bednorz and Müller 1986) worldwide efforts led to the development of oxides exhibiting superconductivity at temperatures above the boiling point of liquid nitrogen (77 K). The best known oxidic high-Tc superconductors (HTSs) are YBa2Cu3O7–x (YBCO) (Wu et al 1987) with a Tc of 92 K discovered in 1987, phases in the Bi—Sr—Ca— Cu—O system (BSCCO) (Maeda et al 1988, von Schnering et al 1988) with a Tc up to 110 K discovered in 1988 and phases in the Tl—Ba—Ca—Cu—O system (TBCCO) (Sheng and Hermann 1988, Sheng et al 1988) with a Tc up to 125 K also discovered in 1988. Since then a huge number of different HTSs has been discovered (an overview for example is given by Rao et al (1993)). Nevertheless basic research and development of technical superconductors up to now has been focused on YBCO, BSCCO and because of its toxicity significantly less on TBCCO. This article deals with some basic physical properties of HTSs and their implications for the application of these materials in technology. The main routes and the present status of technical conductor development will be discussed. B9.2.2 Basic properties of high-Tc superconductors B9.2.2.1 General remarks A common and very important feature of HTSs is their layered structure with copper-oxygen planes separated by different intermediate layers. Figure B9.2.1 shows the structure of YBa2Cu3O7–x (YBCO) in comparison with the structure of Bi2Sr2CaCu2O8+x (Bi-2212) and Bi2Sr2Ca2Cu3O10+x (Bi-2223), these compounds being so far the most investigated ones. Common to all known HTSs are the copper-oxygen planes. This layered structure is reflected by a very anisotropic behaviour of the microscopic as well as the macroscopic properties of HTSs such as, for example, the upper critical field, the critical current density and the pinning properties, but also the grain growth properties. This pronounced anisotropy sets severe implications to conductor design and development. Many experiments have been performed on single crystals or thin films in order to measure basic physical properties and also to determine the ultimate potential and the limitations of this new class of materials. Therefore some of these data will be reviewed for the technologically most important compounds known at present. Table B9.2.1 shows the coherence lengths ξ (0) and the upper critical fields Hc 2 at 77 K and at 4.2 K. These parameters reflect the pronounced anisotropy of this new class of layered materials. As can be seen, the upper critical fields and the coherence lengths are lower for the direction parallel to the c axis of the

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Figure B9.2.1. Structure of (a) YBa2Cu3O7– x , (b) Bi2Sr2Ca1Cu2O8+x and (c) Bi2Sr2Ca2Cu3O10+x . Table B9.2.1. Approximate upper critical fields Bc2 and coherence lengths ξ of YBa2Cu3O7-x and Bi2Sr2CaCu2O8+x for orientation parallel to the c axis (first value) and perpendicular to the c axis (second value) of the structure.

structure. Even for this ‘bad’ direction Hc 2 is of the order of a few tesla at 77 K. At 4.2 K the upper critical fields are extremely high especially compared with the conventional metallic superconductors such as NbTi for which Hc2 is about 11 T or Nb3Sn which has an upper critical field of about 24 T. It is important to note that the coherence length ξc for the direction parallel to the c axis is very small. For YBCO with ξc ≈ 0.5 nm this is roughly the distance between the Cu—O layers (≈0.75 nm) whereas for Bi-2212 with ξc ≈ 0.4 nm this is distinctly smaller than the distance between the Cu—O double layers

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(≈1.2 nm). This immediately tells us that Bi-2212 will behave in a more two-dimensional fashion than YBCO. B9.2.2.2 Y-Ba-Cu oxide Soon after the discovery of superconductivity in YBCO, thin films could be produced on single-crystalline ceramic substrates like MgO or SrTiO3 by different processes such as sputtering or laser ablation. Highquality epitaxial thin films exhibited critical current densities Jc well beyond 106 A cm–2 at 77 K, 0 T. In order to achieve this very high Jc the c axis of the structure had to be aligned perpendicularly to the substrate surface and additionally the films had to be nearly single crystalline with only a small mosaic spread. Figure B9.2.2 shows the magnetic field dependence of the critical current density Jc at 77 K. With the magnetic field B aligned parallel to the film surface (B ⊥ c) very high critical current densities can be obtained up to high magnetic fields. This could be attributed to in trinsic pinning (Tachiki and Takahashi 1989, 1992) of vortices aligned parallel to the layers of the structure. This intrinsic pinning is a direct consequence of the very short coherence length for this direction. As the coherence length ξc is smaller than the c-axis unit cell length a modulation of the order parameter occurs leading to a drop of the potential and thereby to pinning of vortices in the intermediate layers between the Cu—O planes. For the magnetic field B aligned perpendicularly to the film surface Jc drops at a magnetic field of a few tesla because of the lower value of the upper critical field for this orientation. Based on these data one can conclude that YBCO has on principle the potential for application in magnet technology at 77 K up to magnetic fields of a few tesla. Polycrystalline films on the other hand yielded much lower critical current densities even if their c axes were properly aligned. Additionally they show a pronounced magnetic field dependence of the critical current density.

Figure B9.2.2. Critical current density Jc of YBCO (Roas et at 1990) and Bi-2212 (Schmitt et al 1991) singlecrystalline thin films as a function of the magnetic field B at different temperatures. Open symbols refer to measurements with B aligned parallel to the film surface (B ⊥ c). Full symbols refer to measurements with B aligned perpendicularly to the film surface (B || c).

In order to investigate this effect in more detail, sophisticated experiments were performed on bicrystalline thin films (Dimos et al 1988, 1990, Mannhart et al 1988). SrTiO3 single crystals were properly cut and sintered together in order to form a bicrystal with a well defined orientation relationship as a substrate. After depositing YBCO on this substrate thin films with only one grain boundary were obtained. By comparing the critical current within the grains with the value obtained across the grain boundary it could be shown that grain boundaries with misorientations exceeding about 5° severely degrade

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the critical current density. The grain boundaries behave as a weak link limiting the critical current. This behaviour was similarly obtained on grain boundaries of twist and of tilt type. The dramatic consequence is that one has to have at least a biaxial alignment of the grains with only very small angle grain boundaries in order to achieve high critical currents in polycrystalline material. In addition to the above-mentioned experiments work was performed on YBCO bicrystals obtained in procedures similar to the production of single crystals (Larbalestier et al 1991). It could be shown that some special orientational relationships yield strong coupling at the grain boundaries. Detailed transmission electron microscopy work on grain boundaries yielded no direct hint as to the microstructural reason for the weak-link behaviour of most YBCO grain boundaries. It was speculated that this problem may be connected to loosely bound oxygen at grain boundaries widening the disturbed zone to dimensions exceeding the coherence length. Using electron energy loss spectroscopy (EELS) it could indeed be shown that certain grain boundaries are deficient in oxygen and others are not (Zhu et al 1993). B9.2.2.3 Bi-Sr-Ca-Cu oxide With BSCCO the formation of high-quality epitaxial thin films is much more complicated. Nevertheless such films have successfully been prepared. For example the critical current density data of a Bi-2212 epitaxial thin film are included in figure B9.2.2 for comparison. For the magnetic field aligned parallel to the plane of the film, that is perpendicular to the c axis of the structure, Jc is independent of the magnetic field applied. This is due to extremely effective intrinsic pinning in the intermediate layers between the Cu—O planes of the structure. The fact that intrinsic pinning with Bi-2212 is even more effective than with YBCO is connected to the stronger anisotropy of the structure. In Bi-2212 the distance between the Cu—O planes is about 1.2 nm whereas in YBCO this distance amounts to only about 0.75 nm. This pronounced anisotropy has severe implications concerning the pinning properties. With magnetic field B aligned perpendicularly to the film surface (that is, parallel to the c axis of the Bi-2212 structure) Jc is still high at 4.2 K as can be seen from figure B9.2.2. With increasing temperature the critical current density for this orientation of B becomes progressively worse, yielding a very large critical current anisotropy at higher temperatures of e.g. 60 K (see figure B9.2.2). For B aligned parallel to the planes of the structure intrinsic pinning is very effective even at high temperatures; however, no effective pinning centre seems to work for B aligned parallel to the c axis at least at higher temperatures. Even columnar defects introduced by heavy-ion irradiation do not pin the flux lines effectively at 77 K (Neumüller et al 1993). This behaviour supports the idea that the flux lines for B aligned parallel to the c axis fall to pieces of Josephson coupled pancake vortices (Clem 1992). Obviously such a vortex structure can hardly be pinned. At higher temperatures this behaviour results in huge flux creep effects being responsible for the very poor critical current density in the case where the magnetic field has a component parallel to the c axis of the Bi2212 structure. Due to this insufficient pinning BSCCO materials show reversible behaviour during magnetization measurements at higher temperatures. Below a certain threshold temperature Ti r r , dependent on the applied magnetic field B, pinning becomes sufficiently strong and allows for high critical current densities as can be seen from figure B9.2.2. The border line Ti r r (B) between the regime of strong pinning with irreversible magnetization and flux creep with reversible magnetization has been called the ‘irreversibility line’. The irreversibility line Ti r r (B) of BSCCO materials is shifted to lower temperatures than for YBCO. B9.2.2.4 Tl—Ba—Ca—Cu—O The TBCCO system yielded one of the highest transition temperatures Tc so far reported for HTS materials. Tl2Ba2Ca2Cu3O10+x (Tl-2223) exhibits a Tc of 125 K. A variety of superconducting phases exists. As well as Tl-2223 and Tl2Ba2CaCu2O8+x (Tl-2212) and other phases, high-Tc superconducting TlBa2CaCu2O6.5+x

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(Tl-1212) and TlBa2Ca2Cu3O8.5+x (Tl-1223) can also be synthesized. This variety of phases and the problems associated with the high vapour pressure of Tl2O3 complicate the preparation of single-phase material. Nevertheless high Jc thin films could also be produced with TBCCO. As a result of the toxicity of Tl only a few groups are working with this material. The most important point with TBCCO is the abovementioned existence of phases with only one intermediate Tl—O layer (Tl-1212 and Tl-1223) with Tc comparable to those of the respective double Tl—O-layer compounds. These single Tl—O-layer compounds naturally exhibit a smaller distance between the blocks of Cu—O layers. This means a reduced anisotropy which of course positively influences the pinning properties. The irreversibility line of Tl-1212 and Tl-1223 is shifted to higher temperatures than in the respective compounds with two intermediate Tl—O layers (Tl-2212 and Tl-2223). This has been demonstrated on Tl-1223 and Tl-2223 thin films (Nabatame el al 1993). With the magnetic field B aligned perpendicularly to the c axis the critical current density Jc is comparable due to effective intrinsic pinning in both materials whereas for B aligned parallel to the c axis the Tl-1223 thin film distinctly out-performs the Tl-2223 thin film. At the same reduced temperature, T/Tc = 0.6, Tl-2223 becomes reversible at B = 2 T whereas with Tl-1223 this occurs at B = 8 T, showing the improved pinning characteristics of the Tl-1223 material. This point will be discussed in more detail in section B9.2.3.3. B9.2.3 Development of practical conductors In this section work towards real practical conductors will be reviewed with emphasis on processes which seem to be scalable to larger quantities. This is a very important point since HTS production costs must be comparable to the costs of the currently used metallic superconductors if their widespread use in magnet and energy technology is envisaged. Many attempts have been made in order to increase the number of techniques using the critical current density. At present these techniques are far too slow and/or too expensive, e.g. zone melting, directional solidification, some thin-film deposition techniques and others. Some of these techniques will be mentioned throughout this section if their results are outstanding, others will be mentioned in the following section on other applications of HTSs. Nevertheless the following subsections will mainly concentrate on results obtained with processes meeting the above-mentioned requirement. B9.2.3.1 Y-Ba-Cu oxide With the discovery of YBCO worldwide efforts helped start development of practical conductors. The most simple approach seemed to be the so-called ‘powder-in-tube’ technique. As the new HTSs have a variable oxygen content depending in thermodynamic equilibrium on temperature and oxygen partial pressure, silver was often chosen as the sheath material due to its extraordinary high solubility and diffusivity for oxygen. Usually YBCO powder was produced from high-purity Y2O3, BaCO3 and CuO by carefully mixing, grinding and performing multiple annealing steps (‘mixed-oxide technique’). As YBCO is a phase with rigid YBa2CU3O7-x cation composition this repeated processing yields nearly single-phase material. The powder was finally put into Ag tubes and then swaged and/or drawn to its final dimensions. Flat tapes could be produced by cold or even hot rolling. In order to achieve a superconducting transport current a final annealing treatment had to be performed. With YBCO in Ag or Ag alloy sheath this usually means a sintering step at about 900 °C in air or even under inert atmosphere with reduced oxygen partial pressure followed by a slow cooling in oxygen in order to achieve a high oxygen content. The resulting microstructure of course was polycrystalline with no significant degree of texture. As a typical result the critical current density Jc drops markedly with only small applied magnetic fields at 77 K and even at 4.2 K (Tenbrink el al 1989) because of the weak-link nature of the grain boundaries. It was pointed out very early that this weak-link behaviour should lead to a critical current density depending

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on the dimensions of the respective specimen caused by the effect of the self-field yielding critical current densities inversely proportional to the thickness or diameter of the specimen (Dersch and Blatter 1988). This behaviour was indeed observed in YBCO/Ag wires or tapes of comparable quality but with different conductor cross-sections (Tenbrink et al 1989). Utilizing an AgPd alloy sheath, critical current densities up to 5.950 A cm– 2 could be achieved (Fischer et al 1992). Nevertheless the critical current density was distinctly smaller with a magnetic field applied. Higher critical current densities could be achieved by melt processing YBCO (Jin et al 1988). In air the YBCO phase diagram shows a peritectic point at about 1010 °C (Cima et al 1992) where Y2BaCuO5 (due to its colour called the ‘green phase’ ), a liquid and YBa2Cu3O7-x coexist. Using different processing routes through this phase diagram (Cima et al 1992, Murakami 1992) large-grained specimens with high intragrain critical current densities could be produced by applying partial melting of the samples. All these techniques suffer from the fact that here grain boundaries also act as a weak link. Therefore only specimen sizes up to a few centimetres in length can carry high critical currents. Although useless with respect to applications for superconducting wires, such a material may, however, well be applied in other areas as will be discussed in section B9.2.4. In order to improve the critical current density in polycrystalline material, especially in the presence of a magnetic field, texture is required. With the above-mentioned results (see section B9.2.2.2 ) in mind an alignment of all three crystallographic axes is desirable. This has successfully been tried by deposition of a textured yttria-stabilized-zirconia (YSZ) buffer layer on a metallic substrate by means of an ion-beamassisted deposition technique (lijima et al 1992). During deposition of the YSZ buffer onto the metallic substrate a second Ar+ ion beam is used in order to sputter away most of the deposited YSZ. Only those YSZ grains survive and grow which are oriented in such a way that a channelling effect for the Ar+ ions prevents effective sputtering. The texture of the YSZ buffer layer is controlled by carefully adjusting the orientation of the Ar+ ion beam with respect to the substrate. On this textured YSZ buffer layer a YBCO thin film is deposited epitaxially by, for example, laser deposition (lijima et al 1992). Critical current densities exceeding 105 A cm–2 at 77 K have thus been achieved. The texture of YBCO in this case is almost ideal, i.e. a nearly perfect triaxial alignment is obtained. This leads to a magnetic field dependence of Jc similar to the behaviour of the above-mentioned thin films so that the physical requirements for a technical conductor operating at 77 K are met. Obviously this technique is extremely slow and very expensive so that at present one can only think about a short (e.g. 1 m) prototype demonstration conductor. Another approach for texturing of YBCO thick films has been the remelting of the surface of a YBCO pellet sample using a laser (Nagaya et al 1991). A direct crystallization of the orthorhombic superconducting YBa2Cu3O7-x phase was observed yielding a texture with the c axis oriented perpendicularly to the laser scanning direction. The critical current density deduced from magnetization measurements was as high as 1.5 × 104 A cm–2 at 77 K, 0 T. A texturing effect was also noted during laser melting of electrophoretically deposited thick films. The a — b plane of the YBCO structure was seen to align parallel to the temperature gradient (Hofer et al 1992 ). Based on this result YBCO has been deposited on Ag substrates by electrophoresis in the presence of a magnetic field. The paramagnetic anisotropy yields a torque leading to texturing during deposition of the thick film. This pretextured thick film is then sintered, melted using a defocused laser beam and finally annealed to achieve a complete reaction to YBCO. Critical current densities up to 1.7 × 104 A cm–2 could be achieved at 77 K, 0 T (Hofer et al 1993). The results strongly depend on the quality of the pretexturing process. By using DyBa2Cu3O7-x , which has a much higher paramagnetic anisotropy, and by carefully optimizing the powder quality, progress is expected in the future. Obviously the most crucial point is the demand to have a powder which consists of single-crystalline grains in order to yield proper alignment during electrophoresis in a magnetic field. Whether this process allows for the nearly perfect texturing required remains to be seen.

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B9.2.3.2 Bi—Sr—Ca—Cu oxide Unlike the case for YBCO it is very difficult to get single-phase material in the Bi—Sr—Ca—Cu—O system. This is due to the fact that different phases exist and that the phases have a rather large range of solubility (Hettich et al 1991, Majewski et al 1991). Because of this much work was done in the beginning on powder preparation and pellet-type samples in order to study phase formation. Bi-2212 can be produced nearly phase pure by applying the mixed-oxide technique or by pouring a melt into a preform and performing an appropriate annealing treatment (Bock and Preisler 1989). With Bi-2223 one had to substitute about 20–30 at.% Pb for Bi in order to get a phase purity exceeding 90% ( Endo et al 1988 ). Since BSCCO melts at temperatures below 900°C, with a melting point depending on the respective composition and on the oxygen partial pressure (Endo et al 1988), it was then possible to perform partial melting inside an Ag sheath. This has indeed been tried utilizing Bi-2212 in an Ag sheath (Heine et al 1989, Tenbrink et al 1989). The Bi-2212 powder was prepared from high-purity Bi2O3, SrCO3, CaO, and CuO applying the mixed-oxide technique. The precursor powders were intimately mixed and calcined in up to three steps at 800–850 °C in order to remove the residual carbon content and to homogenize the material. After each calcination step the powder was ground carefully. The as-produced powder was put into an Ag tube and drawn to a wire with about 1 mm diameter. The final annealing consisted of a partial melting at 920°C followed by a long-term annealing at 840°C yielding the superconducting Bi-2212 phase with a Tc of 85 K (Heine et al 1989, Tenbrink et al 1989, 1990). At 77 K, 0 T the critical current density of about 1.200 A cm– 2 was comparable to YBCO but the magnetic field dependence of Jc was less pronounced. The same authors found at 4.2 K critical current densities up to about 6 × 104 A cm– 2 in selffield and a very high critical current density of up to 1.5 × 104 A cm– 2 even in a magnetic field of 26 T (Heine et al 1989). This result was a substantial breakthrough as for the first time an apparently weak-link free superconducting current transport was realized in truly polycrystalline HTS material. Obviously this material out-performs the metallic conventional superconductors at 4.2 K in very high fields due to its enormously high upper critical field, coincidentally making possible ultra-high-field superconducting magnets with magnetic fields beyond 20 T operated at 4.2 K. By measuring the temperature dependence of Jc it could be shown that such material carries comparatively high critical currents up to temperatures of 20-30 K (Heine et al 1991, Krauth et al 1991). At higher temperatures the critical current is limited by severe flux creep. The microstructure of these single core wires revealed no hint for texture at least from simple x-ray diffraction experiments performed on longitudinal and transverse cross-sections and also from scanning electron microscopy (SEM) investigations. During processing of ceramic Bi-2212 as well as Bi-2223 it was found that these BSCCO phases had a pronounced grain growth anisotropy leading to platelike grains with a large aspect ratio and thereby very short dimensions of the order of only a few micrometres parallel to the c axis. The other crystallographic directions namely the a and b direction on the other hand grow to the order of several tens of micrometres or even up to 100–200 µm. This behaviour favours the formation of texture in flat thin tapes of Bi2212 as well as of Bi-2223. Different manufacturing schemes have therefore been tried to develop textured tapes with improved critical current density. This work will be discussed in the following sections. (a) Bi-22 212 tapes

A very elegant approach allows the annealing of a Bi-2212-containing layer on an Ag substrate. As the Bi2212 precursor either a green tape is used or the layer is produced by doctor blade casting (DBC), screen printing or dip coating. As a first step an annealing treatment at temperatures of 400-500 °C has to be performed in order to remove the organic binder material. The textured Bi-2212 layer is produced by heating up the material slightly above the point of partial melting (880-900 °C depending on composition and oxygen partial pressure) and then slowly cooling at a rate of about 5-10 K h–1 to about 830-860 °C followed by a relatively short time, e.g. a few hours, at this temperature. On cooling, the platelike Bi-2212 grains nucleate and grow. Microstructural investigations showed the most perfectly textured and single-

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phase material just at the HTS/Ag interface (Feng et al 1992, Hellstrom 1992). As the grain growth is so highly anisotropic only those grains that are aligned parallel to the tape surface will grow to a large size. By this effect nucleation at places away from the HTS/Ag interface will obviously also lead to a situation where the material essentially consists of grains oriented with their large dimension parallel to the tape surface. Whether nucleation takes place at the HTS/Ag interface, in the liquid, or at the free surface remains an open question. A very high degree of texture is obtained using this method. This can easily be seen from figure B9.2.3 showing a cross-sectional scanning electron micrograph of an artificially broken tape. The c axis of the structure is aligned perpendicularly to the plane of the tape yielding the favourable condition for high critical currents— the a—b Cu—O plane is aligned parallel to the direction of current flow. The critical current density can be optimized by quickly cooling to room temperature after the annealing at 830–860°C. This helps to avoid excessive oxygen uptake which would lead to a degraded critical temperature. Additionally quick cooling helps to get around a phase decomposition below a temperature of about 550°C which would lead to a degradation of the critical current density. The loss of Bi2O3 in this open system connected with the relatively high Bi2O3 vapour pressure can be avoided by using Bi2Al4O9 powder and annealing this powder together with the tapes in order to have a higher Bi2O3 partial pressure (Shimoyama J et al 1992). Tapes prepared according to this scheme yield critical current densities exceeding 104 A cm– 2 at 77 K, 0 T. At this temperature flux creep is a severe problem for this material so that with small magnetic fields applied the critical current density drops markedly especially with the magnetic field aligned perpendicularly to the tape surface. Flux creep becomes less pronounced below about 30 K. Because of this, very high critical current densities of up to 2.6 × 105 A cm– 2 at 4.2 K and 12 T could be achieved. This high value is valid for the case where the magnetic field is applied parallel to the plane of the tape. The anisotropy of Jc is surprisingly low so that for the technically more interesting case (at least with respect to magnet technology) of B aligned perpendicularly to the plane of the tape Jc is still slightly higher than 1 X 105 A cm– 2.

Figure B9.23. A cross-sectional scanning electron micrograph of a high-quality Bi-(2212) tape ( Jc at 77 K, 0 T: 1.1 × 104 A cm– 2 ) (Krauth et al 1993), artificially broken perpendicularly to the direction of current flow. The c axes of the grains are aligned perpendicularly to the plates.

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Some disadvantages of such tapes have to be discussed. In order to yield a high degree of texture the Bi-2212 oxide layers have to be rather thin, of the order of only 20 µm thick. For larger thicknesses the texture is lost for that part of the layer which is too far from the HTS/Ag interface. This presents a problem concerning the overall critical current density as the Ag substrate usually has to be thicker so that it can be handled properly. For practical applications the Bi-2212 layer has to be protected from corrosion by moisture etc, a problem which cannot be solved straightforwardly and easily. In addition, such thin tapes are very sensitive and must be handled with extreme care in order to avoid irreversible damage. However, the technique of dip coating a metallic substrate is a comparably cheap and a very simple manufacturing route. Another successful approach is the above-mentioned ‘powder-in-tube’ technique. BSCCO powder of composition Bi:Sr:Ca:Cu equal to 2:2:1:2 was produced from oxide or carbonate precursor powders by annealing at temperatures of about 800°C. The as-produced powder was put into an Ag tube and then swaged and finally rolled into tapes of various thickness (0.1–0.3 mm). After an appropriate final annealing (the annealing conditions have not been published explicitly but are supposed to be equivalent to the conditions applied for the coated tapes mentioned in the preceeding section) critical current densities Jc of up to 3.5 × 104 A cm– 2 were obtained at 77 K and 0 T. In magnetic fields of more than 0.1 T Jc dropped markedly due to insufficient pinning. At 4.2 K very high critical current densities exceeding 105 A cm– 2 at 4.2 K and 0 T and up to 2 × 105 A cm– 2 at 30 T with magnetic field B aligned parallel to the tape surface were achieved (Enomoto 1991). X-ray diffraction experiments and, because of the platelike habit of the grains, optical microscopy revealed a pronounced degree of texture. The grains again align with their c axes perpendicular to the flat HTS/Ag interface. The a and b axes are aligned parallel to the plane of the tape which helps in improving critical current density. The critical current density of such textured tapes is anisotropic with respect to the orientation of the magnetic field. Higher values are obtained with the magnetic field aligned parallel to the plane of the tape. With respect to magnet technology, the value for the magnetic field aligned perpendicularly to the plane of the tape is also relevant. This value is about a factor of two lower (Enomoto 1991). This behaviour is due to anisotropic pinning properties. Very striking is the fact that with magnetic field B aligned parallel to the plane of the tape Jc is independent of the angle between B and the transport current I, that is Jc is not dependent on the Lorentz force (Mimura et al 1991). This behaviour has certain implications for the understanding of the mechanism of current transport in this polycrystalline material and will be discussed in detail in the following subsection. The overall critical current density of such tapes again is a problem. This is because of the large amount of Ag sheath material needed to produce proper thin tapes with a uniform HTS core cross-section. Additionally any insulation material must have a certain thickness. Therefore some tapes would have to be stacked together and this composite insulated as otherwise the amount of insulation in a coil would be too large. (b) Bi-22 212 wires

In order to overcome these problems round wires were produced applying the so-called ‘jelly-roll’ (JR) technique (Mimura et al 1992). A Bi-2212 powder was produced as described above, put into an Ag tube and rolled into tapes of about 0.2 mm thickness. These tapes were wound around an Ag rod so that about ten layers were obtained and then put into an Ag tube. The whole composite was swaged and drawn to a round wire of typically 1.5 mm diameter. This deformation yields a wire with jelly-roll-shaped Bi-2212 oxide layers of about 10 µm thickness. Attempts to produce thinner oxide layers failed as the thickness of the oxide layer became irregular with occasional interuptions. A final annealing treatment is supposed to be similar to that for the above-mentioned coated tapes although the authors do not explicitly report the annealing conditions necessary. The critical current density of such wires is up to 4.700 A cm– 2 at 77 K and 0 T and 3.3 × 104 A cm– 2 at 4.2 K and 10 T (Mimura et al 1992). These high values are due to the texturing of the Bi-2212 phase in the thin oxide layers of the ‘jelly-roll’ during the final annealing treatment. The Ag content of such wires is still rather high giving a distinctly lower overall

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critical current density of about 20% of the original value. The main advantage of these round wires is the easier and more flexible coil technology. With tapes one has to produce stacks of pancake coils whereas wires can be wound in nearly any desired shape. Because of the larger cross-sections and with the benefit of insulation, the handling of material is much easier than for the thin tapes. With these arguments in mind multifilamentary wires were also produced and tested with respect to their properties for application at lower temperatures. As described above for the simple single-core wires a Bi-2212 powder was prepared, put into an Ag tube and drawn to a wire. These wires were cut and bundled into a second Ag tube in order to produce wires with 7, 19, 37 and 85 filaments. By multiple bundling even higher filament numbers are possible. Obviously this technique is simple and can be easily scaled up to larger quantities. It yields wires with an HTS volume fraction as high as 35–40% which is advantageous with respect to the overall critical current density. The final annealing treatment is similar to the two-step annealing described above for the single-core wires. The critical current densities are of course lower than for textured thin tapes. As a result of the lower amount of Ag incorporated into the composite, overall critical current density is quite respectable and values of 6 × 103 A cm– 2 at 4.2 K and 10 T are reached for average specimens (Tenbrink and Krauth 1993). Figure B9.2.4 shows a cross-section of a 1 mm diameter 37-filamentary wire.

Figure B9.2.4. A cross-sectional micrograph of a 37-filamentary Bi-2212/Ag wire of 1 mm diameter.

Figures B9.2.5 and B9.2.6 compile some of the data on critical current density of BSCCO tapes and wires at 77 K and at 4.2 K respectively. It has to be noted that the values given in figures B9.2.5 and B9.2.6 refer to the current density with respect to the cross-section of the HTS and not to the technically more relevant overall critical current density. Furthermore, data for the orientation of magnetic field B aligned perpendicularly to the plane of the tapes often are not explicitly given in the references. Due to the rather sharp anisotropy around the parallel orientation of B this value is valid at least with respect to magnet technology. This current density at 4.2 K is usually about a factor of 1.4 to 2 lower as mentioned throughout the text. At 77 K this anisotropy is much stronger, severely limiting the application of these materials. Wires of course are isotropic with respect to the orientation of the magnetic field. (c) Bi-22 223 tapes

The largest efforts to produce practical conductors so far have been made with Bi-2223. This is because of the relatively high transition temperature Tc of about 110 K and the fact that pinning properties at T > 10 K are superior to those of Bi-2212. The critical current densities achieved so far are of the order of some

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Figure B9.2.5. Critical current density Jc of BSCCO wires and tapes at 77 K. Open symbols refer to measurements with B aligned parallel to the film surface. Full symbols refer to measurements with B aligned perpendicularly to the film surface. Data are from: circles—Yamada et al (1992), squares—Sato et al (1991), diamonds—Enomoto (1991), triangles—Shimoyama J et al (1992), inverted triangles-Krauth et al (1993).

New SC wires

Figure B9.2.6. Critical current density Jc of BSCCO wires and tapes at 4.2 K. Open symbols refer to measurements with B aligned parallel to the film surface. Full symbols refer to measurements with B aligned perpendicularly to the film surface. Data are from: circles— Yamada et al (1992), squares—Sato et al (1991), open diamonds—Enomoto (1991), triangles—Shimoyama J et al (1992), full diamonds—Mimura et al (1992), inverted triangles—Tenbrink et al (1991).

104 A cm– 2 at 77 K and 0 T, at least in short specimens, with a moderate magnetic field dependence of Jc up to magnetic fields of a few tenths of a tesla. The critical current density at this high temperature is limited by flux creep, showing again that pinning properties of BSCCO materials are inferior to those of YBCO. On the other hand the moderate magnetic field dependence of Jc for Bi-2223 at magnetic fields below 1 T (B|| tape surface) allows us to envisage applications in energy technology where only small magnetic fields occur, e.g. in superconducting cables, current bus bars or current limiters. For Bi-2223 different precursor compositions have been used. All compositions given in the literature are slightly off-stoichiometric, while a certain amount (usually 20–30%) of Bi is substituted by Pb. The powders in most cases again are produced from oxide or carbonate precursor powders by applying the mixed-oxide technique. It is important to note that during calcination excessive formation of the Bi-2223 phase has to be avoided in order to yield high critical current densities in tapes produced from the powder. Typical calcination treatments are therefore limited to temperatures below about 820°C. A typical powder therefore mainly consists of the Bi-2212 phase, minor amounts of the Bi-2201 phase, and small amounts of other phases like (Ca, Sr)2PbO4 and alkaline-earth cuprates. The powder is put into an Ag tube, swaged and/or drawn to a wire of typically 1–2 mm diameter, and finally rolled into a tape of 0.1–0.15 mm thickness, the thickness of the ceramic HTS core being about 30–50 µm. A sophisticated thermomechanical treatment of these tapes leads to high critical current densities. A first annealing treatment has to be performed in a temperature range where partial melting occurs, the temperature being in the range 830–850°C (in air), of course depending on the exact composition. In the very initial period of this annealing platelike Bi-2212 forms at the HTS/Ag interface. This can be seen from figure B9.2.7 where the Ag sheath of a tape has been stripped off after only a few hours of annealing. During further annealing this Bi-2212 transforms to Bi-2223 by solid-state diffusion assisted by the presence of a liquid phase which facilitates long-range diffusion. As the plates of the Bi-2223 phase grow, the density of the ceramic core decreases. Therefore the tape has to be densified by a compressive

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Figure B9.2.7. A scanning electron micrograph showing the inner side of the Ag sheath of a Bi-2223/Ag tape.

After about 2 h of annealing the Ag sheath has been carefully mechanically removed. Platelike Bi-2212 grains can be seen attached to the Ag. The steplike features on the Ag surface in all probability are caused by gliding bands formed during the deformation associated with the removal of the Ag sheath.

load. Uniaxially pressing is most effective with respect to critical current density. A second annealing renders possible the healing of the deformation-induced defects and cracks and especially the further growth of the Bi-2223 plates. The highest critical current densities are achieved if the tapes are uniaxially pressed for a second time and afterwards annealed for a third time. A further pressing and annealing cycle usually leads to a decrease of the critical current density due to the fact that the liquid phase is used up. By carefully optimizing the processing, putting special emphasis on the densification of the ceramic HTS core, critical current densities up to 6.6 x 104 A cm– 2 (Yamada et al 1992) could be achieved at 77 K and 0 T in short specimens. Such tapes yield critical current densities slightly above 105 A cm–2 at 4.2 K in very high magnetic fields (B aligned parallel to the tape surface). Some data are included in figures B9.2.5 and B9.2.6 for comparison. If long lengths of such tapes are produced the uniaxial pressing process has to be replaced by rolling. Thereby only lower critical current densities of e.g. about 104 A cm–2 (Malozemoff et al 1992, Mukai et al 1992) could be achieved probably due to the occurrence of additional cracks during rolling or insufficient densification. With the production of longer lengths, multifilamentary tapes are usually preferred as they yield a better uniformity and are superior with respect to their mechanical properties. (d) Current transport in BSCCO

How is it possible to have high critical transport currents in BSCCO whereas in YBCO nearly every type of grain boundary behaves as a weak link? First more speculative ideas are related to the problem of oxygen stoichiometry. In YBa2Cu3O7–x the oxygen stoichiometry may vary between O6 and O7 depending on the respective temperature and oxygen partial pressure during the equilibration of the oxygen content. This huge difference in stoichiometry is adopted in the basal plane of the structure. Full occupancy of the chain sites means the maximum oxygen concentration of O7 and gives optimum superconducting properties. It was speculated that at grain boundaries this relatively loosely bound oxygen is partially lost or that the oxygen occupancy is ‘used’ to reduce strains resulting from structural misfits at the grain boundaries. These effects would lead to a broadening of the disturbed grain boundary zone. In view of the very short coherence lengths of this material this means a loss of superconductivity at this structural defect. Oxygen

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deficiency has indeed been observed at grain boundaries in YBCO (see section B9.2.2.2 (Zhu et al 1993)). In measuring the transport critical current densities Jc of Bi-2212/Ag wires no apparent hint of weak links comparable to those found in YBCO could be detected and the results were discussed in the framework of conventional flux pinning theory (Heine et al 1989, Tenbrink et al 1990). Weak pinning was suggested to be responsible for the low Jc at higher temperatures whereas flux creep becomes less significant at lower temperatures enabling the very high critical current density even in high magnetic fields at 4.2 K. These arguments were further supported by the fact that transport critical current density and critical current density deduced from magnetization measurements are nearly identical in single-core BSCCO wires or tapes measured at low temperatures (Cassidy et al 1992, Heine et al 1992). In BSCCO the variation in oxygen stochiometry is much less pronounced. In Bi2Sr2Ca1Cu2O8+x or Bi2Sr2Ca2Cu3O10+x x lies between +0.1 and +0.4. This variation in oxygen content is distinctly smaller than in YBCO and is supposed to take place in the Bi2O3 layers of the structure. Nevertheless this fact gives no direct explanation of why BSCCO behaves so differently. An important hint towards the underlying mechanism is the fact that the critical current density is observed to be independent of the Lorentz force as discussed above. With a magnetic field B applied parallel to the plane of a thin tape the angle between B and the transport current does not influence the value of the critical current. This is an indication that the transport current does not flow straight but may percolate through the sample in order to flow around barriers. At this point the so-called ‘brick wall’ model was proposed. Microstructural observations revealed that at least in textured thin tapes the BSCCO grains with their very large aspect ratio (large dimension parallel to the a and b axes, very small thickness parallel to the c axis) are aligned with their c axis normal to the plane of the tape. Thus they are stacked like a ‘brick wall’ but with no alignment of the a and b axes ( Bulaevskii et al 1992, Sato et al 1990 ). For a better illustration this is schematically shown in figure B9.2.8(a). In this model it is assumed that the grain boundaries at the short sides of adjacent grains behave as weak links and their contribution to the current transport may be neglected. Therefore the current would have to flow around this obstacle, i.e. parallel to the c axis within a grain and through the c axis twist-type grain boundary between the wide faces of adjacent grains in order to pass the obstacle and move to the neighbouring grains. It was argued that in the case of magnetic field B parallel to the plane of the tape, Jc is in any case limited by the properties of the twist-type grain boundaries giving a Jc independent of the angle between the current and B as indeed observed in transport critical current

Figure B9.2.8. Schematic drawings of the situation of current flow. The ‘brick wall’ model proposes weak-link

behaviour of the grain boundaries at the short faces of the grains and the current flowing around this obstacle as indicated in (a). The ‘railway switch’ model assumes no weak-link behaviour of this type of grain boundary, bad connectivity at the wide faces of the grains as observed by microstructural investigations and a current flow across small-angle grain boundaries as indicated in (b).

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measurements (Mimura et al 1991, Sato et al 1990). For B aligned perpendicularly to the plane of the tape Jc was suggested to be limited by the bulk pinning properties of the material (Sato et al 1990). This model of current flow, i.e. a current percolating through a brick-wall-like structure, was further theoretically elaborated assuming a limitation of Jc by Josephson critical currents across the large-area c-axis twist-type grain boundaries (Bulaevskii et al 1992). This model of current flow is further supported by results on bicrystalline Bi-2212 thin films. A drop of Jc across the grain boundary was found comparable to the results obtained on YBCO thin films mentioned in section B9.2.2.2 (Kawasaki et al 1993). Some doubts remain on the other hand as the Tc values of the films were below 50 K and their quality obviously not comparable to that of the Bi-2212 films mentioned in section B9.2.2.3. It has to be noted that an essential assumption of this model of current transport is that a substantial supercurrent can flow within the grains parallel to the c axis. Based on detailed microstructural investigations, texture analysis and resistivity measurements of isolated Bi-2223 superconductor cores another model was proposed (Hensel et al 1993). Transmission electron microscopy (TEM) investigations revealed that the grain boundaries at the small faces of adjacent grains in most cases are free from extraneous phases and often represent small-angle grain boundaries. The large-area grain boundaries of the wide sides of the grains on the other hand often were seen to be poorly connected and also to contain extraneous phases. The very low normal resistivity of the Bi-2223 cores measured in the experiments favours the argument that the small-angle grain boundaries are strongly linked and that the supercurrent flows parallel to the a—b plane within the grains switching to other grains and thereby percolating through the sample across the small-angle grain boundaries at the small faces of adjacent platelike grains (‘railway switch’ model). This situation is shown schematically in figure B9.2.8(b). This model is further supported by the fact that critical current measurements along the c axes of single crystals were more than one order of magnitude smaller than those needed to explain the high critical transport currents within the framework of the ‘brick wall’ model (Hensel et al 1993). It has to be noted that these points are still controversial. Nevertheless it can be concluded from both arguments that higher critical current densities will be achieved with better textured material although an improved texture may not help to increase Jc with B aligned perpendicularly to a tape. With BSCCO an improved current-carrying capacity requires a better alignment of the c axes of the respective grains whereas the grains may be rotated around the c axis. This is a very distinct difference from YBCO where all crystallographic axes have to be aligned. One has to keep in mind that both BSCCO phases form large platelike grains which facilitate easy formation of texture and provide both types of grain boundary mentioned above. YBCO on the other hand can hardly be textured by this relatively easy method because of its more uniform grain growth properties. (e) Technical aspects

As the sheath material of HTSs has to be permeable to oxygen during the annealing treatment pure Ag has been used during development of BSCCO superconductors. As the typical annealing schedules are relatively close to the melting point of pure Ag (960°C) the sheath material recrystallizes during the annealing treatment and becomes very soft. As during magnet operation large hoop stresses may arise due to the Lorentz force acting on the conductor, this was of course an intolerable condition. A better sheath material for HTSs has to fulfil certain demands, e.g. it must show (a) high workability, (b) oxygen permeability, (c) no detrimental reaction with the HTS core, (d) sufficiently high tensile strength and (e) as the most stringent demand that properties (b), (c) and (d) must be retained during the long-term final annealing treatment. These considerations led to the conclusion that an Ag alloy hardened by a fine oxide dispersion may be more appropriate. Therefore AgNiMg alloys have been investigated with respect to their suitability (Tenbrink et al 1992). Ni leads to an increased deformation hardening because of the finer grain structure of AgNi. The addition of Mg yields a much more pronounced effect. After internal oxidation of Mg (hardening) the yield strength of Bi-2212/AgNiMg wires distinctly increases. After

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hardening and the long-term annealing treatment necessary to form the superconductor, yield strengths Rp 0.1 of Bi-2212/AgNiMg wires of up to 150 MPa could be obtained depending on the type of alloy used. This value nearly compares to that of commercial Nb3Sn bronze wires. This increased strength removes a major obstacle towards the application of HTSs in magnet technology. Another important point is the strain tolerance of superconductor wires. Based on practical experience with metallic superconductors, as a general rule the material should sustain at least 0.2% strain without degradation of the critical current density. This specification has fortunately been met even with simple single-core Bi-2212/Ag wires (Ekin 1992). This type of wire represents the worst case with respect to crack formation and growth. Indeed multifilamentary wires or tapes yielded higher strain tolerances up to values beyond 0.5% (Ekin 1992). The mechanisms allowing for such a high strain tolerance of these ceramic/metal composites are not yet fully understood. (f) Demonstration coils with BSCCO

Small demonstration coils using BSCCO Ag-sheathed tapes or wires have already been manufactured and tested with respect to their superconducting properties. In most cases either Bi-2212/Ag tapes (Shibutani et al 1992, Shimoyama T et al 1992) or Bi-2223/Ag tapes (Sato et al 1992, Kitamura et al 1992) have been used. These tapes were manufactured into several pancake coils which were stacked in order to form the test coil. These test coils were able to achieve self-fields up to 1.6 T at 4.2 K (Shibutani et al 1992). Solenoids have also been prepared with Bi-2212/Ag round wires. Using ‘jelly-roll’ wire a self-field of about 1 T could be obtained (Mimura et al 1992). With a 37-filamentary Bi-2212/Ag wire small solenoids yielded a self-field of 0.13 T ( Tenbrink and Krauth 1993 ). Measurements at 20 K performed on Bi2223/Ag pancake-type coils revealed that the critical current is still 60–65% of the respective 4.2 K value (Sato et al 1992) showing that these coils may well also be applied in this temperature range. These results in principle demonstrate the feasibility of making superconducting magnets with these HTS ceramic/metal composite materials. With respect to a routine application, wire manufacturing technology still has to be scaled up and, in particular, the reproducibility improved. B9.2.3.3 Tl-Ba-Ca-Cu oxide As mentioned in section B9.2.2.4 the irreversibility lines of Tl-1212 and Tl-1223 with their single Tl—O layer are shifted to higher temperatures than for the respective compounds with two intermediate Tl— O layers. This should in principle allow the application of such material in magnetic fields of a few tesla even at 77 K. Applying the usual ‘powder-in-tube’ technique Tl-1223/Ag tapes have been prepared by drawing and finally rolling the composite into a thin tape (Aihara et al 1992). In order to stabilize the Tl1223 phase part of the TI had to be replaced by Pb and part of the Ba by Sr. After annealing at 880°C critical current densities as high as 2.5 × 104 A cm– 2 (77 K, 0 T) could be achieved. Unfortunately the critical current density drops to a level of about 103 A cm– 2 between 0 and 0.5 T, probably due to the presence of weak links at grain boundaries. At B = 8 T Jc is still 500 A cm– 2 because of the effective pinning connected with the reduced anisotropy of this compound. The existence of weakly coupled grains (although much less pronounced than in YBCO) and the fact that Jc at 0 T is lower than in BSCCO tapes may be related to an inadequate formation of texture eventually connected with the smaller grain growth anisotropy of TBCCO compared with BSCCO. As already mentioned only very few groups work on TBCCO especially on bulk material due to the extreme toxicity of Tl. Fortunately Tl-free 1212-type compounds have recently been discovered (Beales et al 1992, Rouillon et al 1989) and other groups are also trying to synthesize similar compounds. Up to now the production of single-phase material has not yet been accomplished. If in the future nearly phase-pure 1212- or 1223-type material with adequate texture is satisfactorily produced this material in all probability will successfully be used in magnet technology at 77 K.

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B9.2.3.4 Possibilities of application in magnet and energy technology Initially HTSs will be applied in areas which in principle are not accessible to the metallic superconductors Obviously this is true for very high-field superconducting magnets beyond 20 T operated at 4.2 K. As the critical current density is sufficient, the mechanical problems seem to be solved by using oxid dispersion strengthened Ag alloys, and small coils have successfully been demonstrated. This application are envisaged in the very near future. In contrast there are applications where HTSs are in competition with well established metallic superconductor technology. In this case HTSs can offer advantages in terms of costs or easier accessibility One such example is superconducting magnets up to magnetic fields of a few tesla operated at 20–30 K. As critical current densities of Bi-2212 at 20 K and Bi-2223 up to 30 K are still quite high this application seems feasible. These magnets may be either conduction cooled using a cryocooler or operated, for example, in liquid neon (27 K). As critical current densities currently are at the lower limit of a meaningful application and perhaps an adapted magnet technology is necessary this issue still require some more work but seems to be feasible also in the near future. At present a straightforward route to the widespread use of superconducting technology at 77 cannot be foreseen. This is due to the severe material problems associated with the anisotropic structur and properties of these layered compounds. If the current-carrying capacity at 77 K of Bi-2223 can b raised to the level of 105 A cm- 2 such a material may well be used in energy technology, e.g. in cables as the magnetic fields involved can be kept sufficiently small. Based on the present knowledge a wide breakthrough for application at 77 K requires materials with reduced anisotropy, i.e. compounds with reduced number of intermediate layers, which in principle exhibit better pinning properties. Tl-free 1212- or 1223-type compounds have been found but phase purity is still a problem. As an additional requiremer easy texturing should be possible which means that the compounds must show a pronounced grain growt anisotropy. YBCO exhibits sufficient pinning to meet the demands for operation at 77 K but unfortunate! it is unclear whether the problem of weak coupling at the grain boundaries may ever be solved. Obviously the issue of 77 K operation requires a lot more work to be done. B9.2.4 Other possible applications of bulk HTS materials With HTSs operating in liquid nitrogen at 77 K new applications of superconductors seem feasible; a few of them should be mentioned. Obviously HTSs may be used as current leads even for conventional superconducting magnets operate at 4.2 K. For this purpose the moderate current densities of untextured Bi-2212 bulk material and even of polycrystalline bulk YBCO are sufficient. As a result of the reduced heat conduction the application of HTSs will help to reduce He losses currently associated with the application of normal-conducting coppe current leads (see, for example, Herrmann et al 1992). Also the application of larger current bus bar between arrays of, for example, accelerator magnets may be envisaged. Another application may be in magnetic shielding technology. Conventional shielding equipmer made of a combination of soft magnetic and aluminium shields shows a lack of shielding ability at relatively low frequencies. HTS tubes on the other hand exhibit very high shielding factors just in thi frequency range (Matsuba et al 1992). It has to be mentioned that a magnetic field present during cooling of the HTSs would be ‘frozen in’ so that some kind of compensation has to be performed in order to reduce the trapped magnetic field. Also flux creep may be a problem concerning the long-term stabilit of the shielding factor. Whether HTS magnetic shields will therefore have wider acceptance in the future remains to be seen. Several techniques have been developed based on partial melting of YBCO, yielding samples up to sizes of some cubic centimetres, consisting of very few large grains. By seeding with a single crystal even

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specimens with essentially one very large grain have been obtained. As YBCO exhibits sufficient pinning at 77 K large intra-grain critical currents are possible leading to high magnetizations (Murakami 1993, Murakami et al 1992). Therefore such materials may be used in magnetic bearing technology. By cooling such samples in the presence of a strong magnetic field the flux is trapped because of effective pinning. Unless the material is warmed up above Tc such a sample behaves as a (superconducting) permanent magnet with magnetic fields of up to 1.5 T already achieved. Such materials therefore may also be used in electric motors and similar applications in the near future. References Aihara K, Doi T, Soeta A, Takeuchi S, Yuasa T, Seido M, Kamo T and Matsuda S 1992 Flux pinning in Tl-(1223) superconductor Cryogenics 11 936–9 Beales T P, Dineen C, Freeman W G, Hall S R, Harrison M R, Jacobson D M and Zammattio S J 1992 Superconductivity at 92 K in the (Pb,Cd)-1212 phase (Pb0.5Cd0.5)Sr2(Y0.7Ca0.3)Cu2O7−δ Supercond. Sci. Technol. 5 47–9 Bednorz J G and Müller K A 1986 Possible high Tc superconductivity in the Ba—La—Cu—O System Z. Phys. B 64 189–93 Bock J and Preisler E 1989 Preparation of single phase 2212 bismuth strontium calcium cuprate by melt processing Solid State Commun. 72 453–8 Bulaevskii L N, Clem J R, Glazman L I and Malozemoff A P 1992 Model for the low-temperature transport of Bi-based high-temperature superconducting tapes Phys. Rev. B 45 2545–8 Cassidy S M, Cohen L F, Cuthbert M N, Laverty J R, Perkins G K, Caplin A D, Dou S X, Guo Y C, Liu H K, Liu F and Wolf E L 1992 High critical currents in Ag-BSCCO(2223) tapes: are the grain boundaries really ‘weak links’? J. Alloys Compounds 195 503–6 Cima M J, Flemings M C, Figueredo A M, Nakade M, Ishii H, Brody H D and Haggerty J S 1992 Semisolid solidification of high temperature superconducting oxides J. Appl. Phys. 72 179–90 Clem J R 1992 Fundamentals of vortices in the high-temperature superconductors Supercond. Sci. Technol. 5 S33–40 Dersch H and Blatter G 1988 New critical-state model for critical currents in ceramic high-Tc superconductors Phys. Rev. B 38 11 391–404 Dimos D, Chaudhari P and Mannhart J 1990 Superconducting transport properties of grain boundaries in YBa2Cu3O7 bicrystals Phys. Rev. B 41 4038–49 Dimos D, Chaudhari P, Mannhart J and LeGoues F K 1988 Orientation dependence of grain-boundary critical currents in YBa2Cu3O7− δ bicrystals Phys. Rev. Lett. 61 219–22 Ekin J W, Finnemore D K, Li Q, Tenbrink J and Carter W 1992 Effect of axial strain on the critical current of Ag-sheathed Bi-based superconductors in magnetic fields up to 25 T Appl. Phys. Lett. 61 858–60 Endo U, Koyama S and Kawai T 1988 Preparation of the high-Tc phase of Bi—Sr—Ca—Cu—O superconductor Japan. J. Appl. Phys. 27 L1476–9 Enomoto N 1991 The transport critical current properties of BSCCO superconducting wires Advances in Superconductivity III ed K Kajimura and H Hayakawa (Tokyo: Springer) pp 625–30 Feng Y, Hautanen K E, High Y E, Larbalestier D C, Ray R II, Hellstrom E E and Babcock S E 1992 Microstructural analysis of high critical current density Ag-clad Bi—Sr—Ca—Cu—O (2:2:1:2) tapes Physica C 192 293–305 Fischer K, Leitner G, Fuchs G, Schubert M, Schlobach B, Gladun A and Rodig C 1992 Preparation andcritical current density of melt processed Y—Ba—Cu—O thick films and AgPd-sheathed tapes Cryogenics 33 97–103 Heine K, Tenbrink J and Krauth H 1989 High-field critical current densities in Bi2Sr2Ca1Cu2O8+x /Ag wires Appl. Phys. Lett. 55 2441–3 Heine K, Tenbrink J and Krauth H 1991 Temperature dependence of critical currents in Bi-based high-Tc superconductor wires High-Temperature Superconductors Materials Aspects ed H C Freyhardt, R Flükiger and M Peuckert (Oberursel: DGM Informationsgesellschaft) pp 1027–32 Heine K, Tenbrink J and Krauth H 1992 Critical current densities and magnetization of Bi-based high-Tc superconductor wires Cryogenics ICMC Suppl. 32 504–7 Hellstrom E E 1992 Phase relations and alignment in bismuth-based high-Tc wires J. Mater. October 48–53

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Hensel B, Grivel J C, Jeremie A, Perin A, Pollini A and Flükiger R 1993 A model for the critical current in (Bi,Pb)2Sr2Ca2Cu3Ox silver-sheathed tapes Physica C 205 329–37 Herrmann P F, Albrecht C, Bock J, Cottevieille, Elschner S, Herkert W, Lafon M O, Lauvray A, Leriche A, Preisler E, Salzburger H, Tourre J M and Verhaege T 1992 European project for the development of high Tc current leads IEEE Trans. Appl. Supercond. AS-3 876–80 Hettich B, Freilinger B, Majewski P, Popp T and Schulze K 1991 Constitution and superconducting properties in the system Bi—Sr—Ca—Cu—O High-Temperature Superconductors Materials Aspects ed H C Freyhardt, R Flükiger and M Peuckert (Oberursel: DGM Informationsgesellschaft) pp 399–404 Hofer G, Kleinlein W and Hein M 1992 A three-step-process for producing a superconducting YBa2Cu3O6.5+x −Ag composite Proc. ICMAS-92 ed C W Chu and J Fink (IITT-International) pp 87–92 Hofer G et al 1993 unpublished result Iijima Y, Onabe K, Futaki N, Tanabe N, Sadakata N, Kohno O and Ikeno Y 1992 In-plane texturing control of Y— Ba—Cu—O thin films on polycrystalline substrates by ion-beam-modified intermediate buffer layers IEEE Trans. Appl. Supercond. AS-3 1510–5 Jin S, Tiefel T H, Sherwood R C, van Dover R B, Davis M E, Kammlott G W and Fastnacht R A 1988 Melt-textured growth of polycrystalline YBa2Cu3O7− δ with high transport jc at 77 K Phys. Rev. B 37 7850–3 Kawasaki M, Sarnelli E, Chaudhari P, Gupta A, Kussmaul A, Lacey J and Lee W 1993 Weak link behaviour of grain boundaries in Nd-, Bi-, and Tl-based cuprate superconductors Appl. Phys. Lett. 62 417–9 Kitamura T, Hasegawa T and Ogiwara H 1992 Design and fabrication of Bi-based superconducting coil IEEE Trans. Appl. Supercond. AS-3 939–41 Krauth H, Heine K, Tenbrink J, Wilhelm M and Neumüller H W 1991 Recent developments in BiSrCaCuO based wires and tapes Advances in Supercond. III ed K Kajimura and H Hayakawa (Tokyo: Springer) pp 613–8 Krauth H, Tenbrink J, Neumüller H W, Wilhelm M, Fischer K, Schubert M, Goldacker W and Keβler J 1993 Processing and properties of Bi—Sr—Ca—Cu—O based wires and tapes Proc. Eur. Conf. on Appl. Supercond. EUCAS’93 (Göttingen, 1993) (Oberursel: DGM Informationsgesellschaft) pp 147–54 Larbalestier D C, Babcock S E, Cay X Y, Field M B, Gao Y, Heinig N F, Kaiser D L, Merkle K, Williams L K and Zhang N 1991 Electrical transport across grain boundaries in bicrystals of YBa2Cu3O7−d Physica C 185–189 315–20 Maeda H, Tanaka Y, Fukutomi M and Asano T 1988 A new high-Tc oxide superconductor without a rare earth element Japan. J. Appl. Phys. 27 L209–10 Majewski P, Freilinger B, Hettich B, Popp T and Schulze K 1991 Phase equilibria in the system Bi2O3–SrO—CaO— CuO at temperatures of 750°C, 800°C, and 850°C in air High-Temperature Superconductors Materials Aspects ed H C Freyhardt, R Flükiger and M Peuckert (Oberursel: DGM Informationsgesellschaft) pp 393–8 Mannhart J, Chaudhari P, Dimos D, Tsuei C C and McGuire T R 1988 Critical currents in [001] grains and across their tilt boundaries in YBa2Cu3O7 films Phys. Rev. Lett. 61 2476–9 Malozemoff A P, Carter W L, Gannon J, Joshi C H, Miles P, Minot M, Parker D, Riley G N Jr, Thompson E and Yurek G 1992 Progress in the development of bismuth-based high-temperature superconducting tapes Cryogenics ICMC Suppl. 32 478–84 Matsuba H, Yahara A and Irisawa D 1992 Magnetic shielding properties of high-Tc superconductor Supercond. Sci. Technol. 5 S432–9 Mimura M, Enomoto N, Uno N, Nakajima M, Kumakura H and Togano K 1991 Bi based high Tc oxide tapes and their high field performance at 4.2 K High-Temperature Superconductors Materials Aspects ed H C Freyhardt, R Flükiger and M Peuckert (Oberursel: DGM Informationsgesellschaft) pp 251–4 Mimura M, Kinoshita T, Uno N, Tanaka Y and Doi K 1992 Solenoid coil using multilayered composite wire of Bi-based superconductor Advances in Superconductivity V ed Y Bando and H Yamauchi (Tokyo: Springer) pp 693–6 Mukai H, Ohkura K, Shibuta N, Hikata T, Ueyama M, Kato T, Fujikami J, Muranaka K and Sato K 1992 Bi-based silver-sheathed high-Tc superconducting wire and application Advances in Superconductivity V ed Y Bando and H Yamauchi (Tokyo: Springer) pp 679–84 Murakami M, Yamaguchi K, Fujimoto H, Nakamura N, Taguchi T, Koshizuka N and Tanaka S 1992 Flux pinning by non-superconducting precipitates in melt-processed YBaCuO superconductors Cryogenics 32 930–5 Murakami M 1992 Processing of bulk YBaCuO Supercond. Sci. Technol. 5 185–203

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Murakami M 1993 Novel Application of high Tc superconductors Appl. Supercond. 1 1157–73 Nabatame T, Saito Y, Aihara K, Kamo T and Matsuda S P 1993 Comparison of transport properties between Tl-(1223) and Tl-(2223) phases of Tl—Ba—Ca—Cu—O systems Japan. J. Appl. Phys. 32 L484–7 Nagaya S, Miyajima M, Hirabayashi I, Shiohara Y and Tanaka S 1991 Rapid solidification of high-Tc superconductors by a laser zone melting method IEEE Trans. Magn. MAG-27 1487–94 Neumüller H W, Gerhaüser W, Ries G, Kummeth P, Schmidt W, Klaumünzer S and Saemann-Ischenko G 1993 Ion irradiation of layered BSCCO compounds: flux line pinning and evidence for 2-D behaviour Cryogenics 33 14–20 Rao C N R, Nagarajan R and Vijayaraghavan 1993 Synthesis of cuprate superconductors Supercond. Sci. Technol. 6 1–22 Roas B, Schultz L and Saemann-Ischenko G 1990 Anisotropy of the critical current density in epitaxial YBa2Cu3Ox , films Phys. Rev. Lett. 64 479–82 Rouillon T, Provost J, Hervieu M, Groult D, Michel C and Raveau B 1989 Superconductivity up to 100 K in lead cuprates: a new superconductor b0.5Sr2.5Y0.5Ca0.5Cu2O7-8 Physica C 159 201–9 Sato K, Hikata T, Mukai H, Masuda T, Ueyama M, Hitotsuyanagi H, Mitsui T and Kawashima M 1990 Advances in Superconductivity II ed T Ishiguro and K Kajimura (Tokyo: Springer) pp 335–40 Sato K, Shibuta N, Mukai H, Hikata T, Ueyama M and Kato T 1991 Development of silver-sheathed bismuth superconducting wires and their application J. Appl. Phys. 70 6484–8 Sato K, Shibuta N, Hikata T, Kato T and Iwasa Y 1992 Critical currents of silver-sheathed bismuth-based tapes at 20 K: small coils and field orientation anisotropy in external fields up to 20 T Appl. Phys. Lett. 61 714–6 Schmitt P, Kummeth P, Schultz L and Saemann-Ischenko G 1991 Two-dimensional behaviour and critical-current anisotropy in epitaxial Bi2Sr2CaCu2O8+x , thin films Phys. Rev. Lett. 67 267–70 Sheng Z Z and Hermann A M 1988 Bulk superconductivity in the Tl—Ca/Ba—Cu—O system Nature 332 138–9 Sheng Z Z, Hermann A M, El Ali A, Almasan C, Estrada J, Datta T and Matson R J 1988 Superconductivity at 90 K in the Tl—Ba—Cu—O System Phys. Rev. Lett. 60 937–0 Shibutani K, Egi T, Hayashi S, Fukumoto Y, Shigaki I, Masuda Y, Ogawa R and Kawate Y 1992 Fabrication of superconducting joints for Bi-2212 pancake coils IEEE Trans. Appl. Supercond. AS-3 935–8 Shimoyama J, Tomita N, Morimoto T, Kitaguchi H, Kumakura H, Togano K, Maeda H, Nomura K and Seido M 1992 Improvement of reproducibility of high transport Jc for Bi2Sr2CaCu2O8/Ag tapes by controlling Bi content Japan. J. Appl. Phys. 31 L1328-31 Shimoyama T, Morimoto H, Kitaguchi H, Kumakura H, Togano K, Maeda H, Nomura K and Seido M 1992 Fabrication of Bi-2212/Ag pancake coils and their properties Applied Superconductivity Conf. ASC’92 (Chicago, 1992) Tachiki M and Takahashi S 1989 Strong vortex pinning intrinsic in high-Tc oxide superconductors Solid State Commun. 70 291–5 Tachiki M and Takahashi S 1992 Effect of intrinsic pinning on critical current in cuprate superconductors Cryogenics 32 923–9 Tenbrink J, Heine K, Krauth H, Szulczyk A and Thöner M 1989 Entwicklung von hoch-Tc -Supraleiterdrähten VDI Ber. 733 399–104 Tenbrink J, Heine K and Krauth H 1990 Critical currents and flux pinning in Ag stabilized high Tc superconductor wires Cryogenics 30 422–6 Tenbrink J, Wilhelm M, Heine K and Krauth H 1991 Development of high-Tc superconductor wires for magnet applications IEEE Trans. Magn. MAG-27 1239–46 Tenbrink J, Wilhelm M, Heine K and Krauth H 1992 Development of technical high-Tc superconductor wires and tapes IEEE Trans. Appl. Supercond. AS-3 1123–6 Tenbrink J and Krauth H 1993 Adv. Cryogen. Eng. 40 305–11 von Schnering H G, Walz L, Schwarz M, Becker W, Hartweg M, Popp T, Hettich B, Müller P and Kämpf G 1988 The crystal structure of the superconducting oxides Bi2(Sr1−xCax )CuO8–δ and Bi2(Sr1− y Cay )3Cu2O10− δ with 0 ≤ x ≤ 0.3 and 0.16 ≤ y ≤ 0.33 Angew. Chem. 100 604 (Engl. Transl. Angew. Chem. Int. Edn. Engl. 27 574) Yamada Y, Satou M, Murase S, Kitamura T and Kamisada Y 1992 Microstructure and superconducting properties of Ag sheathed (Bi, Pb)2Sr2Ca2Cu3Ox tapes Advances in Superconductivity V ed Y Bando and H Yamauchi (Tokyo: Springer) pp 717–20

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Wu M K, Ashburn J R, Torng C J, Hor P H, Meng R L, Gao L, Huang Z J, Wang Y Q and Chu C W 1987 Superconductivity in a new mixed-phase Y—Ba—Cu—O system at ambient pressure Phys. Rev. Lett. 58 908–10 Zhu Y, Wang Z L and Suenaga M 1993 Grain-boundary studies by the coincident-site lattice model and electronenergy-loss spectroscopy of the oxygen K edge in YBa2Cu3O7− d Phil. Mag. A 67 11–2

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B9.3 The case of Bi(2223) tapes

R Flükiger and G Grasso

B9.3.1 Introduction The high-Tc superconductor showing the highest potential for industrial applications is the compound Bi2Sr2Ca2Cu3O10, commonly abbreviated to Bi(2223), which undergoes a superconducting transition at 110 K. This phase is stabilized by the addition of Pb which replaces ≈10% of the Bi, and is then called Bi,Pb(2223). In spite, of the complexity of this system, several manufacturers have already succeeded in fabricating multifilamentary Ag-sheathed tapes of lengths exceeding 1 km based on this compound. The current-carrying capacity of these tapes and their mechanical properties are constantly being improved, so that several applications can already be envisaged, as will be shown in this section. An important advantage of Ag-sheathed Bi,Pb(2223) tapes is that they can be used in a wide temperature range, which extends from 77 K to 4.2 K. It withstands magnetic fields of the order of several tenths of a tesla at 77 K: this is a decisive advantage with respect to a second Bi-based superconductor, Bi(2212), which has a lower Tc value (92 K) and will not be discussed here. Among the known high-Tc superconductors, Bi,Pb(2223) can be considered as a special case. In view of large-scale applications, the other known high-Tc superconductors with Tc > 100 K show additional problems which render their use in industrial devices very difficult. For example, Hg(1223) or Tl(1223) exhibit Tc values of 133 and 125 K, respectively, e.g. higher than Bi,Pb(2223), but are problematic in view of their toxicity. Both compounds, as well as the compound YBa2Cu3O7 (or Y(123), with Tc = 92 K ) show an additional problem: the marked weak-link behaviour at all temperatures, reflected by low critical current densities and a rapid decrease with applied magnetic field. In view of the fabrication of long tapes based on Y(123), it has been shown that high critical current densities can only be obtained for biaxially textured thin films, i.e. films with a very high degree of texturing, both in the direction of the c axis and of the a, b planes. It follows that the uniqueness of Bi,Pb(2223) with respect to the Y-, Tl- or Hg-based systems is its ability to form grain boundaries with a high current-carrying capacity, starting with powder metallurgical methods. In the following, the fabrication of Ag-sheathed Bi,Pb(2223) tapes will be described, starting with monofilamentary tapes. This configuration is not appropriate for industrial use, but allows an easier characterization than for multifilamentary tapes. In particular, the monofilamentary configuration has been used for studying the formation mechanism of the Bi,Pb(2223) phase formation. Another example is the correlation between the initial texturing of the Bi(2212) phase after tape deformation (before reaction) and the texturing of the current-carrying Bi,Pb(2223) grains after reaction. Finally, a comparison is made between the effect of uniaxial pressure and of rolling, thus giving an answer to the question of why the latter in general leads to a higher critical current density. An important part of the present section will be devoted to the fabrication and characterization of multifilamentary tapes. The deformation by the usual rolling with two cylinders (for 37 and 55 filaments)

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and by a new four-roll machine (for 45 and 100 filaments) will be described. It will be shown that the current distribution is more homogeneous for the tapes deformed by four rolls, where the pressure on the filaments is more homogeneous, which is also reflected by more homogeneous filament shapes. The variation of the critical current density as a function of applied field and of its orientation is shown at 77 and 4.2 K. The mechanism of the current transfer between neighbouring grains is briefly reviewed. In view of the applications, it is shown that a high critical current density is not a sufficient criterion for a high quality Bi,Pb(2223) tape, but that this tape should withstand mechanical stresses without damage. For cables or transformers, it should in addition exhibit low a.c. losses. The reinforcement of the Ag matrix is usually obtained by Mg, Sb or Mn additions combined with internal oxidation. A substantial reduction of the a.c. losses in multifilamentary Bi,Pb(2223) tapes is obtained by the concept of the ‘oxide barrier’, which surrounds each filament and leads to an enhancement of the treansversal resistivity, thus causing a drastic decrease of the coupling losses. B9.3.2 Bi(2223) phase formation Numerous studies aimed at understanding the formation mechanism of the Bi,Pb(2223) phase (figure B9.3.1) have been published. However, some details of the process leading to the formation of this phase are still unclear and a better knowledge of the reaction mechanisms would certainly lead to a better control of some of the numerous parameters that are important in the achievement of high-Jc Agsheathed tapes. The formation mechanism of the Bi,Pb(2223) phase will first be discussed for pressed

Figure B9.3.1 . Crystallographic structure of the Bi(2223) phase.

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precursor powders; the more complex case of Ag-sheathed tapes will be discussed later, on the basis of the fewer experimental results. As first proposed by Ikeda et al ( 1988 ) (and confirmed by Jeremie et al (1993) in powder mixtures and by Grivel et al (1993) in Bi,Pb(2223) tapes), the reaction from the phase Bi(2212) to Bi(2223) passes through the intermediate phase Bi,Pb(2212), which temporarily forms during the temperature ramp. The dissolution of a certain amount of Pb in the Bi(2212) phase was evidenced both by direct energy dispersive x-ray (EDX) analysis measurements on single Bi(2212) grains and by differential thermal analysis (DTA) measurements (Grivel et al 1993), when comparing the melting temperatures in calcined powders of otherwise identical composition, the only difference being the presence or absence of Pb. Recently, Grivel and Flükiger (1996) have studied the formation of the Bi,Pb(2223) phase on pressed powders of nominal composition Bi1.72Pb0.34Sr1.83Ca1.97Cu3.13O10+δ . A systematic study of the same specific location on the surface of the pellet was then performed, after several consecutive heat treatments performed in air at 852°C, each one followed by a rapid cooling. After each treatment, the same group of grains was observed by scanning electron microscopy and some of them analysed by microprobe analysis. This method enabled a first observation of the morphological and compositional transformations occurring in the precursor powders during the reaction. From these observations on samples without Ag in air atmosphere (Grivel and Flükiger 1996) it is concluded that, in pressed pellets, the Bi,Pb(2223) phase forms by nucleation and growth. In particular, the decomposition of the Bi,Pb(2212) grains is an essential condition for the nucleation of the Bi,Pb(2223) grains, providing the appropriate local composition for this process. However, from the point of view of industrial applications, the most important case is that occurring inside an Ag sheath, and the question arises as to whether the process is different from the one just mentioned. This is a priori not excluded, the additional element Ag leading to a change of the adjacent multiphase fields in the equilibrium phase diagram. The fact that Pb can leave the Bi,Pb(2212) phase upon heating (Flükiger et al 1996b, Jeremie et al 1993) was also observed in Ag-sheathed Bi,Pb(2223) tapes by Grivel and Flükiger (1994). Recently, Wang et al (1996) proposed an intercalation model, based on transmission electron microscopy (TEM) measurements. A careful examination of the x-ray diffraction (XRD) patterns on tapes quenched from various temperatures shows that in all cases where the 3221 phase is present (500°C to 812°C) and the (200) and (020) peaks of the coexisting Bi(2212) phase are overlapped: the Pb content in this phase has dropped to almost zero. This effect can also be observed by DTA and thermogravity ( TG) measurements: figure B9.3.2 shows the TG measurements performed in air on Ag sheathed Bi,Pb(2223) tapes starting with calcined precursor powders (Luo et al 1993) and shows that the oxygen balance during the heating cycle exhibits a complex behaviour, showing first a slight weight increase ∆m1 = 0.073 mg at T ≈ 500°C for a total powder mass in the tape of 110 mg, followed by a sharp weight decrease at T > 800°C. This behaviour reflects the release of Pb from the Bi,Pb(2212) phase where Pb is believed to be in the 2+ oxidation state and the formation of the Pb-rich 3221 phase (Pb: 4+). During the formation of the 3221 phase, Pb2+ is oxidized resulting in a net overall weight increase of the sample (Luo et al 1993). DTA measurements were performed on a series of single-phased Bi2−x Pbx(2212) samples, with x = 0.2, 0.4 and 0.6, the melting temperature for x ≈ 0.4 being lowered to 876°C (Jeremie et al 1997). This leads to the important conclusion that for powders with overall composition close to Bi,Pb(2223), Pb is dissolved in the Bi(2212) phase during the relatively fast temperature rise of 2°C min-1. For the intermediate phase Bi,Pb(2212), it can be concluded that (i) Pb atoms partially substitute for Bi atoms and (ii) the stoichiometric ratio of the elements with respect to each other is not the same as that in the original Bi(2212) phase, i.e. Ca is also dissolved in the new intermediate phase (see figure B9.3.3). This observation will serve as a basis for the reaction scheme leading to the formation of the Bi(2223) phase. From the arguments developed above, the formation of the Bi,Pb(2223) phase can be described as a sequence of two processes (see figure B9.3.4). However, some questions still remain about the effect of

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Figure B9.3.2. Simultaneous DTA/DTG measurements performed on an Ag-sheathed tape containing the calcined precursor powders. Total sample mass: 504 mg; mass of powder: 110 mg.

Figure B9.3.3. Bi/Pb (Ο), Sr/Ca („) and Cu/Sr (♦) molar ratios determined by EDX as a function of sintering time. Each point represents an average over 12 measurements performed on different grains. All measurements were performed on the same grains. From Grivel and Flukiger (1996).

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Figure B9.3.4. Reaction scheme from initial Bi(2212) in calcined powder with nominal composition Bi1.72Pb0.34Sr1.83Ca1.97Cu3.13O10+δ to final Bi,Pb(2223). From Flükiger et al (1996b).

the Ag sheath on the Bi,Pb(2223) phase formation inside tapes. Two models have been proposed: (i)

in pressed powders, the Bi,Pb(2223) grains form by nucleation and growth (Grivel and Flükiger 1996) (this is the case mentioned above), and (ii) in Ag-sheathed tapes, the Bi,Pb(2223) phase grows from the Bi,Pb(2212) grains by intercalation of Ca and Cu, as proposed by Wang et al (1996) (on the basis of TEM bservations). B9.3.3 The processing steps leading to Bi,Pb(2223) tapes: the powder-in-tube method The most promising technique for producing Ag-sheathed Bi,Pb(2223) tapes is the so-called powder-in-tube (PIT) method (Flükiger et al 1991). At present, the highest critical current densities in industrial Bi,Pb(2223) tapes with lengths > 100 m exceed 22 000 A cm-2 at 77 K, 0 T (calculated for the superconducting cross-section) (Fleshier et al 1996, Hayashi et al 1997, Leghissa et al 1997). On a length of 14.5 m, a value of 28 000 A cm-2 has been attained at the University of Geneva (Marti et al 1997). Values up to 55 000 A cm-2 have already been achieved for rolled tapes of shorter lengths (several centimetres), thus showing the potential of this material (Fleshier et al 1996). Strong efforts are at present being made to reach the high short sample values on tapes of >1 km length. The goal of these developments is to enhance not only the value of Jc , but also the overall or engineering critical current density Je , the latter being taken over the total tape cross-section. Details of the fabrication of Bi,Pb(2223) monofilamentary tapes have recently been reviewed by Flükiger et al (1996b). After a rapid recapitulation of these results, the discussion will be extended to long multifilamentary Bi,Pb(2223) tapes and their properties. B9.3.3.1 The fabrication of monofilamentary Bi,Pb(2223) tapes (a) The powder precursors

Various powder types can be used for the fabrication of Bi,Pb(2223) tapes, but they all have a common feature: the phase Bi,Pb(2223) must be formed by a reaction at the end of the tape deformation process (Flükiger et al 1996b). Attempts starting with pure Bi,Pb(2223) powders have failed to reach critical current densities exceeding 10 000 A cm – 2 at 77 K and 0 T (Flükiger et al 1991), the formation of the high-quality Bi,Pb(2223) grain boundaries requiring the presence of a sufficient amount of liquid phase. Two of the best powder mixtures to be introduced in the Ag tubes are: (i)

mixtures of calcined oxide powders starting with coprecipitated precursors, resulting in a mixture of Bi(2212) (major phase, 70%), Ca2PbO4 , CuO and Bi(2201) (Yamada et al 1991), and (ii) three-compound powders, consisting of Bi,Pb(2212) + Ca2CuO3 + CuO (Dorris et al 1993).

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However, as the Bi,Pb(2212) phase is unstable in air between 400°C and 800°C, both types of powder lead to similar results, as shown by Jeremie and Flükiger (1996). In this work most data have been obtained from a precursor mixture of coprecipitated powders of composition Bi1.72Pb0.34Sr1.83Ca1.97Cu3.13O10+δ (powder mixture (i)). These powders are calcined twice at a temperature of 800–820°C for up to 24 h with an intermediate grinding step. The average grain size of the powder is of the order of 2–5 µm. (b) Cold deformation process

The powders are filled into pure Ag tubes and compacted using a pressure of about 2 kbar, reaching a density of about 5 g cm– 3. The tubes are properly sealed with plugs and then deformed, initially by swaging and then by drawing to an outer diameter of about 1.0–1.5 mm, the cross-sectional reduction for every step being about 10%. At this level, the powder density inside the wire reaches up to 6 g cm–3. Finally the wires are cold rolled to reduce the total thickness to typically 90–100 µm. The oxide core area usually represents about 30%–35% of the total tape cross-section, and the powder density can be higher than 90% of the theoretical density for the Bi(2223) phase. Up to 50 m long monofilamentary tapes have been prepared without major problems (Flükiger et al 1996a). Particular attention has to be given to the deformation steps, in order to avoid the formation of sausaging of the oxide longitudinal section. In general, a reduction of ≈5–10% between two consecutive rolling steps is used, in order to minimize the longitudinal fluctuations. Once the final tape thickness of ≈0.1 mm is reached, a reaction heat treatment at temperatures between 830 and 840°C (in air) is required in order to form the Bi(2223) phase. The highest critical current densities are achieved by a thermomechanical treatment composed of several subsequent sintering/pressing/sintering or sintering/rolling/sintering steps, where each sintering segment lasts about 48 h. So far, all the monofilamentary tapes prepared in our laboratory have always been reacted in a linear shape, in order to avoid the degradation of Jc by bending, as described by Kessler et al (1993) and Ullmann et al (1997). The temperature homogeneity during heat treatments is achieved by using heat pipes. With this procedure, critical current densities as high as 43 000 A cm– 2 at 77 K and 0 T have been achieved in our laboratory on short (2.5 cm long) pressed monofilamentary tapes, while values of 34500 A cm– 2 have been measured on 0.5 m long rolled tapes, cut after the deformation of tapes of 20 m length (Grasso et al 1995b). (c) The reaction temperature

The heat-treatment parameters of both the calcination and the reaction heat treatment play a crucial role in the achievement of high Jc values (Flükiger et al 1996b). In air, a strong correlation between the value of Jc and the reaction temperature has been found by Grasso et al (1996). As shown in figure B9.3.5, the variation of Jc (77 K, 0 T) as a function of the reaction temperature for 200 h heat treatments on rolled monofilamentary Bi,Pb(2223) tapes exhibits a very sharp maximum at about 838°C. The value of Jc decreases by 50% for temperature differences of only two degrees. This effect can cause serious problems in the heat treatment of long tapes, and a very high temperature homogeneity is needed over a big furnace volume in order to reproducibly achieve high Jc values. However, it has been recently found that the maximum shown in figure B9.3.5 is less marked when reacting in an atmosphere with reduced oxygen partial pressure (0.07 bar pO2 ), which has the additional advantage of shortening the reaction time by about 50%. An additional feature resulting from figure B9.3.5 is the importance of the calcination temperature. Both the temperature and the length of the calcination process are known to influence the amount of liquid phase during the subsequent reaction, and thus constitute very important parameters in view of the optimization of Jc . It is a particularity of the Bi,Pb(2223) system that it is not possible to correct an error in the calcination conditions by changing the reaction parameters. The precise calcination conditions depend upon the precursors and the overall initial composition: as an example, 24 h at 800–820°C were found to give satisfactory results (Grasso et al 1995c).

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Figure B93.5. Variation of Jc with reaction temperature for a monofilamentary Bi,Pb(2223) tape. The reaction time was 200 h at all temperatures.

Figure B9.3.6. Bi(2212) texture parameter t = L(115)/[L(115) + L(008)] of unreacted tapes and misalignment angle

φe (determined by anisotropic transport measurements) of the Bi(2223) grains, both as a function of the tape thickness. From Grasso et al (1995b). (d) Deformation-ii nduced texture in Bi,Pb(2223) tapes Recently, Grasso et al (1995c) have furnished proof for a correlation between the texture of the original Bi(2212) grains and the final Bi,Pb(2223) grains in monofilamentary tapes. They found that, at the end of the deformation process, the Bi(2212) grains in the unreacted tapes are already clearly oriented in the tape plane. The texture of the Bi(2212) grains has been investigated by x-ray diffraction analysis of the filament surface after mechanical removal of the Ag sheath. A texture parameter t was defined as

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t = L(115)/[L(115) + L(008)], where L(i j k) is the integral surface of the corresponding peak (i j k). The variation of the texture parameter t of the Bi(2212) phase as a function of tape thickness has been plotted in figure B9.3.6, showing a gradual enhancement of texture. In the same figure, the transport misalignment angle φe of the final Bi,Pb(2223) grains is plotted, showing a marked correlation with t. It will be noted that in the plot of figure B9.3.6 the texture ratio p = H(115)/[H(115) + H(0010)] of the Bi(2223) phase has not been reported, essentially because it is independent of the tape thickness. However, as the real meaning of the parameter φe is the degree of texture of those grains which are carrying the current, we can confirm that the high degree of texture of the Bi,Pb(2212) platelets after deformation is a necessary condition for reaching high critical current densities. B9.3.3.2 The fabrication of multifilamentary tapes The fabrication of multifilamentary Bi,Pb(2223) tapes is not essentially different from the process mentioned above for monofilamentary tapes, except for an intermediate bundling step. In the following, we describe the fabrication technique given by Grasso et al (1997), which may not be very different from the techniques applied by several manufacturers. The deformation of the monofilamentary rod was performed up to a diameter of several millimetres (2–3 mm), after which the latter was drawn to a hexagonal shape. Several hexagonal bars (generally 37 or 55) were bundled and stacked into a second Ag tube of a diameter ranging up to 20 mm, which was deformed again down to the final tape thickness of 200–250 µm, the width varying between 2.5 and 3 mm. The deformation speed for drawing and rolling was around 1.5 cm s–1 for a 5-10% reduction rate between two consecutive rolling steps. The optimized deformation speed depends upon several parameters, e.g. the reduction rate, the diameter of the rolls and the state of the surface of the rolls. The process described here concerns the production of tapes at a laboratory scale. For the fabrication of tapes at industrial lengths (> 1 km), the initial diameter of the rod after bundling must be much larger: manufacturers may introduce an extrusion step, but no details have been published so far. Recently, a new deformation technique for multifilamentary tapes was introduced, based on a motordriven four-roll machine which permits the simultaneous reduction of the thickness and width of the samples (Grasso and Flükiger 1997). A schematic representation of the four-roll machine is given in figure B9.3.7. The main difference between this machine and the common ‘turk’s head’ resides in the fact that the rolling force is much larger in the motor-driven case. As will be shown below, the rolling force on the tape is also more homogeneous than when using a conventional two-roll machine. The role

Figure B9.3.7. A schematic representation of the four-roll machine. The four orthogonal, motor driven rolls simultaneously reduce the thickness and width of a tape. From Grasso and Flükiger (1997).

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Figure B9.3.8. Pressure dependence of Jc for rolled and uniaxially pressed tapes. From Grasso et al (1995b).

of the rolling force was first emphasized by Grasso et al (1995b), who found that there is for each tape configuration an optimum rolling force leading to optimized critical current density values (figure B9.3.8). Square monofilamentary wires of about 1 to 1.5 mm width, prepared by using the four-roll machine, were stacked into a square Ag tube, which was again deformed using the same machine, the final dimensions being the same as for the usual tapes, i.e. 250 µm thickness and 3 mm width. The reaction conditions were the same in both types of Bi,Pb(2223) tape. After the first reaction heat treatment of about 40 h at a temperature of 837 °C in air, the tape thickness was reduced by 15%, by using either the fourroll or the usual rolling technique. A final heat treatment of up to 200 h was given in order to optimize the critical current density. The advantages of the square symmetry are evident when comparing the crosssections of the multifilamentary tapes. The cross-section of figure B9.3.9 represents a four-rolled tape with 100 filaments and a superconducting fraction of about 30%. As shown in figure B9.3.10, the filaments near the centre in standard multifilamentary tapes are more compressed than those at the sides, while the four-rolled ones of figure B9.3.9 show a more homogeneous density. In addition, the distances between the single filaments show higher fluctuations for the standard tape than for the four-rolled tape. A confirmation of the higher homogeneity of the four-roll deformed tapes can be obtained by the measurement of the Vickers microhardness. The measured data have been represented in figure B9.3.11

Figure B9.3.9. Transverse cross-sections of a multifilamentary four-rolled Bi,Pb(2223) tape with 100 filaments.

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Figure B9.3.10. A typical transversal cross-section of a standard 37-filament Bi,Pb(2223) tape.

Figure B9.3.11. Position dependence of the Vickers microhardness of single filaments in the configurations shown in figure B9.3.20 for a standard and a four-rolled tape. From Grasso and Flükiger (1997).

where the Vickers microhardness of single filaments has been plotted as a function of the lateral distance from the filament centre (Grasso and Flükiger 1997). For the standard tapes, a significant variation of the microhardness has been observed between the filaments located at the tape centre and those located at the sides. Typically, the filaments near to the centre exhibit Vickers microhardness values of 130–140 Hv , while the values at the sides are much lower, of the order of 90 Hv. For the four-rolled tapes, the Vickers microhardness is much less position dependent, the decrease between the centre and the sides being smaller, from 145 Hv to 125 Hv, as follows from figure B9.3.11. The higher homogeneity in four-rolled multifilamentary tapes with respect to those produced by the conventional two-roll process is also reflected by the considerably smaller lateral variation of Jc . In figure B9.3.12, it can be seen that the ratio of Jc between the central portion and the sides decreases from 1.6 to 1.15 (Grasso and Flükiger 1997). B9.3.4 Critical current density in Bi,Pb(2223) tapes So far, pressed tapes still have considerably higher critical current densities than tapes prepared by the rolling technique. As already mentioned, the highest reported values for short monofilamentary Bi,Pb(2223) tapes at 77 K, 0 T, Jc = 66 000 and 69 000 A cm– 2, have been published by Yamada et al (1993) and by Li et al (1993). The highest reported values for long, rolled tapes are lower, but have been considerably improved in the last two years, up to values exceeding 20 000 A cm– 2 [11–13] for lengths > 100m, the

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Figure B9.3.12. Lateral Jc distribution for two monofilamentary tapes with Jc = 23 000 and 28 000 A cm– 2, for a standard multifilamentary tape with Jc = 28 000 A cm– 2, and for a four-rolled multifilamentary tape with Jc = 26 000 A cm– 2.

highest value of 55 000 A cm– 2 having been reported for lengths of the order of 2 cm (Fleshler et al 1996). This difference is attributed to the local application of forces during the deformation after the first reaction heat treatment, the rolling causing more damage by perpendicular cracks than the pressing. The damage by deformation-induced transversal cracks can be seen very easily by magneto-optics, a method recently developed by Parrell et al (1996). The importance of internal stress distribution during deformation can best be seen when comparing the lateral distribution of Jc in tapes prepared either by two-roll or by four-roll deformation. B9.3.4.1 Lateral Jc distribution in mono- and multifilamentary Bi,Pb(2223) tapes The question of the lateral distribution of Jc has been studied by Larbalestier et al (1994) and by Grasso et al (1995) on monofilamentary tapes. Grasso et al (1995, 1997) have developed a strip cutting technique, where 0.2 mm wide longitudinal strips were cut and the critical current density was measured without taking away the Ag sheath. The investigation of samples cut from the same tape showed a symmetrical behaviour of Jc on both sides of the central axis, as shown in figure B9.3.12. The Jc distribution is shown for two different tapes with Jc (77 K, 0 T) = 23 000 and 28 000 A cm– 2 respectively. The main difference between the two tapes resides in the starting powder preparation, while the same deformation and reaction parameters have been used for both. The results are in qualitative agreement with those of Larbalestier et al (1994) and can be summarized as follows. (i)

The value of the transport critical current density of the two tapes at 77 K was determined to be 18 000 and 20 500 A cm– 2 at the centre, and increases to 46 000 and 53 000 A cm– 2 for the external portion of the tape (Grasso et al 1995). In both cases, the Jc distribution can be fitted by parabolic-like curves, the local Jc values in the central part being approximately 2.5 times lower than at the sides.

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(ii) The field dependence Jc(B) of individual strips taken from the centre or from the sides does not differ significantly. The effective misalignment angle φe is also found to be essentially constant, regardless of the strip location. It follows that the observed variation of Jc is not due to a variation of the degree of texture. (iii) The reasons for the observed lateral distribution of the critical current density are: (a) the local density, as measured by microhardness, and (b) the distribution of foreign phases, which are mostly concentrated in the central part. First indications of the lateral current distribution in multifilamentary tapes have been recently given by Grasso et al (1995), who extended the strip-cutting technique presented above to multifilamentary tapes (Grasso and Flükiger 1997). These authors found that the distribution of the critical current density in multifilamentary tapes produced by the standard rolling technique is also inhomogeneous. As indicated in figure B9.3.12, a 37-filament tape with Jc (77 K, 0 T) = 28 000 A cm– 2 exhibits at the centre a maximum value of 35 000 A cm– 2, compared with 22 000 A cm– 2 for the filaments at the sides, the ratio being 1.6. For comparison, the lateral variation of Jc in a four-rolled tape with 34 filaments and Jc (77 K, 0 T) = 26 000 A cm– 2 shows a much smaller decrease, from 29 000 A cm– 2 at the centre to 25 000 A cm– 2 at the sides. These observations confirm the observed variation of the Vickers microhardness plotted in figure B9.3.11. It follows that the filaments at the centre have the highest value of Jc . This is in contrast to the distribution in monofilamentary tapes, but is explained by the fact that a multifilamentary tape constitutes a microcomposite, where the pressure is exerted by an Ag/oxide composite rather than by Ag alone, as is the case for monofilamentary tapes. B9.3.4.2 Variation of JC(B) at 77 K for mono- and multifilamentary Bi,Pb(2223) tapes From the point of view of the applications, the case of long multifilamentary tapes is more important than the more academic one of monofilamentary tapes. However, a comparison between these two types of tape gives interesting insights. The variation of the normalized Jc as a function of applied field for a mono-and a multifilamentary tape is represented in figure B9.3.13. In this figure, the monofilamentary tape is characterized by Jc (77 K, 0 T) = 35 000 A cm– 2 over a length of 0.5 m (cut from a total length of 20 m), the superconductor cross-section being 7.5 x 10–4 cm2, and the critical current is 26 A (Marti et al 1997). The overall dimension of this tape was 80 µm in thickness and 2.8 mm in width. The multifilamentary tape is characterized by 55 filaments, Jc (77 K, 0 T) = 28 000 A cm– 2 over a length of 14.5 m (Marti et al 1997). The thickness was 200 µm and the width 3.5 mm, the superconducting cross-section 20% and the critical current Ic = 37 A. It is seen in figure B9.3.13 that the monofilamentary tapes present a steeper drop when the applied field is parallel to the tape surface. At 1 T, the corresponding values for mono- and multifilamentary tapes are 7000 and 10 000 A cm– 2, respectively. There is almost no difference between the transport critical current densities of mono- and multifilamentary tapes when the field is applied perpendicular to the tape surface: in both cases, the current is found to be negligibly small above B ≈ 0.5 T. The fact that the difference in the slope Jc (B) is only observed for B parallel, but not for B perpendicular to the tape surface can be explained in the framework of the ‘railway switch’ model (Hensel et al 1993, 1995). According to this model, the field dependence Jc (B) with B perpendicular to the tape surface is mainly ‘pinning dependent’, and is thus expected to be the same in both mono- and multifilamentary tapes. The measurement of the angle-dependent critical current density in constant applied fields allows us to give information about the average texture of the truly current-carrying Bi,Pb(2223) grains. From the measurements in figure B9.3.14, where the normalized critical currents at a fixed field of 0.15 T are plotted as a function of B cos θe , an average misalignment angle θe = 7° and 5° can be derived for mono-

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Figure B9.3.13. Normalized critical current density of both mono- and multifilamentary Bi,Pb(2223) tapes as a function of applied field and field orientation at 77 K. From Marti et al (1997).

Figure B9.3.14.

Angular dependence of the normalized critical current for both mono- and multifilamentary Bi,Pb(2223) tapes at 77 K at B = 0.15 T. θ is the angle between the field and the tape normal. From Marti et al (1997).

and multifilamentary tapes respectively (Marti et al 1997) (θe being the angle between B and the tape surface). This difference agrees well with that found above by means of x-ray diffraction. It is interesting to compare the highest critical current values achieved in various laboratories. The reported values known to the authors and representing the state of the art at the end of 1996 are listed in table B9.3.1.

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Table B9.3.1. The critical current despite densities of mono- and multifilamentary Bi,Pb(2223) tapes, with their number of filaments, their superconducting ratio and the corresponding references.

B9.3.4.3 Variation of Jc ( B ) for Bi,Pb(2223) tapes at 4.2 K The variation of the critical current density of rolled Bi,Pb(2223) tapes with field at 4.2 K is of particular interest in view of their application for very high-field magnets at temperatures ranging between 4.2 and 27 K, the boiling temperatures of He and Ne respectively. The variation of Jc versus B parallel to the tape surface for the same rolled tape as above with Jc (77 K, 0 T) = 28 000 A cm– 2 is shown in figure B9.3.15 up to B = 14 T. The measurements are taken for both increasing and decreasing magnetic field. The critical current density shows a marked hysteresis, the values being much higher if the field is decreasing. This hysteresis effect has been observed before (Flükiger et al 1996b) and is attributed to the presence of residual weak links being effective at 4.2 K. The critical current density values at 4.2 K in figure B9.3.15 are of the order of 220 000 A cm– 2 and 75–80 000 A cm– 2 at 0 and 14 T, respectively, the ratio Ic (0 T)/Ic (14 T) being 2.5–2.7. The anisotropy of Jc at 4.2 K can be represented by a ratio Jc (B||)/Jc (B⊥ ) ≈ 1.4–1.5 at B = 14 T. It is interesting to compare these values with those of classical superconductors. From the behaviour shown in figure B9.3.15, critical current density values of >75 000 A cm–2 can be extrapolated for B ≥ 20 T. Taking into account a superconducting cross-section of 20–25%, this orresponds to an ‘overall’ value of 16 000–18 000 A cm– 2. The classical systems Nb3Sn and PbMo6S8 reach a value of 20 000 A cm– 2 at 20 T, but at a lower temperature, 2 K. At higher fields, however, the value for Bi,Pb(2223) remains quite constant, while the value for the classical systems decreases rapidly. Since the actual critical current densities for long, rolled Bi,Pb(2223) tapes are a factor of two below the highest reported ones on pressed tapes, it follows that, without any doubt, Bi(2223) is an adequate material for the fabrication of very high field magnets at the operation temperature of 4.2 K. Recently, Sato et al (1995) succeeded in fabricating the first hybrid magnet of 40 mm diameter

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Figure B9.3.15. Critical current densities as a function of magnetic field at 4.2 K for mono- and multifilamentary Bi,Pb(2223) tapes. From Marti et al (1997).

producing 1.5 T in a background field of 22.5 T, thus establishing the record value of 24 T at 4.2 K. From a comparison with the data plotted in figure B9.3.15, it follows that even higher fields can be produced using Bi,Pb(2223) tapes. It is noteworthy that the main problem arising when producing very high-field magnets based on Bi,Pb(2223) tapes is not the critical current density, but the mechanical strength, necessary to withstand the important Lorentz forces at high fields. By alloying Mg to the Ag sheath, the mechanical properties of the tapes have been enhanced in recent years (Goldacker et al 1997). It is thus expected that the construction of high-field magnets will be one of the first large-scale industrial applications of high-Tc superconducting compounds. B9.3.4.4 Reinforcement of Bi,Pb(2223) tapes by dispersion hardening It is obvious that the problem of mechanical stability of Bi,Pb(2223) tapes has to be solved when envisaging magnets that will reproducibly provide magnetic fields well above 20 T. It is known that the irreversible strain in Bi,Pb(2223) tapes is enhanced from 0.2% to 0.6% when going from the mono- to the multifilamentary configuration (Kessler et al 1993, Ueyama et al 1996, Ullmann et al 1997). Several ways have been proposed, all being based on dispersion hardening of Ag by alloying small quantities of other elements, e.g. Mg, Mn or Sb. As recently published by Kessler et al (1993), Ueyama et al (1996) and Ullmann et al (1997), the tensile stress of reinforced multifilamentary Bi,Pb(2223) tapes increased from 80–100 MPa for pure Ag to 200–250 MPa for the dispersion-hardened Ag, which is close to the value for multifilamentary Nb3Sn wires. However, a compromise has always to be found between good deformability and low reaction with the Bi,Pb(2223) phase, which limits the amount of additive to the total contents to about 1 wt%. The reported values are thus 1–2 wt% Mg (Kessler et al 1993, Ullmann et al 1997), 0.5 wt% Mn (Goldacker et al 1997, Ueyama et al 1996), 0.3 wt% Sb (Ueyama et al 1996) or 2.5 wt% Au–1.0 wt% Mg (Yoo et al 1997).

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B9.3.4.5 Enhancement of the matrix resistivity A second reason for dispersion hardening of Ag is the low electrical resistivity of the latter, which leads to high a.c. losses, which is particularly important for industrial applications, e.g. cables, transformers, current limiters or motors. The problem is that all the alloyed elements mentioned above oxidize during the reaction heat treatment, either in air or in a 7% oxygen atmosphere. The consequence of this internal oxidation is that the additive transforms partly into an oxide, while its content in the Ag is lowered. This leads in all cases to a ‘cleaning effect’ and to a lowering of the electrical resistivity of Ag, which in turn causes higher a.c. losses. At present, the enhancement of the electrical resistivity of Ag at 77 K between neighbouring filaments as a consequence of dispersion hardening has not exceeded a factor of 2–3, with one exception, Au, which has to be excluded for economical reasons. It can be said that no satisfactory solution has been found so far that leads to a substantial reduction of a.c. losses in Bi,Pb(2223) tapes. In our laboratory, a very promising possibility has recently been introduced, by which each filament in a multifilamentary tape is surrounded by an ‘oxide barrier’, as shown in figure B9.3.16. This thin barrier consists of an oxide that does not react with the superconducting core and leads to considerably enhanced electrical resisitivities between the filaments. Among the materials studied so far, not only BaZrO3, the new crucible material for single-crystal growth of Y(123) or R.E.(123) single crystals, where R.E. stands for rare earth (Erb et al 1996), but also MgO were found to be inert to the superconducting Bi,Pb(2223) core (Huang et al 1997). Actually, this new ‘oxide barrier’ technique has already been proven to lower the a.c. losses in multifilamentary Bi,Pb(2223) tapes (Huang et al 1997) and a grat deal of effort is being put into further development.

Figure B9.3.16. Partial cross section of a multifilamentary Bi,Pb(2223) tape with an Ag matrix: the filaments are electrically decoupled by a 1.5 µm thick BaZrO3oxide barrier.

B9.3.5 The current-transport mechanism in Bi,Pb(2223) tapes Two models have been proposed for explaining the microscopic mechanism of current transport in polycrystalline filaments of anisotropic high-Tc superconductors. These models have been discussed in detail in several papers by Malozemoff (1992), Bulaevskii et al (1992) (brick wall model) and Hense et al (1993, 1995) (railway switch model). They have been recently reviewed by Flükiger et al (1996b), and will thus only briefly be presented here. B9.3.5.1 The microstructure of the filaments in Bi,Pb(2223) tapes The notion that the platelets in the superconducting filament should be stacked like bricks in a wall led to the ‘brick wall’ model (Bulaevskii et al 1992, Malozemoff 1992). This model relies essentially on the idea that an interruption of the direct current path, i.e. a weak link or no connection at all between two platelike grains along their thin edges, can be bypassed if the supercurrent meanders through neighbouring grains.

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However, by detailed microstructural investigations of Bi,Pb(2223) tapes with critical current densities Jc (77 K, 0 T) > 15 000 kA cm− 2, it was found that the fundamental elements of the microstructure contributing to the current transport are described by small-angle grain boundaries or ‘railway switches’ that connect adjacent grains (Hensel et al 1993, 1995). The elements of the microstructure in Bi,Pb(2223) tapes and in particular the different types of grain boundary have been the object of many TEM investigations (Eibl 1990, 1995, Grindatto et al 1996, Umezawa et al 1994, Yan et al 1994, 1996) and are schematically represented in figure B9.3.17. The different types of grain boundary are: twist boundaries and colony boundaries. The colonies consist of several stacked grains with a common c axis. The grains within a colony are piled up along the common c axis and are separated by [001] twist boundaries. Evidence for coherent twin boundaries was reported by Eibl (1990) and Grindatto et al (1996), the latter observing rotation angles around 29°, possibly arising from Σ17 boundaries (according to the coincidence site lattice model).

Figure B9.3.17. The elements of the microstructure in Bi,Pb(2223) tapes: (a) colony, consisting of several stacked grains with a common c axis (twist boundaries); (b) small-angle c-axis tilt (SCTILT) boundary (or ‘railway switch’); (c) edge-on c-axis tilt (ECTILT) boundary. For simplicity, the twist between the grains within the colonies is not shown in (b) and (c).

Another type of boundary is the colony boundary, formed between colonies having slightly tilted c axes with respect to each other. Following the notation of Grindatto et al (1996), one can distinguish between edge-on c-axis tilt boundaries (or ECTILT boundaries) and small-angle c-axis or SCTILT boundaries. These different types of colony boundary are illustrated by figures B9.3.18 and B9.3.19. The misorientation between two colonies connected via a symmetrical ECTILT boundary is accommodated by partial edge dislocations (Eibl 1990). The type being more frequently encountered is that of the asymmetrical ECTILT boundaries, where the two grains forming this colony boundary are also twisted around the c axis with respect to each other. Also shown in figure B9.3.19 are the SCTILT boundaries, which are often compositionally and structurally modulated in order to accommodate the mismatch. In addition, the SCTILT boundaries in figure B9.3.19 appear to be free from intergranular phases. The ‘railway switch’ model was first proposed by Hensel et al (1995) on the basis of scanning electron microscope (SEM) observations, which did not give sufficient details about microstructural details at the grain boundaries. As shown in the preceding paragraph, high-resolution TEM (HRTEM) observations give more weight to the main conclusions of this model (Grindatto et al 1996). However, it must be stressed that even if the HRTEM results are consistent with the railway switch model, they can neither prove nor disprove the electromagnetic data. A definitive answer will only be given once the results from Bi,Pb(2223) bicrystals are known. B9.3.5.2 Current-limiting processes in Bi,Pb(2223) tapes Armed with the microstructural observations made above, one can now address the question of which dissipation mechanism limits the critical current density of the Bi,Pb(2223) tapes. Based on: (i) the slow

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Figure B9.3.18. (a) A TEM image showing the various microstructural elements in Bi,Pb(2223) tapes, (b) The corresponding schematic drawing of the twist boundaries and colony boundaries.

Jc (B) dependence at low temperature; (ii) the fact that for arbitrary orientation between H and tape it is only the perpendicular field component which determines Jc ; (iii) the reasonable agreement between transport and magnetization data, Maley et al (1992) suggested that in Bi,Pb(2223) tapes Jc is limited by thermally activated flux motion, and not by weak-link effects at the grain boundaries. This was soon confirmed by Caplin et al (1993), who showed that the hysteretic magnetization of tapes displays a temperature and field dependence which is essentially identical to that of the Bi,Pb(2223) powder extracted from these tapes and ground. Since the powder magnetization contains no intergranular current contribution, this implies that intragranular flux motion is indeed the current- limiting mechanism. Dhallé et al (1994) pointed out that one can distinguish two different regimes of Jc (B) behaviour throughout the whole superconducting temperature range, depending on the magnetic field that is applied. At low fields the critical current exhibits a relatively fast and power-like drop-off, Jc ∝ 1/Bn, which in higher fields changes to a slower exponential decay, Jc ∝ exp(−B/Bp ). Magneto-optical measurements indicate that the microstructure is indeed inhomogeneous across the Bi,Pb(2223) core section, with variationsin texture, density, grain size

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Figure B9.3.19. HRTEM image of an SCTILT boundary (‘railway switch’). The incident electron beam is parallel to [110] in the lower grain and a few degrees away from the [100] direction in the upper grain. Some BiO layers are are marked by arrows.

and phase purity leading to a colony boundary network of varying coupling strength. In low field this leads to a ‘magnetic fragmentation’, observed in magneto-optics (Pashitski et al 1995) and magnetization (Dhallé et al 1997a) experiments, where weakly coupled junctions rapidly cease to carry significant current. The disappearing current contribution of these weakly coupled paths leads to the power-law Jc (B) behaviour. At low temperatures Bi,Pb(2223) tapes are capable of carrying quite high currents up to very high magnetic fields. Comparing Jc (B,T) data obtained from transport experiments and magnetization data, the latter for both intact tapes and oxide powder extracted from them, over a range of tapes with widely varying self-field current densities, Dhallé et al (1997a) showed that in the high-field regime Jc is limited by flux motion within the superconducting grains only, regardless of the overall degree of connectivity. In this regime the critical current density of the grains themselves falls below Jc of the strongly coupled colony boundaries which survived the ‘magnetic fragmentation’ process and the macroscopic Jc (B, T) dependence becomes determined by the pinning strength within the Bi,Pb(2223) grains. We can make some important general observations concerning the pinning potential in Bi,Pb(2223) tapes. (i)

Regardless of its absolute value, the Jc (B) variation at high fields is identical from tape to tape and depends only on temperature (Dhallé et al 1997a). This implies a sample-independent pinning strength which is not influenced by the tape preparation details. ( ii ) The Jc (B) dependences at different temperatures can be scaled on each other (Hensel et al 1993, Dhallé et al 1997a), showing that a single pinning mechanism describes the whole superconducting temperature range. ( iii ) Whereas the intergranular Jc field dependence of Bi,Pb(2223) tapes, hot rolled bars and sintered pellets shows important variations from system to system, the magnetization data on powders extracted from these samples do not reveal significant differences in the pinning potential for these systems (figure B9.3.20) (Dhallé et al 1997c). The implication is that the pinning potential in tapes is inherent to the Bi,Pb(2223) phase. However, these properties can be improved by artificial pinning centres, as illustrated by introducing columnar defects after irradiation with heavy ions (Civale et al 1993) or high-energy protons (Dhallé et al 1997b, Safar et al 1995). However, irradiation is costly and leaves the tapes radioactive—two important factors disfavouring its use for commercial applications. Hence other methods will have to be developed to increase the pinning energy in the Bi,Pb(2223) system.

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Figure B9.3.20. The variation with temperature of the high-field Jc (B) decay, as given by the exponential attenuation field Bp , for various Bi,Pb(2223) systems: (a) shows this decay for the intact samples, while in (b) the same data are plotted for the ground powders extracted from them.

B9.3.6 Conclusions In spite of the progress accomplished in the last few years, the Bi,Pb(2223) system might still offer considerably greater potential, and substantially higher critical current densities are expected, compared with the highest values reported at present, as represented in table B9.3.1. It is clear that for applications, the relevant value will not be the critical current density in the superconductor, but the critical current density over the whole tape cross-section, also called the ‘engineering’ critical current density, Je . The achievement of ‘engineering’ values of the order of Je = 100 00–20 000 A cm2 will need a substantial enhancement of the superconducting cross-section, from 25%, which is actually reached, to values between 30 and 50%. This seems to be beyond the possibilities given by the usual stacking procedures, followed by conventional rolling (two-roll machine), but may be achieved by the newly introduced four-roll processing (Grasso and Flükiger 1997). We have shown that most problems connected with the industrial fabrication of Bi,Pb(2223) tapes can be solved or still bear a potential for improvement: the ‘engineering’ critical current density, Je , the mechanical reinforcement, the reduction of a.c. losses and the production of long tape lengths, of the order of > 1 km. The reasons for the limitation of Je have also been recognized and can be summarized as follows: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

the inhomogeneity of the precursors the details of the phase diagram in the vicinity of the region of interest the mechanism of Bi,Pb(2223) phase formation the homogeneity of the deformation process the microstructure at the grain boundaries the variation of the critical current density with applied magnetic field parallel to the ab planes the anisotropy of the critical current density dependence a high superconductor to Ag ratio.

Among these problems, there are two which will be particularly difficult to solve i.e. points (v) and (vi). Point (v) depends on the average misalignment angle of the Bi,Pb(2223) grains with respect to the

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tape surface, which is actually 5° for the best tapes. It is expected that further improvements of the deformation technique will lead to even lower misalignment angles. The real limitation for Bi,Pb(2223) tapes will be the anisotropy of the critical current density with magnetic field orientation, which is an inherent physical property of this phase. After having demonstrated the strong effect of proton irradiation on the anisotropy of Bi,Pb(2223) tapes, the question is whether chemical or thermodynamical processes can be found which introduce a partial disorder in the Bi,Pb(2223) structure. Acknowledgments The work was funded by the Swiss National Foundation (PNR30), by EC-Brite/Euram, Contract No BRE2 CT92 029, by OFES, Contract No BR060 and by the Swiss Priority Programme for Materials (PPM). References Bulaevskii L N, Clem J R, Glazman L I and Malozemoff A P 1992 Phys. Rev. B 45 2545 Caplin A D, Cassidy S M, Cohen L F, Cuthbert M N, Laverty J R, Perkins G K, Dou S X, Guo Y C, Liu H K, Lu F, Tao H J and Wolf E L 1993 Physica C 209 167 Civale L, Marwick A D, Wheeler R IV, Kirk M A, Carter W L, Riley G N Jr and Malozemoff A P 1993 Physica C 208 137 Dhallé M, Cuthbert M N, Johnston M D, Everett J, Flükiger R, Dou S X, Goldacker W, Beales T and Caplin A D 1997a Supercond. Sci. Technol. 10 21 Dhallé M, Cuthbert M N, Perkins G K, Cohen L F, Caplin A D, Guo Y C, Liu H K and Dou S X 1994 Proc. 7th IWCC (Alpbach, 1994) p 553 Dhallé M, Marti F, Grasso G, Hensel B, Paschoud E, Victoria M and Flükiger R 1997b M 2S-HTSC-V Conf. (Beijing, 1997) Dhallé M, Marti F, Grasso G, Perin A, Grivel J-C, Walker E and Flükiger R 1997c M 2S-HTSC-V Conf. (Beijing, 1997) Dorris S E, Prorok B C, Langan M T, Sinha S and Poeppel R B 1993 Physica C 212 66 Eibl O 1990 Physica C 168 239 Eibl O 1995 Microscopy Res. Technol. 30 218 Erb A, Walker E and Flükiger R 1996 Physica C 258 9 Fleshier S, Li Q, Parrella D, Walsh P J, Michels W J, Riley G N Jr, Carter W L and Kunz B 1996 Critical Currents in Superconductors Proc. 8th IWCC ed T Matsushita and K Yamafuji (Singapore: World Scientific) p 81 Flükiger R, Graf T, Decroux M, Groth C and Yamada Y 1991 IEEE Trans. Magn. MAG-27 1258 Flükiger R, Grasso G, Grivel J C, Hensel B, Marti F, Huang Y and Perin A 1996a Proc. 8th Int. Workshop on Critical Currents in Superconductors ed T Matsushita and K Yamafuji (Singapore: World Scientific) p 69 Flükiger R, Grasso G, Hensel B, Däumling M, Gladyshevskii R, Jeremie A, Grivel J C and Perin A 1996b Bismuth Based High Temperature Superconductors ed H Maeda and K Togano (New York: Dekker) pp 319–56 Goldacker W, Mossang E, Quilitz M and Rikel M 1997 Phase formation in Ag and AgMg sheathed Bi(2223) tapes IEEE Trans. Appl. Supercond. AS-7 Grasso G and Flükiger R 1997 Improvement of the microstructure and homogeneity of long Ag sheathed multifilamentary Bi(2223) tapes Advances in Superconductivity vol IX, ed S Nakajima and M Murakami (Berlin: Springer) p 835 Grasso G, Hensel B, Jeremie A and Flükiger R 1995a EUCAS Conf. (Edinburgh, 1995) (Inst. Phys. Conf. Ser. 148) (Bristol: Institute of Physics) p 463 Grasso G, Jeremie A and Flükiger R 1995b Supercond. Sci. Technol. 8 827 Grasso G, Marti F, Huang Y and Flükiger R 1997 Long lengths of mono- and multifilamentary Ag sheathed Bi(2223) tapes Advances in Superconductivity vol IX, ed S Nakajima and M Murakami (Berlin: Springer) Grasso G, Marti F, Jeremie A and Flükiger R 1996 Advances in Superconductivity vol VII (ISS’95), ed H Hayakawa and Y Enomoto (Tokyo: Springer) p 855 Grasso G, Perin A and Flükiger R 1995c Physica C 250 43

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Grindatto D P, Hensel B, Grasso G, Niessen H-U and Flükiger R 1996 Physica C 271 155 Grivel J-C and Flükiger R 1994 Physica C 235–240 505 Grivel J-C and Flükiger R 1996 Supercond. Sci. Technol. 9 555 Grivel J-C, Jeremie A, Hensel B and Flükiger R 1993 Supercond. Sci. Technol. 6 725 Hayashi K, Hahakura S, Saga N, Kobayashi S, Kato T, Ueyama M, Hikata T, Ohkura K and Sato K 1997 IEEE Trans. Appl. Supercond. AS-7 Hensel B, Grivel J-C, Jeremie A, Perm A, Pollini A and Flükiger R 1993 Physica C 205 329 Hensel B, Grasso G and Flükiger R 1995 Phys. Rev. B 51 15456 Huang Y, Grasso G, Marti F, Erb A, Flükiger R, Kwasnitza K and Clerc S 1997 SPA ‘97 (Xi’an, 1997) to be published Ikeda S, Ichinose A, Kimura T, Matsumoto T, Maeda H, Ishida Y and Ogawa K 1988 Japan. J. Appl. Phys. 27 L999 Jeremie A and Flükiger R 1996 Physica C 267 10 Jeremie A, Alami-Yadri K, Grivel J-C and Flükiger R 1993 Supercond. Sci. Technol. 6 730 Jeremie A, Grasso G and Flükiger R 1997 J. Therm. Anal. 48 685 Kessler J, Blüm S, Wildgruber U and Goldacker W 1993 J. Alloys Compounds 195 511 Larbalestier D C, Cai X Y, Feng Y, Edelman H, Umezawa A, Riley G N Jr and Carter W L 1994 Physica C 221 299 Leghissa M, Fischer B, Roas B, Jenovelis A, Wiezorek J, Kautz S and Neumuller H W 1997 IEEE Trans. Appl. Supercond. AS-7 Li Q, Broderson K, Hjuler H A and Freltoft T 1993 Physica C 217 360 Li Q, Riley G N Jr, Parrella R, Michels B, Walsh P J, Carter W L and Rupich M W 1997 IEEE Trans. Appl. Supercond. AS-7 Luo J S, Merchant N M, Escorcia-Aparicio E, Maroni V A, Gruen D M, Tani B S, Riley G N Jr and Carter W L 1993 IEEE Trans. Appl. Supercond. AS-3 972 Maley M P, Kung P J, Coulter J Y, Carter W L, Riley G N and McHenry M E 1992 Phys. Rev. B 45 7566 Malozemoff A P 1992 AIP Conf. Proc. No. 251 ed Y H Kao et al (New York: AIP) p 6 Marti F, Grasso G, Huang Y and Flükiger R 1997 IEEE Trans. Appl. Supercond. AS-7 Parrell J A, Larbalestier D C, Riley G N Jr, Li Q, Parrella R D and Teplitsky M 1996 Appl. Phys. Lett. 69 2915 Pashitski A E, Polyanskii A, Gurevich A, Parell J A and Larbalestier DC 1995 Physica C 246 133 Safar H, Cho J H, Fleshier S, Maley M P, Willis J O, Coulter J Y, Ulmann J L, Lisowski P W, Riley G N Jr, Rupich M W, Thompson J R and Krusin-Elbaum L 1995 Appl. Phys. Lett. 67 130 Sato K, Ohkura K, Hayashi K, Ueyama M, Fujikami J and Kato T 1995 Int. Workshop on Advanced High Magnetic Fields (Tsukuba, 1995) Sokolowski R S, Hazelton D, Walker M and Haldar P 1996 Development of HTS device applications at intermagnetics Advances in Superconductivity vol VIII, ed H Hayakawa and Y Enomoto (Berlin: Springer) p 1241 Ueyama M, Ohkura K, Hayashi K, Kobayashi S, Muranaka K, Hikata T, Saga N, Hahakura S and Sato K 1996 Physica C 263 172 Ullmann B, Gäbler A and Goldacker W 1997 IEEE Trans. Appl. Supercond. AS-7 Umezawa A, Feng Y, Edelman H S, Willis T C, Parrell J A, Larbalestier D C, Riley G N Jr and Carter W L 1994 Physica C 219 378 Wang Y L, Bian W, Zhu Y, Cai Z X, Welch D O, Sabatini R L, Thurston T R and Suenaga M 1996 Appl. Phys. Lett. 69 580 Yamada Y, Obst B and Flükiger R 1991 Supercond. Sci. Technol. 4 165 Yamada Y, Satou M, Murase S, Kitamura T and Kamisada Y 1993 Proc. 5th Int. Symp. on Superconductivity (ISS’92) eds Y Bando and Y Yamauchi (Tokyo: Springer) p 717 Yamasaki H, Endo K, Kosaka S, Umeda M, Yoshida S and Kajimura K 1993 IEEE Trans. Appl. Supercond. AS-3 1536 Yan Y, Evetts J E, Soylu B and Stobbs W M 1994 Phil. Mag. Lett. 70 195 Yan Y, Evetts J E, Soylu B and Stobbs W M 1996 Physica C 261 56 Yoo J, Chung H, Ko J and Kim H 1996 Long-length processing of Bi(2223) tapes made by using Ag sheath alloys IEEE Trans. Appl. Supercond. AS-7 Zhou R, Huilts W L, Sebring R J, Bingert J F, Coulter J Y, Willis J O and Smith J L 1995 Physica C 255 275

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C1 Basics of superconducting magnet design

F Zerobin and B Seeber

C1.0.1 Introduction Superconductive magnets are used for a variety of tasks. There are different kinds of magnet from small and simple solenoids to sophisticated 3D shaped coils of large volume (e.g. for fusion). Before any classification of the type of magnet can be made, the required magnetic field must be specified. The following parameters should be known. • • • •

•

Magnetic field strength and direction: it is common to specify the magnetic induction B in tesla, rather then the magnetic field H in amperes per metre. Volume of the magnetic field: the spatial extension of the useful magnetic field. Uniformity of the magnetic field: many applications require a particular field profile in a defined volume. Variation of the field with time: depending on the application, the stability of the magnetic field with time may be important. This can be controlled by the power supply but there are applications where this is not sufficient and the magnet must operates in the persistent mode (e.g. magnetic resonance imaging—MRI—or magnetic resonance spectroscopy). In such a case the magnet terminals are short circuited by a superconducting switch. Access to the magnetic field: Some applications need a special access to the magnetic field (e.g. a radial access to a solenoid magnet). This is of specific interest for the mechanical design of the magnet.

According to these specifications one may distinguish between solenoid magnets, multipole magnets and special (3D-shaped) magnets. C1.0.2 Survey on field calculations The calculation of magnetic fields in theory and practice is described in many books (e.g. Montgomery and Weggel 1980, Wilson 1983). Here the most important methods for the design and the calculation of superconducting magnets are outlined and discussed. The theoretic basis for the calculation of static fields is Maxwell’s law

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where H is the magnetic field, J the current density, B the magnetic induction and µ is the permeability. These relations are also valid in the presence of non-linear material, for instance magnetic iron, where B is a strongly non-linear function of H.. The Maxwell equations are not practical for the calculation of real problems. The most important methods follow the law of Biot—Savart which describes the dependence of the magnetic induction on the current flowing through a (infinitely) thin filament.

where I is the magnitude of the current in the loop C, ds is the unit vector of the current and r is the distance to the point where the magnetic induction is calculated. The integral may be calculated in the 3D space for a straight, current-carrying filament with defined ends A and B (figure C1.0.1). The magnetic induction at point P is as follows.

with

The windings of a magnet can be modelled to any accuracy by splitting it up into individual straight current filaments. An example of how a dipole magnet can be modelled is given in figure C1.0.2. The field contributions of all straight parts are then individually calculated and summed up. The method is well suited for computer codes and often applied in practice. A special case is an infinitely long current filament (figure C1.0.3). There is only one, azimuthal, field component outside the wire

where I is the current and r is the distance from the centre of the wire to the point of interest.

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Figure C1.0.1. Calculation of the field at point P generated by a finite straight current filament AB

Figure C1.0.2. An example of the modelling of a dipole magnet by straight current filaments.

Another important case is a current loop (figure C1.0.4). Supposing an ideal loop (the conductor cross-sectional area is zero) the induction can be calculated according to the following equations (Smythe 1950):

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Figure C1.0.3. The generated field (azimuthal) of an infinitely long circular wire carrying a current.

Basics of superconducting magnet design

Figure C1.0.4. A current loop for the calculation of fields.

K (Φ) and E (Φ) are elliptical integrals of the first and second kind, respectively.

The z-component of the on-axis induction can be calculated according to

The calculation of the field of a conductor with rectangular cross-sectional area (figure C 1.0.5) and an uniform current (constant current density) can be carried out by

Another method for the calculation of magnetic fields is the method of finite elements (FE method or FEM). This method enables us to take into account non-linear materials (e.g. iron). As a summary it should be mentioned that the current-carrying cross-sectional areas, the non-current-carrying areas (air) and the iron areas are divided into coherent elements (triangles, trapezoids and rectangles). Taking all knots and boundary conditions into account, a system of equations is established which gives, after being solved,

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Figure C1.0.5. An infinitely long current-carrying bar with constant current density.

the magnetic induction in the knots. Figure C1.0.6 illustrates the method in the case of a dipole magnet. For symmetry reasons only one quadrant of the magnet is shown. In figure C1.0.6(a) a possible mesh for the calculations is shown. Figure C1.0.6(b) and (c) gives the obtained contour lines for the magnetic flux and the magnetic induction (flux density), respectively. It should be mentioned that software for the calculation of magnetic circuits according to the FEM is commercially available (e.g. TOSCA from Vector Fields, UK). Computer programs using the FEM allow us to calculate not only the magnetic field, but also the stored energy, inductances and magnetic forces. Moreover this method is indispensable when non-linear materials are present. According to the magnet specification the generated field must satisfy uniformity in a restricted part of the bore or aperture. Because there is no real standardization of how to define the homogeneity of a field the latter has to be defined depending on the application. Although often not mentioned, one should also specify the uniformity at the required strength of the field. A quite common practice is to define a spherical volume where the uniformity is specified. For instance ordinary laboratory superconducting magnets are commercially available between typically 1 × 10− 2 and 1 × 10−5 field homogeneity in 10 mm dsv (diameter spherical volume). MRI magnets need ∼1 × 10− 6 in 500 mm dsv and magnetic resonance spectroscopy requires a field uniformity of approximately 2 × 10−10 in a measurement volume of 0.2 cm3, i.e. 7.2 mm dsv (see section G2.1). Other definitions of field uniformity may be on a straight line, a circle or a cylindrical surface inside the aperture of e.g. a dipole or quadrupole magnet. There are essentially two possibilities to calculate uniformity. One method is to take the highest and lowest values of the field in the considered space and to calculate ∆B = |Bm a x − Bm i n |/B0 with B0 the field in the centre. The other method uses integrated fields, e.g. along a line, Bm e a n = [1/(z2 - z1)] ∫ z1z2 Bdz which allows the calculation of the local deviation from this average value ∆B = Bm e a n − B or the mean value of the local deviation ∆Bmean = [1/(z2-z1)] ∫ z1z2 ∆Bdz. Finally producing a magnetic field means storage of energy. Because the stored energy in a superconducting magnet can be high, the knowledge of the stored energy is important for the design of the quench protection system. The stored energy of a magnet can be expressed by

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Figure C1.0.6. Modelling one-quarter of a dipole magnet, (a) A possible mesh for calculations according to the FEM; (b) contour lines for constant magnetic flux; (c) contour lines for constant magnetic induction (flux density).

A method to determine the amount of stored energy may be deduced from this equation. The considered volume, V, is divided into elements where the magnitude of the magnetic induction is calculated and then summed up. Another way to calculate the stored energy is to use the following equation.

where L is the inductance of the magnet and I is the magnet current. Inductances can be calculated with common formulae of electrical engineering.

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Figure C1.0.7. Winding package of a solenoid magnet (without winding bobbin or flanges).

C1.0.3 Solenoid magnets Solenoid magnets are the simplest and most used magnet types. The superconductor (round or rectangular wire) is wound layer by layer on a cylindrical coil body. According to figure C1.0.7 the mean current density of the windings is

where N is the number of windings, I is the current of the conductor, 2l is the length of the solenoid and a and b are the inner and outer radius, respectively. The current density of the conductor, J , is higher due to the electrical insulation and an eventual mechanical reinforcement. The filling factor of a coil winding can be expressed by the ratio Kf = Jc o i l /J and depends on • • • • • • •

the geometric form of the conductor (e.g. round wire or edge rounded rectangular wire) the thickness of the electrical insulation of the conductor (e.g. enamel or heat-resistant glass insulation) the thickness of the electrical insulation between the layers (if any) the configuration of the winding (four-pack or six-pack configuration, see figure C2.0.5) the positioning of the conductor (winding tension, accuracy of conductor positioning, pressing of the conductors) the mechanical supporting structures (e.g. bandages) the measures for additional cooling (e.g. tubes).

Figure C1.0.8 illustrates calculated filling factors for various configurations of a round wire (figure C1.0.8(a), (b)), as well as for a wire with rectangular cross section (figure C1.0.8(c)). Practical filling factors are less, depending on the precision of the winding process. By the knowledge of the filling factor it is possible to find the relation between coil current density and conductor current density. Then the required conductor cross-section can be defined. The magnetic field of a solenoid magnet cannot be calculated at any point in a closed manner. One has to employ the method of Biot—Savart or FEs. However, the on-axis field can be calculated analytically by a simple equation and with high accuracy. One of the most interesting points is the centre of the magnet. At this point the magnetic induction is B0 (figure C1.0.7). For the calculation of B0 the coil is

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Figure C1.0.8. Calculated filling factors for (a) a round wire in six-pack configuration, (b) a round wire in four-pack configuration and (c) a wire with rectangular cross-sectional area.

subdivided into a number of circular current loops. Integrating the total coil volume gives (Montgomery and Weggel 1980, p 4)

where a is the inner radius of the winding package (figure C1.0.7) and F (α , β) is a function depending only on the geometry of the coil.

with α = b/a and β = l/a.

The function F (α , β) is shown in figure C1.0.9. Note that for the same field in the centre of the solenoid different combinations of α and β are possible. For a particular central field B0 there is a combination of α and β which gives a minimum of the coil volume and therefore a minimum of the necessary superconductor. The minimum coil volume can be calculated by the equation

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Solenoid magnets

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Figure Cl.0.9. Shape factor F (a, b) for the calculation of the on-axis field of a solenoid magnet as a function of a and β. The line of minimum winding volume is also indicated.

It turns out that a solenoid magnet with a minimum coil volume is rather short and bulky. There are essentially two major drawbacks. One is the poor uniformity of the field and the other is a high ratio between the maximum field Bm (peak field), seen by the windings, and the central field B0. For an evaluation of the field uniformity the on-axis field profile (r = 0) can easily be calculated by the following consideration. For symmetry reasons the flux density B0 at the centre of a 2l long coil may be thought of as the sum of two coils having each length l. Then the field profile B(z , r = 0), where z = 0 is in the centre of the coil, can be calculated by varying the lengths of both coils so that one coil is (l — z ) long, and the other one is (l + z ) long.

For points of interest which are off axis, but still inside a sphere not touching the bore of the solenoid (central zone), the field is (Montgomery and Weggel 1980, p 232)

where r and ϕ are the radial and azimuthal coordinates, respectively. Pn are Legender polynomials and P′n is their first derivative. The En coefficients are defined as follows.

The restriction of these equations to the central sphere is due to its divergence for r = a. If the field must be known outside the central sphere, but still inside a sphere containing the whole coil, the method

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Figure C1.0.10. The dependence of the ratio Bm /B0 on α and β. Bm is the maximum field (peak field) in the winding package and B0 is the field in the centre of the solenoid.

of summation of elemental loops must be applied (see equations (C1.0.1) and (C1.0.2) and Montgomery and Weggel 1980, p 237). This method can also be applied for the central zone but is less practical with respect to equations (C 1.0.5) and (C 1.0.6). The maximum field (peak field) in the windings Bm can now be calculated (numerically). As indicated in figure C1.0.7, this is at point (z = 0, r = a). It was found that the ratio Bm /B0 depends on α and β, which is shown in figure C1.0.10 (Boom and Livingstone 1962). Because the current density in the superconductor is reduced by a higher peak field, the magnet designer has an interest in working with a Bm /B0 ratio near unity. The line for a minimum coil volume in figure C1.0.9 has been calculated with the assumption of a constant current density in the coil. Such a design does not employ the superconductor efficiently because the coil current density is determined by the peak field in the winding package. There are windings, especially the outer sections of the solenoid, which are exposed to a much weaker field corresponding to a higher critical current in the superconductor. A clever magnet design subdivides the magnet into sections with optimized critical current density in the superconducting wire. Supposing that all sections are connected in series, which means the same operating current, the superconducting wire in the low-field regime can have a smaller cross-section with respect to the wire in the high-field regime. This technique is known as ‘grading’ and allows us to optimize the amount of superconductor, as well as to build compact magnets. A more detailed discussion can be found in the work of Montgomery and Weggel (1980 p 146) and Wilson (1983 p 23). The simplest measure to achieve good axial field uniformity of a solenoid magnet is to increase its length, which means that the β value must be increased (equation (C1.0.4)). The next possibility for further improvements is to introduce so-called ‘notches’ in the winding. Notches may have a half or zero current density. An example is shown in figure C1.0.11 where the cross-section of a typical magnet for magnetic resonance spectroscopy is illustrated (section G2.1). Such an arrangement allows the construction of magnets of the sixth order, which means that E2 = E4 = 0 in equations (C1.0.5) and (C1.0.6). To reach a field uniformity of up to 2 in 10−10 several additional measures must be undertaken: series correction coils at the end of the magnet and superconducting and room- temperature shim systems (see section G2.1).

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Multipole magnets

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Figure C1.0.11. A cross-section of a solenoid magnet for magnetic resonance spectroscopy applications. This magnet is of sixth order, which requires the introduction of notches with half and zero current density.

The inductance of a solenoid magnet, which is required for the calculation of the stored energy and the quench protection system, can be evaluated by

where N is the number of windings and a, b and l are dimensions of the solenoid (figure C 1.0.7). For shorter solenoids the inductance can be evaluated by

where D = (a + b)/2. The function K (λ , ϕ) is depicted in figure C1.0.12 where λ = l/D, ϕ = h/D and h = (b — a)/2. C1.0.4 Multipole magnets There are applications where the direction of the magnetic field must be perpendicular to the bore of the magnet. An example is a dipole magnet for the bending of electrically charged particles in an or for the rotor winding of a superconducting generator. Another example is a quadrupole magnet which allows the generation of a constant field gradient perpendicular to the bore. Such magnets are indispensable for the focusing of particles, again in an accelerator, or for magnetic lenses. The magnetic field of a multipole magnet can be calculated analytically when end effects are neglected (infinitely long magnet) and no material with non-linear behaviour (e.g. iron) is present in the considered space. The problem is reduced to a 2D one. From a theoretical point of view, an ideal dipole field can be obtained by a current distribution on the surface of a circular cylinder (radius a) which obeys the equation I = I0 cos ϕ. The variable ϕ is the azimuthal angle of a cylindrical coordinate system (figure C1.0.13(a)). The field components inside the cylinder can be calculated according to

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Figure C1.0.12. The function K (λ , ϕ) for the calculation of the inductance of a solenoid.

A quadrupole field requires a current distribution I = I0 cos 2ϕ (figure C1.0.13(b)) and the field components inside the cylinder are

Such a field is exactly zero for x = y = 0 but there is a constant field gradient across the aperture equal to µ0I0/2a2. Multipole fields with higher order can be generated by a general current distribution of I = I0 cos (nϕ), where n is the order of the multipole (n = 3 for a sextupole, n = 4 for an octupole etc). The current distribution I = I0 cos (nϕ) can be obtained by a specially shaped conductor with constant current density. The simplest case for a dipole field is the intersection of two cylinders (radius a) where the cylinder axis are separated by a distance < a (figure C1.0.14(a)). In the intersection area (aperture) there is no current. However in both remaining both outer sections the current flows perpendicularly in and out of the drawing plane. More general is the intersection of two or more conductors with elliptical cross-section (figure C1.0.14(b) and (c). This allows to build up multipole fields with the order of n. The field inside an elliptical conductor with the semi-axes a and b, and with constant current density J, can be calculated (Beth 1968).

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Multipole magnets

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Figure C1.0.13. The generation of multipole fields can be modelled by a current distribution of I = I0 cos (nϕ) on the surface of a circular cylinder (aperture of the magnet), (a) The the current distribution for a dipole field (n = 1); (b) the distribution for a quadrupole field (n = 2).

In practice, especially when dealing with superconductors, the above-mentioned specially shaped conductors are not very suitable. The required current distribution I = I0 cos (nϕ) is approximated by a winding in sector blocks, winding of concentric shells or winding in horizontal layers (figure C1.0.15). For instance the dipoles for the large hadron collider (LHC) at CERN are a combination of sector blocks and concentric shells (see chapter G4). Some of the arguments which have to be considered for selecting the kind of winding are • • • •

dimension and cross-section of the available conductor ratio of the peak field in the winding and the field in the aperture (normally < 1.1) construction of the end region of the multipole, for instance the smallest possible bending radius at the coil ends mechanical supports for taking over magnetic forces.

Mainly for cost reasons one should attempt a design which requires only one type (cable) of superconductor. The design of the coil configuration follows the 2D Biot-Savart law assuming a current flowing in an infinitely long filament, perpendicular to the x—y plane. Current filaments can be grouped together into a ‘current sheet’ or a ‘current block’. The ordinary procedure is to start with a practically feasible current distribution outside the aperture and to calculate analytically the resulting magnetic field inside the aperture. Then the current distribution is modified until the required field quality is obtained. The

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Figure C1.0.14. Multipole fields can be obtained by specially shaped conductors with a constant current density: (a) intersection of two circular cylinders for the generation of a dipole field; (b) intersection of two elliptical cylinders for the generation of a dipole field; (c) crossing of two elliptical cylinders for the generation of a quadrupole field. Note that the current flows in one section into the drawing plane and in the adjacent one out of the drawing plane.

calculation of the field components in the x—y plane inside the aperture can be carried out with the help of a vector potential A and with B = curlA. Outside the aperture the method is limited to regions without any conductor (regions with zero current). If the current is perpendicular to the x—y plane, the vector potential has only one component parallel to the direction of current flow. If the field must be known inside a current-carrying conduct the above-mentioned method fails because a potential inside the conductor cannot be defined. In such a situation the method of complex variables extended to multipole fields by Beth is convenient (Beth 1968). Any point in the x—y plane is described by z = x + iy and the field is defined by B = By + iBx. In what follows the field of the most important current distributions are given. Further details of the calculation can be found in the work of Mess et al (1996) and Beth (1968). Generally the situation of an infinitely long current-carrying conductor perpendicular to the x—y plane can be described as in figure C1.0.16. If the current, I, is intersecting the plane at point A (a, ϕ0 ) with the cylindrical coordinates a and ϕ0, the field at point P (r, ϕ) inside the aperture (r < a) can be calculated according to

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Multipole magnets

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Figure C1.0.15. Practical winding of superconducting cables for the approximation to a I = I0cos(nϕ) current distribution or that of overlapping ellipses. (a) A cos ϕ distribution (dipole field) can be approximated by ‘sector blocks’ or by ‘concentric shells’; (b) an overlapping ellipse distribution for a dipole field can be achieved by ‘concentric shells’ or by ‘horizontal layers’.

Figure C1.0.16. A coordinate system for the calculation of the magnetic field at point P in the x—y plane which is generated by an infinitely long current-carrying conductor perpendicular to this plane (point A).

The expansion into a series is a consequence of the analytic description of the problem. Because the summation goes from n = 1 to infinity one says that a pure line current generates a multipole field of any order. If one considers now an arrangement of line currents so that the current distribution on a circular

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cylinder with radius a is I = I0 cos (nϕ) (figure C1.0.13) the field components are

For n = 1 the field components of a pure dipole field are found (equation (C1.0.7)) and for n = 2 the field is a pure quadrupole field (equation (C1.0.8)). Consequently n = 3 is a sextupole field and so on. As already mentioned, in practice it is not possible to construct a multipole magnet with the exact current distribution of I = I0cos (nϕ). There are several ways to approximate such a current distribution such as the winding in shells and sector blocks (figure C1.0.15). The basis of all calculations is the general multipole expansion of the field components.

The prefactor B0 is the field magnitude at the reference radius r0. The radius r0 is a design parameter and has to be fixed according the magnet specification. For instance multipole magnets for an accelerator require a particular field quality inside this reference radius r0 where particles are circulating. The reference radius of the Tevatron and HERA magnets is 25 mm, about two-thirds of the radius of the aperture. The coefficients bn and an are the ‘normal’ multipole coefficients and the ‘skew’ coefficients, respectively. Note that these coefficients depend on the reference radius. It can be shown that for symmetry reasons in a dipole magnet all skew coefficients are zero and only odd values of n exist (n = 1, 3, 5,…). In a quadrupole magnet all skew coefficients are also zero but n = 2, 6, 10,… With the above-mentioned definition of B0 the coefficient for the lowest order is normalized to unity (e.g. for a dipole b1 = 1, for a quadrupole b1 = 1 etc). Coefficients with higher order are zero for an ideal multipole current distribution I = I0cos (nϕ) but are non-zero in the case of an approximation (figure C1.0.15). However the better the approximation the smaller the higher-order coefficients, which become then a measure for the field quality. Note that in American literature the counting of multipoles often starts with n = 0 (not with n = 1); this means that a dipole has n = 0, 2, 4,… and a quadrupole n = 1, 5, 9, …. In the case of a simple current shell with current density J, thickness ∆a = a1–a2 « a = (a1 + a2)/2 and limiting angle ϕl (figure C1.0.15) the generated field is

or the magnitude of the multipole field of the order n is

so by selecting current shells (blocks) with the right thickness, current densities and angles multipole components of the magnet can be reduced to values < 10– 4 (see also G4).

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Design criteria

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The design of the ends of multipole magnets is much more difficult. One has to take into account the kind of bending of the conductor. For instance the cable can be bent so that the wide side is always perpendicular to the cylindrical aperture. Another possibility is quasi-isoperimetric bending, which follows the natural form of a thin tape. Needless to say the conductor must be fixed mechanically very well in order to prevent any movement under magnetic forces. Finally it should be mentioned that the peak field of the magnet is located at the end region. By the insertion of spacers, which spread out the windings at the end, the peak field can be reduced. The same technique is used to suppress undesired higher multipoles generated at the ends. Figure C1.0.17 shows the end spacer for the inner shell of the LHC dipole. Due to the complexity of the problem, calculations are performed mainly numerically. A more detailed discussion of the end region of multipole magnets can be found in the book by Mess et al (1996, p 58).

Figure C1.0.17. End spacers of the inner shell of the LHC dipole magnet at CERN. Reproduced by permission of R Perin.

In the presence of iron one has to combine the above-mentioned analytical calculation with FE analysis. For very accurate field calculations this is not sufficient and the following trick is commonly used. In a first step the field and its uniformity are calculated analytically with the assumption that the susceptibility of the iron is µ = ∞ (unsaturated iron). The same calculation is carried out with the FEM once with unsaturated and once with saturated iron. The difference between both FE calculations is added as a correction to the analytical one. This practical approach has been proven in practice and the result is in good agreement with measurements. C1.0.5 Design criteri Because the current density of a superconductor depends on the magnetic field and the temperature (see section B8.1) one has to fix the operating current of a magnet so that a sufficient current and temperature margin for safe operation exist. In a superconducting magnet the maximum current (or current density) is limited by the peak field in the windings and not by the field in the centre of the magnet. The situation can be illustrated by figure C1.0.18 where the critical current against field is shown schematically at two temperatures. Taking into account a temperature margin of ∆T = T2 – T1, the temperature of the magnet is at T2. Then for a specified central field the load line of the magnet is defined by the line OA . However there are parts of the winding which are exposed to a higher field (peak field) and the critical current in the superconductor is reduced. If the temperature margin is maintained one obtains a new load line OB with an operating current Io p < Ic (T2). As a consequence the originally designed central field can no longer be achieved.

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Figure C1.0.18. The definition of load lines for the operation of a superconducting magnet.

A possible design sequence for a solenoid with one winding section is as follows. For a given central field, B0, the required current is obtained from measurements of the critical current of the superconducting wire used against field and temperature (point A in figure C1.0.18). If not completely available, missing data can be obtained by extrapolation using scaling laws (section B7.3). Note that critical currents are also strain dependent and in the case of a superconductor with rectangular cross-sectional area Ic shows an anisotropic behaviour (section B7.3). Then, according to equation (C1.0.3), the shape factor F(α , β) is obtained. Depending on the selection of α and β, the ratio Bm /B0 can be found from figure C1.0.10 which defines the new load line (OB in figure C1.0.18). After a few iterations the optimum design parameters can be fixed. The situation is more complex in graded solenoids where the peak field in different sections depends on other sections (Montgomery and Weggel 1980, p 146). Typical operating currents are in the range of 70–85% of the critical current of the superconducting wire or cable (e.g. laboratory magnets). This value can rise to 90% for accelerator magnets. For largevolume magnets, however, values of approximately 50% are common. In general the operating current must also take into account all kinds of uncertainty such as the temperature margin, the variation of the critical current over the length of a wire or cable and strain effects. For persistent-mode operation with extremely low drift of the field, the operating current must also be adjusted with respect to the abruptness of the V—I curve (i.e. the n value) of the critical-current measurement. The higher the n value of the considered superconductor (see section B7.3) the higher can be the operating current. C1.0.6 Mechanical design In a superconducting magnet the following acting forces must be considered: • • •

forces induced by the winding tension during the manufacturing of the magnet; forces due to differential thermal contraction between different materials in the winding package and the winding support (bobbin); electromagnetic forces (Lorentz forces) due to the simultaneous presence of high current densities and high magnetic fields.

In order to insure a precise positioning of the conductor winding is performed under tensile stress, typically between 10 and 40 MPa. This stress must be maintained during the whole winding process and also for the vacuum impregnation, so often special clamping tools are indispensable. Note that ‘wind-and-react’ Nb3Sn magnets lose this winding stress during and after the heat treatment for the formation of the Nb3Sn phase. When cooled from room temperature to the temperature of liquid helium/nitrogen, any material is subject to thermal contraction. The linear thermal contraction is material specific and may have a strong

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anisotropy as in the case of fibre-reinforced plastics (FRPs). The volume coefficient of thermal expansion at constant pressure is β = (1/V)dV/dT|p , where V is the volume and T is the temperature. For isotropic materials one can calculate the linear thermal expansion coefficient by α = β/3 = (1/L)dL/dT|p , where L is the length of the considered material. In table C1.0.1 the thermal contraction of different materials for cooling from room temperature (293 K) to 4.2 K, as well as the E modulus (Young modulus), is summarized. For further details see chapter F6.

Table C1.0.1. Material parameters for the mechanical design.

Differential thermal contraction may be an important issue because the induced forces between different materials can be considerable. For instance this is commonly used for collars in multipole magnets which compress the winding package at 4.2 K. They are designed so that the windings are still under compression at nominal current in the magnet, improving substantially the quench stability. Another example is a superconducting wire with a copper matrix containing NbTi filaments. Upon cooling, copper contracts more then NbTi with the consequence that at 4.2 K the copper is under tensile stress and the NbTi under compression. Because the critical current and the critical field of NbTi are not essentially influenced by stress, the differential thermal contraction can be neglected. However in Nb3Sn the superconducting parameters depend strongly on stress and any change of the stress state has important consequences regarding the performance of the wire (see sections B7.3 and B8.1). Electromagnetic forces will act on the superconductor when the magnet is generating a field. According to Lorentz the force per unit conductor volume is F = J x B . Because the current density, J, and the magnetic field, B , can be extremely high in superconducting magnets, magnetic forces become also very high and must be considered in the design. For this purpose the field distribution inside the windings must be known. The radial field profile in the central plane of a solenoid magnet with α = 3 and β = 2 is illustrated in figure C1.0.19. Note that the radial field decreases linearly from the inner radius a (r/a = 1) of the magnet to the outer radius b (r/a = 3) and even becomes negative. An individual turn with a current density J in an axial field B is now subject to a radial Lorentz force which is directed outwards from the winding (figure C1.0.20). Then the superconducting wire comes under tangential (hoop) stress σϕ = I Ba/Ac where I is the current and Ac is the cross-sectional area of the conductor, respectively. The tangential stress has been calculated at different radii for the above-mentioned solenoid with α = 3 and β = 2 and the result is depicted in figure C1.0.21. With increasing radius first the tangential stress increases, which indicates that the turns have a tendency to separate from each other. A further increase of the radius yields to a decreasing tangential stress. This means that the turns can pile up and the outer turns may support the inner ones. Mainly for this reason the approach of individual, non-mutually-interacting, turns cannot be very precise. The next step is to take into account the connection of turns via the electrical insulation and the impregnation. Useful analytical equations for the tangential and radial stress have been given for the case

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Figure C1.0.19. The normalized field inside the windings of a solenoid magnet with α = 3 and β = 2 (after Montgomery and Weggel 1980).

Figure C1.0.20. A turn of a solenoid magnet with radius a and current density J exposed to the field B of the magnet. The resulting Lorentz force, F, is directed outwards.

for a infinitely long solenoid where the field outside the coil is Bb = 0 and inside Ba = µ0 J a (1 – α) (field at radius r = a). In addition isotropic insulation and impregnation is assumed (Wilson 1983).

where α = b/a of the coil. In figure C1.0.22 the results of such a calculation are shown for a thin coil (α = 1.3) and a thick one (α = 4). In the thin coil the tangential stress is rather high with respect to the thick coil. However the radial stress is always σr < 0, which means that it is compressive. In the case of α = 4 the radial

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Figure C1.0.21. Normalized tangential stress (hoop stress) of a solenoid magnet with α = 3 and β = 2 (after Montgomery and Weggel 1980).

Figure C1.0.22. The normalized tangential and radial stress distribution in the windings of a long solenoid magnet with isotropic components: (a) thin coil with α = 1.3; (b) thick coil with α = 4. Reproduced by permission of M Wilson.

stress as a function of the radius changes sign. For a normalized radius of r /a < 3 this stress is positive and therefore tensile. Such a situation is essential to avoid because mechanically weak components of the windings, such as the insulation and the impregnation, behave better under compression. For this reason, in the infinitely-long-solenoid approximation, α should be α < 1.85. In practice this can be obtained by subdividing the coil into sections. At the ends of a solenoid magnet there are radial field components. Such a radial field is at the origin of axial Lorentz forces. These forces are compressive and sum up so that the maximum is reached at the central plane of the magnet. Because the axial stress is generally les than 25% of the tangential stress, it can often be neglected (Gersdorf et al 1965). In multipole magnets the superconducting cable must be mechanically supported by an appropriate structure. As in solenoid magnets, the design must take care that no tensile stress acts on the constituents of the windings. This can be achieved by a compressive pre-stress of the magnet obtained by differential thermal contraction. For instance the frequently used aluminum alloy collars have an important thermal contraction upon cooling. The design and stress analysis of the support structure follows classical mechanical engineering methods. Then the results are refined with respect to the particularities of the magnet (Leroy et al 1988, Perin et al 1995). As an example in figure C1.0.23 the force distribution of the

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Figure C1.0.23. Distribution of radial and tangential forces per meter in the two-shell HERA dipole magnet at about 5 T. (a) Arrangement of conductors and definition of forces; (b) magnetic forces (Lorentz forces) as a function of conductor number. Reproduced by permission of World Scientific Publishing.

two-shell dipole magnet for HERA at 5 T is shown (Mess et al 1996). The inner shell has 32 conductors and the outer 20 (figure C1.0.23(a)). For simplicity wedges are neglected. In the medium plane (y = 0) the inner shell is pushing radially outwards with 12 kN m–1 metre length of the magnet. In contrast, the outer shell has an inward-directed force of 6 kN m–1. For an increasing conductor number (increasing angle) the radial force of the inner shell goes to nearly zero but the force of the outer shell stays constant (figure C1.0.23(b)). For both shells tangential forces are directed to the medium plane and increase with increasing angle. These forces are finally much higher than the radial ones. References Beth R A 1968 Analytical design of superconducting multipolar magnets Proc. 1968 Summer Study on Superconducting Devices, BNL Report 50155, vol 3, pp 843–59 Boom R W and Livingstone R S 1962 Superconducting solenoids Proc. IRE pp 275–85 Gersdorf R, Muller F A and Roeland I 1965 Design of high field magnet coils for long pulses Rev. Sci. Instrum. 36 1100–9 Leroy D, Perm R, deRijk G and Thomi W 1988 Design of a high field twin-aperture superconducting dipole model IEEE Trans. Magn. MAG-24 1373–6 Mess K H, Schmüser P and Wolff S 1996 Superconducting Accelerator Magnets (Singapore: World Scientific) Montgomery D B and Weggel R J 1980 Solenoid Magnet Design (Huntington: Krieger) Perin R, Perini D, Salminen J and Soini J 1995 Finite element structural analysis of LHC bending magnet IEEE Trans. Magn. MAG-32 2101–4 Smythe W R 1950 Static and Dynamic Electricity (New York: McGraw-Hill) Wilson M W 1983 Superconducting Magnets (Oxford: Clarendon)

Further reading Brechna H 1973 Superconducting Magnet Systems (Berlin: Springer) Ceilings E W 1986 Applied Superconductivity, Metallurgy, and Physics of Titanium Alloys vols 1 and 2 (New York: Plenum) Evetts J 1992 Concise Encyclopedia of Magnetic and Superconducting Materials (Oxford: Pergamon) Foner S and Schwartz B B 1974 Superconducting Machines and Devices (New York: Plenum) Suenaga M 1980 Filamentary A15 Superconductors (New York: Plenum) Turner S (ed) 1996 Superconductivity in particle accelerators CERN Accelerator School CERN 96-03

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C2 Practical aspects of superconducting magnet manufacturing F Zerobin and B Seeber

C2.0.1 Wires and cables In this chapter basic concepts of wires and cables, as they are used for the fabrication of superconducting magnets, are briefly discussed. The most important are compiled in table C2.0.1. Monolithic conductors, or strands, are multifilamentary wires with round or rectangular cross-section. Typical dimensions and currents are given in table C2.0.1. They are employed for magnets with operating currents of up to ∼500 A. One of the advantages of a monolithic conductor is the high packing factor, especially for a rectangular conductor, which allows a high current density of the winding package and therefore compact magnets. Also this type of conductor is relatively easy to wind and winding tensions are moderate. A special type is the so-called wire in channel, where a superconducting wire is soldered in a Cu profile allowing more freedom for the design of particular magnets. In order to achieve higher magnet currents (lower inductance of the magnet), round monolithic wires are assembled to form a cable or composite conductor. This allows more freedom for the stabilization of the conductor as well as for the mechanical design and the cooling. The most important cable types are included in table C2.0.1. From the point of view of the magnet manufacturer, cables have the advantage that several wires (strands) are connected in parallel and the current is distributed over them. If one strand has a local imperfection, or even an interruption, the other strands can take over the current. In such a case it is still possible to carry the nominal current but with a reduced current or temperature margin. Taking this effect into account one can design magnets with high reliability. A disadvantage of cables is the reduced packing factor which can partially be improved by a rectangular or a trapezoidal cross-section of the cable. A very special type of conductor is the cable-in-conduit conductor (CICC). This type was developed for very high currents, especially for coils for fusion magnets (Tokamaks, Stellerators). Due to the high magnetic fields, the magnetic forces become extremely large, and an extra mechanical reinforcement of the superconductor is required. This function is taken over by the outer jacket (stainless steel or aluminium alloy). In this jacket the cabled strands are assembled with a certain void fraction, so liquid helium can circulate through the cable for direct cooling. For further details see chapter B6.

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Practical aspects of superconducting magnet manufacturing Table C2.0.1. Typical dimensions and currents for superconducting wires and cables.

C2.0.2 Electrical insulation Although under d.c. conditions the current flows only in the superconducting filaments without any voltage drop, electrical insulation of the wire or the cable is required for any change of the magnet current, in particular in the event of a quench. The electrical insulation of the conductor and to ground is an important issue as in every conventional electrical device. The voltage across the terminals of a magnet is

where R is the resistance of the current leads and of internal joints (interconnections), I is the current and L is the self-inductance of the coil. This voltage usually is only a few volts, typically 5 V to 10 V. In special applications, e.g. superconducting magnetic energy storage, voltages can be much higher due to fast charge and discharge times (up to 3000 V). The insulation system must also be designed to withstand a quench. Therefore electrical insulation between the individual conductor and ground is absolutely necessary

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Figure C2.0.1. Different kinds of electrical insulation of a superconductor: (a) varnish insulation of a round and rectangular wire; (b) wrapping of a conductor with an insulating tape and a pre-impregnated tape (prepreg); (c) insulating the conductor with spacers.

and has to be designed and assembled carefully. Three basic types of turn-to-turn insulation are in use (figure C2.0.1): • • •

varnishing the surface of the conductor wrapping the conductor with insulating tapes or braids spacers between conductors.

The design and the choice of the electrical insulation for superconducting magnets has also to be evaluated with respect to mechanical requirements of the magnet. Varnishing conductors is used for round or rectangular wires with a smooth surface. The advantage of such an insulation is the small thickness of the insulating layer which allows high packing factors for the windings. In addition, the tolerances of the conductor dimensions are small which is important for the manufacturing of magnets with a high field homogeneity, e.g. magnets for magnetic resonance spectroscopy (section G2.1). It is important to note that the coating of the wire with enamel requires a short heating up to approximately 250°C. Without precise control the superconducting wire may overheat which can reduce the critical current density (e.g. Nb—Ti). Varnishing is employed for Nb—Ti conductors. Sometimes it can be used also with Nb3Sn conductors when the thermal treatment for the formation of the Nb3Sn has been carried out before the coil winding (react and wind technique). However, due to the brittle nature of Nb3Sn this is a risky operation. Normally Nb3Sn wires are wrapped with a glass insulation so the heat treatment for the formation of Nb3Sn can be carried out after the magnet has been wound (wind and react technique). Wrapping the surface of conductors with electrical insulation tapes or braids is a frequently used process. In table C2.0.2 the most commonly used materials are listed with typical thicknesses and

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Practical aspects of superconducting magnet manufacturing Table C2.0.2. Most commonly used materials for the insulation of superconductors.

Figure C2.0.2. Different overlapping methods for a wrapped insulation.

Figure C2.0.3. A typical cable insulation line. 1—cable supply spool, 2—dimensional control, 3—cleaning (brushes, ultrasonic bath etc), A—wrapping head for insulation tape, 5—high-voltage insulation control (e.g. pinhole detector, 1 kV, 50 Hz), 6—wrapping head (optional if a prepreg is used), 7—puller, 8—repair and inspection station, 9—cable take-up spool or coil bobbin (on-line insulation line).

breakdown voltages. Note that a glass-type insulation requires vacuum impregnation with an epoxy resin system for full electrical insulation and for mechanical robustness. Tapes are applied in an overlapping manner to ensure at least one insulation layer. Usually the overlapping is between 50% and 66% (figure C2.0.2). The wrapping process can be applied either on-line to the winding of the magnet or off-line. A typical insulation line for wrapping conductors consists of the following elements (figure C2.0.3) • • • • • •

conductor supply spool conductor cleaning dimensional control wrapping head electrical insulation check (pinhole detector, e.g. 1 kV/50 Hz) cable storage spool.

Conductor cleaning before wrapping with the insulation tape is required to remove all chemical residuals, or residues of lubricants, from the conductor/cable manufacturing process as well as all kinds of metallic and nonmetallic chip. Mechanical cleaning with brushes or sandblasting or/and ultrasonic cleaning is an efficient procedure. Wrapping the tapes is done on special rotating wrapping machines. Such devices allow the control of the wrapping speed and the overlapping factor of the tape. The tapes to be wrapped

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are mounted on spools onto the machine. The tapes must be under a certain tension, typically in the range between 10 g and 50 g per millimetre width of the tape. However, these values should be tested before starting any insulation process. For Nb3Sn magnets manufactured by the wind and react method special insulation techniques are in use or in development. For instance one method uses heat-resistant tapes made from glass (E, R or S glass) which supports temperatures of 600°C to 700°C for 50 h to 200 h. Note that the mechanical strength of such tapes decreases significantly with increasing temperature. Furthermore, it is very important to burn off all organic additives used by the glass manufacturer before one goes up to the temperature for the reaction heat treatment. This can be achieved by heating up to about 300°C to 500°C under air or a particular partial pressure of oxygen. Without this intermediate heat treatment, electrically conducting graphite particles are formed which greatly reduce the insulation resistance of the magnet. There are also mica tapes where a glass tape acts as a carrier. Here the organic binder is used for the connection of both tapes and must be removed before the final heat treatment of the magnet. The burnoff process is similar to that of a pure glass insulation. The glass-mica tape is normally used with the mica on the conductor side. After the heat treatment to obtain the Nb3Sn superconductor, the glass or glass— mica insulation is extremely fragile and care is required during handling. In fact, the glass insulation acts as a thin spacer and the gap between must be filled with paraffin or epoxy resin by vacuum impregnation. Uninsulated conductors separated by electrical insulating spacers are used when the surface of the conductor should not be covered totally by an insulator. In such a case a direct contact of the conductor surface to the liquid helium bath is possible. Obviously such an arrangement requires additional space and the current density of the winding package is reduced. The design of the electrical insulation by spacers has to take into account several breakdown voltages: • • •

electrical breakdown of the spacer itself sliding breakdown of the surface of the spacer electrical breakdown of the cooling medium.

It should be emphasized that liquid helium has a relatively good breakdown voltage whereas helium gas is a bad insulator. Further details are given in chapters D2 and F7. Other parameters to check are: • •

that the conductor is not supported mechanically between the spacers, the spacer materials which can be sensitive to humidity during assembly. If humidity is absorbed, the breakdown voltage decreases.

A drawback of this insulation process is its sensitivity to any kind of dirt during the winding and assembling of the magnet, as well as in operation. In practice it is rather difficult to assure clean helium without any electrically conducting particles so the described insulation method is not used very much. After the insulation is completed tests have to be performed in order to check the quality. For instance, after wrapping the tape, the insulated conductor runs through a set of brushes (e.g. bronze), or better still through a bath of balls or rolls, where a high voltage is applied. The metallic part of the conductor is connected to ground. There are two test systems currently in use. D.c. systems work between 300 V and 2500 V and after the measurement the insulation is electrically charged. Before further handling, measures must be taken to discharge the latter. An a.c. system (50/60 Hz) has the advantage that the insulation is not charged. Typical test voltages are in the range of 200 V to 2000 V. Both test methods require a current limiter which has to limit the current in the case of a fault. The optimal test voltage has to be found experimentally. It depends mainly on the thickness of the tape and as a rule of thumb one can take ∼500 V per 0.1 mm. The correct functioning of the whole setup may be checked by putting artificial holes in the insulation.

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Finally it should be mentioned that most of the insulating materials are sensitive to the storing conditions. For instance temperature and humidity can change the insulation properties. For this reason it is imperative to respect the specification for the storage given by the manufacturer. C2.0.3 Winding techniques Depending on the type of magnet, different winding techniques have been developed. Magnets can be classified as follows: • • • •

solenoid magnets flat coils (racetrack, D-type coils, etc) multipole coils (dipoles, quadrupoles, etc) special magnets (e.g. three-dimensional (3D) shaped coils).

There are a few fundamental techniques that are adapted to the specific application. The experience of the personnel as well as the available tools are other parameters to be taken into account. The conductor is spooled off from the storage spool and wound under tension onto (or into) the coil bobbin or a temporary mandrel. A certain winding tension on the conductor to be wound is required to ensure a good and accurate positioning of the conductor onto its support. Typical values are between 10 MPa and 40 MPa (1 MPa = 1 N mm-2 ). This range holds for copper- and aluminium-stabilized conductors. For special configurations, e.g. with stainless steel jackets, winding tensions can be much higher and depend mainly on the stiffness and the required bending radius. Current tools and auxiliary equipment for winding magnets are: • • • • • • •

conductor supply spool electrical insulation device (if on-line process) conductor guiding and positioning coil bobbin or winding mandrel mechanical brake (caterpillar, spring load, etc) for the conductor tension winding machine or rotating table conductor fixation clamps.

These are the most important items for the winding of magnets, independent of its complexity (solenoid coil, 3D shaped coils, etc). Solenoid coils are the most common and widely used for laboratory magnets (chapter G1), magnetic resonance spectroscopy (section G2.1) and magnet resonance imaging (section G2.2). The geometry is simple and the technology is well proven. The conductor is wound layer by layer (figure C2.0.4) on a cylindrical former (bobbin) made of stainless steel, aluminium or a reinforced epoxy tube with flanges at both ends. For metallic formers it is important that they are electrically insulated before winding. The same insulation materials as mentioned in table C2.0.2 can be used. The electrical insulation between turns is provided by the conductor and the insulation between the individual layers can be improved by extra layers of insulation material. For round wires one has to distinguish two winding configurations, as shown in figure C2.0.5. The six-pack filling factor is between 0.7 and 0.8 and the four-pack filling factor typically between 0.5 and 0.65. These theoretical values can be achieved in practice when careful winding is performed. The most critical regions for winding are the ends of the coil where the conductor has to go from one layer to the next. For larger conductors a tapered wedge, made from an insulating material, is inserted in order to support the conductor. Needless to say that the winding should be as tight as possible and the space between windings should be a minimum. The latter is particular important for the mechanical performance of the magnet after vacuum impregnation with epoxy resin. After winding, the coil is wrapped usually with insulating tapes in order to create an outer insulation and a mechanical protection of the conductor.

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Figure C2.0.4. The principle of the winding of a solenoid magnet.

Figure C2.0.5. Winding configurations for a round wire. The six-pack configuration allows a filling factor between 0.7 and 0.8. The filling factor is reduced to 0.5–0.65 for the four-pack configuration.

Because Nb3Sn magnets are normally built according the wind and react technique, they have to be heated up to ∼700°C after the winding in order to form the superconducting phase. Stainless steel formers are commonly used in order to match the thermal expansion with that of the Nb3Sn conductor. The former is electrically insulated by glass tapes or mats. Another possibility is an Al2O3 coating obtained by plasma spraying. A particularly critical region of such a solenoid is the outlets of the coil where the conductor has to go through the flange. Care has to be taken not to over-bend the conductor and to fix it well against any movement. It is worth designing the outlets carefully because most of the problems of superconducting magnets, not only those using Nb3Sn, have their origin at the ends of the coil. There are situations where the conductor length is limited (e.g. high-current cables, high-Tc superconductors, etc). It is still possible to build magnets but they must be subdivided into sections. If the terminals of one section are at the outer diameter one deals with a double-pancake coil (figure C2.0.6). Several sections are put together to build up the magnet. All sections are connected in series outside the coil. The methods of interconnecting the conductors is described later. An advantage of this winding method is that the pancakes can be tested individually before the final connection. Large solenoids (e.g. detector magnets) with diameters up to 6 m are usually wound onto an external support cylinder (figure C2.0.7). Because the electromagnetic forces are mainly directed outwards, no inner mandrel is necessary. The winding process must assure that all windings are tight and no empty space is left between them. Some applications require flat coils. Such a coil can be considered as a short solenoid, but not necessarily with a circular shape. Winding is carried out on a mandrel as described earlier. Often the

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Figure C2.0.6. The principle of the winding of a double-pancake coil.

Figure C2.0.7. The principle of the winding of a large-bore solenoid (detector magnet) onto an outer coil bobbin.

Figure C2.0.8. The support for the winding of a dipole magnet.

method of pancake coils is used. Taking two flat racetrack coils and bending them around a cylinder leads to a dipole coil. If there are more than two such coils one speaks of a multipole magnet (e.g. quadrupole, sextupole, etc). Dipole or quadrupole coils are wound on winding mandrels which can, in addition to the winding rotation, be turned in the dipole case by ±90° (figures C2.0.8 and C2.0.9). In the case of a quadrupole this angle is ±45°. The conductor, in general a cable which is often keystoned (i.e. trapezoidal

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Figure C2.0.9. The arrangement for the winding of a dipole magnet showing the movement of the cable supply spool and the winding post.

cross-section), is wound around a winding post turn by turn. In the straight part the cable has to be clamped (compressed) to the exact winding position. The coil heads (ends of the coil) are rather complex and difficult to wind. In order to achieve good precision and reproducibility in a series production, the winding process should be computer controlled. C2.0.4 Interconnections In a magnet different types of wire connection are required. One can classify them as follows: • • •

coil terminals and outlets, in graded coils two different types of conductor have to be connected; one has to connect either the same type of conductor, but with different diameters, or different conductors, e.g. Nb—Ti and Nb3Sn, if the length of the conductor is too short for winding all turns, two conductors of the same type have to be connected together.

Interconnecting superconductors deals with many special problems in the design, the manufacturing and the final operation of the magnet. The following areas have to be considered: • • • • • • • •

joint type and dimensions electrical joint resistance power loss temperature increase, cooling stability against quench magnetic forces fixation against movement connection techniques (soldering, etc).

Any interconnection is a special design effort and an additional risk in the manufacturing and the operation of the magnet. For these reasons they should be kept to a minimum. The best solution for an interconnection would be one which is fully superconducting and therefore without any power loss (no heating). Such connections can be made, but very special techniques and experience are required. In contrast many applications allow a small electrical resistance and common methods are clamping and soldering. Other techniques, such as ultrasonic welding, hard brazing or explosion welding, are very specialized and not currently used. Figure C2.0.10 shows some basic types of joint geometry. Note that there are connections

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Figure C2.0.10. Basic configurations of different conductor interconnections.

Figure C2.0.11. The model used for the calculation of the interconnection electrical resistance.

where the interconnection has the same cross-section as the conductor. The overlap-type connection, however, requires much more space. Figure C2.0.11 shows the cross-section of a typical connection. The current of one conductor has to go into the second one. The current flows through different resistive materials. To calculate the distribution of the current is complex because the I—B characteristic of the superconductor and the magnetoresistivity of resistive materials must be taken into account at every local point of the interconnection. The basic equation for the calculation of the total resistance of an interconnection is

where i denotes the different materials and ρ is the resistivity, l is the current path length and A is the cross-sectional area. Note that the resistivities of the involved materials mostly vary with the applied magnetic field (see also chapter F3). Comparing the resistivity of copper with a residual resistivity ratio (RRR) of 50 (ρ = 3 x 10-10 Ω m) with that of a solder (e.g. PbSn40, ρ = 3 x 10-9 Ω m) shows that the most important contribution comes from the solder. From this it is concluded that the amount of solder between the conductors must be a minimum. The following equation allows an estimate of the resistance R of the interconnection shown in figure C2.0.11 (Wilson 1983).

where ρ is the resistivity of the solder, L is the length of the soldered interconnection and f = h/d (see figure C2.0.11). This equation is very useful for the design of interconnections and the dependence of RL/ρ on the ratio h/d is shown in figure C2.0.12 (Wilson 1983). Typical values of the resistance are between 1 × 10-9 Ω and 1 × 10-8 Ω. To achieve better values special techniques, as already briefly mentioned, must be applied.

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Figure C2.0.12. Calculated resistance of a soldered interconnection according the model in figure C2.0.11 (after

Wilson 1983).

When a current is flowing through the interconnection, power is dissipated according to P = I 2 R. This loss has to be added to the heat load of the magnet which must be taken over by the cooling system. Although losses in well designed interconnections are rather small, they can locally increase the temperature. It is recommended to make an estimate of this temperature increase which can be calculated to a first approximation (adiabatic case) by

where R is the resistance of the interconnection and L is the length of the joint. L has to be chosen in accordance with the superconductor itself and the allowed temperature increase. The minimum length should be the current transfer length (see section B7.3). In the case of a cable the minimum length of the interconnection is the transposition length of the strands (individual wires) because the current must be fed to all of them. In a bath-cooled system, with direct access of liquid helium, the allowed temperature increase can go up to ∼0.2 K. If cooling is not as good ∆T should be below 0.05 K. The mechanical design of an interconnection must fulfil the same requirements as the superconductor itself within the coil. The design principle that the conductor must not move under electromagnetic forces has also to be applied here. Although interconnections are normally made outside the magnet in a low-field region, mechanical forces cannot be neglected and must be known. The mechanical stability can be improved by adding a stabilization material, e.g. copper, in the form of bars or profiles (figure C2.0.13). These profiles may also be of help for the manufacturing and assembly of a joint. Magnet terminals are made with an excess length of the superconductor going in and out of the magnet. For small conductors, where it is dangerous to handle them, a parallel shunting of Cu and/or a second superconductor is recommended.

Figure C2.0.13. Different kinds of mechanical and electrical reinforcement of conductor interconnections.

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A good solution for joints is soft-soldering where the copper stabilizers of the superconductors are soldered together. This technique is well developed and uses most of the standard soldering procedures of electrical machinery. Mechanically rigid joints can be built with very low electrical resistance. The solidification temperature of the solder alloys is between 180°C and 250°C. These temperatures are relatively low and do not reduce the critical current of conventional superconductors like Nb—Ti and Nb3Sn. Various alloys can be used for this purpose and the most common are compiled in table C2.0.3. The appropriate flux is supplied by the manufacturer of the solder. It should be noted that the content of acid may vary from flux to flux and particular care is required after soldering to remove any excess flux. Table C2.0.3. Common solders for interconnections with their solidification temperatures.

To obtain superconducting joints, the Cu stabilizer of the superconductor has to be removed e.g. by etching. Then the individual filaments can be connected by ultrasonic welding, cold welding or other techniques. After that, a stabilizing material should again be brought into contact with the filaments. This procedure is very specialized and is used in magnets where an ultra-low electrical resistance is required, e.g. in magnets with a low specified current decay rate when running in the persistent mode (e.g. NMR spectroscopy and MRI magnets). These joining techniques are then, of course, often kept as technological secrets in companies. In some cases the connection should be detachable. Here only a heavy clamping with a fixation by screws can be made. The joint resistance will also be affected by the surface-contact resistance. Special care should be taken with the surfaces, their roughness, cleanliness and especially the removal of layers of oxides. Protection of the surface, e.g. by coating with Ag, is recommended. The mechanical design of such a configuration has to take into account the magnetic forces of the currents and forces coming from the cool-down. In particular, it is critical that the tensile force in the joint is fixed because any movement under friction should be avoided. C2.0.5 Impregnation As already mentioned in chapter C1, the windings of a superconducting magnet must be fixed against any movement because movement could cause a quench. This fixing can be achieved by an impregnation with paraffin or epoxy, preferably under vacuum. The basic idea of impregnation is to fill all voids in the coil volume, so every location of a conductor within a winding package is well supported against magnetic and other forces. Furthermore, after impregnation, the coil is well protected and can be handled more easily. The simplest method is to dip the coil into liquid wax (e.g. bees wax, paraffin, etc). To get wax with the right viscosity, it must be heated up to about 60°C. There is a risk that gas bubbles cannot be removed by this method. Wet winding is an alternative. During the winding process the insulated conductor is covered with a wet epoxy resin. When the coil winding is terminated, the magnet will be heated up to higher temperatures to cure the epoxy. A drawback is that personnel are exposed to liquid epoxy and its vapour.

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The most used method is vacuum impregnation because all the empty space (voids) between windings can generally be filled with wax or epoxy without gas bubbles. After having been wound, the coil is put into a form (mould) which gives the final precise dimensions. The form itself, which is frequently of stainless steel or aluminium, should be constructed in such a way that easy disassembly is possible. It must be covered with a release agent to allow the coil to be removed from the mould after impregnation. If a precise mould is not possible, too expensive or the dimensions are not so critical, an open system can be made. Here the coil is covered by a tape which keeps the wax or the epoxy inside the coil during the impregnation process. Then the whole is put into a vacuum vessel where a pressure of about 1 mbar is sufficient to remove air and other gases between the windings. It must be possible to heat the vessel in order to remove humidity and to control the viscosity of the wax/resin during the impregnation process. The heating must also be designed to cure the epoxy resin, if this is used. Before impregnation starts, the wax/resin is prepared in a separate vessel which is also under vacuum. The material used for impregnation must be outgassed (no gas bubbles) and well mixed. Because an epoxy resin can have many components, good mixing is of particular importance. The impregnation of the coil should be done slowly until the mould is filled. Fast filling favours the formation of bubbles. Once the coil is completely impregnated one can apply a small over-pressure to the coil with an inert gas (e.g. N2 ). In the case of an epoxy resin curing is the next step, requiring temperatures around 120°C for ∼24 h. In general, all temperatures (outgassing, impregnation and curing) must be quite precise, of the order of ±5°C. One problem which may occur with epoxy, when the temperature is locally too high, is that curing can start too early and eventually interrupt the impregnation process. Figure C2.0.14 shows a schematic diagram of a vacuum impregnation system.

Figure C2.0.14. A schematic diagram of an epoxy resin vacuum impregnation installation.

The epoxy resin for impregnation consists of several components: • • • •

epoxy resin hardener accelerators (one or more) special components, e.g. glass filler.

The following considerations may help in choosing the right epoxy system: • • •

void size to be filled; an epoxy system with high viscosity fills small-sized voids more easily than a low-viscosity one; volume shrinking during curing and cool-down to room temperature; volume shrinking during cool-down to cryogenic temperatures;

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Practical aspects of superconducting magnet manufacturing mechanical characteristics (can be improved with glass as a filler); flexibility; epoxy systems can be brittle, adding appropriate components decreases the brittleness; curing conditions; sensitivity to radiation.

For some special cases (accelerator dipoles, quadrupoles, detector magnets) pre-impregnated tapes are used as an alternative to vacuum impregnation. A glass tape is impregnated with an epoxy resin but not cured fully. The epoxy is kept in a so-called B-stage. This pre-impregnated tape, also known as a prepreg, can be handled and wrapped easily because it is dry. After winding, the coil is heated up to about 40°C and then to 80°C where the epoxy gets soft. If required, the coil is compressed. Then a curing at approximately 150–170°C for typically 2 h polymerizes the epoxy. Advantages of prepregs are that no mould is required, the epoxy is located where it is needed and the mechanical behaviour, as well as the cooling conditions, can be controlled by the amount of epoxy. It is even possible to allow liquid helium to penetrate and to be in direct contact with the superconductor. Drawbacks are that the mechanical strength is reduced with respect to vacuum impregnation and it cannot be used for Nb3Sn coils built by the wind and react technique. References Wilson M N 1983 Superconducting Magnets 1st edn (Oxford: Clarendon) pp 314–5

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C3 Quench propagation and magnet protection

K H Meß

C3.0.1 Introduction Under steady-state conditions the heat W(T) dissipated to the surroundings by a conductor carrying the current density J equals the Joule heating, Q (T) = ρ(T)J 2. In general the resistivity ρ depends on the temperature T and the cooling also varies with temperature. There may be more than one temperature T at which the solution of the steady-state condition W(T) = Q(T) is stable. That depends on how much resistivity or cooling vary with temperature. For example, a wire immersed in water can operate at two temperatures. Either the liquid carries the heat away or, at a higher temperature, a vapour film cools the perimeter of the wire. Obviously the poorer heat conduction of the vapour forces an equilibrium at a higher temperature. If for some reason the water temperature rises above the lower stable point, the temperature will increase locally until it reaches the upper stable point where vapour bubbles form. Furthermore, the vapour front will propagate along the wire because heat conduction raises the temperature of the wire and the nearby coolant. The velocity of the vapour front will depend on the resistivity, on the current density, on the surface conditions of the wire and on the temperature. A metal wire with rapidly varying resistivity, for example at low temperatures, behaves similarly. In this case the zone of higher resistivity, i.e. the hot zone, will eventually expand. Of course, the wire can also become too hot locally and melt. These kinds of bistable system are omnipresent. This chapter, however, will concentrate on one very special case of bistability, namely the transition from superconductivity to normal conductivity in type II superconducting wires and cables. The argumentation would also hold in principle for high-temperature superconductors, but the thermal behaviour of all materials depends highly on the temperature. There is no need to discuss the quite different conclusions for high-temperature superconductors, because no largescale applications of high-temperature superconductors are known at present. Incidentally, helium cooling exhibits some irregularities at the low temperatures of ordinary superconductivity. They complicate the issue substantially. We will ignore the details here, nevertheless, as they bear little influence on the quench protection. The loss of superconductivity, usually called a ‘quench’, is in itself not dangerous; however, the stored energy is released in a quench and it can destroy the device. In particular, magnets can store sizable and dangerous amounts of energy that can easily destroy the coil. This is possible because the current densities in superconductors are usually much higher than in normal conductors. Hence the heat production is high in the case of a quench and therefore the upper ‘stable’ temperature may mean that the cable insulation, solder joints or even the alloy melt. The possible danger and the extremely small energy

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needed to initiate a quench make quench propagation and magnet protection important issues on which numerous researchers have worked for many years. This chapter can only give a short summary of the technical aspects of how to avoid damage. It is hardly possible to add ‘quench protection’ to an existing magnet system. It has to be an integral part of the design. Almost unprotectable magnets are easily built. For an effective quench protection it is necessary to influence the basic design parameters. Hence the material that is covered in section B3.1 is of utmost importance. The most effective quench protection is to avoid quenches. This topic is closely related to the minimum propagating zone and the energy needed to create it. It is reasonable to match this energy, which is a design parameter, to the expected spectrum of possible causes for temperature disturbances. If, however, the initiating disturbance is too large the maximum energy density deposited in a magnet coil during a quench has to be minimized. Minimizing the energy density will also minimize the highest temperature in the coil and also the danger of destruction. The highest temperature, and hence the possible damage, depends on the deposited Joule heating energy (that is on the Joule heating integrated over time ∫ ρJ 2dt ) and on the affected volume. Therefore it is necessary to detect quenches as early as possible to influence the energy deposition by some active means and to spread the deposited energy over a large volume. Again the design of the superconducting cable, the support structure, the method of cooling and the electrical insulation determine how fast the quench has to be detected and whether active measures to spread the quenching zone are necessary. Quench detection methods are mentioned and schemes to divert the stored magnetic energy into external dumps or to absorb it are discussed. C3.0.2 The transition to the normal-conducting state C3.0.2.1 Minimum propagating zone As mentioned above and explained in section B3.1 a quench is the transition from the superconducting to the normal state. Such a transition will invariably occur if any of the three parameters, temperature, magnetic field or current density exceeds a critical value. If the superconducting compound wire has for some reason a local temperature above the bath temperature, as indicated in figure C3.0.1, heat will flow along the strand and into the helium bath. Hence the temperature disturbance can be cooled away. If, however, the temperature is high enough Joule heat generation will take place. Still, the wire can be cooled and the heat conducted away provided the quenching volume is small enough. What could create such a normal-conducting zone? Knowledge about the possible reason for a particular temperature excursion is usually quite limited, because many energy sources are strong enough to overcome the very low specific heat at low temperatures. For example, all magnets have welds or solder joints somewhere in the cable. A good solder joint has a resistance of 10-9 Ω and hence a typical current of 5000 A creates a steady heat load of 5 mW. Anything significantly beyond this can already present a stability problem. Accelerator magnets, fusion reactor magnets or detector magnets for high-energy experiments will always be exposed to some level of radiation. A bunch consisting of 1011 protons of 1 TeV deposits an enormous energy density of 10 J cm-3 in a typical copper-stabilized Nb—Ti coil. Moreover, as explained in section B3.1, very short energy pulses, typical for sudden beam loss, are particularly effective in quenching. To avoid the mechanical and electrical stress of a quench on the magnet the beam should be safely dumped in case of excessive beam loss. In the early operation of HERA (High Energy Ring Accelerator at the Deutsche Elektronen Synchrotron, DESY, Hamburg, Germany) the beam loss detection system was not in operation. This resulted in a number of quenches caused by beam loss. Once it was connected to the beam abort system the number of quenches went down by 80%.

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Figure C3.0.1. A sketch of the heat balance in a quenching cable and of a possible temperature profile.

Other heat sources are related to the Lorentz forces in the magnet. These mechanical quenches presumably account for so-called magnet ‘training’. Magnets ‘learn’ by repeated quenching to achieve a higher field until a plateau is reached. The Lorentz forces deform the coil. This may result in movements or bending of conductors and hence in friction (Maeda et al 1982a). For the first magnets designed for the SSC (Devred 1992) a 10 mm movement over a conductor length of 500 mm was sufficient to release 10 mJ. (The SSC (Superconducting Super Collider, Dallas, TX, USA) was cancelled by a decision of US Congress in October 1993.) Unfortunately, the obvious cure for wire motion, epoxy impregnation of the coil, does not work very well. Epoxy becomes brittle at low temperatures and may develop microcracks or debonding (Maeda et al 1982b). Links can only break once, of course. At the next excitation, the field will rise to a higher value until a critical stress value is reached somewhere else. Such a mechanism can explain training. Well designed magnets, however, should reach their design values without intensive training. Strong clamping with a high pre-stress on the coil and careful gluing suppresses mechanically initiated quenches quite effectively. But the conductor itself may also micro-yield if it is not treated properly during manufacture. Varying magnetic fields present a very special case. They will induce eddy currents in the matrix material of the cable and in the support structure. Hence heat is generated in a large volume. If the rate of change of field is high enough, the heat cannot be conducted away. A large volume quench will be the consequence. This phenomenon is called quench-back and can be both a nuisance and a useful tool for protection. How large is a stable normal-conducting zone, the so-called minimum propagating zone? Obviously, the created Joule heat must be balanced by a temperature increase (enthalpy) of cable and coolant and by heat conduction. The current density in the matrix Jm = (Im /xn )A is zero, if the temperature is low enough. The current will entirely flow in the superconductor with a density J = (I/xs )A. Im stands for the current in the nonsuperconducting fraction xn of the compound wire of cross-section A. For convenience the average current density is called Ja v g = I/A. At higher temperatures all the current will flow through the matrix because the resistivity of Nb—Ti is much higher than that of copper. The so-called current-sharing model states that the superconductor carries as much current as possible up to the critical current density Jc . Any additional current flows through the matrix. In many superconductors the critical current density depends almost linearly on the temperature for a given field,

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at least in the technically interesting region. Hence the current-sharing (or heat-generation) temperature is (compare equation (B3.1.14))

At the critical temperature Tc , the current density in the superconductor equals zero. T0 is the bath temperature. Below the current-sharing temperature the Joule heating is zero because the superconductor carries the current. Above the critical temperature the superconductor is practically free of current and the average resistivity is ρa v g = ρm /xn because the resistance of Nb—Ti exceeds the resistance of good copper by a factor of 2000 ( ρm = ρc o p p e r ≈ 3 × 10-10 Ω m, ρs = ρN b - T i ≈ 6.5 × 10-7 Ω m). Using the linear approximations for the matrix current density, as mentioned above, one calculates the Joule heating term for a given magnetic field B as a function of the temperature (equation (B3.1.16))

In section B3.1 the case of a single composite wire, immersed in an infinite coolant, is treated. In this case the Biot parameter (equation (B3.1.1)) is small, that is the wire cross-section is at an almost constant temperature except at the boundary to the coolant. Under these assumptions the minimum energy needed to trigger a quench can be calculated and likewise the longitudinal extension of the minimum propagating zone. The lateral dimension is given by the wire dimension. Cable-in-conduit magnets could correspond to this picture if the asymmetric support structure and thermohydraulic effects could be ignored. Most magnets, however, are not wound from individually cooled wires. In particular many smaller research magnets or correction magnets in accelerators have vacuum impregnated coils. Magnets or magnetic devices with ‘rapidly’ changing fields often rely on a clamped coil made of Rutherford-type cable (Wilson 1983). In both cases the small (if any) amount of helium at the surface of the single strand in the cable or wire, respectively, cannot be treated as an infinite heat sink. It acts as a bad heat conductor. Hence the dimensions of the whole coil package enter the equation for the Biot parameter. In summary, inside the coil package the temperature can vary considerably and the minimum propagating zone has a shape quite different from the one calculated for a single wire in a helium bath. To calculate the size of a minimum propagating zone we will follow the argumentation of Wilson (1983) and assume a homogeneous conductor with axial heat conductivity Kz and radial heat conductivity κ⊥ . As explained by Wilson, the heat equation can be solved in a transformed coordinate system scaling the radial dimension by αw = pk⊥/k . The temperature in the region with Joule heating (i.e. inside a ‘sphere’ of radius Rg ) is z

with

The factor (1 – ε) takes into consideration that a fraction ε of the coil consists of helium-filled voids that do not contribute to the heat creation. The temperature exceeds of course the currentsharing temperature and sometimes also the critical temperature. Equation (C3.0.2) overestimates the

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heat production in the latter case. Thus equations (C3.0.3) and (C3.0.4) describe an upper limit only. In real space the minimum propagating zone is an ellipsoid, oriented along the quenching wire, with a total length of 2Rg and a transverse diameter of 2Rg pκ⊥/κ = 2αWRg. The value of the constant Θ in equation (C3.0.3) determines the maximum temperature and the minimum propagating energy belonging to a particular minimum propagating zone (Meβ 1996a). z

C3.0.3 Heating of the coil during the quench C3.0.3.1 Quench propagation The quench will expand if the disturbance is larger than the minimum propagating zone. The propagation velocity of the transition between normal and superconduction depends not only on the material properties of the cable but also on the cooling and the electrical insulation. In general all cable properties that are useful to stabilize the superconductor will also slow down the quench propagation. One would like the largest possible minimum propagating zone for magnet protection, because a large zone also means a ‘large’ energy to trigger the quench. One may hope that such a kind of disturbance is rare and hence the likelihood of quenches is small. However, once a quench has started, one would like the normal-conducting zone to expand quickly. Often the magnet has to absorb the stored magnetic energy or at least a sizable fraction of it. It is primarily, of course, the normal-conducting part that absorbs the energy. Fast quench propagation in all directions is the most effective way to spread the deposited energy. A large normal-conducting volume results in a low energy density and hence a low maximum temperature. Once a normal zone has started to expand it will continue to grow as long as the current density and the magnetic field are high enough. The low heat conduction of the insulation and the latent heat of the helium content in the cable impede the transverse expansion. Therefore the normal zone will expand predominantly along the cable. Nevertheless, in most cases the transverse quench expansion affects the increase of resistivity very importantly. The transverse propagation occurs namely in two dimensions. The front may propagate slowly in the transverse direction but it is very broad. (a) Measurement of the quench velocity

Quench velocities are easy to measure in a piece of wire or cable in an external magnetic field. Voltage taps distributed along the wire are sufficient to detect the arrival of the normal zone by measuring the resistive voltage over the normal zone. Figure C3.0.2 shows a typical example. The voltage rises linearly when the normal zone expands between two voltage taps. Thereafter the voltage rises slowly because the resistance is almost constant below some 25 K. Measurements at real magnets are more difficult. Basically four methods exist to measure the quench velocity in a magnet. One can try to attach voltage taps to the coil during assembly. This does not work too well because the attached wires are always a hindrance during the coil compression under curing. At best a few connections in the magnet heads, where access is somewhat easier, are possible. Another method was used to observe quenches in a short HERA prototype magnet (Bonmann et al 1987). Devices that carried needles (as a porcupine does) were inserted into the coil aperture. The needles could be expanded and pierced into the inner coil layer by turning a key from one of the magnet ends. Several porcupines could be inserted. However, it is both difficult and dangerous to insert such a device because insufficient alignment results in shorts between the coil windings. A developing quench is rather violent and produces ultrasonic noise. It was observed that noise due to the flux redistribution (Nomura et al 1980) can be picked up by microphones. This method of monitoring has some advantages because the acoustic emissions are not electromagnetic and are thus immune to electromagnetic noise. It is particularly useful when the magnetic field changes rapidly. Acoustic measurements with several microphones can also reveal the origin and propagation of a quench.

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Figure C3.0.2. Measurement of the voltage drop over consecutive cable sections during a quench as a function of time.

The drawback is, however, that quite often acoustic signals are emitted that do not correspond to a quench or vice versa (Chikaba el al 1990). The most elegant way to observe, measure and trace back quenches has recently been invented by Krzywinski (Leroy et al 1993)] and has since been used at CERN (Siemko et al 1994) and SSCL (to be built at CERN, Geneva, Switzerland and France) (Ogitsu 1994, Ogitsu et al 1993). A moving wire or the current redistribution between strands at the front of an expanding normal zone creates field distortions, which induce signals in pickup coils, properly located inside the free aperture of the quenching magnet. Such antennae can be made insensitive to changes of the main dipole field by quadrupolar or sextupolar coil arrangements. This works very much in the same way as in the measurement of the magnetic multipole components. The twin aperture magnets for the Large Hadron Collider (LHC) (the Superconducting Super Collider Laboratories, Dallas, TX, USA were discontinued) offer the favourable possibility of subtracting signals of corresponding pickup coils in the two apertures. Radial pickup coils are easier to produce to precisely the same dimensions and the results are easier to interpret. To detect quenches and observe the quench propagation it is sufficient to insert four radial coils, rotated by π/2, preferentially all mounted on the same shaft and covering the length of the magnet under investigation. To cancel dipole contributions either a corresponding coil in the second aperture or the properly weighted average of the other three coils can be used. The latter solution, however, is less attractive, because the number of independent signals is reduced. Alternatively, sets of coils can be made that measure quadrupolar and sextupolar field components, both regular and ‘skew’, i.e. shifted by π/4 and π/6 respectively. For quadrupole magnets, of course, sextupolar and octupolar pickups are necessary. The starting point (radius and azimuth), the direction and the change of the magnetic strength characterize the transverse motion of a magnetic moment. Four different coils are sufficient to measure this. At SSCL this technique allowed the quench origin to be located at the inner edge of a particular winding turn. The longitudinal position of the quench origin can be deduced from the development of the signal with time. To achieve this at least two, preferably many, sets of pickup coils are stacked along the length of the magnet. Depending on the circumstances, many identical sets of coils can be mounted on a common shaft or the sets are positioned individually, employing the technique that is used to position the devices to measure the field quality. If more than one set of coils detects the quench and if the quench velocity is

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reasonably constant the axial position of the quench origin can be determined to better than 1 cm. To measure the velocity of the quench front two methods can be used. In the simplest case one uses the pickup coils as voltage tap replacements. The distance between the pickup coils is known and times of the arrival of the quench front can be detected. Figure C3.0.3 shows an example from SSCL (Ogitsu 1994). The aim of the experiment was to find the reason for the ramp-rate dependence of the apparent critical current. Hence the various ramp rates given in the figure correspond in reality to different current densities and fields as indicated. Note that the measured velocities are as large as 100 m s-1.

Figure C3.0.3. Measurements of the time of detection of a local field distortion in ‘quench antenna’ coils as a function of the coil position. The four sets of data correspond to quenches occurring at different ramp rates (that is currents and magnetic fields) in the SSC dipole prototype DCA 312. The lines are fits to the data (from Ogitsu 1994).

In the second approach, the magnetic flux in the pickup coil is calculated by integrating the induced voltage. In figure C3.0.4 this is done for four consecutive pickup coils in a 1 m long LHC model magnet (Siemko et al 1994). The time intervals at which the propagating normal zone passes by can easily be measured and compared with the slope. Again about 60 to 80 m s-1 is observed. One may even see that the velocity increases slightly. In figure C3.0.5 (Siemko et al 1994) the signals in three pickup coils at the same longitudinal position are shown for a somewhat longer time. The second bump (or dip respectively) corresponds to the current redistribution in one of the adjacent turns. The quench obviously needs 14 ms to propagate azimuthally by one turn. Unfortunately, a complete and consistent set of quench velocity measurements in a large magnet has not so far been published. The quench antenna method is also useful to study other phenomena. It has been observed that sharp signals are accompanied by mechanical oscillations. The damping of the oscillations depends on the absence or presence of iron. Obviously the potential of the method has not yet been fully exploited. (b) Estimate of the adiabatic quench velocity Measurements on cables showed that the normal zone expands at a constant velocity, except in the very beginning, where the manner in which the quench has been initiated has some influence. In real magnets, however, the magnetic energy has to be dumped and the current has to be decreased in order to protect the coil from melting. The rapidly changing magnetic field causes movements of the coil and eddy current heating. (This may even happen at normal ramp rates, as was demonstrated by the measurements of

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Figure C3.0.4. The development of magnetic flux in a series of ‘quench antenna’ coils as a function of time shows the longitudinal propagation of the quench in an LHC model magnet (from Siemko et al 1994).

Quench propagation and magnet protection

Figure C3.0.5. The superimposed voltages of three ‘quench antenna’ coils of the same longitudinal section show signals when the quench front of one cable passes by and, after 14 ms, when the normal zone has expanded to the adjacent winding.

figure C3.0.3.) The helium may be blasted along the wire, preheating the wire ahead of the front. All these effects are difficult to take into account. Needless to say, approximate explicit formulae can only be calculated under the assumption that all material parameters depend on temperature in very simple ways. Often some averaged values have to be assumed. Even the definition of the edge of the normal zone is somewhat arbitrary because the current-sharing region dominates the front of the expanding normal zone and the temperature profile changes with time. To obtain an estimate of the quench-zone expansion it is tempting to apply the same technique as used in deriving the minimum propagating zone and the temperature profile. This leads, however, to integrals that, to the author’s knowledge, do not have a closed-form representation. We will therefore start the analysis at a time at which the quench is already well in progress and approximate the normalzone ellipsoid by a cylinder. The quench propagation velocity is calculated under this assumption in section B3.1. For our purpose of illustration and for a rough guess the following simplifications will be sufficient: (i) there is no coolant at the conductor (adiabatic limit); (ii) heat conductivity κ, resistivity ρ and the heat capacity C do not depend on temperature; (iii) all heat is created at the current-sharing temperature Tr . (Any temperature Ts between Tr and Tc seems a reasonable choice.) This yields the adiabatic quench velocity (see equation (B3.1.43))

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where all constants are averaged over the conductor. Using the Wiedemann-Franz-Lorentz law, which relates electrical and thermal conductivity, equation (C3.0.5) can be written as (L0 = 2.45 × 10− 8 W Ω K-2 )

(c) Transverse quench propagation

In the calculations for the longitudinal quench velocity the cable was always imagined as a homogenous piece of metal with direction-dependent heat conductivity. The longitudinal heat conductivity is high, while the radial conductivity is low and not well known. The superconducting material, the ‘dirty’ copper in the centres of the strands, the poor contact between the strands and finally the helium in the voids contribute to the lateral conductivity in a complicated manner. Nonetheless, one can schematically write the radial quench velocity inside the cable as

Once the normal zone has reached the cable insulation the expansion stops. The warm cable has to heat up both layers of insulation between adjacent cables. Eventually the insulation reaches the critical or current-sharing temperature on the surface of the adjacent cable. Now the quench can also proceed in this cable. The transport of heat energy into the insulation layer takes time. Meanwhile the temperature in the first cable winding will continue to grow. As will be shown in section C3.0.3.2 the temperature increases roughly as J 4t 2, or, for short times, approximately like (1 – cos(Javg2 t/constant)). The temperature gradient in the insulation exceeds by far the longitudinal gradient, if the normal zone is much longer than the cable’s lateral dimensions. That is usually the case. Thus, the two adjacent cables are at two distinct temperatures and heat flows solely through the insulation from the hot to the cold cable. This corresponds to the heat flow in an isolated rod, albeit a very short and very broad ‘rod’. If one end of such a rod of length l is kept at a harmonically oscillating temperature of some amplitude the temperature at the other end will follow with some delay, as can be found in textbooks on mathematical physics. In our case the delay is proportional to (l /Javg)pC/2κ⊥ and the time lag is inversely proportional to the current density. It grows with the square root of the heat capacity. One may of course ignore the details and describe the delay globally as a reduced velocity. This makes sense because the heat capacities of epoxy insulator and copper do not differ very much at low temperatures. Therefore one may define an effective transverse velocity vr ∝ J p2K . One may indeed write vr vz =pk k , because the longitudinal quench velocity is approximately proportional to the current density. The remaining deviations can undoubtedly be buried in our insufficient knowledge of the radial heat conductivity. ⊥

z

C3.0.3.2 Hot-spot temperature The temperature in the quenching region varies from the bath temperature up to a maximum temperature. The cable reaches the highest temperature where most energy has been deposited, which is found at the point where the quench was initiated. (We disregard the special case of inhomogenous matrix resistivity along the wire. Highly resistive sections of the cables or a large magnetoresistance may alter the picture.) The hottest spot is also the spot most in danger and hence one has to be concerned mostly with this peak temperature. As a simplification local adiabaticity is assumed, because it is always a conservative assumption. Furthermore a quench lasts only about one second which is an order of magnitude less than large-scale heat exchange in a cryostat. Under this assumption the locally produced heat results in a local temperature rise

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All quantities are averaged over the winding cross-section, including insulation and helium in the voids. Equation (C3.0.8) reads after rearrangement and integration

The rearrangement implies that ρ(T) does not depend explicitly on the time. This is not exactly true for copper-stabilized cable. During a magnet discharge the current density and hence the magnetic field changes. Because of the magnetoresistance the resistivity depends on the current density. An effective resistivity should be used as an approximation

For a given coil the function F(TH ) can be used to estimate the maximum temperature TH . F(TH ) depends only on known material constants and can be easily calculated if the magnetoresistance is ignored. The integral over the squared current density is easily measured in the case of a single magnet. In fact, often only the integral over the square of the current is calculated and quoted in units of 106 A2 s, sometimes called MIITS which stands for Mega I*I*t, invented at the Teratron, Fermi National Laboratory (FNAL), Batavia, IL, USA. A direct measurement of the temperature is cumbersome and requires many temperature sensors in the coil. Alternatively, one can determine the average temperature between two voltage taps from the resistance, that is from the voltage drop. Figure C3.0.6 shows such a set of measurements (Bonmann et al 1987) plotted against the integral over the squared current density. To relate that to currents, one has to know that the cable area was A = 1.32 x 10-5 m-2. The fully drawn curve shows TH (F), which is the inverse of the function F(TH ) calculated for a HERA-type coil. The equivalent curve for a pure copper coil (ρ4.2K = 10-10 ω m) is drawn as a broken line. The case of pure Nb-Ti cannot be illustrated in the same figure. The curve would essentially coincide with the vertical axis because the large resistivity results in a dramatic heating. Despite the fact that both C and ρ are complicated functions of the temperature, the function TH (F) can usually be approximated fairly well as a parabola with some offset. In the case of a HERA-type cable one finds

For pure copper the relation is

C3.0.3.3 Resistance The growth of the resistance during a quench depends of many details. Even disregarding the temperature dependence of material properties the calculation has to be based on the limited knowledge of the quench

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Figure C3.0.6. Measurement of the hot-spot temperature in a short test dipole for HERA as a function of ∫ J2 dt. The full curve represents a calculation, the broken curve applies for pure copper (ρ4.2K = 10-10 Ω m).

propagation. The section on stability demonstrates how sensitively the result depends on the assumptions. Hence it might seem fruitless to try to describe the quench resistance RQ analytically and one is tempted to rely more on numerical methods (see below). However, Wilson (1983) has demonstrated that an analytic ‘solution’ may help to get some physical feeling and thereby serve the purpose of a guideline. Wilson made the very simplifying assumptions that (i) (ii) (iii) (iv)

the magnet is short circuited, hence all stored energy is dissipated in the coil; the temperature rises as described by the simplified formula TH (F) ∝ F2 (see equation (C3.0.10)); the resistivity rises linearly with temperature; the current density stays constant until all stored magnetic energy is dissipated.

In particular, the last condition is never fulfilled. However, in all practical cases the current stays almost constant for some time and drops thereafter very rapidly to zero. The quench develops as a growing ellipsoid. The temperature on the outer surface of the ellipsoid is the generating temperature Tr . Inside, the temperature rises to the hot-spot temperature in the centre and concentric ellipsoids are isothermal surfaces. The length of the quenching zone at time t is 2vt and the width is 2αWυt. The resistance is

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calculated by adding the contributions of the ellipsoidal shells. Thanks to the special assumption about the temperature dependence of the resistivity the integral can be solved easily. Once the resistance is known the characteristic time tQ for the magnetic energy to be dissipated can be found and the hot-spot temperature, the current decay and the voltage can be estimated. In reality, however, the quench may reach boundaries and it cannot propagate further. Only the temperature will continue to rise as long as magnetic energy is dissipated. The quench may reach one boundary (edge of the coil), two boundaries (filling the area of the winding), or even three boundaries once the quench has propagated along the cable until it hits itself again. Table C3.0.1 tries to summarize the relevant formulae. The constant A describes the wire or cable cross-section. The ratio of the transverse and the longitudinal quench velocity is written as αW. I0 and J0 are current or current density respectively at the start of the quench. The current density is averaged over the cable cross-section. The resistivity ρ0 and the inverse of the hot-spot integral (equation (C3.0.9)) F0 are taken at a suitable intermediate temperature Θ0. A convenient choice would be 100 K or 150 K. The quench velocity v is taken in the adiabatic limit, as mentioned above. Table C3.0.1. Approximate quence development (overview).

C3.0.3.4 Voltage In table C3.0.1 the voltage is calculated disregarding the mutual inductance. Referring to figure C3.0.7 the voltage can be written as UQ = IRQ − (LQ + M )(dI/dt ) and L(dI/dt ) = IRQ , because the power supply is assumed to be an ideal voltage source that has been switched to zero. L is the inductance of the coil, M the mutual inductance between the normal-conducting and the superconducting parts of the magnet. RQ denotes the resistance and LQ the inductance of the normal-conducting part. Combining the two equations yields

which is always less than or equal to

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Figure C3.0.7. Equivalent electrical circuit of a quenching magnet and the voltage distribution.

C3.0.3.5 Example Wilson (1983) illustrates the use of the approximations with the example of a medium-sized solenoid build at the Rutherford Laboratory. We refer to this example. However, to get some additional insight let us apply the formulae to a medium-sized accelerator dipole. Let us take a somewhat simplified HERA main dipole. In the first step we recall the geometry. The dipole consists of two layers that approximate the cosine theta current density distribution required for a dipole field (see e.g. Meβ et al 1996, Wilson 1983). Figure C3.0.8 shows a simplified sketch of the coil’s cross-section. The outer winding layer is generally at a very low field and will be disregarded here. The conductor has the shape of a 10 mm high trapezoid with a width of 1.67 mm at the top and 1.28 mm at the base. However, the 24 strands make up for only 13.3 mm2 of the 14.75 mm2, leaving 10% voids. The bath temperature is 4.6 K on average. The ratio of copper to Nb—Ti is 1.8. Secondly we have to calculate the current density and the critical temperature at the operating point. To achieve a field of 5 T a current density of Ja v g = 3.85 x 108 A m- 2 is required. Previously, the critical current density had been measured to be Jc (B = 6.69 T, T = 4.72 K) = 4.72 x 108 A m- 2. To interpolate, Morgan’s formula, as quoted by Devred (1992), can be used.

It follows that Jc (5 T, 4.6 K) = 8.33 × 108 A m- 2. The critical temperature for 5 T can be found using Lubell’s formula (Lubell 1983)

The critical temperature is 7.17 K for 5 T in Nb—Ti.

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Figure C3.0.8. A cross-section of the HERA coil (simplified sketch).

Next, we have to calculate the generating temperature (equation (C3.0.1)) Tr = 6.4 K. Using a longitudinal heat conductivity of κz = 350 W m-1 K-1, a guess for the transverse heat conductivity of κ⊥ = kzαW2, αW = 1/40, and a resistivity of 1.25 × 10−10 Ω m at bath temperature and 5 T, we arrive at a minimum propagating zone of length 2 × 10 mm and width 0.5 mm. To calculate the adiabatic quench velocity we need the heat capacity. The heat capacity of the compound cable can be derived from the equation found in Lubell’s publication (Lubell 1983)

and the adiabatic longitudinal quench velocity amounts to 9 m s-1, much less than the (not adiabatic) values measured for SSC and LHC magnets, but it agrees with measurements on short pieces of HERA cable (Meβ et al 1983). There are no measurements for the HERA magnets at 5 T. At 6 T the velocity is around 50 m s-1. Hence, all values estimated using the adiabatic velocity are likely to be on the safe side. Now we know enough to make use of table C3.0.1. The characteristic time (table C3.0.1) is calculated to be tQ = 0.29 s. However, at a velocity of 9 m s-1 the quench reaches the opposite side of the cable (10 mm) in 43 ms. For the other dimension, where the cables are stacked together, we calculate 7 ms per winding, corresponding to 420 ms for the 64 windings of the inner shell. The 7 ms agree fairly well with measurements on the thicker LHC cable shown in figure C3.0.5. Measurements at HERA indicate a transversal quench velocity of 50 to 80 windings s-1 (at 6 T). Note that a purely longitudinal quench propagation needs 142 s to transverse the full cable length of about 64 × 20 m. We conclude that the quench will hit a radial boundary. Hence the ‘one-boundary’ column has to be used. Let us now assume that the quench starts at the upper, inner corner of the coil package where the field is highest. This is the most likely spot for a quench and the most dangerous because the quench can propagate only longitudinally in both directions. In the radial direction the quench reaches the other side of the cable after 43 ms and in the azimuthal direction it can develop only downwards, as indicated in figure C3.0.8. Obviously, the quench does not develop like an expanding ellipsoid but rather like a quarter

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of it. Hence the resistance will be only a quarter of the number following the calculations in table C3.0.1. Likewise the real characteristic time, ignoring the radial stop after 42 ms, will be affected (RQ c = RQ1/4, tQ = tQ6p4. The characteristic time, i.e. the time needed to absorb the stored energy, is now td = 0.55 s and c the maximum temperature is Tm 1 = 290 K, which still sounds acceptable. However, the conclusion depends critically on the least known parameter α and on the velocity. In this rough guess the velocity was assumed to be constant. In reality the magnetic field changes from winding to winding. Moreover, inserts, made of copper or G10 to shape the current density distribution, present obstacles for the azimuthal quench propagation not covered by the simplifying formulae. In conclusion, numerical calculations, based on measurements, will be necessary for a serious design. In this particular case Rodriguez-Mateos calculates values around 230 K (private communication 1997) using the QUABER program (Hagedorn et al 1991). Crude measurements indicated temperatures around 250 K, in rough agreement with early QUENCH calculations (see below). C3.0.3.6 Numerical calculations The complete heat equation, as quoted in section B3.3, cannot be integrated analytically even if all material properties are assumed to be constant. Also approximate solutions fail if external actions, like firing of heaters or bypass thyristors (diodes) to control the hot-spot temperature, are to be taken into account. A series of numerical programs has been written to simulate quenches on the computer. The first program published was ‘QUENCH’ (Wilson 1968). Starting with a current I0 at the time t0 the quench velocity is calculated using the calculated magnetic field and the corresponding critical temperature to determine the material properties. Assuming a constant expansion speed the volume V1 at the time t1 is calculated. Now the average temperature T1 in the normal-conducting volume V1 and the current decay, if applicable, are determined. In the next step the material properties are recalculated using the new temperature, a new normal-conducting layer is added, and the temperatures in the inner layers are updated. Each layer keeps its own record of temperature history. Magnetic field and temperature distribution determine the total coil resistance and hence the resistive voltage drop at any time interval. From the external protection resistors or diodes, the inductance and the coil resistance the coil current can be calculated which is then used for the next time interval. At this point also inductive couplings and other complications can be taken into account. For the transverse expansion either of the two approximations mentioned above can be used. The simple Euler algorithm with variable material constants converges acceptably provided the time steps are small enough. In a similar fashion Koepke predicted successfully the behaviour of the Tevatron magnets (Koepke 1980). These programs as well as the adopted version of QUENCH for HERA (Otterpohl 1984) or QUENCH-M (Tominaka et al 1992) are bulky Fortran programs specially tuned for a particular magnet type and protection circuit. The ideas led Pissanetzky and Latypor (Pissanetzky et al 1994) to a modern version, applicable to magnets with a single or with multiple coils, with or without iron, operating in the persistent mode or from external power. The method assumes again that a quench starts at an arbitrary point of the coil and propagates in three dimensions. Multiple independent fronts can coexist. Local magnetic fields and inductive couplings of the coils are calculated by the finite-element method. All properties, like fields and temperatures, are obtained by solving the corresponding equations at each point in space and time. It is worth noting that the authors also face problems in describing the current sharing. For the LHC magnets Hagedorn and Rodriguez-Mateos designed a different generally applicable and versatile simulation package. The general simulation tool, called QUABER (Hagedorn et al 1991), is based on a professional tool, called SABER (trademark of Analogy Inc.). Bottura and Zienkiewicz (Bottura et al 1992a, 1992b) developed a finite-element program for magnets with ‘cable-in-conduit’, i.e. magnets with a forced helium flow (see chapter C4). All programs mentioned above and many variants of them

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are able to describe the ‘typical’ quench fairly well. However the result of the model calculation depends essentially on the assumptions. It is advised that as many different input assumptions as reasonable are investigated to determine the uncertainty in the calculations. Good engineering practice would call for 100 K as the upper limit, because the thermal expansion and the mechanical stresses in the coil and support structure start to increase above this temperature. Common practice is, however, to go far beyond this point in order to save coil volume, conductor and hence cost. It is actually surprising how high a temperature a coil can sustain. Experiments at the Brookhaven National Laboratory (BNL, Brookhaven, NY, USA) showed that coils tolerated theoretical hot-spot temperatures of 800 K. However, 450 K, well below the melting temperature of solder joints, seems to be a safer choice. In the HERA coil at 5000 A, 450 K would be reached after 0.75 s. C3.0.4 Quench detection and external safety circuits C3.0.4.1 Quench detection If the magnet is connected to an external power source the minimum response to a quench necessary to prevent conductor burnout must be to shut down or to bypass the power supply. This requires of course that a quench has been detected. Several signals can be used to sense a quench. Acoustic emissions precede a quench and follow it. This signal, however, is not very specific because noise emission is not always accompanied by a quench. A resistive voltage UQ = RQ I builds up when a normal zone grows and expands, as explained above. The rising resistance leads also to a change in current which in turn induces an inductive voltage. In the same fashion an inductive voltage arises when the current changes for other reasons or whenever the coil is magnetically coupled to a coil with changing current (Hilal et al 1994). Somehow the inductive voltages have to be cancelled. In the simple case of a single coil, as indicated in figure C3.0.9(a), a single bridge circuit will be a reliable solution. For this purpose a centre tap on the magnet coil is needed and the bridge has to be balanced to better than about 0.5%. Once set properly, which may be tedious, the bridges can stay unchanged for years as experience shows. Of course, this method can never detect a quench that develops in both half-coils identically. Some additional measures exclude this rather exotic case. In a large system the bridge method can be repeated for groups of magnets, for example.

Figure C3.0.9. Quench detection by (a) measuring the current through a bridge or (b) comparing the total voltage with the inductive voltage.

Instead of subtracting the inductive voltage by a bridge directly, one can also measure it by some additional device and subtract it from the coil voltage electronically. Figure C3.0.9(b) indicates as an example the measurement with an additional pickup coil. Alternatively the average voltage of a large set of identical magnets in series can serve as a measurement of the inductive voltage. In either case, problems may arise with the dynamic range, the initial adjustment and eventual drifts with temperature.

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Magnetically coupled coils, such as stacked correction coils, the windings in fusion reactor magnets or simply subdivided coils (see below), present a particular problem. Current changes in any of the coupled coils induce voltages in all other coils too. Hence the subtraction method has to be expanded to all combinations of all coils. This procedure is somewhat akin to the Gaussian elimination in matrix diagonalization (Halil et al 1994). During a quench the magnet coils float at an unpredictable potential with respect to ground. The measurement technique has to take care not to destroy the magnets by more than one unintended short to ground. Thus it is advisable to add a resistor in series with the potential tap as close to the coil as practical in order to limit the possible current to ground and hence the damage. However, the series resistors, the cable capacity and the input impedance of the amplifier act as a low-pass filter. The signal distortion can in principle be corrected for, if it is measured once. However, this requires a precise measurement and introduces a delay for the signal reconstruction. If one applies the bridge detection method high-valued series resistors will decrease the sensitivity. In this case a protection with high-voltage fuses is possible. In fact the simplest high-voltage fuse for this purpose is a piece of wire-wrap wire, as used in electronics, at a sufficient distance from conducting material. Note that the fuses have to be tested continuously as parts of the quench detection circuit. Figure C3.0.10(a) shows the quench detection system of the Tevatron, the first large superconducting accelerator located at the Fermi National Accelerator Laboratory (FNAL), Batavia, IL, USA. It is based on the measurement of voltage differences. Average voltage differences are calculated, including the inductive voltages during ramps, and compared with the measured values. A significant discrepancy indicates a quench. The large values for the resistors, chosen for safety reasons, together with the cable capacity introduce a sizable signal distortion that has to be corrected. The system developed for HERA (figure C3.0.10(b)) is based on bridge circuits for each magnet. Additional bridges over groups of magnets increase the redundancy. A radiation-resistant magnetic isolation amplifier detects the bridge current. It is insensitive to noise pickup. All these bridges had to be adjusted by hand, applying a 30 Hz a.c. current to the magnets. Nowadays, of course, automatic procedures would be considered. An interesting solution for a simultaneous bridge and current difference measurement has recently been proposed for the UNK project (Bolotin et al 1992). The UNK (accelerating storage complex) project was under construction in 1997 at the Institute for High Energy Physics in Serpukhov, Russia. The current through the centre of the bridge is amplified with a radiation-insensitive magnetic amplifier that could be embedded in the cryostat, reducing the danger of shorts to ground. By its nature the magnetic amplifier decouples the potentials. The detection system enables a certain threshold to be set and the bridge current to be recorded at a distance of up to 1 km, outside the hostile accelerator environment. A bridge circuit is also under discussion for the LHC. However, isolation amplifiers with semiconductors are at present favoured. C3.0.4.2 Quench protection Most of the ideas about how to ‘protect’ superconducting coils have been well known for many years (Smith 1963). The hot-spot temperature and the maximum voltage must both be minimized to a tolerable level. Energy extraction, energy spreading or problem isolation by subdivision are the keywords. We will discuss these topics one by one, but it must be clear that a proper combination will always achieve the best result. At the end of this chapter we will apply the methods to a few typical types of superconducting magnet and collect the results in a table. (a) Protection by internal energy absorption

Some magnets are disconnected from external power supplies during normal operation. Internally, inside the cryostat, the electrical circuit is closed such that the current can flow undisturbed for long periods.

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Figure C3.0.10. Quench detection circuits used in large systems: (a) Tevatron; (b) HERA.

In these cases the stored magnetic energy has to be absorbed internally. The rough formulae given in table C3.0.1 may help to decide whether the natural quench propagation is sufficient to keep the temperature and the voltage low enough. If this seems not to be the case it is advisable to consider a subdivision of the coil, inductive coupling, active quench spreading by heaters or any combination thereof as discussed below. (b) Protection by external dump resistors

If the magnet has power connections towards the warm environment there exists the possibility to extract at least a fraction of the stored energy into an external dump. This device is not necessarily a pure resistor nor does it need to be outside the cryostat. For example, it could also be a diode instead of the superconducting short mentioned in the previous paragraph. Rephasing the power supply to pump the energy back into the mains is possible but usually too slow. For a single magnet, one can switch off the power supply and dissipate a large fraction of the stored energy by one of the circuits sketched in figure C3.0.11. In circuit (a) in figure C3.0.11 the current continues to flow through a diode (‘free wheel diode’) and

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Figure C3.0.11. Possible energy extraction schemes for a single magnet: (a) external resistor; (b) internal diode and resistor plus external resistor; (c) magnetically coupled resistor.

a load resistor Re . In this case the power source is symbolized as an ideal current source that can be set to 0 A. The high internal resistance is bypassed with a diode. The load resistor determines the equivalent decay time τ = L(Re + RQ )-1 and hence the hot-spot temperature even if Re is not very large compared with RQ at t = tQ . The reason is that Re is always large compared with RQ at the beginning of the quench. Hence some energy is already extracted at a very early time. The choice of the external resistor depends mainly on the quality of the electrical insulation of the magnet. The possibility of thermal damage inside the coil can be weighed against the danger due to overvoltage. The adopted solution will depend on the kind of electrical insulation. A large external resistor seems adequate for a vacuum-impregnated coil that contains practically no coolant and has a reasonable insulation. The external resistor must presumably be smaller and the energy extraction is less effective for Rutherford-type cables. In circuit (b) in figure C3.0.11 an extra diode has been added. As an example it has been installed inside the cryostat as a so-called ‘cold diode’. In the case of a quench and after the current source has been switched to zero the current will commute into the diode branch because the diode knee voltage is reached instantaneously. This is a few tenths of a volt at room temperature and a few volts at 4.2 K. In the worst case the diode will melt, but not the magnet. Of course, the clamping force that connects the diode with its cooling blocks must assure that a melting diode presents a short circuit. (c) Protection by inductive coupling Circuit (c) in figure C3.0.11 contains, in addition to the energy extraction circuit, an inductively coupled resistor R’s. This can be the support cylinder of a large solenoid or some other structural element. A

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change of the primary current couples magnetically to the secondary, heating up both Rs and R’s. The heat can be used to accelerate the quench propagation. This solution, termed ‘quench-back’, is preferred for slowly ramped magnets where the heat produced during charging of the magnet can be tolerated. The eddy current heating can be very annoying for fast-ramping superconducting devices. In fact, it is present to some extent in any case. The changing magnetic flux will always couple to the metallic structure but in particular to the matrix material in the compound cable and produce some heat. Depending on the interstrand insulation this can be a nuisance (Ogitsu 1994). The HERA magnets quench back if the magnetic field change exceeds about 0.3 T s-1. (d) Protection by subdivision

Often, the coil consists of several parts which are wound separately. The parts may be coupled magnetically. In this case it seems natural to subdivide also the quench protection. Figure C3.0.12(a) sketches the situation for a short-circuited magnet divided into n sections. It might have been charged up by a flux pump and now a current I is flowing through the coil and back through the superconducting bypass. Let us discuss the case of negligible magnetic coupling first. The reduced inductance L1 = L/n has to be used to calculate the characteristic time tQ . This shortens the characteristic time to tQS = tQ/6pn and lowers the hot-spot temperature by 1/3pn. A simple division into two parts decreases the hot-spot temperature by 20%. The effect is much larger if the coils are magnetically coupled. In order to gain some insight let us assume that all resistors Ri , and all inductors Li , respectively, in figure C3.0.12(a) are equal. Let us further assume that the mutual inductance between two subcoils can be written as Mij = kp LiLj.. If now subcoil 1 develops a quench with resistance RQ (t), symmetry allows us to combine all other coils to give the equivalent circuit shown in figure C3.0.12(b) with Rs = (n — 1)R1 , Ls = (n – 1)(1 + k(n – 2))L1,

Figure C3.0.12. Quench protection of a subdivided coil: (a) with n subdivisions; (b) equivalent circuit for (n – 1) equal subcircuits.

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M = (n - 1)k L1 and L = L1(1 +(n-1)k)n. Kirchoff ’s rule yields the set of equations.

Let us discuss the simple case of n = 2, R1 =0 first, because the differential equation, which results from the three equations above

shows explicitly that only the stray inductance enters the equation for the characteristic time. In the idealized case the magnetic energy, stored in subcoil 1, is redistributed into the other subcoil. The relevant time is given, as in a transformer, by the stray inductance and by the resistance. Of course, the current density in the second half-coil will now, in general, exceed the critical current density and the coil will quench gently everywhere provided the coupling is strong enough. For vanishing values of the resistors and an arbitrary number of subdivisions we arrive at the equation

which can be solved for the approximate development of the quench resistance given in table C3.0.1. The result is

A subdivision into eight subcoils with couplings of 90% will reduce the characteristic time by 55% and the hot-spot temperature by 80%. Incidentally, the equation for the current describes almost a step function as we required for the calculation of the quench resistance. Of course, it is unrealistic to operate a coil with many shorts. Even resistors would hardly allow a change of field. Consequently in a real application the subdivision of a set of coupled coils requires an external switch and resistor. The internal (or external) bridges over the subcoils can be diodes (or thyristors). Single diodes allow unipolar operation only. If bipolar operation is required sets of antiparallel diodes (or thyristors) can be employed. The subdivision of a magnetic coupled coil seems a very useful tool to control hot-spot temperatures. However, care must be taken not to couple different current circuits magnetically. Otherwise the result could be a coupled quench or persistent eddy currents, which alter the field quite considerably. (e) Summary on the protection of a single magnet In summary, a number of reliable methods have been developed to protect a single magnet after a quench. The quench signal has to be detected and discriminated from noise signals. The power supply has to be switched off without interrupting the magnet current. The stored energy has to be dissipated in suitable devices. If necessary, the quench can be spread artificially by activating heaters in or at the windings (see section C3.0.4.3(b)). Which combination of quench-back, subdivision and energy extraction is best applicable depends on other boundary conditions, in particular on the general type of coil. In table C3.0.2 the advantages and disadvantages are summarized for a few typical types of superconducting magnet. Undoubtedly other considerations will have to be taken in account in addition.

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Quench propagation and magnet protection Table C3.0.2. Options for the quench protection of a single magnet.

C3.0.4.3 Protection of a string of magnets An accelerator consists of large number of magnets in series. A fusion reactor magnet, even worse, consists of a large number of magnetically coupled coils. The protection of such a string or group of coils is a challenge. For example, the inductance in the HERA ring adds up to L = 26.5 H. At 5.5 T, 470 MJ are stored in the ring; an energy sufficient to melt 780 kg of copper. Unfortunately, a simple switch, as in the case of one magnet, cannot work. A magnet is barely able to absorb its own stored energy without active quench spreading. Simply switching off dumps almost all stored energy into the quenching magnet and destroys it. On the other hand, energy extraction with external resistors would require an enormous resistance and hence a voltage of more than 300 kV. The recipe is therefore a combination of the known methods with one addition (i) (ii) (iii) (iv)

detect the quench; isolate the quenching magnet; spread the energy; subdivide the inductance (if possible).

The principle of quench detection has been discussed above. (a) Energy bypass The energy of the unquenched magnets has to be kept away from the quenching magnet. Basically guiding the main current around the magnet achieves this effect. Figure C3.0.13 shows an equivalent circuit diagram. Note that now the protection diode or thyristor has the same polarity as the magnet

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Figure C3.0.13. Equivalent circuit of the bypass of quenching magnet.

during charge-up. Its purpose is to feed the current forward and not back as with a single coil. The total inductance L of the magnet string is much larger than the inductance L1 of a single magnet. Hence the main current I decays with a much larger time constant than the current IQ in the quenching magnet. The differential equation for IQ is (neglecting diode voltage drops)

To minimize the current remaining in the coil, the resistor Rb , in the bypass line should be made as small as possible. Since RQ(t) grows with time an analytic solution is not available. But once the whole coil has become normal one arrives at a steady-state solution

Two basic solutions exist. Thyristors act as fast switches in fast-ramping machines like the FNAL Tevatron or UNK. Figure C3.0.14 shows part of the electrical circuit for the Tevatron. In the Tevatron magnets the return bus is an integral part of the coil and has to be protected. Therefore half the magnets are fed by one bus with half a winding powered by the return bus. The interleaving other half of the magnets is connected in the opposite way. Also shown are the heaters that are needed to distribute evenly the stored energy in a magnet group. The energy of the rest of the ring is bypassed by thyristors. They have to be mounted outside the cryostat and therefore current feed-throughs are needed. These require a very careful design since their electrical resistance (which is the main contribution to Rb) should be small. Their thermal resistance, on the other hand, should be large to avoid a heat load on the liquid-helium system. During a quench the safety current leads heat up considerably which means the connection points to the superconductor are also in danger of quenching. In addition a fast recooling time is an important design criterion. The development of high-temperature superconductor current feed-throughs may alter the situation considerably. Diodes can replace thyristors in storage rings that have a low ramp rate and hence small inductive voltages during normal operation. Cottingham (1971) working at BNL on the discontinued superconducting proton storage ring project, Isabelle, first proposed to mount diodes inside the liquid-helium cryostat. This solution has several advantages. The bypass resistance is normally much smaller than with external safety leads. Each magnet can have its own bypass diode, a fact that effectively improves the subdivision. There is no permanent heat load on the cryogenic system due to the safety leads. Finally, the cryostats are easier

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Figure C3.0.14. The magnet connections in the Tevatron. Dipoles and quadrupoles share two buses. Groups of magnets are protected against the total stored energy by bypass thyristors.

and cheaper to build if the current feed-throughs are missing. This concept has since successfully been adopted or is proposed for HERA, RHIC and LHC. Figure C3.0.15 shows the voltages over a quenching HERA dipole protected by one cold diode per half-coil. In the beginning, both half-coils develop a resistive (positive) voltage. However, the quench expands asymmetrically: only in one half-coil does the voltage stays positive. The current starts to change,

Figure C3.0.15. Voltages over the half-coils of a quenching HERA dipole with two protection diodes.

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and the mutual inductance drives the voltage in the other coil negative. When the protection switch is opened and hence the external current starts to change an additional negative voltage is added. The resistive voltage in the quenching half-coil overcomes the inductive voltage very quickly and drives the diode into conduction. At 4 K the required forward voltage is around 4 V for this type of diode. However, the current through the diode warms it up rapidly thereby lowering the forward voltage drop. The voltage drop rises after 9 s because the current has almost vanished and the helium has cooled the diode below 30 K again. The subdivision of the inductance of one HERA coil with two diodes reduces the maximum voltage and the hot-spot temperature simultaneously. This has been explained in section C3.0.4.2(d). Note that the magnet is part of a large system. The surplus energy that was stored in the mutual inductance when the quench started can then be quickly and easily absorbed by the other magnets by a small increase in the total current I. The bypass diode has to be selected carefully. Firstly, the diode should have a low dynamic resistance and it should not change as a result of aging or neutron bombardment. Secondly, the proper reverse voltage has to be selected carefully. High-voltage diodes have a thick p—n junction that is susceptible to radiation damage and which has a large forward resistance that heats the conducting diode. On the other hand, not all magnets in a string will quench simultaneously. The inductive voltage that develops during the ramp down of the external current will concentrate on the still superconducting magnets and may exceed the reverse voltage of the diode. Note that the reverse voltage depends on the operating temperature. The extremely high current in the LHC magnets and the expected level of radiation pose severe constraints on the cold diodes. Nevertheless promising solutions have been found (Hagedorn and Nagele 1991). A stack of four protects a group of four double dipoles of the LHC. Alternatively, a solution with two diodes per double magnet has been proposed recently, which needs only two connections to the power bus. The dipoles have to be heated to distribute the energy evenly.

Figure C3.0.16. The HERA magnet connection scheme.

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Figure C3.0.17. The proposed connection scheme for one of the RHIC magnet rings. The second ring is almost identical to the one shown.

(b) Heaters

Relatively conservative magnets, such as the HERA and RHIC magnets, do not need artificial quench spreading, in principle. The energy density is sufficiently low, in particular if every magnet has its own bypass. Quench heaters need some energy storage, some firing electronics and feed-throughs into the cryostat. The heater band has to be in close thermal contact with the coil, because the heat must reach the coil as fast as possible. In fact, the best place is between the two coil layers (Ganetis and Stevens 1984). This is of course hazardous. Good heat conduction means little electrical insulation and hence the risk of shorts to the coil. In summary, quench heaters are costly and a potential danger themselves. However, for safety against quenches in the coil heads, even the HERA dipole magnets are equipped with heater strips, very much as in the Tevatron magnets. Artificial quench spreading is essential to avoid excessive energy densities in the LHC magnets. At the Tevatron and at HERA the heaters consist of simple steel strips attached to the outer coil layer. This is not very effective because the field is quite low there (not to

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Figure C3.0.18. The proposed electric circuit of the two magnet rings of UNK.

mention the problems with the burr of the strip). Moreover, the heater bands are warmed up everywhere. This is hardly efficient, because the electric energy for the heaters has to be stored on costly, high-quality capacitors. As shown in section B3.1 a short heat pulse is most efficient. The required energy must therefore be stored on relatively ‘small’ high-voltage capacitors. At HERA and similarly at the Tevatron the stored energy per heater band is around 500 J and the voltage is above 500 V. Smaller capacitors would have been better but would have required even higher voltages. Likewise an increase of stored energy requires higher voltages. In conclusion, the electric energy at the heaters is limited. The new approach at LHC is to employ limited energy at a number of spots by using partially copper-coated steel bands which will reduce the required voltage, reach higher temperatures at those spots and respond faster. (c) Independent current circuits

Finally, it is necessary to subdivide the machine into as many independent current circuits as feasible. This can be achieved in two ways. As shown in figure C3.0.16, at HERA, all magnets are fed by one power supply. This results in good tracking of bending power and focal strength. In total ten mechanical switches break the circuit in the case of a quench into nine pieces separated by resistors. In fact, the resistors are just steel pipes suspended from the ceiling. The resistors are matched to the inductance such that the centres of the resistors and of the nine magnet strings are virtually at ground potential. Hence the ring breaks up virtually into nine independent subcircuits. If a switch fails to open, the symmetry is broken; therefore the installation of an additional equalizing line is necessary. The solution for RHIC, shown in figure C3.0.17, is similar. One twin power supply feeds the two rings. The subdivision follows, of course, the geometry of the tunnel and the circuit breakers consist of thyristors. Very large rings contain so much energy that the virtual subdivision is not safe enough. The LHC

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(Coull et al 1993) will, according to present planning, be divided into eight independent units each of which stores more magnetic energy than HERA. Sixteen circuit breakers and dump resistors, that is two of each for each sector, are required. The fact that each half octant can be switched off independently is clearly an advantage. A further division, which the geography of the Large Electron—Positron (LEP) tunnel at CERN would allow, has been abandoned. Presumably the adopted solution optimizes the costs. The proposed solution at present for the UNK (Afanasiev et al 1992) is akin to the classical FNAL circuit for the Tevatron. Figure C3.0.18 shows the principle of the one double ring of magnets with many circuit breakers and dump resistors. However, 24 power supplies contribute to the voltage needed to ramp the magnets. C3.0.5 Summary Effective quench protection cannot be added afterwards. It is an integral part of the magnet and system design from the very beginning. The costs for cumbersome and complicated protection should be weighed against the costs of structural improvements such as more copper for stabilization, a clever support structure, or better electrical insulation. Moreover aspects of reliability also play an important role. Quenches can and will always happen. Consequently the quench protection has to be reliable and fail-safe. Heaters should be added if necessary, and the overall layout of the power circuit has to be planned and simulated carefully. Magnetic coupling can add to the problems but it can also be turned into an advantage by spreading quenches over large volumes in a short time. Quench protection is basically a combination of measures to assure a low energy density in the case of a quench while keeping voltages low. References Afanasiev O V et al 1992 The protection system for the superconducting electromagnet ring of the UNK Supercollider 4 ed J Nonte (New York: Plenum) p 867 Bolotin I M, Erokhin A N, Enbaev A V, Gridasov I V, Rriyma M V, Sychev V A and Vasiliev L M 1992 The quench detector on magnetic modulator for the UNK quench protection system Supercollider 4 ed J Nonte (New York: Plenum) Bonmann D, Meβ K H, Otterpohl U, Schmueser P and Schweiger M 1987 Investigations on Heater Induced Quenches in a Superconducting Test Dipole Coil for the HERA Proton Accelerator (DESY-HERA) (Hamburg: DESY) p 87-13 Bottura L and Zienkiewicz O C 1992a Quench analysis of large superconducting magnets. Part I Cryogenics 32 659 Bottura L and Zienkiewicz O C 1992b Quench analysis of large superconducting magnets. Part II Cryogenics 32 719 Chikaba J, Irie F, Takeo M, Funski K and Yamafuji K 1990 Relation between instabilities and wire motion in superconducting magnets Cryogenics 30 649 Cottingham J G 1971 Magnet fault protection Internal Report BNL-16816, BNL, Brookhaven, NY, USA Coull L, Hagedorn D, Remondino V and Rodriguez-Mateos F 1993 LHC magnet quench protection system 13th Int. Conf. on Magnet Technology (MT13) (Victoria, 1993) LHC Note 251 Devred A 1992 Quench Origins, The Physics of Particle Accelerators vol 2, ed M Month et al (New York: American Institute of Physics) Dresner L 1995 Stability of Superconductors (New York: Plenum) Ganetis G and Stevens A 1984 Results of quench protection experiment on DM1-031 SSC Technical Note No 12 Brookhaven National Laboratory Hagedorn D and Nägele W 1991 Quench protection diodes for the large hadron collider LHC at CERN Cryogenic Engineering Conf. (Huntsville, AL, 1991) LHC Note 148 Hagedorn D and Rodriguez-Mateos F 1991 Modelling of the quenching process in comp superconducting magnet systems 12th Int. Conf. on Magnet Technology (Leningrad, 1991) LHC Note 159 Hilal M A, Véscey G, Pfotenhauer J M and Kessler F 1994 Quench detection of multiple magnet system IEEE Trans Appl. Supercond. AS-4 10

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Koepke K 1980 TMAX Program (Batavia, IL: Fermilab) Leroy D, Krzywinski J, Remondino V, Walkiers L and Wolf R 1993 Quench observation in LHC superconducting one meter long dipole models by field perturbation measurements IEEE Trans. Appl. Supercond. AS-3 781 Lubell M S 1983 Empirical scaling formulas for critical current and critical field for commercial NbTi IEEE Trans. Magn. MAG-19 754 Maeda H, Tsukamoto O and Iwasa Y 1982a The mechanism of frictional motion and its effects at 4.2 K in superconducting magnet winding models Cryogenics June 287 Maeda H and Iwasa Y 1982b Heat generation from epoxy cracks and bond failures Cryogenics 473 Meβ K H 1996 Quench protection 1995 CERN Accelerator School (CAS), Superconductivity in Particle Accelerators ed S Turner (Geneva: CERN) 96-03 Meβ K H, Otterpohl U, Schneider T and Turowski P 1983 Measurements of the Longitudinal Quench Velocity in the HERA Cable (DESY-HERA) (Hamburg: DESY) p 83-05 Meβ K H, Schmüser P and Wolff S 1996 Superconducting Accelerator Magnets (Singapore: World Scientific) Nomura H, Sinclair M N L and Iwasa Y 1980 Acoustic emission in a composite copper NbTi conductor Cryogenics 283 Ogitsu T 1994 Influence of cable eddy currents on the magnetic field of superconducting particle accelerator magnets Internal Report SSCL-N-848; Thesis Institute of Applied Physics, University of Tsukuba Ogitsu T, Devred A, Kim K, Krzywinski J, Radusewicz P, Schermer R I, Kobayashi T, Tsuchiya K, Muratore J and Wanderer P 1993 Quench antenna for superconducting particle accelerator magnets IEEE Trans. Magn. MAG-30 2773 Otterpohl 1984 Untersuchungen zum Quenchverhalten supraleitender Magnete DESY-HERA 84/05; Diploma Thesis University of Hamburg Pissanetzky S and Latypov D 1994 Full featured implementation of quench simulation in superconducting magnets Cryogenics 34 795 Rodriguez-Mateos 1997 private communication Siemko A, Billan J, Gerin G, Leroy D, Walckiers L and Wolf R 1994 Quench localization in the superconducting model magnets for the LHC by means of pick-up coils IEEE Trans. Magn. AS-5 727 Smith P F 1963 Protection of superconducting coils Rev. Sci. Instrum. 34 368 Tominaka T, Mori K and Maki N 1992 Quench analysis of superconducting magnet systems IEEE Trans. Magn. MAG-28 727 Wilson M N 1968 Computer simulation of the quenching of a superconducting magnet Internal Report RHEL/M 151, Rutherford High Energy Laboratory, UK Wilson M N 1983 Superconducting Magnets (Oxford: Oxford Science—Clarendon)

Further reading Wilson M Superconducting Magnets (Oxford: Oxford Science—Clarendon) (The book is excellently written and provides an outstanding overview along with many detailed calculations.)

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C4 Quench propagation and protection of cable-in-conduit superconductors L Bottura

C4.0.1 Introduction In the general discussion on quench propagation and protection in chapter C3, a brief mention was made of the characteristics of cable-in-conduit conductors (CICCs). Because of their importance for large projects (e.g. thermonuclear fusion and energy storage applications) we give here a more complete overview of the quench behaviour and quench protection of this class of cables, and for the magnets wound using them. As already discussed in section B3.4 and chapter B6 CICCs are cooled by a forced flow of helium, which provides a limited heat sink for the stabilization of the cable. When a quench is originated in a CICC, the helium is heated violently and is blown out of the initial normal zone. This flow of hot helium, in excellent thermal contact with the cable, is the main mechanism that propagates the normal zone in a CICC. Therefore, in a first suggestive approximation, the normal zone grows in CICCs with the expansion of the heated helium mass contained originally in the initial normal zone. This fact makes quench propagation in CICCs peculiar compared with other types of conductors. The aspects of concern in the quench protection design in a CICC are, as for other types of conductor, the maximum—hot spot—temperature, the maximum voltage induced during the quench (and the following magnet discharge or dump) and the propagation speed (for detection purposes). In addition, maximum pressure and helium expulsion must be considered in order to guarantee the structural integrity of the conduit and to size the venting from the coil. In the next section we review jointly the experimental and analytical work on quench propagation in CICCs, starting with helium expulsion, followed by hot-spot temperature and finally covering quench propagation and quench-back. A brief overview of existing models for quench simulation and related issues is reported thereafter. Finally a summary section collects the relevant expressions for the of interest in the design of the quench protection system specific to CICCs. C4.0.2 Quench propagation in CICCs The early experimental and analytical work on quench transients in a CICC was mainly motivated by the activity within the US Large Coil Task (LCT) Program (Beard et al 1988). Three of the six coils that were designed, built and assembled in the test facility at ORNL were based on internally (force-flow) cooled

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cables, and in particular the Westinghouse LCT coil (IEA 1988) was wound with a Nb3Sn-based CICC. As clearly discussed by Dresner (1985), one of the concerns of the designers of large-sized force-cooled coils was the internal pressure rise and helium expulsion in case of quench. Initial experiments and theories were therefore aimed at giving conservative estimates for both these parameters, without putting much emphasis on the quench initiation and propagation. C4.0.2.1 Maximum pressure and helium expulsion Miller et al (1980) produced experimental data on maximum pressure and expulsion from a dummy (copper) cable-in-conduit conductor as a function of the operating current. Maxima of both quench pressure and thermal expulsion are reached when the full length is normal, and the advantage of using a dummy cable in the experiment was that the results did not depend on the actual details of a quench initiation and propagation. The helium pressure in the cable centre increased during the initial phase of the experiment, corresponding to the temperature increase. In this phase the velocity profile was established along the cable length. Mass ejection from the cable ends caused the density to drop until eventually the pressure increase caused by heating was balanced by the flow. The pressure reached a maximum and dropped afterwards. The maximum pressure results for this experiment are summarized in figure C4.0.1 (note that there the cable half-length l = L/2 is reported as a parameter), while expulsion velocity traces are shown in figure C4.0.2.

Figure C4.0.1. Peak pressure in the experiment of Miller et al (1980), plotted as a function of the factor x = Q2(L/2)3/Dh , (see text for details of the symbols). The circled W marks the value of the parameter x for the Westinghouse LCT coil. The lines are the values computed with equation (C4.0.2) using either f = 0.0044 or f = 0.013. Note the weak dependence on f. Reproduced from Dresner (1991b) by permission of Elsevier Science Ltd.

Based on these data, and assuming that a quenching CICC can be modelled as one dimensional (1 D) along its length, Dresner (1981) developed an expression for the maximum pressure in the case where inertia can be neglected. This assumption can be justified in view of the large friction force in CICCs, as discussed in section B3.4. The following implicit expression for the maximum pressure pmax was obtained (see Miller et al (1980) for details on its derivation):

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Figure C4.0.2. Helium expulsion velocity v for the experiment of Miller et al (1980): (a) plotted as a function of time t and (b) product vt 1/3 plotted as a function of the product I 2t of squared operating current and time. The solid line is the value computed with equation (C4.0.3). Reproduced from Dresner (1991b) by permission of Elsevier Science Ltd.

where f is the space- and time-averaged friction factor, Q is the heating rate per unit helium volume (assumed constant), L is the length of the CICC, Dh its hydraulic diameter and p0 the initial pressure. When the maximum pressure is much larger than the initial pressure, equation (C4.0.1) can be further simplifed

The scaling predictions of equation (C4.0.2) have been shown to be in excellent agreement with the experimental data reported in figure C4.0.1. In fact, both the scaling and the quantitative predictions match very well the measurements over a wide range of heating and lengths, as shown in figure C4.0.1. For short times after the beginning of the transient, the helium expulsion velocity v at the ends of a completely normal-conducting cable was also computed by Dresner (1981)

where β is the thermal expansion coefficient of helium in a constant-pressure process, c is the is entropic speed of sound, ρ is the helium density, Cp is the helium specific heat at constant pressure and t is the time elapsed. All helium properties above are intended as evaluated at the initial conditions. Equation (C4.0.3) has been derived neglecting frictional heating (compared with Joule heating) and based on the assumption of constant helium properties. Therefore it can only be regarded as a first-order approximation. Still, for low values of t , the agreement of the predicted expulsion with measured values is satisfactory, as shown in figure C4.0.2 where equation (C4.0.3) is compared with measured data from the experiment of Miller et al (1980).

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C4.0.2.2 Hot-spot temperature In the experiment quoted above (Miller et al 1980) it was also shown that the maximum temperature reached in a CICC during a quench (and following dump) could be estimated adequately using an adiabatic heat balance as for other conductor types (see chapter C3). If we introduce the fractions of the cable components fi referred to the cable space cross-section (i.e. the cross-section enclosed by the conduit) their densities ρi and heat capacities Ci , we can write the adiabatic local heat balance as follows

where we have defined the function γ (T) as

and T is the cable temperature, ρC u is the copper resistivity, JC S is tne cable space current density and the index i runs over all components in the cable ( fC u is in particular the copper fraction). Once the geometry of the cable is given, the function γ only depends on the temperature T. Note incidentally that equation (C4.0.4) gives the local and instantaneous heating rate of the CICC. We have chosen the normalized form of the heat balance equation (C4.0.4) because the actual geometry of the cable (its size) disappears, and universal expressions can be used to design for a given current density. As explained in chapter C3, we can obtain a universal expression integrating equation (C4.0.4) as follows

where the function Γ(T0, Tm a x ) only depends on initial temperature T0 and the maximum allowed temperature Tm a x . The time integral of the current density is generally known for given detection and discharge time constants. Therefore the hot-spot temperature can be calculated from the material integral Γ (see appendix B of this chapter for an example). In fact, for the accurate determination of the hot-spot temperature, we must take into account two characteristics of CICCs. Firstly the helium undergoes a complex thermodynamic process, flowing out of the heated region, and secondly the large heat capacity of the conduit is only in loose thermal contact with the cable. This results in uncertainties in the effective contribution of the heat capacities of both components to the adiabatic balance. Indeed, the actual weight of the contributions can change during the temperature evolution, thus invalidating the assumption that the function γ only depends on T. To avoid complex simulation of the complete process, we prefer an approximate treatment based on a parametric study of the influence of helium heat capacity and of the effective conduit fraction contributing. The effect of helium can be easily bounded by taking the two extreme processes that can take place, i.e. either under constant pressure or constant density conditions. This is an easy exercise that shows that helium has a marginal effect on the heating rate once the temperature is above 20 K, while the effect on the hot-spot integral Γ is at most 10%. The situation with respect to the conduit is different. This is shown in figure C4.0.3, where the functions γ and Γ have been plotted for a typical CICC cable space design (40% void fraction, 60% cable fraction) and different typical ratios of conduit to cable space cross-sections, in the range 0–2. We see there that because the heat capacity of structural materials (e.g. steel) is large, the conduit contribution can be significant to both the local heating rate and the final temperature.

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Figure C4.0.3. Calculation of the hot-spot-related quantities γ (in MA2 s K–1 m– 4 ) (a) and Γ (in MA2 s m– 4) (b) for a typical CICC design and different ratios of the conduit-to-cable space cross-section (indicated as parameter on the curves). A field of 12 T and a residual resistivity ratio (RRR) of 100 have been assumed for the calculation of the copper resistivity.

From a practical point of view upper and lower bounds must be taken for design purposes to explore the sensitivity of the design, and the results are verified against simulations and experiments. Note finally that although the copper resistivity and the material heat capacities are changing by orders of magnitude over the temperature range of interest for hot-spot temperature evolution, the temperature increase rate

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at constant operating current, which is inversely proportional to the quantity γ, only changes by a small factor (five at most) over the full temperature span, so that an easy estimate can be made based on its initial value (computed in the known initial operating conditions). C4.0.2.3 Normal-zone propagation As understanding improved, it became clear that the conditions of a fully normal CICC simulated in the experiment of Miller et al (1980) were pessimistic for the actual operation of a CICC-based coil. In fact the most common case is when a normal zone initiated in a short portion of the coil propagates along the cable length until a significant resistance is detected and the coil is discharged. By this time the normal length is generally only a fraction of the total cable length, and the maximum pressure and helium expulsion will be much smaller than the upper conservative estimates given by equations (C4.0.1) and (C4.0.3). However, in this case, other questions become important, namely, those concerning the quench propagation, speed and the time needed to detect a normal zone and safely discharge the magnet. Interest shifted therefore to the quench propagation mechanism in a CICC. It was clear from the very early studies, and has been proven experimentally on coil samples (Ando et al 1988, 1990, Lue et al 1991), that the quench propagation in a CICC does not scale as in other types of conductor, where conduction along the cable provides the main mechanism for normal-zone growth. As recognized by Dresner (1983), the main quench propagation mechanism in CICC is hot helium expulsion. The helium in the initial normal zone receives Joule heating from the cable, its temperature rises and it expands in the (still) superconducting region driving it into the normal state through convection heat exchange. Dresner postulated that ‘…the velocity of normal zone propagation equals the local velocity of expansion of the helium’ (Dresner 1983). The result of this approximation is that ‘…the normal zone engulfs no new helium, or in other words that the heated helium comprises only the atoms originally present in the initial normal zone. We are thus led to the picture of a bubble of hot helium expanding against confinement by the cold helium on either side of it’ (Dresner 1985). This statement has since been the basis for most of the analytical work on quench propagation in CICCs. The most complete model of a quenching CICC is at present the one developed by Shajii and Freidberg (1994a) who have given approximate expressions for quench propagation speed and pressure increase by neglecting the inertia in the equation of helium motion, taking perfect gas properties for the helium and assuming that the cable has a perfect thermal coupling to the helium (i.e. equal temperature in helium and cable). Finally, they have assumed that the current is constant throughout their analysis. Owing to the fact that friction dominates in the momentum balance for the helium motion, it can be shown that the pressure profile evolves along the cable following a nonlinear diffusion equation, rather than through the more common sound wave propagation (section B3.4). The characteristic diffusion length for the pressure profile in a quench is given by (Shajii et al 1995)

where vq(tM ) is the quench propagation speed—equal to the helium velocity at the front—at a time tM and c0 is the speed of sound at the initial conditions. In general we use the subscript 0 above and in the following to indicate properties computed at the initial conditions. For short times tM we intuitively expect no effect from the values of pressure at the boundary of the cable (the coil manifolds), as the pressure diffusion has not yet reached the cable ends. At long times the diffusion wave reaches the ends and the pressure profile is necessarily influenced by the boundary values. In reality the time elapsed

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during a quench is not an arbitrary quantity, being limited by the time needed to reach the maximum allowed hot-spot temperature Tm a x . In practical applications the time tM will therefore be of the order of the coil discharge (dump) time constant. Taking this value for tM we can distinguish two regimes of quench propagation, namely that of a short coil, for which the total cable length L is much smaller than the diffusion length, and that of a long coil, when the opposite condition is satisfied

Note that the distinction between these two regimes depends on the assumption on the maximum time tM , and that the propagation speed, so far unknown, enters the expression of the characteristic length, equation (C4.0.6). A second distinction identified by Shajii and Freidberg (1994a) depends on the strength of the quench and the corresponding pressure increase. Based on the pressure increase ∆p, so far unknown, and the initial pressure p0, they identified a low- and a high-pressure-rise regime as

In summary, the combination of the two criteria given above leads to the differentiation of four regimes of quench. For each possible combination the following asymptotic expressions for the propagation velocity and pressure increase have been obtained (see Shajii and Freidberg (1994a) for details) short-coil, low-pressure-rise regime

long-coil, low-pressure-rise regime

short-coil, high-pressure-rise regime

long-coil, high-pressure-rise regime

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The meaning of the additional symbols used is the following: R is the perfect gas constant (for helium R = 2077 in SI units), Lq is the initial quenched length and γ0 is the value of the parameter γ at the initial conditions. This value is actually chosen such that the group JCS2 /fCuγ0 gives a good approximation to the temperature growth rate throughout the transient (see equation (C4.0.4) and figure C4.0.3). At this point it is interesting to examine the typical scalings of the different regimes. Firstly we note that, as expected, quench propagation and pressure in a long coil do not depend on the coil length L, while this is the case for a short coil. The low-pressure-rise regime is generally more sensitive to the changes in operating current density and initial quenched length. With respect to current density the propagation scales with power 2, the pressure with power 3–4, while in the high-pressure-rise regime the scaling of both is of the order of 1. Similarly for the scaling with initial quenched length Lq the power is 1 for the propagation speed and of the order of 1.5–2 for the pressure rise, compared with powers of the order of -12 to -13 in the high-pressurerise regime. Finally, in a short coil the propagation speed and pressure rise do not depend on time, while they do so weakly in the long coil. The set of equations above gives an almost complete picture of quench propagation, which only lacks the additional case of thermal hydraulic quench-back (see next section). The only drawback is that the validity conditions on length and pressure increase contain the unknown parameters of quench speed and pressure increase. However, a coherent way of overcoming this problem has been shown by Shajii et al (1995) representing the four regimes of quench in a plot of universal scaling laws. This is obtained by introducing the two following dimensionless variables

where the definitions of the two parameters λ and η are the following

In terms of the two dimensionless quantities l and q the four regimes of quench are then given by short coil, low pressure rise long coil, low pressure rise short coil, high pressure rise long coil, high pressure rise and they can be conveniently reported on a single q—l plot as shown in figure C4.0.4 (note that the expressions defining the boundaries do not join exactly as they are obtained as limiting case approximations). Depending on the cable characteristics, on the operating conditions and the initial quenched length, a quench will be represented by a single point on the graph, thus defining the appropriate expressions

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Figure C4.0.4. A universal quench diagram for the determination of the quench regime. The nondimensional parameters q and l are obtained from the cable geometry, operating point and allowables (see text and appendix C).

holding for the quench propagation and pressure increase. According to the groups in the parameters l and q, we see that an increase of the allowable maximum temperature Tm a x shifts the operation towards the short-coil regime (consistently with the longer time needed to reach the maximum temperature). Similarly a longer cable length L shifts the quench towards the long-coil regime. Finally, a higher operating current density JC S produces a shift towards the high-pressure regime (consistently with the stronger heating). An example of the use of the diagram is given in appendix C. C4.0.2.4 Quench-back In recent experiments (Ando et al 1985, Lue et al 1993), quench propagation in CICCs has been observed to accelerate rapidly from an initial conventional phase, with propagation velocity of the order of 1–10 m s–1, up to velocities exceeding 100 m s–1. This phenomenon was also observed in earlier simulations (Cornellissen and Hoogendoorn 1985, Luongo et al 1989). The reason for the acceleration is the heating of the dense helium column in front of the propagating normal zone through compression and friction work. When the helium temperature reaches the current-sharing limit Tc s , the strands become resistive and suddenly large lengths of conductor transit to the normal state. The propagation speeds up, with an upper limit set only by the sound speed in helium—a thermohydraulic quench-back has taken place. Dresner (1991a) initiated analytical work on this subject, developing expressions for the time of onset of quench-back and the asymptotic time at which the full conductor length becomes normal. A more comprehensive analytical theory of quench-back has been presented by Shajii and Freidberg (1996). The two fundamental parameters of quench-back derived there are the initiation time tq b and its propagation speed vq b . Limiting ourselves to the case of a long coil and small temperature margin Tcs–T0 compared with the initial temperature T0, their values are given by

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where φ introduced here is the Grüneisen parameter of helium, defined as

and generally tabulated with the other helium properties. As discussed above, a small helium temperature increase at the front, compared with the temperature rise during quench, is sufficient to initiate the quenchback. Therefore all helium properties in the two equations above are evaluated at the initial conditions, as indicated by the subscript 0. Note that the initiation time depends strongly on the temperature margin (to the fifth power) and that once a quench-back is initiated it propagates at approximately constant speed (there is no time dependence in equation (C4.0.20). It is possible to translate the condition on the initiation time for which a quench will evolve to a quench-back before the maximum allowed temperature is reached, using the normalized parameters l and q (Shajii et al 1995). For this purpose we must define a new parameter M

and the quench-back conditions are (see Shajii et al (1995) for details on the derivation) long coil, high pressure rise short coil, low pressure rise short coil, high pressure rise The above relations define a lower-bounded region in the q–l plot of figure C4.0.4. An example of their use is given in appendix C. C4.0.2.5 Normal voltage A final parameter of interest for the design of the protection system of a CICC based coil is the voltage in the normal zone. As CICCs are mainly used in large-scale applications, the coil discharge relies on an external resistor which dominates the voltage drop during coil discharge. Therefore the maximum voltage is always attained at the coil terminals and is known from the characteristics of the discharge system. Here the main concern is the detection of the quench through a measurement of the normal voltage in the coil operating in a system that can be pulsed or subject to large electromagnetic perturbation. For this purpose we can give here an estimate of the normal voltage development based on the results of the previous sections. Firstly we recall two features that have been demonstrated in the previous discussion, namely that both the temperature growth rate and the quench velocity are approximately constant or weakly dependent on time. The consequence is that a quench initiating over a length Lq will develop an approximately piecewise linear temperature profile. The temperature is flat in the initial quenched length and linearly decreasing over the remaining length. The total normal length at a time t after the quench initiation will be approximately Lq + 2υq t. The voltage drop V(t) (a function of time) can be found by integrating the electric field along the length of this zone, or

In general the above integral requires detailed knowledge of the temperature as a function of space and time. If we take a linear temperature profile, with maximum temperature Tm in the initial quenched length

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and temperature T0 in the superconducting region, we can approximate the voltage as follows

where, according to equation (C4.0.4), the central temperature Tm is approximately given by

and the function Ω(Tm ) gives a measure of the weight of the new normal length vqt compared with the initial length Lq , taking into account the differences in the cable temperature and its distribution. The definition of Ω and a suitable approximation for copper are

Using expressions (C4.0.23) and (C4.0.24) in equation (C4.0.22) gives an explicit relation for the voltage as a function of time that can be used to estimate the typical initial development of quench. C4.0.3 Numerical simulation It should be clear from the previous sections that initiation and propagation of quench in CICCs is a highly nonlinear problem. The analytical solutions and approximations presented in the previous section are extremely useful for design and scaling purposes, but often numerical solution is the only viable tool for reliable analysis of a large magnetic system. In this spirit, several models have been developed in the past (Arp 1980, Cornellissen et al 1985, Bottura 1996, Bottura and Zienkiewicz 1992, Hoffer 1979, Luongo et al 1989, Marinucci 1983, Shajii and Freidberg 1994b), of which the references quoted are only a sparse sample. The basic model for all of them is represented by the set of equations (B3.4.2)–(B3.4.5) of section B3.4, namely flow equations for helium coupled to 1D conduction along the length of the cable components. Through years of experience, it has become clear that in addition to the general complication of the problem, involving coupled heat transfer and fluid flow in a regime of highly nonlinear material properties, the main issue to be solved to guarantee adequate simulation is proper tracking of the propagating front (Bottura 1995, 1996, Bottura and Shajii 1995). Numerical algorithms suffer from the largest approximation errors in the regions of strong gradients. Such regions are typical of a propagating front in a CICC, characterized by a temperature gradient induced by the localized heating onset in the normal zone and maintained by the poor conduction in the cable along its length—compared with helium convection and cooling. Now, it can be shown that errors generated at the normal front will affect not only the sharpness of the front resolution, but also the speed of the propagation itself, through a numerical, artificial heat flux across the front. This effect, owing to the positive feedback of the normal length on the propagation speed, will tend to be amplified as the simulation proceeds, leading in some extreme cases to wildly overestimated normal zones and pressure increases (Bottura and Shajii 1995). Interestingly enough, as hot-spot temperature is a quantity determined by local heat balance, even a low-accuracy solution of quench propagation will lead to satisfactory approximations for the maximum temperature. Therefore this last is not a suitable parameter to confirm the accuracy of a quench simulation. In summary, much care is needed to confirm a solution, either through a priori accuracy analyses and error estimators or by performing systematic convergence studies, changing the space and time step of the

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solver. As shown by Bottura (1996) and Shajii and Freidberg (1994b), satisfactory results and reasonable computer processing efficiency can be reached through adaptive schemes. This is in fact the basis for most modern simulation codes. C4.0.4 Summary of relevant expressions for design We recall here the major definitions and the basic expressions extracted from the previous discussion. The equations below are direct consequences of the ones already given. In the form given here it is easy to translate them into a design manual for CICCs. C4.0.4.1 Hot-spot temperature (adiabatic heat balance) The hot-spot temperature equation is

where Γ(T0, Tm a x ) must be calculated based on the cable geometry, and τd e t e c t i o n and τd u m p are respectively detection delay and dump time constant (for an exponential current discharge). C4.0.4.2 Maximum pressure The maximum pressure for a completely normal CICC, based on a typical average CICC friction factor of 0.014 is

C4.0.4.3 Helium expulsion velocity The helium expulsion velocity for a completely normal CICC (valid only for early times and based on similarity solutions, neglecting inertia and frictional heating and assuming constant helium properties) C4.0.4.4 Quench propagation regimes Quench propagation regimes on the universal q— l graph of figure C4.0.4 short coil, low pressur rise long coil, low pressure rise short coil, high pressure rise long coil, high pressure rise

where the l and q parameters are defined in equations (C4.0.15) and (C4.0.16). For each propagation regifne the appropriate expressions in the set of equations (C4.0.7)–(C4.0.14) must be used to estimate the propagation velocity and pressure increase.

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C4.0.4.5 Quench-back initiation on universal q—l graph of figure C4.0.4 For quench-back initiation

where M is defined by equation (C4.0.21). Once quench-back is initiated the propagation velocity can be approximated using equation (C4.0.20). As the quench-back propagation is generally much faster than a conventional quench, the full cable can be considered to become normal conducting at once, and the upper estimates for the helium pressure and expulsion velocity in equations (C4.0.26) and (C4.0.27) become valid. C4.0.4.6 Normal voltage A rough estimate of the normal voltage at a time t for a CICC can be obtained from the initial length Lq , the propagation speed vq and the maximum temperature Tm reached at a given time in the cable

The quench propagation velocity is computed as described in the previous two sections. The temperature Tm is approximated by

Appendix A Case study 1—maximum pressure rise during quench We use here the approximate expression of Dresner (1981), equation (C4.0.26), to determine the maximum pressure in one of the experimental runs of Miller et al (1980). The data of the conductor are derived from Miller et al (1980). The cable had a length L of 69 m and a hydraulic diameter Dh of 0.56 mm, with a copper fraction fC u of 0.53, and a helium fraction fH e , of 0.47. The copper resistivity was approximately 2x 10–10 Ω m. We choose a run at 940 A, under a cable space current density JC S of 60.7 A mm–2. We can now compute the heating per unit helium volume as Q = ρC u J C2S /fC u fH e ≈ 2.95 × 106 W m-3. Finally, we insert these data into equation (C4.0.26), which gives a maximum pressure of 43 x 105 Pa. This value is much larger than the initial pressure (that was approximately 5 × 105 Pa), so that the expression should be well within the validity range. As a final check, we compute the parameter x = Q2 (-L2)3/Dh ≈ 6.4 × 1020, and we see from figure C4.0.1 that indeed in the actual experiment a maximum pressure of the order of 40-50 × 105 Pa was obtained (open triangles). Appendix B Case study 2—hot-spot temperature We try to determine the maximum allowable operating current density for a typical CICC for fusion applications. We take in particular a cable of the ‘ITER’ class, with a copper to noncopper ratio of 2, void fraction of 40%, and equal conduit and cable space cross-sections This results in fC u = 0.4, fH e = 0.4, fs = 0.2 (the noncopper fraction). We assume arbitrarily that a quench can be detected within 2 s, and we c take a time constant for the external dump circuit of 25 s. Finally, we set a maximum of 150 K for the hot-spot temperature. The orders of magnitude of these parameters are typical of a large magnetic system,

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such as the toroidal field magnet of a tokamak. Now, from equation (C4.0.25) we can derive a condition on the cable space current density once the hot-spot integral Γ is known. To obtain this value from the plot of figure C4.0.3 we need an additional assumption on the contribution of the conduit to the heat balance. The two extremes are of no contribution, for which we obtain Γ(150 K) ≈ 4.5 × 1016 A2 s m–4, and full contribution, for which we obtain Γ(150 K) ≈ 1 × 1017 A2 s m–4. At the two extremes we finally obtain that the allowed cable space current density is JC S ≈ 35 A mm–2 when we neglect the conduit, or JC S ≈ 52 A mm–2 when we take its contribution into full account. The difference is not dramatic, but worth a deeper investigation through detailed simulation. Finally, we can estimate the temperature growth rate (at constant current density) using equation (C4.0.4). An average value of γ (see figure C4.0.3) is in the range of 5 × 1014 A2 s K–1 m–4. For these values, and with JC S ≈ 52 A mm–2, we can compute that dT/dt ≈ 13 K s–1. Therefore at constant current the limit of 150 K is reached in 11 s, while room temperature is reached in 23 s. Appendix C Case study 3—quench propagation regimes We study here the quench behaviour of the conductor of case study 2, with fC u = 0.4 and fH e = 0.4. We have to compute first the l and q parameters of the possible quench points. We take an initial operating condition at T0 = 5 K and p0 = 6 × 105 Pa. According to the results of case study 2, we choose an operating cable space current density of JC S ≈ 45 A mm–2. We then assume that the coil length L is of the order of 720 m. The additional parameters entering equations (C4.0.15)–(C4.0.18) are: R ≈ 2077 J kg–1 K, ρ0 ≈ 133 kg m– 2, Tm a x = 150 K, c0 ≈ 223 m s–1. For a typical large CICC we can choose Dh ≈ 1 mm and f ≈ 0.014. Finally, we take an average value for the parameter γ0 of 5 × 1014 A2 s K–1 m–4. With this choice, we obtain that

where Lq is the initial quenched length. This finally gives a relation between q and l

Figure C4.0.C1. The quench line on the universal q — l diagram (dot—dashed) and quench-back boundary (dashed).

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References

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which we can plot on the universal quench diagram of figure C4.0.4 as shown in figure C4.0.C1 for a typical range of short (0.1 m) to long (40 m) values of Lq . We see from there that for Lq in the range 0.1–0.5 m the quench is in the low-pressure-rise regime, while it shifts towards the high-pressure-rise regime as the initial normal length is chosen longer than 0.5 m. Using equation (C4.0.7) we can estimate the propagation speed for a short Lq =0.1 m, for which we obtain υq ≈ 0.2 m s–1. At the opposite extreme, for Lq = 40 m, we have using equation (C4.0.13) that vq slowly depends on time according to υq ≈ 10.4t –1/5. After 5 s the quench would have a speed of υq ≈ 7.5 m s–1. We note, however, that the quench will evolve to a quench-back before the maximum temperature is reached for q > 10.7, i.e. for Lq > 4.7 m. For the case of Lq = 40 m, the quench-back will be initiated after a time tq b ≈ 0.03 s, given by equation (C4.0.19), i.e. nearly instantaneously. Thus the actual propagation velocity for this value of Lq will be υq b ≈ 28 m s–1, according to equation (C4.0.20) where a copper resistivity of 7 × 10–10 Ω m has been used. References Ando T, Nishi M, Hoshino M, Oshikiri M, Tada E, Painter T, Shimamoto S, Vede T and Itoh I 1988 Experimental investigation of pressure rise of quenching cable-in-conduit superconductor Proc. ICEC-12 (Southampton, 1988) (Guildford: Butterworths) pp 908–12 Ando T, Nishi M, Kato T, Yoshida J, Itoh N and Shimamoto S 1990 Propagation velocity of the normal zone in a cable-in-conduit conductor Adv. Cryogen. Eng. 35 701–8 Ando T, Nishi M, Kato T, Yoshida J, Itoh N and Shimamoto S 1994 Measurement of quench back behavior on the normal zone propagation velocity in a CICC Cryogenics 34 599–602 Arp V D 1980 Stability and thermal quenches in force-cooled superconducting cables Proc. 1980 Superconducting MHD Magnet Design Conf. (Boston, MA: MIT) pp 142–57 Beard D S, Klose W, Shimamoto S and Vecsey G 1988 The IEA large coil task Fus. Eng. Des. 7 1–2, 23–4 Bottura L 1995 Numerical aspects in the simulation of thermohydraulic transients in CICCs J. Fusion Eng. 14 13–24 Bottura L 1996 A numerical model for the analysis of the ITER CICCs J. Comput. Phys 125 26–41 Bottura L and Shajii A 1995 On the numerical studies of quench in cable-in-conduit conductors IEEE Trans. Appl. Supercond. AS-5 495–8 Bottura L and Zienkiewicz O C 1992 Quench analysis of large superconducting magnets Cryogenics 32 659–67 Cornellissen M C M and Hoogendoorn C J 1985 Propagation velocity for a force cooled superconductor Cryogenics 25 185–93 Dresner L 1981 Thermal expulsion of helium from a quenching cable-in-conduit conductor Proc. 9th Symp on Engineering Problems of Fusion Research (Chicago, 1981) pp 618–21 Dresner L 1983 The growth of normal zones in cable-in-conduit superconductors Proc. 10th Symp. on Fusion Engineering pp 2040–3 Dresner L 1985 Protection considerations for force-cooled superconductors Proc. 11th Symp. on Fusion Engineering pp 1218–22 Dresner L 1991a Thermal hydraulic quenchback in cable-in-conduit superconductors Cryogenics 31 557–61 Dresner L 1991b Superconductor stability ‘90: a review Cryogenics 31 489–98 Hoffer J K 1979 The initiation and propagation of normal zones in a force-cooled tubular superconductor IEEE Trans. Magn. MAG-15 331–6 Lue J W, Schwenterly S W, Dresner L and Lubell M S 1991 Quench propagation in a cable-in-conduit force-cooled superconductor—preliminary results IEEE Trans. Magn. MAG-27 2072–5 Lue J W, Dresner L, Schwenterly S W, Wilson C T and Lubell M S 1993 Investigating thermal hydraulic quench-back in a cable-in-conduit superconductor IEEE Trans. Appl. Supercond. AS-3 338–41 Luongo C, Loyd R J, Chen F K and Peck S D 1989 Thermal hydraulic simulation of helium expulsion from a cable-in-conduit conductor IEEE Trans. Magn. MAG-25 1589–95 Marinucci C 1983 A numerical model for the analysis of stability and quench characteristics of forced-flow cooled superconductors Cryogenics 23 579–86

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Miller J R, Dresner L, Lue J W, Shen S S and Yeh H T 1980 Pressure rise during the quench of a superconducting magnet using internally cooled conductors Proc. ICEC-8 (Genova, 1980) (Guildford: Butterworths) pp 321–9 Shajii A and Freidberg J P 1994a Quench in superconducting magnets II. Analytic solution J. Appl. Phys. 76 3159–71 Shajii A and Freidberg J P 1994b Quench in superconducting magnets I. Model and numerical implementation J. Appl. Phys. 76 3149–58 Shajii A and Freidberg J P 1996 Theory of thermal hydraulic quenchback Int. J. Heat Mass Transfer 39 491–501 Shajii A, Freidberg J P and Chaniotakis E A 1995 Universal scaling laws for quench and thermal hydraulic quenchback in CICC coils IEEE Trans. Appl. Supercond. AS-5 477–82

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C5 Radiation effects on superconducting fusion magnet components H W Weber

C5.0.1 Introduction It is generally agreed that economic operation of future fusion reactors based on the magnetic confinement principle requires the application of superconducting magnets. Because of their complicated shape (torus coils for tokamak reactors and Yin-Yang, or end-plug, coils for mirror devices) and the performance requirements, which are close to the limits of present superconductor fabrication technologies, extensive test programmes on large coils have been made. All of them (Large Coil Task (LCT), Oak Ridge; Mirror Fusion Test Facility (MFTF), Livermore; Tore Supra, France) have demonstrated in a highly successful way that strong magnetic fields of the order of 9 T can-be produced reliably under the typical force configurations of tokamak or mirror devices. Therefore, even higher magnetic fields, the inclusion of superconducting poloidal magnets, is under consideration for fusion plants of the next generation (e.g. NET, TIBER-II, ITER, etc). In support of these developments, numerous material test programmes have been carried out. They are mainly aimed at an improvement of critical current densities in superconductors, the assessment of strain effects in these materials and the evaluation of the electrical and mechanical properties of insulating materials. One aspect which deserves particular attention in view of the burning of deuterium—tritium (DT) plasmas in fusion plants of the next generation and, of course, in future fusion reactors, is the effect of a radiation environment on the properties of all the magnet components. The most important issues which have to be addressed are: (i)

the radiation and strain tolerance of ‘commercial’ high-field superconductors preferably studied under the operating conditions of the magnet, including possible synergistic effects† and repeated thermal cycles to room temperature (to simulate plant shut-downs during the reactor lifetime) (ii) the strain- and radiation-induced changes of the resistivity of the stabilizer material (usually copper) (iii) the performance of insulating materials (iv) the stability of the whole composite forming the magnet windings

† ‘Synergistic’ means that two or more ‘abnormal’ conditions prevail simultaneously (e.g. radiation and strain). Their combined effect may lead to property changes different from those of an individual exposure.

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(v) to a certain extent, questions of radiation-induced radioactivity in the structural materials of the magnet. Of course, not all of this information has become available so far, in particular because of the scarcity of suitable (low-temperature) irradiation facilities. However, considerable progress was made during the 1980s and has led to a more basic understanding of the physics involved in radiation damage of highfield superconductors (Gregshammer et al 1988, Hahn et al 1986a, b, 1991, Nardai et al 1981, Snead and Parkin 1975, Söll et al 1972, Weber 1982, 1986, Weber et al 1982, 1988, 1989), some information on synergistic strain-radiation effects in Nb3Sn (Okada et al 1988), data on stabilizer performance under reactor simulation conditions (Brown 1981, Coltman 1982, Hahn et al 1986a, Klabunde and Coltman 1984) and preliminary results on various types of insulating material (Brown 1981, Coltman 1982, Evans and Morgan 1982, Weber et al 1983, Yasuda et al 1989). More recent activities pertain almost exclusively to insulating materials and address the assessment of intrinsic mechanical material parameters as well as some specific property changes needed for ITER design purposes (cf section C5.0.6). In this chapter we will discuss the relevance for fusion magnet applications of irradiation studies carried out at existing neutron and γ-sources, and survey our current knowledge on radiation effects in superconductors, stabilizer materials and insulators. In view of their relatively small significance, radiation effects in structural materials will not be considered (Guess et al 1975).

Figure C5.0.1. The neutron flux density distribution for four different neutron sources: fission reactor (TRIGA, Vienna), spallation source (IPNS, Argonne), DT source (RTNS-II, Livermore), fusion spectrum at the magnet location (STARFIRE design). After Hahn et al (1986).

C5.0.2 The radiation environment at the magnet location—operating conditions In order to evaluate the possibilities of meaningful material tests in neutron sources available at present, a careful characterization of these sources with regard to their neutron energy distribution as well as an analysis of expected neutron flux density distributions and γ-doses at the magnet location in fusion reactor designs are required. A comparison of this type is shown in figures C5.0.1 and C5.0.2, where the neutron flux density per unit lethargy† is plotted versus neutron energy for three irradiation sources (TRIGA reactor, Vienna;

† Dimensionless quantity ( ‘ logarithmic energy decrement ’ ) relating the initial neutron energy E0, i.e. the neutron energy produced in the fission process, to the actual ( moderated ) neutron energy at a certain position in the neutron spectrum, i.e. In (E0 /E).

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Figure C5.0.2. Normalized flux density per unit lethargy versus neutron energy (a) for three irradiation sources: TRIGA Mark-II reactor; Intense Pulsed Neutron Source, spallation source; Rotating Target Neutron Source, DT neutron source and (b) for two fusion reactor designs; STARFIRE, Tokamak reactor; Mirror Advanced Reactor Study. The latter two spectra refer to the magnet location. After Hahn et al (1986).

Intense Pulsed Neutron Source (IPNS), Argonne; Rotating Target Neutron Source (RTNS), Livermore) and two reactor designs (STARFIRE (Baker and Abdou 1980); Mirror Advanced Reactor Study (MARS) (Donohue and Price 1984)). No reference is made to the ITER design, because the neutronics have not been completed yet and the radiation conditions will be less stringent anyway than in a full reactor design. For the presentation of figure C5.0.2 the spectra were normalized in such a way that the sum of all group flux densities is equal to 1.0, which is most useful when the percentage of neutrons within a particular energy range is to be compared for different neutron sources, regardless of their total flux density. A summary of total and fast-neutron flux densities is given in table C5.0.1. In addition to fast neutrons a significant amount of γ-radiation will be present at the magnet location.

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Radiation effects on superconducting fusion magnet components Table C5.0.1. Flux densities at various neutrons sources and fusion reactor designs (at the magnet location.)

This can be totally ignored in the case of metals, since the changes in their physical properties are almost exclusively determined by fast-neutron damage, but it plays an important role in the case of insulating materials, where the physical properties, especially of the various resins but also of the resin—reinforcement interfaces, are affected by both kinds of radiation. Apart from the radiation environment, material tests also need to allow for the specific operating conditions of the magnet, which is expected to operate successfully over the entire plant lifetime (30 years). Obviously, the radiation damage will be introduced into the material at low temperature, but several thermal cycles to room temperature will occur when plants are shut down for routine service work. This leads to annealing effects in the metals and to gas release in the insulators as well as to additional stress during warm-up and subsequent cool-down. These conditions must be included in a ‘final’ simulation test programme, but are obviously quite difficult to meet from an experimental point of view and also to predict reliably from an operational point of view. C5.0.3 Damage energy scaling—absorbed energy From an inspection of figures C5.0.1 and C5.0.2 it seems obvious that reactors and spallation sources provide adequate radiation environments for magnet material tests. However, in order to put these comparisons on a more quantitative basis, an attempt can be made to scale the damage effect on a certain physical property by an appropriate radiation-related quantity. This has been tested successfully for superconductors and insulators with different radiation-related scaling quantities, as will be described in the following. In the case of superconductors, we restrict ourselves to the fast-neutron spectrum and scale, e.g., the transition temperature Tc or the critical current density Jc by the displacement energy cross-section 〈σ T〉 and the total energy transferred to each atom of the material (damage energy, ED ) in the following way (Greenwood 1982, 1987, Greenwood and Smither 1985, Hahn et al 1986)

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σ (E) and T (E) are the neutron scattering cross-section and the primary recoil energy distribution, respectively, Φ is the neutron flux density and t is the exposure time to the neutron flux. Because a whole spectrum of neutrons with largely varying energies interacts with the solid, the energy of the primary knockon atom is not a unique quantity but forms a spectrum, the primary recoil energy distribution, which reflects the energy distribution of the neutrons. An evaluation of equations (C5.0.1) and (C5.0.2) requires detailed knowledge of material parameters, elaborate computer codes, and the availability of the exact flux density distribution of the neutron source. If the irradiated material contains more than one element, a linear scaling of individual damage energy cross-sections with atomic percentage ci of the constituents i and summation have usually been employed

Although only minor errors are expected to stem from this simplification (Parkin and Coulter 1979) in simple binary alloys, more refined computer codes have become available recently which take the compound nature into account (Greenwood 1987, Gregshammer et al 1988). As an example, displacement energy cross-sections calculated from equations (C5.0.1) and (C5.0.3) are listed in table C5.0.2. Table C5.0.2. Displacement energy cross-sections (in keV b), scaled to neutron energies above 0.1 MeV, for various neutron sources and fusion reactor designs (at the magnet location). b = barn - 10-28 m2.

Calculations of this type confirm the feasibility of meaningful irradiation tests in an unambiguous way and specify the range of damage energies to be covered by these tests. However, damage energy scaling of certain physical properties of the superconductor can only be expected if the nature of the defects does not change within the relevant range of neutron energies. Low-energy neutrons (E < 0.1 MeV) lead to Frenkel pair production, in which an atom is displaced from its regular lattice site to create a vacancy at the original site and an interstitial at a new site, and to transmutation, in which a new atom is created through a nuclear reaction. Such neutrons play only a minor role through their small effect on normal state resistivity. By contrast, fast neutrons will produce sufficiently energetic primary knock-on atoms (E ≥ 10 keV) which will be able to displace a series of further atoms to produce a displacement cascade, and can also lead to collapsed cascades and point defect clusters (in particular following room temperature reactor irradiation). These effects have to be considered as the main origin of radiation-induced property changes in superconductors. The neutron energies of interest are, therefore, specified to range from 0.1 to 14 MeV. Based on these considerations, a test of damage energy scaling has been made for the critical current densities of NbTi wires by subjecting identical pieces of this material to ambient temperature irradiation in

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Table C5.0.3. Lifetime (30 years of operation) neuron fluence (E > 0.1 MeV) and damage energy per atom (calculated for Nb-46.5 wt% Ti) at the magnet location of two fusion reactor designs.

a reactor, a spallation source and a 14 MeV neutron source (for the corresponding source characterization, cf figure C5.0.1 and tables C5.0.1–C5.0.3). The results on the fractional change of Jc with damage energy ED , measured at 4.2 K and in a magnetic field of 5 T, are shown in figure C5.0.3. Although the actual highest neutron fluences differed by nearly as much as a factor of five, perfect scaling of the data with damage energy is observed. In addition to confirming that the nature of defects does not change with neutron energy in the range from 0.1 to 14 MeV, this result is considered most valuable from a practical point of view, since any type of fusion reactor spectrum can be used to evaluate the effects on critical current densities from graphs like figure C5.0.3, if appropriately converted to damage energies according to equations (C5.0.1) and (C5.0.2). Similar work has recently been done on insulating materials. As mentioned above, the damage

Figure C5.0.3. Fractional change of critical current densities with damage energy ED for three multiflamentary NbTi superconductors irradiated at ambient temperature at the neutron sources of figure C5.0.1.

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production is not restricted to fast neutrons. Hence, the low-energy part of the neutron spectrum as well as the γ -dose rate in the irradiation facility have to be included in damage considerations. With regard to the neutrons, the total flux density is readily available from reactor dosimetry experiments, such as the one shown in figure C5.0.1, and easily correlated to the fast-neutron flux density (E > 0.1 MeV) discussed so far. The γ -dose rate has to be assessed separately, which is usually done by calorimetry on high-purity aluminium. As the relevant radiation-related scaling quantity, we consider the absorbed energy in the compound, which consists of the total absorbed energy produced by the entire neutron spectrum plus the absorbed energy produced by the γ environment. The first can be calculated from the same computer codes as discussed previously: a typical example for the TRIGA reactor is listed in table C5.0.4 (Weber et al 1986). The latter is obtained simply by multiplying the experimental γ-dose rate (Gy h–1 )† by the exposure time and then adding it to the neutron contribution. (This simple procedure neglects the energy deposited by the neutrons in Al in the course of the calorimetry experiment, but this contribution is very small compared with the energy deposited by the neutrons in compounds consisting mainly of hydrogen, cf table C5.0.4.) Table C5.0.4. Calculated damage parameters of selected elements exposed to a total neutron fluence of 1.1 × 1023 m-2 (equivalent fast-neutron fluence: 4 × 1022 m-2, E > 0.1 MeV).

An example of these calculations is summarized in table C5.0.5, where we assume that two resins of different composition are exposed to exactly the same irradiation conditions in the TRIGA reactor (fast fluence: 5 × 1022 m–2 (E > 0.1 MeV), corresponding total fluence: 1.38 × 1023 m–2, γ dose: 1.83 × 108 Gy, i.e. 106 Gy h–1 multiplied by the irradiation time of 183 h). The results show not only that the absorbed energy is different (by ~40%!) because of the differences in composition, but also that the relative amounts of energy deposited by the neutrons and by the γ-rays, respectively, are very different. In order to check for a possible scaling behaviour with the absorbed energy, various types of glassfibre-reinforced epoxy were subjected to four different radiation environments (60Co γ-source, Takasaki; 2 MeV electrons, Takasaki; TRIGA reactor, Vienna; IPNS, Argonne) and their ultimate tensile strengths measured as a function of dose at 77 K ( Humer et al 1994a, b). The results are shown in figure C5.0.4,

† Unit of absorbed dose, 1 J kg−1 = 1 Gy = 100 rad.

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Table C5.0.5. Calculation of absorbed energy in two different plastics exposed to the same raidation environment. The neutron contribution is calculated from the data of table C5.0.4. Absorbed energy in grays: irradiation in the TRIGA reactor to a total neutron fluence of 1.38 × 1023 m-2 (irradiation time, 183 h; gamma dose rate, 106 Gy h-1).

Figure C5.0.4. Fractional change of the ultimate tensile strength with absorbed energy for three different types of glass-fibre-reinforced plastic irradiated at ambient temperature in various radiation environments (ZI-005: threedimensionally reinforced bismaleimide; ZI-003: three-dimensionally reinforced epoxy; CTD-101: two-dimensionally reinforced epoxy). Radiation sources: O—TRIGA Vienna, •—2 MeV electrons, G—60Co γ-rays, „—IPNS Argonne.

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in which the fractional change of the ultimate tensile strength (UTS) is plotted as a function of absorbed energy as calculated according to the above procedures. Excellent scaling is observed for all materials, a fact which is quite surprising in view of the previous literature on the subject. At present, we attribute this remarkable agreement to the following two facts. Firstly, the computer codes and associated procedures to calculate the absorbed energy were employed for the first time, whereas previously rather rough estimates of the energy deposited by the neutrons had to be used. Secondly, all of the reactor irradiation experiments made in this programme so far, were carried out on materials containing boron-free glass. Hence, the specific damage process related to the boron neutron–α reaction did not contribute to the degradation of the tensile properties, even in the case of neutron irradiation. This aspect of the radiation environment is currently being investigated in a separate programme by comparing the radiation response of two ‘identical’ composites, one having E-glass (optimized for electrical insulation properties) and the other S-glass (optimized for mechanical strength properties) reinforcement (Spieβberger et al 1996). In summary, scaling properties as reported for Jc versus ED in superconductors and as indicated to exist for UTS versus absorbed energy in some glass-fibre-reinforced plastics are considered to represent most valuable assets for radiation testing of magnet components, since they provide us with good estimates of the material degradation for arbitrary fusion reactor spectra. C5.0.4 Superconductors C5.0.4.1 Niobium-titanium At present, NbTi is still the most commonly used superconductor for magnet fabrication. This is based on the fact that advanced technologies for the production of excellent-quality multifilamentary wires have become available and, in particular, that because of the ductility of this material the winding of the magnet does not impose any major problems. Of course, the achievable fields are limited to 8 to 9 T because of the relatively low transition temperatures (8.5–9.5 K) and critical fields (10.5–12 T at 4.2 K). In view of the metallurgy of this alloy and its relation to the primary superconductive properties, i.e. the transition temperature Tc and the upper critical fields HC 2 (Hampshire and Taylor 1972, Larbalestier 1980, Maix 1974, Neal et al 1971, Pfeiffer and Hillmann 1968), an intermediate range of compositions (40–65 wt% Ti) is most suitable for applications. Whereas earlier, especially in Europe (Hampshire and Taylor 1972, Maix 1974, Neal etal 1971, Pfeiffer and Hillmann 1968), work was concentrated on high-Ti materials (≥49 wt% Ti), a standard alloy with nominally 46.5 wt% Ti emerged later. More recently, considerable effort has been put into researching high-Ti and ternary materials (e.g. Nb—Ti—Ta alloys) in the United States (Larbalestier 1981, Lee etal 1989), in view of the field and critical-current-density requirements of the Superconducting Supercollider. Extensive work on metallurgical microstructure, especially the formation of normal-conducting α-Ti precipitates ( Lee et al 1989, West and Larbalestier 1980, 1982), has led to a fairly clear picture of flux pinning mechanisms and, hence, the interrelation between microstructure and achievable critical current densities. Irradiation work on these materials has been pursued for a long time (for a review of early work, cf Sekula 1978). Systematic research on superconductors of varying metallurgical microstructure was initiated by the group at the Atominstitut der Osterreichischen Universitaten, Vienna in the late 1970s and led to a satisfactory understanding of the physics of radiation damage in this alloy. The main steps of this development may be summarized as follows. Ambient temperature reactor irradiation of a wide spectrum of NbTi superconductors, which varied not only in their Ti content (42, 46.5, 49, 54 wt%) but also in the final stages of their thermomechanical treatment (final annealing and cold-working conditions), and subsequent measurement of critical current densities at 4.2 K and in fields up to 6 T, has established clear correlations between metallurgical microstructure and radiation response. Although Jc was found to decrease with the neutron fluence in

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all cases, this degradation was always considerably smaller if flux pinning by precipitates dominated over other pinning mechanisms (e.g. pinning by dislocation cells), i.e. the radiation response of high-Ti materials was, in general, superior (Nardai et al 1981, Weber 1982, Weber etal 1982). The same observations were made in the course of ‘simulation’ experiments, in which the superconductors were irradiated at low temperatures (5 K) and subjected to repeated thermal cycles to room temperature (Hahn et al 1986a, b, Weber 1986), and, to a certain extent, in low-temperature irradiation and in situ critical-current-density measurements (Söll et al 1972). An example of these results, obtained after irradiation at 5 K and thermal cycling, is shown in figure C5.0.5(a), where the fractional change of Jc , measured in a magnetic field of 5 T, is plotted versus the neutron fluence for alloys containing 42, 49 and 54 wt% Ti.

Figure C5.0.5. Fractional change of critical current densities versus neutron fluence at 5 T and at 8 T. The data refer to three NbTi superconductors prepared under identical annealing and final cold-working conditions, but differing in their Ti content (Ο—42 wt%, +—49 wt%, ×—54 wt%).

However, if the same measurements are taken at higher magnetic fields (≥7 T), this clear correlation of Jc degradation with microstructure disappears and a nearly uniform radiation response is obtained (figure C5.0.5(b)). This observation, which has been interpreted tentatively as being a consequence of a different pinning mechanism operative near Hc 2 , could be fully explained by a series of experiments on the radiation-induced change of upper critical fields Hc 2 and transition temperatures Tc (Weber et al 1988). As an example, the evaluation of Hc 2 , which is based on a flux-pinning theory of plastic flux-line flow near Hc 2 (Schmucker 1977), is shown for a superconductor in the unirradiated and the irradiated state in figure C5.0.6. In this case, as well as in all other NbTi superconductors investigated so far, the upper critical field was found to decrease by 2 to 6% with neutron irradiation up to a fluence of 3 × 1022 m–2 (E > 0.1 MeV). Although this result may be surprising at first sight, because radiation-induced defects are expected to increase the normal-state resistivity and, hence, Hc 2 , direct measurements of resistivity on bare filaments have shown that no resistivity changes were detectable, presumably because of the high pre-irradiation resistivity of these materials and the repeated thermal cycling to room temperature. The physical origin of the Hc 2 reduction was identified through careful Tc measurements, as being the result of a small reduction of Tc with neutron fluence. The corresponding results (figure C5.0.7) confirm the

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Figure C5.0.6. Extrapolation of critical current densities to determine the upper critical field Hc 2 according to the Schmucker theory (Ο—unirradiated NbTi, •—after neutron irradiation to 3 × 1022 m–2, E > 0.1 MeV).

Figure C5.0.7. Correlation between Hc 2 and Tc for several NbTi superconductors (10 and 19 with 42 wt% Ti, 12 and 21 with 54 wt% Ti). Sample numbers with a bar refer to the irradiated state (3 × 1022 m– 2, E > 0.1 MeV).

well-known Tc and Hc 2 dependence on Ti concentration and show excellent Hc 2 – Tc correlations prior to and following neutron irradiation. From these results, we conclude that at very high fields flux pinning by the metallurgical microstructure becomes less important than the elastic or plastic flow properties of the flux-line lattice. In fact, if we

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assume that only Hc 2 changes with neutron fluence in the equations for the pinning force at high fields, all flux-pinning models result in the same expression for the fractional change of Jc

An evaluation of equation (C5.0.4) in the field range from 8 to 10.5 T at 4.2 K proved highly successful and enabled us to describe the experimental results on the Jc degradation at high fields in general to an accuracy of ±5%. In summary, critical current densities in NbTi are always found to decrease with neutron fluence by up to 20 to 30%. Results on the Swiss LCT conductor, following irradiation at 5 K and thermal cycling, are shown in figure C5.0.8. Clear correlations of the Jc degradation with microstructure were established at low fields, where samples with predominant precipitate pinning are least radiation sensitive. On the other hand, at high fields a rather uniform radiation response was observed. In this case, the reduction of critical current densities is explained by the radiation-induced decrease of Hc 2 , which in turn is caused by a decrease of the transition temperature Tc .

Figure C5.0.8. Fractional change of critical current densities with neutron fluence for the Swiss LCT conductor at 5 and 8 T (irradiation conditions: 5 K, thermal cycling after each irradiation step).

C5.0.4.2 Niobium-tin To achieve magnetic fields above 10 T, the A15 superconductor Nb3Sn was developed for large-scale applications. As a result of the requirement for a final heat treatment at around 700°C, which is needed to form the stoichiometric A15 phase, and because of the brittleness and very small strain tolerance of this material, the fabrication of large magnets is much more difficult than in the case of NbTi. Nevertheless, magnets made from multifilamentary Nb3Sn wires will be employed for the next generation of fusion devices. Radiation damage studies on advanced commercial A15 superconductors are scarce. Most of the earlier work was done under ambient-reactor-temperature irradiation conditions (e.g. Sweedier et al 1979), which can be summarized as follows. The transition temperature Tc of most of the A15 materials (figure C5.0.9) decreases in a uniform way with neutron fluence and saturates at a level of about 0.2Tc 0 at very high fluences (≤1024 m–2, ED ≈ 100 eV). This significant decrease of Tc , which would lead to a transition temperature of 15 to 16 K in Nb3Sn after the lifetime fluence in a fusion reactor compared with 18 K prior to irradiation (Tc 0 ), is ascribed primarily to a reduction of the degree of long-range order and an expansion of the lattice parameter in the originally highly ordered crystal lattice of this compound. On the other hand, at low fluences this radiation-induced disorder leads to an increase of the

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Figure C5.0.9. Transition temperatures Tc versus damage energy ED for three A15 superconductors.

upper critical field Hc 2 through the radiation-induced increase of normal-state resistivity ρn , which in turn results in an increase of critical current densities at fixed fields, especially in the high-field region. An experimental result of this type obtained on a 19-filament wire following ambient temperature irradiation is shown in figure C5.0.10. Finally, the drastic decrease of Jc at higher damage levels is correlated with the deterioration of the primary superconductive properties, especially Tc . Low-temperature irradiation, including thermal cycling to room temperature, has been reported for various (Nb1–xTix)3Sn superconductors (Hahn et al 1986b, Weber 1986). Much interest has been concentrated on this class of compounds with substitutions of the order of a few weight per cent Ti or Ta, because of the significant increase of upper critical fields (e.g. Smathers et al 1985), which is caused again by an increase of normal-state resistivity. The irradiation results obtained on a compound with 1.5 wt% Ti are also shown in figure C5.0.10. Although a small increase of critical current densities occurs at low fiuences, the drastic Jc degradation starts at damage energy levels which are by about a factor of four smaller than for nonsubstituted Nb3Sn. Considering the fact that the pre-irradiation enhancement of Hc 2 amounts to ~13% in this alloyed compound (Hc 2 at 4.2 K has increased to 21 T compared to 18.5 T in pure Nb3Sn) and, furthermore, that an increase of approximately 12% has been measured at the peak of the Jc –ED dependence in pure Nb3Sn (Snead private communication), the almost immediate decrease of Jc with further disorder is not unexpected for the alloyed compound. This result is, of course, disappointing for applications in fusion reactors, but useful from a fundamental viewpoint, and provides evidence for the significant role of disorder in the damage process. It also suggests that the interaction of neutron damage with the pinning microstructure, as well as the influence of irradiation temperature, should be small in these materials.

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Figure C5.0.10. Fractional change of critical current densities vesus damage energy for Nb3Sn (ambient-temperature irradiation) and Nb3Sn with 1.5 wt% Ti (irradiation at 5 K plus thermal cycling after each irradiation step). Data taken from Hahn et al (1986a) in a modified form.

Experimental proof of the latter aspect was obtained only recently by comparison of ambient- temperature and 12 K irradiations on the same conductors with 14 MeV neutrons (Hahn et al 1991). These commercial multifilamentary conductors (one containing 1.2 wt% Ti) showed identical results in Tc degradation with neutron fluence under these two irradiation conditions and nearly identical results for the critical current densities. In conclusion, brief reference should be made to strain effects in irradiated Nb3Sn. The effects of synergisms between strain and ambient-temperature irradiation on the transition temperature Tc were investigated (Snead and Suenaga 1980) by irradiating monofilamentary Nb3Sn with a bronze overlayer (prestrain effect) and measuring Tc with and without this overlayer. The Tc difference between these two states, which amounted to 0.5 K in the unirradiated state, was found to double at high fluences, thus indicating a degrading influence of strain on the radiation performance. Concerning critical current densities, irradiated multifilamentary Nb3Sn conductors were subjected to strain within the reversible range (∈ < ∈i r r e v ) (Okada et al 1988). These data indicate that irradiation does not appreciably change the strain sensitivity of Jc or the pinning forces if the comparison is made under appropriate scaling conditions, i.e. the known effects on Hc 2 are taken into account. In summary, most of the experimental evidence accrued so far seems to indicate that magnets made of Nb3Sn will be able to sustain the radiation environment in a fusion reactor, although with a very small safety margin (the limit is generally considered to be reached when Jc /Jc 0 drops to 1 after the radiation-induced peak). C5.0.4.3 Other materials Having discussed NbTi and Nb3Sn superconductors, the list of ‘commercial’ materials is exhausted. However, with the quest for still higher magnetic fields for applications on laboratory scale and even for advanced fusion reactor designs, extensive material development programmes have provided us with some new promising superconductors. As examples, sputtered NbN films (Gavaler et al 1971, Capone et al 1986), multilayer sandwich structures (Gray et al 1988) and layers deposited on thin fibres (Dietrich

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and Dustman 1984), wires made from Chevrel-phase superconductors (Seeber et al 1989) and Nb3(A1,Ge) tapes (Kamakura et al 1989) should be mentioned. Radiation testing of these materials is in a very early stage. However, the results available on singlelayer magnetron-sputtered NbN films (Gregshammer et al 1988, Weber et al 1989) revealed excellent performance under high-level neutron irradiation. Considering, in addition, the known strain tolerance (Ekin et al 1982), the high upper critical fields (∼24 T at 4.2 K) and critical current densities of the order of 108 A m– 2 at 20 T, this material represents a promising candidate for high-field applications in a radiation environment. The same holds for NbN/AIN multilayer films (Herzog et al 1990). The first irradiation experiments (ambient reactor temperature) were made on four types of film (Gregshammer et al 1988, Weber et al 1989) and extended to a fluence of 1023 m– 2 (E > 0.1 MeV). Over this range, the transition temperatures were found to decrease by 4% to 7%, in close agreement with previous measurements on bulk NbN (Dew-Hughes and Jones 1980 ), whereas the upper critical fields were affected only slightly (±0.5% to +3.4%). Concerning the critical current densities, a very interesting fluence dependence was observed (figure C5.0.11). Whereas in low fields (B/µHc2 < 0.5) a radiationinduced decrease of Jc occurs, the high-field results (B/µHc 2 > 0.7) show an increase of Jc by 10% to 90% depending on sample preparation.

Figure C5.0.11. Fractional change of critical current densities versus neutron fluence (ambient-temperature irradiation) for an NbN film produced under high-rate sputtering conditions.

An extension of this work to low-temperature irradiation (4.6 K, 5.3 × 1022 m–2, E > 0.1 MeV ) and to Jc measurements up to 23 T has been reported more recently ( Herzog et al 1991). Two sets of measurements were made. In the first, the samples were warmed up from 4.6 to 77 K and transferred into the measuring rig at this low temperature, whereas in the second the samples were warmed up to room temperature prior to the Jc measurements in order to simulate the magnet operating conditions as closely as possible. The results show practically no change of Jc under any condition (except for enhancements at fields above 16 T, which were successfully correlated with radiation-induced enhancements of Hc 2 ). In summary, the data available so far demonstrate that the overall prospects for applying this material in fusion reactor magnets are encouraging. They also emphasize the necessity for testing these high-field superconductors up to very high magnetic fields. Of course, commercial production techniques must still be developed, in order to render NbN a viable alternative to present day materials. At this point, brief reference should be made to radiation effects on high-temperature superconductors (Weber and Crabtree 1992), although their application in large-scale magnets is not likely until far in the future mainly for technological reasons. It turns out that in all cases fast-neutron irradiation, but also irradiation with certain ions, in particular with gigaelectronvolt heavy ions producing amorphous

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tracks, results in considerable enhancement of the critical current densities, thus indicating that the pre-irradiation defect structures are far from optimal. Apart from this problem, which could be solved in the future by improved processing technologies, more fundamental problems associated with the physics of these materials should be addressed briefly. All high-temperature superconductors show very high critical current densities at low temperatures, i.e. 4.2 K, which are nearly field independent due to the enormously large values of the upper critical field Hc 2. They are, therefore, considered as suitable for insert coils to extend the accessible d.c. field range up to, say, 40 T at 4.2 K, once the technological problems of fabricating sufficiently long tapes or wires have been overcome. However, for ‘real’ applications of high-Tc materials an operating temperature of 77 K seems mandatory. At these high temperatures the flux pinning capability of the as-grown as well as of artificially introduced defects becomes very small and in some cases negligible, which has led to the definition of the so-called irreversibility line, i.e. a line in the H—T plane, which separates the domain of flux pinning (Jc ≠ 0) from the one with Jc = 0. The location of the irreversibility lines (figure C5.0.12) is strongly material dependent and presumably correlated with the ‘dimensionality’ of the material, i.e. the distance between the ‘superconducting’ CuO2 planes and, hence, the thickness of the insulating or normal-conducting interlayer volume. In strongly two-dimensional systems, such as Bi-2223 or Tl-2223, the flux lines break up into pancake structures, which are only weakly coupled, resulting in a drastically reduced pinning capability and, hence, in low values of the irreversibility line. This detrimental effect is reduced in ‘more three-dimensional’ materials, such as Y-123 and Tl-1223, where strongly coupled pancakes or even flux lines exist and consequently higher values of the irreversibility lines are observed. Hence, under all circumstances a huge research and development effort will be required before superconducting magnets made of high-Tc superconductors will become operational at 77 K, even at low fields (~2 T).

Figure C5.0.12. ‘Irreversibility’ lines for three high-temperature superconductors (H parallel to the c axis of the single crystals): Tl-2223—G, Y-123—∆, Tl-1223—Ο). The broken line indicates the approximate temperature dependence of the upper critical field for a Tl-2223 single crystal.

C5.0.4.4 Summary This section has summarized our current knowledge of radiation effects on high-field superconductors for fusion magnets. A survey of results on the fractional change of critical current densities with damage energy per atom is shown in figure C5.0.13.

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Figure C5.0.13. Overview of results for the fractional change of critical current densities with damage energy. The data on NbTi and (NbTi)3Sn refer to irradiation at 5 K and thermal cycling, the other data refer to ambient temperature irradiation. S and M indicate the lifetime fluences of the STARFIRE and MARS design at the magnet location.

Clearly, NbTi remains the obvious choice for magnet fabrication, if the intrinsic field limitation of 8 to 9 T can be tolerated. For higher magnetic fields, which are required in the ITER design for example, Nb3Sn or (NbTi)3Sn will have to be employed. It should be borne in mind, however, that the existing data base indicates that the radiation load for a real reactor design is either close to or exceeding the tolerance of these materials. Therefore, and in view of the demand for still higher magnetic fields, development and radiation testing of new materials are certainly worthwhile. Even for the ITER design, radiation testing with a simulation character (i.e. low-temperature irradiation and in situ testing, thermal cycling) suitable for the chosen type of conductor should be recommended. C5.0.5 Stabilizer materials Very little work has been done on this magnet component, which determines the layout of the magnet in view of quench stabilization through its resistivity, and none in recent years. Aluminium has been considered briefly (Birtcher et al 1975, Blewitt and Arenberg 1968, Brown 1981, Brown et al 1974, Horak and Blewitt 1975, Klabunde et al 1979), but will not be discussed in the following because all real fusion magnet designs rely on copper as the stabilizing material. Usually the final preparation step of superconducting wires and cables consists of a thermal treatment to anneal most of the defects in the Cu introduced during the various cold-working stages and to achieve residual resistivity ratios ( RRRs) of the order of 100 or better. For practical reasons, the RRR is defined here as the resistivity at room temperature divided by that slightly above Tc . For commercial conductors, e.g. the Swiss LCT conductor, to which reference will be made in the following several times, the resistivity at 10 K (NbTi superconductor) amounts to ~0.15 nΩ m. This typical value for ‘magnet copper’ is enhanced in two ways during magnet opeation. Firstly, magnetoresistivity increases ρ by an amount which depends on starting purity and, of course, temperature

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and magnetic field strength. Secondly, radiation-induced defects (transmutations, point defects, defect clusters and cascades) enhance ρ, but anneal partly or even completely upon warming the magnet to room temperature. Early work on high-purity copper and its alloys (Böning et al 1970, Lengeler et al 1970) has shown some deviations from Matthiessen’s rule (Matthiessen 1860) and nonuniform Kohler plots (Kohler 1949), but more recent work has confirmed the existence of a unique Kohler relation for Cu with RRRs of interest in the present context (Guinan and van Konynenburg 1984). Matthiessen’s rule (discovered experimentally) states that the partial resistivities arising from the scattering of conduction electrons at different types of scatterer are additive and that the increase of resistance due to a small concentration of another metal in solid solution, is in general independent of temperature. Kohler’s rule states that the quantity [ρ(B) − ρ(0)]/ρ(0) remains unchanged upon increasing the impurity concentration c and the field B by the same factor. This dependence, i.e. [ρ(H) − ρ(0)]/ρ(0) versus H/ρ(0), where ρ(0) is the zero-field resistivity, is of course most valuable in predicting the evolution of ρ under various field and irradiation conditions, even if the pre-irradiation RRRs vary. The most complete set of data pertaining to low-temperature irradiation and thermal cycling can be found in the work by Hahn et al (1986a), Klabunde and Coltman (1984) and Nakata et al (1985). Hahn et al (1986a) report on a comprehensive study of radiation effects in NbTi superconductors, in which numerous different wires with different final thermomechanical treatments were sequentially neutron irradiated at 5 K, subjected to a room-temperature annealing cycle and then experimentally characterized before the next irradiation step. In all cases the zero-field resistivity of the copper stabilizer was measured as well. Klabunde and Coltman (1984) report on in situ 5 K irradiation and annealing of comparable magnet copper, although only up to a neutron fluence of ∼0.8 × 1022 m–2 (E > 0.1 MeV ), and resistivity measurements in magnetic fields up to 6.5 T. The results were as follows. Firstly, the pre-irradiation data on the ‘as-produced’ wires display a very uniform behaviour. Those materials that were subjected to final cold work as the last preparation step showed resistivity ratios around 60 (ρ (10 K) ∼ 0.30 nΩ m), which remained constant in the course of the irradiation programme. On the other hand, those materials subjected to a final heat treatment showed initial resistivity ratios around 120 (ρ (10 K) ∼0.15 nΩ m), but degraded continuously towards the ‘cold-work limit’ of 60 as they were irradiated. Secondly, based on the results of Guinan and van Konynenburg (1984) on similar ‘magnet’ copper, which established the applicability of a Kohler plot over a wide range of starting resistivities and irradiation/annealing conditions, the data taken at zero field can be converted to project the increase of stabilizer resistivity with neutron fluence at high fields, e.g. 8 T. (It should be borne in mind, however, that this representation is typical of copper only in this resistivity range and is not universal because of the known deviations from Matthiessen’s rule (e.g. Lengeler et al 1970). The results of this evaluation are shown in figure C5.0.14, which includes both the measured data and the high-field predictions. It will be noted that the inclusion of magnetoresistivity drastically reduces the amount of resistivity increase with fluence because of the decrease of magnetoresistivity with increasing zero-field resistivity. Since the data presented refer to the post-annealing conditions only, the crucial question about the resistivity increase at 8 T prior to the annealing cycle has still to be answered. Here, the in situ 5 K irradiation and annealing experiments on comparable magnet copper (ρ (4.2 K) ~ 0.16 nΩ m) can be used (Klabunde and Coltman 1984). Perfect agreement between the two sets of data is found in the overlapping fluence range (figure C5.0.15). Therefore, the extrapolation for the Swiss LCT conductor operating at 8 T, shown by the broken line in figure C5.0.15, represents a reasonable approximation to the actual operating behaviour of a fusion magnet, if we assume a lifetime fluence of 4 × 1022 m–2 (E > 0.1 MeV) over 30 years and plant shut-downs (annealing cycles) after 2, 6, 9, 12.5, 15, 20 and 25 years of operation. Although extrapolation errors should be allowed for, it is quite obvious that present design limits for the permissible resistivity increase of the stabilizer at 8 T (25%) cannot be reconciled with a reasonable operating schedule of a fusion plant.

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Figure C5.0.14. Increase of stabilizer (copper) resistivity with fast-neutron fluence (E > 0.1 MeV) normalized to the pre-irradiations values at B = 0 and 8 T respectively. Zero-field data are measured and the 8 T data projected. Experimental points pertain to 5 K irradiation and measurement after an annealing cycle to room temperature. Multifilamentary NbTi superconductors; stabilizer resistivity ratios prior to irradiation: ~60 (No 34), ~120 (Nos 35, 36).

Figure C5.0.15. Complete simulation cycle for the change of stabilizer resistivity with neutron fluence (E > 0.1 MeV) over the plant lifetime. The conductor is NbTi (Swiss LCT conductor). The in situ low-fluence data of Klabunde and Coltman (1984) are used to extrapolate the post-annealing data to a fluence of 4 × 1022 m2 (E > 0.1 MeV).

What can be done to overcome this problem? Further annealing cycles are clearly ruled out for economic reasons, but the remaining two possibilities are also quite costly. The first consists of using more stabilizer material to reduce the overall resistance and the second is based on a larger radiation shield, thus reducing the lifetime fluence at the magnet location. Since both of these alternatives lead to larger and less efficient magnets, a material problem has been identified, which certainly deserves more attention in future reactor designs. C5.0.6 Insulators Another area of concern is the radiation response of insulating materials regarding both their electrical and their mechanical degradation processes, the latter being considered to be more serious (Sisman and Bopp 1956). Consequently, almost all of the more recent work in this field has been concentrated on an

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assessment of the mechanical properties of various materials, in particular glass-fibre-reinforced epoxies, polyimides and bismaleimides, and their change under irradiation. Proper testing conditions are extremely difficult to achieve, mainly because of the small volumes available in the few existing low-temperature irradiation facilities, which do not accommodate standard test geometries for tensile, shear or fracture mechanical tests. Hence, research has proceeded along two lines (Weber and Tschegg 1990). In the first, some small test geometry is devised, which is suitable for low-temperature irradiation but is known to be not completely representative of the desired load conditions, e.g. shear sample with two displaced notches (Nishijima et al 1988) or the cylindrical uniaxial rod sample subjected to torsional loads (Kasen 1986). Of course, this approach is perfectly well suited for material screening purposes, because tests on various fibre-matrix composites in exactly the same sample geometry will allow one to decide on the general suitability of a certain material (cf e.g. Munshi and Weber 1992). Very specific test arrangements, such as special lap-shear tests (Spindel et al 1994) or tests under a specific combination of different load conditions, e.g. the shear/compression sample developed for the ITER magnets (Simon et al 1994), also fall into this category. The second line of research follows a more fundamental approach and is aimed at an assessment of intrinsic material parameters for these highly anisotropic compounds, which would allow engineers to design magnets on the basis of the ultimate tensile strength and the fracture behaviour in mode I and II (crack opening and intralaminar shear mode; Tschegg et al 1991, 1993, 1995, and their dose dependence (Humer et al 1994a, b, 1995). This programme has required extensive pre-irradiation development work, especially ‘scaling’ experiments starting from standard test geometries (according to DIN or ASTM standards) to ensure the suitability of small samples for obtaining intrinsic material parameters, as well as

Figure C5.0.16. Sample geometries developed for irradiation experiments on glass-fibre-reinforced plastics by scaling from standard test geometries. Values in brackets refer to the scaled-down geometry for 5 K irradiation.

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the introduction of novel concepts such as the fracture energy concept. In all cases (Tschegg et al 1991, 1995, Humer et al 1992), test geometries could be established, which are very well suited for irradiation experiments; a schematic view of three types of these test samples is shown in figure C5.0.16. A few results of both types of experiment will be discussed here. Typical results pertaining to a material screening programme are shown in figure C5.0.17, where the torsional ‘shear strength’ at 77 K is compared for various S-glass-fibre- (boron-free!) reinforced plastics at various steps of radiation exposure to a combined neutron and γ field at ambient reactor temperature (Munshi and Weber 1992). Clearly, some of the composites (e.g. G-11 CR, CTD-100) show a catastrophic breakdown of their mechanical properties, whereas others (e.g. the polyimide CTD-300 or the bismaleimide CTD-200) display considerable radiation tolerance. On the other hand, the pre-irradiation ‘shear’ strength of these materials is considerably lower than that of some epoxies and may even be comparable to the post- irradiation properties of some ‘radiation-hard’ epoxies (e.g. CTD-101). Similar selection criteria can be found from ambient temperature electron irradiations of tensile test samples (figure C5.0.18; Humer et al 1994a, b). Two of the materials (Epo HGW and Orlitherm N, both glass-fibre-reinforced epoxies) are or quickly fall below the minimum design requirement for fusion magnets (~600 MPa), whereas other materials perform much better up to doses of ∼2 × 108 Gy. In particular the three-dimensionally reinforced bismaleimide ZI-005 shows almost no change in ultimate tensile strength, a fact which can also be observed in the normalized presentation of figure C5.0.4, where the scaling properties of the ultimate tensile strength with absorbed energy were discussed. In summary, ambient temperature irradiation of various materials in various test geometries is certainly most valuable for selecting materials for complex and time-consuming low-temperature irradiation work.

Figure C5.0.17. (Torsional) shear strength for various S-glass-fibre-reinforced plastics at neutron fluences up to 5 × 1022 m–2 (E > 0.1 MeV). The code numbers for the resin systems are as follows: CTD-100, 101, 110–epoxies; CTD-200—bismaleimide; CTD-300, 310—polyimides; G-11 CR—epoxy. Ambient temperature irradiation was used and the mechanical tests were carried out at 77 K.

The data base for the second class of experiments is still very small and currently limited to the ultimate tensile strength. Results pertaining to 5 K irradiation in a combined neutron and γ field and tensile tests at 77 K are shown in figure C5.0.19, which contains two sets of data. The first refers to the ultimate tensile strength measured at 77 K upon direct cold transfer of the samples into the testing machine at this

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Figure C5.0.18. Ultimate tensile strength of ZI-005 (Ο), ZI-003 (•), CTD-101 (G), ORLITHERM N („) and EPO-HGW (+) as a function of absorbed dose following room-temperature irradiation with 2 MeV electrons and fracture at 77 K.

Figure C5.0.19. Ultimate tensile strength of ZI-005, ZI-003 and CTD-101 as a function of total absorbed dose following 5 K irradiation (neutrons and γ - rays) with (•) and without (Ο) a warm-up cycle to room temperature before testing at 77 K.

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temperature (open symbols), and the second refers to the same kind of measurement, but made after a well defined annealing cycle to room temperature. No systematic influence of the annealing cycle is detected. In addition, the degradation of the ultimate tensile strength is almost identical to that observed upon room-temperature irradiation of the same materials (cf figure C5.0.4). This result is somewhat surprising in view of earlier work (for a review see Brown 1981) and also in view of the expected gas production and release mechanisms (table C5.0.4). According to these considerations, it would be expected that the gas production (mainly hydrogen and helium) remains locally stabilized at low temperatures and is then suddenly released upon warming up to room temperature, which would lead to distinctly different defect configurations and hence to different mechanical failure situations. The lack of such a difference, which is consistent with the observed agreement between low-temperature and ambient-temperature irradiation conditions, is tentatively ascribed to material specific properties such as the boron-free reinforcement and the corresponding absence of significant contributions to the gas production rate. It is futhermore restricted to the tensile properties of the materials. More serious problems are expected for load conditions involving shear stresses, in the worst case intralaminar shear with crack initiation within the fibre cloth. First results on this new test (ambient- temperature irradiation in a combined neutron and γ field) are shown in figure C5.0.20, where the specific fracture energy GF for crack initiation in mode II (Humer et al 1994a) is plotted versus absorbed energy. The fracture energy is defined as the total energy absorbed during the whole fracture process and is assessed experimentally from the area under the load—deformation curve. (GF can be related to the shear strength τ by finite-element calculations.) The specific fracture energy is defined as the fracture energy divided by the area of the fracture surface. The results differ remarkably from those reported above for the tensile properties in several ways. Firstly, the material specific radiation resistances seem to play only a minor role. Secondly, some kind of plateau seems to be exceeded above ∼2 × 107 Gy, resulting in a catastrophic loss of strength at higher doses. This change of behaviour is seen more clearly in the actual load—deformation curves, which can be evaluated only with great difficulty in terms of the usual assumptions of the fracture energy concept and which seem to indicate that the fibres are slack at these dose levels rather than elastic or plastic. In any case, this first set of data indicates that all materials fail under these intralaminar shear conditions at dose levels, which is unacceptable for the performance of a fusion magnet. Results of this kind clearly demonstrate that further research is badly needed. The following issues are being addressed in current irradiation programmes: low-temperature (77 K) irradiation with and without room-temperature annealing cycles and testing in the crack-opening and in the intralaminar shear mode; low-temperature (77 K) irradiation with and without room-temperature annealing cycles and testing in

Figure C5.0.20. Specific fracture energy for crack initiation versus total absorbed dose for four different glass-fibrereinforced plastics: O—ZI-005, •—ZI-003, G—CTD-101, „—ISOVAL10/S.

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the interlaminar shear mode (Spieβberger et al 1997); testing of pure resins under the same conditions and accompanying acoustic emission analysis as well as optical and scanning electron microscopy of the fracture surfaces. Furthermore, the encouraging results on the tensile properties of some materials are analysed in terms of the application of boron-free glass and the corresponding reduced gas production in the composites. (A reinforcement with S- or T-glass is not considered to represent a problem commercially, if really needed for this purpose.) In summary, the current knowledge of the radiation response of glass-fibre-reinforced plastics is still not satisfactory for realiably predicting the lifetime performance of this fusion magnet component. Very promising results are available regarding the scaling of the ultimate tensile strength with absorbed energy, which could greatly facilitate radiation testing in the future. On the other hand, substantial information, in particular on the behaviour under shear loads, is still missing, which seems to warrant additional research efforts with radiation testing and materials development. C5.0.7 Summary The radiation environment at the location of a superconducting magnet in a nuclear fusion reactor has been specified. It should be borne in mind that these specifications refer to ‘standard’ well-shielded parts of the magnet, but they could be exceeded considerably at design-specific locations such as ports or feedthroughs. It has been shown, furthermore, that the 14 MeV neutrons originally produced are ‘moderated’ considerably and transformed into a flux density distribution at the magnet location, which is not too different from what occurs in a fission reactor or even better in a spallation source. In addition to the neutrons, γ-radiation produced from all kinds of nuclear reaction will prevail at the magnet location. The impact of this radiation environment on various components of the superconducting magnet has been discussed. It was shown that the superconductor is affected mostly by fast neutrons (E > 0.1 MeV) with minor (less direct) contributions by thermal neutrons (mainly through transmutations and their influence on the normal-state resistivity and, hence, the upper critical field Hc 2 ). In the case of the stabilizing materials, the entire neutron spectrum will contribute to the enhancement of their resistivity, but most of the damage anneals upon warming the magnet to room temperature. Finally, both the entire neutron spectrum and the γ-radiation contribute to the degradation of the insulating materials. Operation of the magnet implies that the damage is introduced at low temperatures, but periodic service work on the fusion reactor will lead to repeated warm-ups of the entire magnets to room temperature. These operational constraints have strongly differing effects on the various magnet components, as will be summarized below. Radiation effects on superconductors have been investigated in the most comprehensive way under conditions relevant for fusion magnet design purposes. NbTi superconductors do not seem to cause any problem. Although in situ testing at high fields has not been performed in the irradiated state, it is safe to conclude that the degradations in this case will not be much larger than those observed on many NbTi superconductors of different metallurgical compositions (including technical wires) upon low-temperature irradiation and thermal cycling. The decrease of Jc after the lifetime exposure to the radiation environment will be of the order of 30%. Of course, NbTi is an unlikely candidate for fusion magnets of the next generation because of its intrinsic field limitations to ∼9 T. For higher fields, Nb3Sn or alloyed Nb3Sn superconductors will have to be used. The latter show quite an unfavourable radiation tolerance both under in situ irradiation and testing conditions as well as when thermal cycles are included. Again, the differences between these two states are only marginal. For pure Nb3Sn superconductors, the lifetime fluence comes close to the precipitous degradation of Jc , which makes testing of ‘final’ conductors mandatory. Excellent radiation hardness is found in the high-field superconductor NbN, which is not available for commercial production. Practically all data indicate that the influence of the irradiation temperature and of thermal cycling is very small in all of these technical materials and that spectral

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differences of the neutron environment can be scaled successfully for Jc (and Tc ) by the damage energy concept. Radiation effects on the stabilizing materials have been investigated in some detail, although a comprehensive and complete in situ irradiation/testing/thermal-cycling/testing programme is still missing. The presently available data base indicates, however, that the conditions specified for stable operation of an NbTi fusion magnet (8 T) cannot be reconciled with an economic reactor schedule, since the radiation-induced increase in resistivity, i.e. the deterioration of the stabilizing capability, exceeds the permissible range (+25%) already after a fraction (~-14) of the plant lifetime, even if several room-temperature anneals are made. However, since fusion magnets of the next generation will operate at higher fields, the considerable decrease of magnetoresistivity with increasing (radiation-induced) zerofield resistivity may help to reduce costly alternative solutions such as an increase of stabilizer mass or amount of radiation shielding. Again, appropriate test programmes seem to be highly desirable. In conclusion, results on the ‘weakest spot’ of the fusion magnet will be summarized. Progress has been made with the assessment of spectral influences of the radiation environment on the tensile properties of glass-fibre-reinforced plastics by establishing scaling properties with the absorbed energy in the resins. Materials could be identified (especially a three-dimensionally reinforced bismaleimide) which showed remarkable radiation hardness and would be able to survive the lifetime dose of the magnet under tensile load by a considerable safety margin. These materials, which are all made with boron-free glass reinforcement, furthermore show good stability to low-temperature irradiation and to thermal cycling, again under tensile load conditions. In contrast, the intralaminar shear properties degrade rapidly and fall below acceptable limits at a fraction of the lifetime dose. However, testing is at a very early stage with regard to proper simulation conditions—and progress will be quite slow, because only one 5 K irradiation facility is operational worldwide, which—on top of that—cannot be used for several desirable mechanical test samples because of the extremely small cold bore (16 mm diameter). Hence, most of the current research is carried out at larger 77 K irradiation facilities, which should, however, be sufficient for the present purpose as can be concluded from the very small property changes of plastics between 4.2 and 77 K. Acknowledgments HWW wishes to express the following acknowledgments related to the radiation effects programme. This work is based on a series of diploma and PhD thesis research programmes carried out to a major extent at the Atomic Institute of the Austrian Universities during the last decade. I wish to acknowledge in particular the contributions of Dr F Nardai, Dr P A Hahn, Dr K Humer and Dr R Herzog as well as the continuous expert technical support by Mr H Niedermaier and Mr E Tischler. Thanks are due to the Federal Ministry of Science and Research and the Austrian Academy of Sciences for partial financial support. The cooperation of numerous colleagues and companies in Japan, Germany, France, Switzerland and the US is gratefully acknowledged. Parts of this manuscript have been published before by Hahn et al (1986) and Weber (1989). References Baker C C and Abdou M A (eds) 1980 STARFIRE—A Commercial Tokamak Fusion Power Plant Study ANL/FPP-80-1 Birtcher R C, Blewitt T H, Brown B S and Scott T L 1975 Proc. Conf. on Fundamental Aspects of Radiation Damage in Metals (Gatlinburg, TN, 1975) p 138 Blewitt T H and Arenberg C 1968 Trans. Japan. Inst. Met. 9 Supplement 226 Böning K, Fenzl H J, Welter J M and Wenzl H 1970 Die Kohler-Regel fur den longitudinalen Magnet- widerstand von neutronenbestrahltem Kupfer bei 4.6 K Phys. Kondens. Materie 12 72–80 Brown B S 1981 Radiation effects in superconducting fusion magnet materials. J. Nucl. Mater. 97 1–14 Brown B S, Blewitt T H, Scott T L and Klank A C 1974 J. Nucl. Mater. 52 215

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Capone D W II, Kampwirth R T and Gray K E 1986 High field properties of NbN ribbon conductors Adv. Cryogen. Eng. 32 659–61 Coltman R R 1982 Organic insulators and the copper stabilizer for fusion reactor magnets J. Nucl. Mater. 108&109 559–71 Dew-Hughes D and Jones R 1980 The effect on neutron irradiation upon the superconducting critical temperature of some transition metal carbides, nitrides and carbonitrides Appl. Phys. Lett. 36 856–9 Dietrich M and Dustman C H 1984 High field NbN superconductor on carbon fibers Adv. Cryogen. Eng. 30 683–97 Donohue M and Price M (eds) 1984 Mirror Advanced Reactor Study UCRL-53 480 University of California Ekin J W, Gavaler J R and Gregg J 1982 Effect of strain on the critical current and critical field of B1 structure NbN superconductors Appl. Phys. Lett. 41 996–8 Evans D and Morgan J T 1982 A review of the effects of ionising radiation on plastic materials at low temperature Adv. Cryogen. Eng. 28 147–64 Gavaler J R, Janocko M S, Patterson A and Jones C K 1971 Very high critical current and field characteristics of NbN thin films J. Appl. Phys. 42 54–7 Gray K E, Kampwirth R T, Murdock J M and Capone D W II 1988 Experimental study of the ultimate limit of flux pinning and critical currents in superconductors Physica C 152 445–55 Greenwood L R 1982 Neutron source characterization and radiation damage calculations for material studies J. Nucl. Mater. 108&109 21–7 Greenwood L R 1987 SPECOMP calculations of radiation damage in compounds Proc. 6th ASTM-Euratom Symp. on Reactor Dosimetry (Jackson Hole, WY, 1987) Greenwood L R and Smither R K 1985 SPECTER: Neutron Damage Calculations for Materials Irradiations ANL/FPP/TM-197 Argonne National Laboratory Gregshammer P, Weber H W, Kampwirth R T and Gray K E 1988 The effects of high-fluence neutron irradiation on the superconducting properties of magnetron sputtered NbN films J. Appl. Phys. 64 1301–6 Guess J F, Boom R W, Coltman R R and Sekula S T 1975 ORNL/TM-5187 Oak Ridge National Laboratory Guinan M W and van Konynenburg R A 1984 Fusion neutron effects on magnetoresistivity of copper stabilizer materials J. Nucl. Mater. 122&123 1365–70 Hahn P A, Guinan M W, Summers L T, Okada T and Smathers D B 1991 Fusion neutron irradiation effects in commercial Nb3Sn superconductors J. Nucl. Mater. 179–181 1127–30 Hahn P A, Hoch H, Weber H W, Birtcher R C and Brown B S 1986a Simulation of fusion reactor conditions for superconducting magnet materials J. Nucl. Mater. 141–143 405–9 Hahn P A, Weber H W, Guinan M W, Birtcher R C, Brown B S and Greenwood L R 1986b Neutron irradiation of superconductors and damage energy scaling of different neutron spectra Adv. Cryogen. Eng. 32 865–72 Hampshire R G and Taylor M T 1972 Critical supercurrents and the pinning of vortices in commercial Nb–60 at-% Ti J. Phys. F: Met. Phys. 2 89–106 Herzog R, Weber H W, Gray K E, Kampwirth R T, Miller D J and Murdock J M 1990 Radiation effects in superconducting NbN/NbAl multilayer films J. Appl. Phys. 68 6327–30 Herzog R, Weber H W, Kampwirth R T, Gray K E and Gerstenberg H 1991 Low temperature neutron irradiation of magnetron sputtered NbN films J. Appl. Phys. 69 3172–5 Horak J A and Blewitt T H 1975 Nucl. Technol. 27 416 Humer K, Tschegg E K and Weber H W 1992 Specimen size effect and fracture mechanical behavior of fiber reinforced plastics in the crack opening mode (mode I) Cryogenics 32 (ICMC Suppl.) 14–7 Humer K, Weber H W, Tschegg E K, Egusa S, Birtcher R C and Gerstenberg H 1994a Tensile and shear fracture behavior of fiber reinforced plastics at 77 K irradiated by various radiation sources Adv. Cryogen. Eng. B 40 1015–24 Humer K, Weber H W, Tschegg E K, Egusa S, Birtcher R C and Gerstenberg H 1994b Tensile strength of fiber reinforced plastics at 77 K irradiated by various radiation sources J. Nucl. Mater. 212–215 849–53 Humer K, Weber H W and Tschegg E K 1995 Radiation effects on insulators for superconducting fusion magnets Cryogenics 35 871–82 Kamakura H, Togano K, Dietderich D R and Tachikawa K 1989 Structure and superconducting properties of Nb3Al and Nb3(Al,Ge) tapes by high energy beam irradiation Mater. Res. Soc. Int. Symp. Proc. Series 6 (Pittsburgh, PA: Materials Research Society)

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Kasen M B 1986 High quality organic matrix composite specimens for research purposes J. Comp. Technol. Res. 8 103–6 Klabunde C C and Coltman R R 1984 The magnetoresistivity of copper irradiated at 4.4 K by spallation neutrons US Department of Energy Report DOE/ER-0113/3 Klabunde C E, Coltman R R and Williams J M 1979 J. Nucl. Mater. 85&86 385 Kohler M 1949 Z Phys. 126 495 Larbalestier D C 1980 NbTi alloy superconductors—present status and potential for improvements Adv. Cryogen. Eng. 26 10–36 Larbalestier D C 1981 Superconducting materials—a review of recent advances and current problems in practical materials IEEE Trans. Magn. MAG-17 1668–86 Lee P J, McKinnell J C and Larbalestier DC 1989 Microstructure control in high-Ti NbTi alloys IEEE Trans. Magn. MAG-25 1918–21 Lengeler B, Schilling W and Wenzl H 1970 Deviations from Matthiessen’s rule and longitudinal magnetoresistance in copper J. Low Temp. Phys. 2 59–86 Maix R K 1974 Fluβverankerung und kritische Stromdichten in Niob-Titan-Supraleitem Thesis Technical University of Vienna Matthiessen A 1860 Ann. Phys. Chem. 110 190 Munshi N A and Weber H W 1992 Reactor neutron and gamma irradiation of various composite materials Adv. Cryogen. Eng. A 38 233–9 Nakata K, Takamura S, Toda N and Masaoka I 1985 Electrical resistivity change in Cu and Al stabilizer materials for superconducting magnets after low temperature neutron irradiation J. Nucl. Mater. 135 32–9 Nardai F, Weber H W and Maix R K 1981 Neutron irradiation of a broad spectrum of NbTi super-conductors Cryogenics 21 223-33 Neal D F, Barber A C, Woolcock A and Gidley J A F 1971 Structure and superconducting properties of Nb–44 at-% Ti wire Acta Metall. 19 143–9 Nishijima S, Okada T, Miyata K and Yamaoka H 1988 Radiation damage of composite materials at cryogenic temperatures Adv. Cryogen. Eng. 34 35–42 Okada T, Fukumoto M, Katagiri K, Saito K, Kodaka H and Yoshida H 1988 Effects of neutron irradiation on the critical current of bronze processed multifilamentary Nb3Sn superconducting composites J. Appl. Phys. 63 4580–5 Parkin D M and Coulter C A 1979 Displacement functions of diatomic materials J. Nucl. Mater. 85&86 611–5 Pfeiffer I and Hillmann H 1968 Der EinfluB der Struktur auf die Supraleitungseigenschaften von NbTi50 und NbTi60 Acta Metall. 16 1429–39 Schmucker R 1977 The influence of plastic deformation of the flux line lattice on flux transport in hard superconductors Phys. Status Solidi b 80 89–97 Seeber B, Hermann P, Zuccone J, Cattani D, Cors J, Decroux M, Fischer O and Kny E 1989 Recent advances of Chevrel phase superconductors Mater. Res. Soc. Int. Symp. Proc. Series 6 (Pittsburgh, PA: Materials Research Society) Sekula S T 1978 Effects of irradiation on the critical currents of alloy and compound superconductors J. Nucl. Mater. 72 91–113 Simon N J, Drexler E S and Reed R P 1977 Shear/compressive tests for ITER magnet insulation Adv. Cryogen. Eng. 40 977–83 Sisman O and Bopp C D 1956 ASTM STP 208 119 Smathers D B, Marken K H, Lee P J, Larbalestier D C, McDonald N K and O’Larey P M 1985 Properties of idealized designs of Nb3Sn composites IEEE Trans. Magn. MAG-21 1133–6 Snead C L 1986 private communication Snead C L and Parkin D M 1975 Effect of neutron irradiation on the critical current of Nb3Sn at high magnetic fields Nucl. Technol. 29 264–7 Snead C L and Suenaga M 1980 Synergism between strain and neutron irradiation in filamentary Nb3Sn conductors Appl. Phys. Lett. 37 659–61 Söll M, Wipf S L and Vogl G 1972 Change in critical current of superconducting NbTi by neutron irradiation IEEE Publication 72 CH 0682-5-TABSC pp 434–9

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Spieβberger S, Humer K, Tschegg E, Weber H W and Gerstenberg H 1996 Bending and interlaminar shear strength of fiber reinforced plastics at 77 K after room and low temperature reactor irradiation Adv. Cryogen. Eng. 42 105–12 Spindel A, Reed R P, Tupper M, Darr J and Pollock D 1994 Low temperature electron irradiation of insulating films and adhesives Adv. Cryogen. Eng. 40 1169–76 Sweedler A R, Snead C L and Cox D E 1979 Irradiation effects in superconducting materials Treatise on Materials Science and Technology vol 14, eds T Luhman and D Dew-Hughes (New York: Academic) pp 349–26 Tschegg E, Humer K and Weber H W 1991 Influence of test geometry on tensile strength of fibre reinforced plastics at cryogenic temperatures Cryogenics 31 312–8 Tschegg E K, Humer K and Weber H W 1993 Fracture tests in mode I on fibre-reinforced plastics J. Mater. Sci. 28 2471–80 Tschegg E K, Humer K and Weber H W 1995 Mode II fracture tests on fibre-reinforced plastics J. Mater. Sci. 30 1251–8 Weber H W 1982 Neutron irradiation effects on alloy superconductors J. Nucl. Mater. 108&109 572–84 Weber H W 1986 Irradiation damage in superconductors Adv. Cryogen. Eng. 32 853–64 Weber H W 1989 Neutron damage of superconductors for fusion magnets Kerntechnik 53 189–96 Weber H W, Böck H, Unfried E and Greenwood L R 1986 Neutron dosimetry and damage calculations for the TRIGA Mark-II reactor in Vienna J. Nucl. Mater. 137 236–40 Weber H W and Crabtree G W 1992 Neutron irradiation effects in high-Tc single crystals Studies of High Temperature Superconductors vol 9, ed A V Narlikar (New York: Nova) pp 37–79 Weber H W, Gregshammer P, Kampwirth R T and Gray K E 1989 High-fluence neutron irradiation of superconducting NbN films Mater. Res. Soc. Int. Symp. Proc. Series 6 (Pittsburgh, PA: Materials Research Society) p 57 Weber H W, Khier W, Wacenovsky M and Hoch H 1988 Radiation-induced changes of critical fields in NbTi superconductors Adv. Cryogen. Eng. 34 1033–9 Weber H W, Kubasta E, Steiner W, Benz H and Nylund K 1983 Low temperature neutron and gamma irradiation of glass fiber reinforced epoxies J. Nucl. Mater. 115 11–5 Weber H W, Nardai F, Schwinghammer C and Maix R K 1982 Neutron irradiation of NbTi with different flux pinning structures Adv. Cryogen. Eng. 28 239–335 Weber H W and Tschegg E 1990 Test program for mechanical strength measurements on fiber reinforced plastics exposed to radiation environments Adv. Cryogen. Eng. 36 869–75 West A W and Larbalestier D C 1980 Transmission electron microscopy of commercial filamentary NbTisuperconducting composites Adv. Cryogen. Eng. 26 471–8 West A W and Larbalestier D C 1982 Alpha-titanium precipitation in NbTi alloys Adv. Cryogen. Eng. 28 337–44 Yasuda J, Hirokawa T, Uemura T, Iwasaki Y, Nishijima S, Okada T, Okuyama H and Wang Y A 1989 Cryogenic and radiation resistant properties of three dimensional fabric reinforced composite materials New Developments in Applied Superconductivity ed Y Murakami (Singapore: World Scientific) pp 449–54

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D2 Dielectric properties of cryogens

J Gerhold

D2.0.1 Introduction Reliable electrical insulation is essential for the successful use of superconductivity. High voltages have to be controlled in power applications, and any field adjustment in a magnet causes voltage pulses. In particular, adverse conditions may prevail in the case of a magnet quench. Helium, hydrogen and nitrogen are excellent insulating fluids. Many insulation systems in classical superconducting applications are based on helium 4, either in its liquid state (LHe), in its supercritical state (SHe) or simply using the cold gas (GHe). Gaseous or liquid hydrogen (GH2 , LH2 ) and nitrogen (GN2 , LN2 ), respectively, will be used in future high-temperature superconducting (HTS) applications. Neon has become of some interest for machines provided supply can be guaranteed at a reasonable price. Dielectric data are scarce at present but some extrapolation can be made by analogy. Any cryogenic insulating fluid has to meet four main requirements: (i) (ii) (iii) (iv)

it must show a very high resistivity, i.e. the fluid must contain very few free charge carriers dielectric losses must be extremely low in the case of a.c. applications the dielectric strength must be fairly high the fluid must be a well-behaved insulator; degradation from extrinsic effects such as stressed volume or interface phenomena must be precalculable. Accommodation with solid insulators is also of concern.

The cryogenic fluids meet all these points when utilized in an adequate thermodynamic state. A high density is mandatory. However, degradation can be severe and some precautions are needed against it. D2.0.2 Dielectric properties The fluids are normally free from charge carriers. Leakage currents can be caused only by ionizing radiation, which may be shielded in practice inside a cryostat. Nonself-maintained discharge current densities are below 10− 9 A m− 2. Extremely high resistivities have been found in cryogenic liquids for instance, e.g. >1016 Ω m−1 in LHe. Leakage currents are often not detectable unless charge carriers are emitted from the electrodes; this needs a high local field strength of more than 107 V m−1.

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The molecules of cryogenic fluids are nonpolar. This makes the evaluation of the relative permittivity εr very easy, since the Clausius-Mossotti formula can be used

where Nυ is the density number (i.e. the number of molecules per unit volume), ε0 the permittivity of space and αi the polarizability of the molecule. The polarizability does not depend on the fluid state. Data for αi /3ε0 are indicated in table D2.0.1. It is evident that permittivities in the gasous state are very close to unity since Nυ is limited. The permittivity for the liquids, εr L , at the normal boiling temperature Tn b p is also given in table D2.0.1. There is a strong correlation between the data, because the internal van der Waals forces which are responsible for liquefaction are due to fluctuating dipole moments (Gerhold 1987). Table D2.0.1. Polarizability and permittivity of cryogenic fluids.

Liquids may be inhomogeneous when bubbles are present. Any stressing electrical field within the bubble increases from its mean value by an order of εr L : 1. This effect must be borne in mind, especially since vaporization is easy in cryogenic liquids. A field distortion is also caused by frozen foreign gas particles, especially in LHe. These particles are pulled to high stress points by dielectrophoretic forces. Dealing with solids is more difficult. Solid insulators show a permittivity from 2 up to more than 5 (see chapter F7). A considerable permittivity mismatch must be put up with, resulting often in critical local field distortions. This favours partial discharges. On the other hand, nonpolar molecules are a prerequisite for very low dielectric losses. Dielectric absorption is not relevant below some tens of kHz. In fact, no losses could be found in cold GHe, GH2 or GN2. However, surprisingly high losses have often been measured in the liquids, even at power frequencies, i.e. 50–60 Hz. The reason is not fully clear. Predischarges have been claimed sometimes, but these can be excluded definitely in LN2 for instance. The most probable source may be foreign impurity particles which are charged during electrode striking. The critical stress for loss increase is often of the order of 5 MV m−1, as can be seen from figure D2.0.1. Losses may vary with time (Jefferies and Mathes 1970). It is easy to suppress the losses by using intermediate solid barriers. For instance, lapped tape insulation packages show lower losses than would be obtained from the bulk impregnating fluid ( Kahle and Frosch 1987). D2.0.3 Dielectric strength The dielectric strength of insulators is never purely a function of materials. ‘Intrinsic strength’ is fictitious; however, it can be taken as a kind of asymptotic approximation. Materials with a low strength may come close to this approximation even under practical conditions. For instance gases show only limited degradation until pressurized. Paschen’s law is valid in uniform fields. In general, breakdown or discharge voltages can only be measured as a function of the overall electrode—fluid system. The corresponding

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Figure D2.0.1. Dielectric losses in liquefied gases at power frequencies; the liquids are close to the respective normal boiling point.

strength Eb is often simply defined by the breakdown voltage to gap length ratio, Vb /d. Figure D2.0.2 shows a test device for near-uniform field breakdown measurements in a cold fluid. D2.0.3.1 Gas breakdown To break down a gas, the stressing field E must intersect a critical number of gas molecules. Electron multiplication must reach a critical amount k so that

where α is the first Townsend ionization coefficient (Dakin et al 1977). None of the cryogenic gases is electron attaching. Multiplication of random electrons emitted from the cathode can only take place by inelastic collisions with the much heavier molecules or atoms. The ionization energy eVi must be ‘collected’ during free flights with a mean free path length λ (e is the unit charge; Vi the ionization voltage). The cumulative mean free path voltage Vcu is accumulated in n precursory random flights with elastic collisions so Vc u = nEλ. Figure D2.0.3 illustrates a random electron moving as energy is collected. Vc u may be a multiple of the mean free path voltage Eλ (Gerhold 1987). This can be interpreted in terms of an electron temperature according to kTe l = Vc u e where k is the Boltzmann constant. Te l is much higher in the breakdown regime than the gas temperature, e.g. of the order of 104 K in nitrogen or

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Figure D2.0.2. Test cryostat for breakdown experiments: S indicates the near-uniform field electrode system; the gap length d is set by means of the microscrew device M; C is the cooled tube spiral; T the temperature sensor. Reproduced from Gerhold (1972) by permission of Elsevier Science Ltd.

Figure D2.0.3. A random electron moving in helium gas. The broken line shows the actual electron path between collisions; the length of arrow indicates the kinetic energy of an electron.

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hydrogen and 105 K in helium. The gas temperature has practically no influence on electron movement or avalanching. The electron mean free path cannot be measured directly. However, λ is inversely proportional to the density number Nυ which is well known for any particular thermodynamic state. The first Townsend ionization coefficient is therefore usually written as

Figure D2.0.4 indicates the amount of ionization versus

which is proportional to ( Eλ )−1; U indicates the total gap voltage.

Figure D2.0.4. Impact ionization, α/Nυ , versus the density number to field strength ratio, Nυ /E.

Equation (D2.0.2) reads simply exp(αd ) = k in a uniform field; Eλ is now constant across the gap. This implies a constant α value, and relates α to d −1. The intersection number d/λ ∝ dNυ is now obviously the breakdown controlling term for any actual α, see equation (D2.0.3). The general Paschen law claims therefore that the breakdown voltage Vb , i.e. the summed intersection voltages, is a pure function of the density number-spacing product Nυ d. Figure D2.0.5 indicates the Paschen curves for nitrogen, hydrogen, neon and helium. The accuracy of these results in the cryogenic temperature regime has been confirmed experimentally in nitrogen and helium, respectively, by many workers in the field. The gases can be assumed to be very pure at low temperatures. However, the Paschen curves indicate a typical parallel shift. The molecular gases can limit Vc u very effectively by means of inelastic collisions leading to rotation or vibration of atoms in a molecule. The monatomic and very perfect helium-4 atom on the other hand cannot take any energy from electrons much below Vi so Vc u becomes very important. Therefore, helium is the weakest of all gaseous insulators at ambient conditions. A strong correlation with the normal boiling point is again evident. Only in the left-hand part of the Paschen curve, i.e. where the total number n of precursory random flights with elastic collisions is limited, does helium break down at a higher stress since d/λ = n. The

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Figure D2.0.5. Paschen curves for nitrogen, neon, hydrogen and helium. After Gerhold (1987).

electrons cannot be ‘thermalized’. This regime provides a steady transition to vacuum breakdown; it is sometimes called a ‘semivacuum’. An important density increase takes place when the gases are cooled down. Near Tn b p , nitrogen is concentrated by a factor of almost 4 and helium by about 70. The Paschen curves for an identical spacing, e.g. 1 mm, come close together (Gerhold 1987). The dielectric strength of the vapours is of the order of that of SF6 , a gas commonly used in pressurized gas insulated systems. Example 1 A superconducting coil has to be tested during assembly. A quench condition is being simulated. A voltage stress of 10 kV has been found for the most critical gap d = 2 mm, this gap being filled with GHe at 40 K–0.5 MPa during a real quench. The Paschen curve yields a breakdown voltage of 20 kV for the helium density number times spacing product Nυd = 4 × 1024 m−2 a safety margin of two seems reasonable against the quench voltage. The simulation test is based on GN2 , i.e. on an Nυd -product value of 1.2 x 10−23 m−2. A nitrogen density number of 6 × 1025 m−3 is required. This yields a pressure of almost 0.23 MPa at ambient temperature (Nυ o = 2.66 × 1025 m−3 at 20°C-0.1 MPa). Paschen’s law is no longer valid and Vb remains below the Paschen level if the field strength at the cathode exceeds 107 V m−1. This is typical for very high gas densities. Additional electron emission or micro-gas discharges may be responsible. The effect is well known in nitrogen and hydrogen at ambient temperature; it is also typical in cold GHe. Such a high stress breakdown is weak-link dominated. Area effects have to be watched. Figure D2.0.6 shows the general course in cold GHe. Increasing statistical scattering will also be encountered. Of course, actual electrode surface quality is important as an extrinsic parameter (Dakin et al 1977). Example 2 A voltage of 10 kV across a 1 mm gap filled with GH2 requires a density number Nυ = 1.4 × 1026 m−3. The corresponding stress is just 107 V m−1. This can be achieved by, for instance, cooling the nonpressurized gas down to 55 K. Thus, hydrogen is sensitive to electrode surface conditions at lower temperatures where

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Figure D2.0.6. Uniform field breakdown in high-density helium: 1—extrapolated Paschen curve; 2—breakdown in a 1 mm gap. The stressed area is 28 cm2 . After Meats (1972).

Eb > 107 V m−1. Nonpressurized helium on the other hand may be cooled down to 4.5 K without severe degradation. Breakdown in very dense gases is sensitive to the gap configuration, too. High-stress regions are normally shielded by space-charges arising from the corona. Nitrogen and hydrogen are nonattaching gases, i.e. electrons cannot be caught easily by the molecules. However, negative ions of limited lifetime can occur, and negative ion clouds can shield high-stress regions in a nonuniform gap, for instance the point electrode in figure D2.0.7. GHe on the other hand builds up no negative ions at all, and negative space-charge shielding normally is impossible. The corona losses heat up the stressed gas and decrease the local density; gas thinning may be moderate at ambient temperature since heat diffusion is very effective. However, corona is often suppressed in pressurized gases so breakdown is immediately initiated above the threshold voltage. Corona is much more doubtful at low temperatures. Thinning of the surrounding gas is much more severe and the corona losses must be cooled by a refrigerator. Corona may also be very harmful to the nonself-healing solids in a composite insulation. Polarity effects due to weak negative ion shielding compound the problem. Figure D2.1.8 illustrates divergent field corona and breakdown in nitrogen, and figure D2.0.9 shows the same effects in helium gas. A higher density leads to higher voltages. However, corona onset voltages Vc often show a distinct saturation for larger gap lengths. The breakdown voltages increase regularly with gap length when the point is positive. In the case of a negative point, a delayed increase is found in a near-saturated vapour (Hara et al 1990). This effect may be attributed to electron trapping. Corona onset can be calculated in general from equation (D2.0.2) if the field strength course is known as well as k; k may be derived from uniform field breakdown data. The adverse effect of divergent fields also leads to an increasing sensitivity against solid contaminants. Particles are a very common source of degradation in any compressed gas insulation system and must be eliminated. A special kind of corona is partial discharge (PD) in fluid-filled voids. These voids occur easily within a solid/fluid compound, e.g. an impregnated taped insulation package (see, for instance, chapter F7). Voids

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Figure D2.0.7. (a) Test cryostat for a nonuniform field corona and (b) breakdown experiments with a point-plane electrode system. Reproduced from Hara et al (1990) by permission of Elsevier Science Ltd.

also may be formed by cracks in an GFRP (glass fibre reinforced plastic) insulation for instance. PD onset may be evaluated from the Paschen curve according to the actual density number and the void depth. However, PDs do not result in insulation breakdown unless the solid is destroyed by aging effects. D2.0.3.2 Liquid breakdown Condensation of molecules can be greatly increased by liquefying the gas. Breakdown is still being initiated in gaseous volume; a precursory vapour bubble formation has to be assumed. Some workers have correlated liquid breakdown directly with the vapour strength since vaporization of cryogenic liquids needs only a small heat input. This is especially true in LHe. However, this idea seems to be too simple a picture. Liquid breakdown as a result of the vapour strength is only to be anticipated in the case of considerable external heat input. In a slightly subcooled liquid, a heat pulse must be fed in to generate a vapour zone by electrical energy dissipation; the local field strength must be >108 V m−1. The generation of heat pulses and initiating bubbles via electrical energy dissipation needs the motion

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Figure D2.0.8. Corona onset voltage, Vc , and sparking voltage, Vb , in a point to plane gap for gaseous nitrogen at 78 K: (a) Nυ = 2.28 × 1025 m–3 for a negative point; (b) Nυ = 9.13 x 1025 m–3 for a negative point; (c) Nυ = 2.28 × 1025 m–3 for a positive point; (d) Nυ = 9.13 × 1025 m–3 for a positive point. After Hara et al (1989).

of charge carriers in the stressing field. Electron emission from the cathode is an obvious phenomenon so the cathode surface condition is a very important parameter. No interference with the anode takes place in LHe; this has been verified by many experiments. The electrons cannot penetrate the bulk liquid immediately, because of a strong repulsive force. However, field ionization near the anode has been claimed to be a concurrent mechanism in LN2 . The anode conditions may be relevant for a large spacing. No detailed information is known in the case of LH2 and LNe breakdown. Macroscopic breakdown fields of the order of 40–180 MV m–1 are typical in any of the nonboiling cryogenic liquids under well-controlled laboratory conditions (Gallagher 1975). These high strength values also can be relevant in practice, especially with pulse voltages (Bobo et al 1987). Figure D2.0.10 gives measured impulse breakdown voltages versus spacing in a plane-sphere gap. To guarantee a high strength level at d.c. or a.c. stress, any severe degradation has to be prevented: careful filtering out of any field-distorting particles should be provided for instance. The fluids must not be accepted with arbitrary ‘technical purity’ from the supplier; these fluids may contain frozen impurity gases and often show a considerably degraded breakdown; compare figure D2.0.11 with figure D2.0.10. The data shown in figure D2.0.11 can also be used in the case of d.c. stress in fluids with technical purity. Degradation in LHe may often be more severe than that in LH2 or LN2. The behaviour of spontaneous bubbles in a bulk boiling liquid is under discussion. These bubbles may be produced by external heat input. The field which stresses the liquid is enhanced within the bubble as mentioned already. Bubble discharge can be calculated by the Paschen curve. The discharge fields in helium, hydrogen and nitrogen come close together in extended bubbles: > 20 kV mm–1 at 0.1 mm size

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Figure D2.0.9. Corona onset voltage, Vc , and sparking voltage, Vb , in a point to plane gap; gaseous helium: (a) Nυ = 1.956 × 1026 m−3 at 40 K for a negative point; (b) Nυ = 2.710 x 1027 m−3 at 4.2 K for a negative point; (c) Nυ = 1.956 × 1026 m−3 at 40 K for a positive point; (d ) Nυ = 2.710 x 1027 m−3 at 4.2 K for a positive point. After Hara et al (1990).

Figure D2.0.10. 1/50 µs impulse breakdown with a plane-sphere gap; liquids are close to the respective normal boiling point. After Fallou (1975).

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Figure D2.0.11. A.c. breakdown in a 62.5 mm diameter sphere to sphere gap; the liquids are close to the respective normal boiling point. After Fallou et al (1971).

and ∼13.6 kV mm−1 at 1 mm size for the normal boiling point (Gerhold 1987). Surface tension may enhance the internal vapour pressure well above the external liquid pressure in the case of microbubbles. The high density number then causes a higher strength to the right of the Paschen minimum. It is not clear whether a discharge can be initiated within a nonelectrode-adjacent bubble. However, electrode approach is impeded by strong repulsive dielectrophoretic forces in any divergent field. These forces can be calculated from

where a is the bubble radius. Fd is often much higher than buoyancy by orders of magnitude, even in LHe with its very low permittivity. It depends on the actual gap geometry whether bubbles collect in a stagnation region. Figure D2.0.12 shows a nice photograph taken in LN2; the nominal maximum electrode surface stress is about 15 kV mm−1. No bubbles can be observed near the highest-stressregion. The liquid pressure has a strong influence on breakdown. Figure D2.0.13 shows the typical normalized course in LHe, LH2 and LN2. A kind of saturation can often be observed. However, no extrapolation should be made to the very low-pressure region. Breakdown normally does not continue down to the saturated vapour level. This has been investigated especially in LHe in view of its relevance for superconducting magnets. Even in saturated LHeII with its extremely low vapour strength, liquid-breakdown strength is typically of the order of >20 MV m−1. Area and gap effects are an important general source of degradation. Increasing the stressed area, for instance, generates more active sites for discharge initiation. In LHe, only the cathode area is of relevance. The area effect has been directly demonstrated in LN2 and LHe. A more detailed evaluation could be carried out for LHe with sphere gaps. The stressed area is synonymous with the sphere radius times the spacing product, rd (Gallagher 1975). Eb ∝ (rd)−0.33 has been found to cover more than three orders of area magnitude. The critical field strength Eb must be taken with its maximum value at the electrode surface. However, there may be an inherent spacing effect. For instance, Eb ∝ d −0.2 has been found in LHe with bubbles in a plane—plane gap, the same formula also being valid in LN2 .

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Figure D2.0.12. Vapour bubbles in a plane to cylinder gap for boiling liquid nitrogen; V = 11 kV, d = 0.8 mm. Reproduced by permission of M Hara, Kyushu University, Fukuoka, Japan.

It is likely that area as well as gap length may be a source of degradation in practical insulation systems, the bulk fluid being involved via particles or bubbles which can come close to a stressed electrode surface. A more general formula should therefore indicate a volume degradation according to (Gerhold et al 1994)

The relevant critical volume υc may include all of the volume with a stress higher than 80% of the nominal field strength Eb . Eb = E(x = 0) refers to the maximum field at the cathode. The exponent v may be taken as 0.1 in the case of nonboiling LHe. The total gap voltage Ub can be found from

Example 3 The coil already mentioned is stressed across the 2 mm gap with total active area of 1 x 106 mm2 and a running voltage of 5 kV rms. Testing must be performed at 10 kV rms and follow a particular regime. The active volume of 2 × 106 mm3 is then stressed with ∼14 kV crest voltage. Figure D2.0.11 yields a crest breakdown voltage of 46 kV for a 2 mm gap. However, an active electrode area of A = 60 mm2 must be assumed; 80% of the nominal strength Vb /d corresponds to a surface angle of 8°. Thus, the stressed volume must be taken as Ad = 120 mm3 . Assuming a degradation exponent v = 0.1 in equation (D2.0.5), which fits the LHe characteristics in figure D2.0.11, yields a strength decrease of as low as 38% within the coil. Thus, a crest breakdown voltage of 17 kV comes out as a safe figure. However, even filling the gap completely with saturated vapour would result in a crest breakdown voltage of >20 kV, taking into account the particular density number times spacing product value of 5.1 × 1024 m−2. Breakdown below the Paschen limit seems very unlikely since degradation in a gas at

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Figure D2.0.13. Breakdown in pressurized liquids. After Gerhold (1979).

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only 107 V m−1 stress is marginal. Therefore, testing the coil with 10 kV rms may be reasonable, without taking special care about the LHe purity. The field configuration is another important extrinsic gap parameter. LHe breakdown is controlled by the cathode field in moderately divergent field gaps. The often low cathode area in a divergent field must be borne in mind. For point to plane configurations, an empirical formula

has been found in boiling LHe. In the case of normal boiling LN2 the relation

may be used (note: d in mm; Vb in kV). These formulae may be used up to d < 20 mm. Corona may arise before breakdown, especially when stressing the gap with an a.c. voltage. Note that negative ion shielding also occurs in LHe since the electrons are virtually trapped (Kara et al 1990). No formula is known for LH2 or LNe. Electrode surface roughness is also critical. The divergent field arguments may be valid for very rough surfaces, e.g. in the >10 µm regime. This is close to a multiple-needle configuration. Degradation by a factor of almost three against a smooth surface has been claimed in the case of LHe for instance. Note that surface roughness can increase drastically when an oxidized electrode is cooled down, due to oxide cracking. Careful preparation before cooling equipment down does not guarantee a matching electrode surface during operation. Finally, statistical scattering has to be considered. Breakdown frequency distributions are often skewed, or may be double headed. Estimation of a practically ‘no-breakdown’ condition needs some care therefore. Standard deviations >20% can often arise. However, scattering is lowered when severe degradation is dominant. Less than 10% may be taken as an upper limit in large-gap systems. D2.0.3.3 Breakdown in supercritical fluid Breakdown in the supercritical state is somewhat in between gas breakdown and liquid breakdown. Density numbers are of the liquid order. On the one hand bubbles cannot be generated but on the other hand density fluctations arise easily. Very few fluids have been investigated. Helium is the best known candidate, since SHe is often used in practice, e.g. in superconducting magnets or superconducting transmission lines. The supercritical state may be out of practical scope in hydrogen, neon and nitrogen, because of the very high pressures needed. There is a steady crossover from GHe breakdown to SHe breakdown to saturated LHe breakdown. This is illustrated in figure D2.0.14 by breakdown frequency distributions. These data have been measured under well-controlled conditions. Obviously, helium density is the main controlling factor in a superficial view only. It is evident that the mean breakdown values give very incomplete information. In LHe, breakdown may scatter to considerably lower voltages than in SHe, in spite of similar mean values. Any diagram such as that shown in figure D2.0.15, which has been often used as a standard (Meats 1972), suffers from this lack of information. Area and gap effects which are involved as in LHe breakdown make this standard even more arbitrary. However, the contour plot may be very useful to estimate the relative general course of SHe breakdown when varying pressure and temperature. SHe breakdown has also been investigated extensively with divergent field gaps. A corresponding contour plot shows the sparking voltages in a negative point to plane gap (see figure D2.0.16). Scattering up to 40% can be encountered in the high-density range. Sparking voltages increase with spacing in a similar manner as in LHe so equation (D2.0.6) may be used again for an estimation.

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Figure D2.0.14. Breakdown frequency distributions in cold helium with a 0.5 mm gap: 1—GHe, Nυ = 2.73 × 1027 m– 3; 2—SHe, Nυ = 1.20 X 1028 m– 3; 3—LHe, near the normal boiling point, Nυ = 2.0 × 1028 m– 3. After Gerhold (1988).

Figure D2.0.15. A contour plot of uniform field breakdown voltages in helium with a 1 mm gap and a stressed area of 28 cm2. Reproduced by permission of Meats (1972).

The cryogenic fluids can be used profitably as basic insulants. Resistivities are extremely high, and the density-dependent permittivities are modest. Dielectric losses may be caused by contaminant particles but can be eliminated in multiple-barrier packages. Paschen’s law gives a precise description for gas breakdown in the moderate-stress regime. All gaseous fluids show a reasonable strength at cryogenic temperatures. Breakdown in highly densified or in liquefied fluids must be seen as a phenomenon of the overall

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Figure D2.0.16. A contour plot of divergent field breakdown voltages in helium with a 1 mm negative point to plane gap. Reproduced from Hara (1989) by permission of The Cryogenic Society of Japan.

electrode-fluid system. Near-intrinsic breakdown strength is very high, but degradation due to various weak-link effects prevails in practice. This is similar to the results in common insulating fluids such as pressurized SF6 or oil. Semi-empirical formulae and graphs must be referred to. This is especially true in divergent fields where corona may preceed a breakdown. Corona stabilization should be used with care because of its inherent heat production and the premature aging of adjacent solid insulators. Hydrogen corona must be considered with the utmost caution; decomposition of adjacent solids might cause oxygen production which can become extremely dangerous. References Bobo J C, Poitevin J and Nithart H 1987 Tenue dielectrique des isolants a 4.2 K CICRE Symp. 05-87 (Vienna, 1987) (Paris: CIGRE) pp 100–4 Dakin T W, Gerhold J, Krasucki Z, Luxa G, Oppermann G, Vigreux J, Wind G and Winkelnkemper H 1977 Breakdown of Gases in Uniform Fields: Paschen Curves for Nitrogen, Air, Sulfur Hexafluoride, Hydrogen, Carbon Dioxide and Helium (Paris: CIGRE) Fallou B 1975 A review of the main properties of electrical insulating materials used at cryogenictemperatures Proc. 5th Int. Conf. on Magnet Technology (Rome, 1975) (Frascati: Laboratori Nationali del CNEN) pp 664–8 Fallou B, Bobo J C, Burnier P and Carvounas E 1971 Les isolants electriques aux tres basses temperatures Congres SFE (Nice, 1971) (Societe Francaise des Electriciens) Gallagher T J 1975 Simple Dielectric Liquids (Oxford: Clarendon) Gerhold J 1972 Dielectric breakdown of helium at low temperatures Cryogenics 12 370–6 Gerhold J 1979 Dielectric breakdown of erogenic gases and liquids Cryogenics 19 571–84 Gerhold J 1987 Dielectric strength of gaseous and liquid insulants at low temperatures CIGRE Symp. 05-87 (Vienna, 1987) (Paris: CIGRE) pp 100–1 Gerhold J 1988 Helium breakdown near the critical state IEEE Trans. Electr. Insul. EI-23 765–8 Gerhold J, Hubmann M and Telser E 1994 Gap size effect on liquid helium breakdown Cryogenics 34 579–86 Hara M 1989 Electrical insulations in superconducting apparatus Cryogen. Eng. Japan 24 72–81

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Hara M, Suehiro J, Matsumoto H and Kaneko T 1989 Breakdown characteristics of cryogenic gaseous nitrogen and estimation of its electrical insulation properties IEEE Trans. Electr. Insul. EI-24 609–17 Hara M, Suehiro J and Matsumoto H 1990 Breakdown characteristics of cryogenic gaseous helium in uniform electric field and space charge modified non-uniform field Cryogenics 30 787–94 Jefferies M J and Mathes K N 1970 Dielectric loss and voltage breakdown in liquid nitrogen and hydrogen. IEEE Trans. Electr. Insul. EI-5 85–91 Kahle M and Frosch P 1987 Application problems of LN2 /paper-insulation CIGRE Symp. 05-87 (Vienna, 1987) (Paris: CIGRE) pp 100–6 Meats R J 1972 Pressurized-helium breakdown at very low temperatures Proc. IEE 119 760–6 Further reading Gerhold J 1989 Breakdown phenomena in liquid helium IEEE Trans. Electr. Eng. EE-24 155–66 Gerhold J 1994 Liquid helium breakdown as a function of temperature and electrode roughness IEEE Trans. Dielectr. Electr. Insul. DEI-1 432–9 Goshima H, Hayakawa N, Hikita M and Okubo H 1995 Weibull statistical analysis of area and volume effects on the breakdown strength in liquid nitrogen IEEE Trans. Dielectr. Electr. Insul. DEI-2 385–93 Hara M, Honda K and Kaneko T DC electrical breakdown of saturated liquid helium at 0.1 MPa in the presence of thermally induced bubbles Cryogenics 27 567–76 Hara M and Kubuki M 1990 Effect of thermally induced bubbles on the electrical breakdown characteristics of liquid nitrogen Proc. IEE 137 209–16 Ishii I and Noguchi T 1979 Dielectric breakdown of supercritical helium Proc. IEE 126 532–6 Meek J M and Craggs J D 1978 Electrical Breakdown of Gases ( New York: Wiley ) Menon M M, Schwenterly S W, Gauster W F, Kernohan R H and Long H M 1976 Dielectric strength of liquid helium under strongly inhomogeneous field conditions Advances in Cryogenic Engineering vol 21 ed K D Timmerhaus and D H Weitzel (New York: Plenum) pp 95–101 Nelson R L 1974 Dielectric loss of liquid helium Cryogenics 24 345–6 Peier D 1979 Breakdown of LN2 by field induced microbubbles J. Electrostat. 7 113–22 Suehiro J, Amasaki K, Matsuo H, Hara M and Gerhold J 1994 Pulsed electrical breakdown in liquid helium in the ms range IEEE Trans. Dielectr. Electr. Insul. DEI-1 403–6 Yoshino K, Fujii H, Takahashi R, Inuishi Y and Hayashi K 1979 Electrical breakdown in cryogenic liquids J. Electrostal. 7 103–12

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D3 The thermodynamics of cryocycles

J Weber

D3.0.1 Introduction In order to cool a material it is necessary (i)

to withdraw an amount of heat from this material at a temperature level T which is below ambient temperature TU (ii) to transport this amount of heat to ambient level by means of a refrigerant (iii) to reject the heat at ambient level to the surroundings. This procedure is called production of cold and needs a convertible type of energy, e.g. work. According to Carnot, for stationary processes this work amounts to

and is independent of the method of production. Convertible types of energy, such as potential and kinetic energy and, especially, electrical energy can be expressed with the generic term ‘exergy’. By means of a temperature—entropy (T—s) diagram, as shown in figure D3.0.1, the exergy of an open, stationary process can easily be derived (Frey and Haefer 1981). By definition, the exergy of a system is the work W which is necessary to convert the system from ambient condition to the desired final conditions. Thereby, as described above, the exchange of heat with the surroundings is only possible at ambient temperature. As a consequence of this restriction, the change of the specific entropy ds = 0 for all temperatures T not equal to TU. Starting from condition U it is therefore necessary to go via condition 1—characterized by T1 = TU and s1 = s2—if one intends to reach condition 2. The first law of thermodynamics defines the way from U to 1 as

From the second law of thermodynamics it follows that

Applying the first law of thermodynamics to the way from 1 to 2 leads to

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Figure D3.0.1. T—s diagram for the He heating (compression) process.

as per restriction q12 = 0. A summary of equations ( D3.0.2 )-( D3.0.4 ) results in the following expression of the specific exergy

With equation (D3.0.5) it is now possible to calculate the reversible work which is necessary to convert a system of mass M from any condition 1 to any condition 2. Noticing the restriction that heat can be exchanged with the surroundings at ambient temperature only, the system has to pass condition U on its way from condition 1 to 2. From 1 to U, there is a gain in work of w1u = e1 , whereas the work wu 2 = e2 has to be performed from U to 2. The total reversible work needed amounts to

Substituting condition 1 by U (surroundings) and condition 2 by L ( boiling liquid ), one will get the minimum work (usable exergy) which is necessary to liquefy a mass flow m in the case of a lossless, reversible process. The quotient of this value and the actual necessary work Wc defines the thermodynamic efficiency, which is generally used as a criterion of the effectiveness of liquefiers

The thermodynamic efficiency of a refrigerator, which offers cold at constant temperature is expressed by the known equation of the Carnot cycle

D3.0.2 Refrigeration processes D3.0.2.1 The Joule-Thomson process Refrigeration technology started in 1895 with the liquefaction of air by Carl von Linde. Until that time, gases which were to be liquefied had to be compressed to high pressures and then throttled in a valve

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(isenthalpic expansion). However, gases with an enthalpy that is higher at final compression pressure and ambient temperature than the enthalpy of the saturated vapour at expansion pressure could not be liquefied using this method. Carl von Linde arranged a heat exchanger downstream of the cycle compressor, thereby precooling the compressed air before throttling. The air which was not liquefied in the throttle valve was used as precooling refrigerant. A refrigerator corresponding to this concept, is shown schematically in figure D3.0.2. The gas is compressed at ambient temperature with compressor C and by means of cooling air or cooling water the main part of the heat of compression is carried away to the atmosphere. The high-pressure gas is fed to the heat exchanger E1 thereby cooling down to condition 2 in counterflow to the low-pressure return gas. Finally, the high-pressure gas is throttled to condition 3 within the Joule—Thomson (JT) valve V.

Figure D3.0.2 A scheatic view of cryogenic plants according to the JT process.

In the case of liquefaction mode and stationary operation, the mass flow rate mL , is fed into the process at point Z, liquefied and decanted from vessel D. In condition 4, the part which has not been liquefied returns to heat exchanger E1, warms up in counterflow to the high-pressure gas and leaves the plant in condition 5. The balance equations are as follows

Using these two equations one can calculate the fraction of gas which liquefies in the throttle valve

In refrigeration mode, all liquid produced by throttling the high-pressure gas in valve V is evaporated due to the heat input Q. In contrast to the liquefaction mode, no liquid leaves the system limits; mass flow rates are equal at all balance points

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The thermodynamics of cryocycles The fraction of liquid produced in refrigeration mode is given by

By means of the JT process all gases with a temperature of inversion above ambient temperature can be liquefied without external precooling, i.e. all gases with the exception of neon, hydrogen and helium. If one combines a JT stage with precooling stages it is even possible to liquefy these gases. Most of the helium and hydrogen liquefaction/refrigeration plants are equipped with a JT stage. In liquefaction mode, the thermodynamic efficiency of the liquefier results in (see figure D3.0.2(a))

Figure D3.0.3. Thermodynamic efficiency ηt d of cryogenic plants as a function of the inlet pressure p1 according to equation (D3.0.15). From Streich (1977).

In figure D3.0.3, the thermodynamic efficiencies ηt d of liquid nitrogen and hydrogen plants are shown as a function of precooling temperature T1 and inlet pressure p1 the critical temperature and the critical pressure are shown too. One can see that in the case of supercritical temperatures higher efficiencies can only be achieved if the actual pressure is significantly higher than the critical pressure; even then the efficiency maximum is only in the region of 10%. Furthermore it is noticeable that efficiency increases with decreasing temperature T1. What are the reasons for these rather poor efficiencies? Using the equations for exergy calculation, the exergy loss of each single component can easily be calculated and starting points for process improvements can be evaluated. As an example, figure D3.0.4 schematically shows a nitrogen liquefier based on a process similar to the Linde principle. It is obvious that within the cold part of the plant the highest exergy losses arise in the throttle valve. Even in the case of external precooling—the example figure D3.0.4(b), shows an NH3 vaporizer—only a minor improvement can be achieved. The specific characteristics of different refrigerants as well as the selected process allow us to reach temperatures close to or even lower than the two-phase-region temperature at the JT stage/throttle valve inlet. In these cases, the throttle losses are significantly reduced as the real gas factor z , which describes

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Figure D3.0.4. N2 liquefaction: a schematic view and exergy flow diagram: (a) without precooling and (b) with NH3 precooling. From Streich (1977).

the divergence of the real gas from the law of an ideal gas, decreases towards the two-phase region (inside the two-phase region as well as in the region of subcooled liquid to values << l). The single-stage propane refrigerator of figure D3.0.5 shows this effect; throttling of subcooled liquid results in low exergy losses. So far the loss and exergy calculations have concentrated on the process and the components inside the cold box. To get an idea of the efficiency of a complete plant, the losses of the cycle compressor have also to be taken into consideration. As can be seen from figures D3.0.4 and D3.0.5, these losses are significant. Besides specific, compressor-type dependent parameters the compressor discharge pressure determines the energy consumption of a refrigerator. At fixed boundary conditions, e.g. refrigeration and/or liquefaction capacity and compressor suction pressure, only one optimum inlet pressure p1 exists which makes it possible to reach the minimum ratio of energy consumption to refrigeration power. This is again caused by the real gas factor z , which then shows a minimum thereby reducing the throttle losses. The exergetic optimization is limited by the second law of thermodynamics. For example, it is not possible to reach throttle valve inlet temperatures which are lower than or equal to the gas temperature at the decanter outlet (see T2 and T4 in figure D3.0.2). Heat can only be transferred from higher to lower temperature levels. Arranging several JT stages with different refrigerants of different boiling temperatures in a cascade

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Figure D3.0.5. The JT cycle with propane (compression refrigerator). From Streich (1977).

it is possible to reach high refrigeration capacity (e.g. megawatts at the liquid nitrogen (LN2 ) level) in an economic manner. Usually the cycle with the higher boiling temperature condenses the cycle with the lower boiling refrigerant. JT cascades with up to a dozen isothermal evaporation stages work, for example, in large natural gas liquefaction plants using propane/propylene, ethane/ethylene, methane and nitrogen as refrigerant. JT cascades using nitrogen, hydrogen and helium also produce cold in miniature refrigerator plants of small capacity (in the region of several watts) to reach very low temperatures. Such an arrangement is shown schematically in figure D3.0.6.

Figure D3.0.6. A JT cascade for lower temperatures.

D3.0.2.2 The Brayton process In contrast to the JT process the refrigerant in a Brayton process remains gaseous throughout the complete cycle; heat input from the cooling object therefore results in a temperature rise of the refrigerant. A further difference to the JT process arises from the method of the production of cold: work performed during expansion by means of expansion in turbines or machines. Processes like this are also known as ‘gas refrigeration cycles’. A schematic view of a Brayton cycle to reach temperatures of about 20 K is shown in figure D3.0.7(a). After compression within compressor C and recooling to about ambient temperature the cycle gas enters heat-exchanger E1 and cools down in counterflow to the returning low-pressure gas. The work Wx is extracted from the system by means of the expansion machine/expansion turbine X; the gas is expanded to operational pressure thereby cooling down to operational temperature. After having absorbed the heat Q from the cooling object, the cycle gas is warmed up to approximately ambient temperature within E1 and then fed to the suction side of compressor C.

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Figure D3.0.7. The Brayton circuit: (a) schematic view and (b) specific energy consumption. From Frey (1991).

The enthalpy balance is as follows

whereas Wx = m(h2 – h3 ) and Q = m(h4 – h3 ). It is evident from enthalpy balance that cold can only be produced in the case where Wx reaches a higher value than the exchange losses at the warm end of the refrigerator. The exchange losses will be discussed in more detail in section D3.0.3; mainly they result because heat exchange needs temperature differences and different pressures of the cycle gas. The specific energy demand Wc /Q of the single-stage Brayton cycle as shown in figure D3.0.7(a) is illustrated in figure D3.0.7(b) as a function of inlet pressure p1 and fixed boundary conditions. From this it follows that: for a given temperature difference T1 – T5 (warm end E1) Wc /Q increases with decreasing T4 (the temperature which is provided by the cooling power): among other reasons, this is caused by the fact that the specific refrigeration capacity of an expander decreases with temperature (ii) for fixed T4 , Wc /Q increases with increasing T1 - T5 (see section D3.0.3) (iii) Wc /Q increases with increasing pressure drop of the compressed gas along its way through the components of the plant (tubing, valves, heat exchangers, etc) as part of the compression work is not available for expansion. (i)

For economical reasons Brayton cycles below 20 K are driven with two precooling stages in most cases. Figure D3.0.8 describes the principal alternatives

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with m1 = m3 + m5.

Figure D3.0.8. Brayton circuits for temperatures below 20 K: (a) with LN2 precooling, (b) with two expansion machines in series (plants with high cooling power at high p1 ) and (c) with two expansion machines in parallel (independent control of Q1 and Q2 ).

D3.0.2.3 The Claude process This process—applied to the liquefaction of gas by Claude in 1902—consists of a Brayton cycle followed by a JT stage. It is also known as the ‘liquid refrigeration cycle’ as the gas is liquefied during the cycle. Figure D3.0.9 shows the process in a version for nitrogen liquefaction. After compression to process pressure by means of compressor C, the gas is recooled to approximately ambient temperature (E0). Cool-down to inlet temperature of the expansion machine X is performed by E1 and the NH3 precooler. Part mx of the gas is expanded; the remaining gas is cooled down further (E2, E3) and finally throttled in valve V. Part of the throttled gas liquefies; the liquid is drawn out from decanter D. The part m7 not liquefied warms up (E3) to the outlet temperature of the expander, then being mixed with mx. Further warm-up to approximately ambient temperature occurs by means of E1 and E2. The mass balance of the complete system can be expressed as

The corresponding enthalpy balance is as follows

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Figure D3.0.9. A schematic view of the Claude process and exergy flow diagram. From Streich (1977).

The balances of the complete system are composed from the single balances of the JT cycle

and the Brayton cycle

In the above example, the liquid produced is withdrawn from the decanter. When the liquid is used to fill this vessel the liquefaction capacity of the plant increases. The volumetric flow mL/ρL entering the decanter replaces cold gas (mD /ρD ) which is usable for precooling of the high pressure flow. This allows a reduction in the flow mX via the expander and an increase in the flow through the throttle valve. The gain in liquefaction rate is in the region of 15–20% in the case of a helium liquefier. Comparing the Claude process of figure D3.0.9 with the JT process of figure D3.0.4(b) shows the following advantages.

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Figure D3.0.10. (a) Claude cycle and (b) thermodynamic efficiency ηt h as a function of T2. From Streich (1977).

(i)

As a result of the splitting of the high-pressure flow (approximately 70% is fed to the expander) the flow through the throttle valve decreases. Furthermore, a lower valve inlet temperature can be reached resulting in a higher liquid fraction. The real gas factor z decreases and exergy losses by throttling are reduced from 43.6% to 2.7%. (ii) The temperature difference within the cold part of the plant can be reduced significantly. The exergy losses at this point (E2) decrease from 12.9% to 1.5%. (iii) The work produced by the expansion machine can be used to reduce the energy need of the cycle compressor. Piston expanders can be coupled with a generator to produce electrical energy; an expansion turbine, equipped with a gear, can drive a blower thus increasing the suction pressure of the compressor. The energy need of the compressor is reduced by the amount of energ recovery. However, one has to notice that besides mechanical losses within the gear or generator, the quotient of energy gained by expansion to achieve the necessary energy decreases with the operating temperature of the expander. Consequently energy recovery below LN2 temperature is not economic. Figure D3.0.10(b) shows the thermodynamic efficiency of the Claude cycle as a function of the inlet temperature T2 (see figure D3.0.10(a)). A monotonic ideal gas is assumed. In addition the process is idealized, because harmful influences such as heat input from the surroundings to cold parts of the plant, compression losses, the above-mentioned losses of the energy recovery of expansion engines as well as pressure drops have not been considered. The efficiency ηa d of the expander, the temperature difference T1 – T5 of E1 and the pressure ratio p1 /p5 are the parameters which have been varied. From figure D3.0.10 it follows that:

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the negative effect of temperature differences T1 – T5 increases with decreasing pressure ratio p1/p5 (curves f and g) (ii) the negative effect of temperature differences T1 - T5 increases with decreasing temperature T2 (curves c and d as well as curves e and f) (iii) ηt d reaches only very low values at efficiencies ηa d < 100% and temperatures close to ambient temperature. (i)

An exergy analysis for a process according to figure D3.0.10(a) for a real, loss-affected expansion engine (in advance of section D3.0.3) results in

and

for an ideal, lossless machine. Using these expressions, one can calculate the exergy loss ev of the engine

The first term of the equation shows the exergy loss due to a higher than ideal outlet temperature; the second term corresponds to the loss of expander work. With decreasing temperature (T2 → 0) the latter approaches zero as the enthalpy drop from the expander inlet to the outlet approaches zero too. When T2 is close to ambient temperature (high enthalpy drops) this term, however, causes a strong decrease of ηt d . The question of an optimum pressure ratio is difficult to answer. In ideal cycles high-pressure ratios can be advantageous whereas in practice this statement has to be restricted. (i)

High pressure ratios result in a high temperature decrease in the expander. The cold thereby produced may not suit the refrigeration demand (it follows from the second law of thermodynamics that from an energetic standpoint cold should be available at the level at which it is needed). (ii) A high pressure ratio reduces the efficiency of expansion turbines (e.g. high speed) and of piston expanders (long piston stroke). (iii) High absolute pressures complicate the use of moderate plant components (plate-fin heat exchangers, screw compressors with high flow). In most cases Claude liquefaction cycles with low boiling refrigerants (He, H2 ) are equipped with more than one precooling stage to save energy. Precooling can be provided with expansion engines either alone or in combination with exchange units for external cold (e.g. LN2 vaporizer). The following table shows the decrease of the specific energy demand of a helium liquefier as a function of the number of stages. Using one single stage, the specific energy need amounts to 4.8 kWh 1–1 liquid helium (LHe); this value is approximately 22 times higher than the theoretical value (of an ideal process) of about 0.22 kWh 1–1 He at the assumed boundary conditions. The application of a second precooling stage leads to a reduction of more than 50%, whereas further precooling stages show substantially lower savings, as shown in table D3.0.1. It is obvious that the installation of a second stage is worthwhile. The energy consumption as well as the necessary high-pressure flow rate are reduced by a factor of approximately two. The exchange surface of the heat exchangers is reduced although the temperature differences become smaller with an increasing number of precooling stages. Moreover, the savings at the compressor side come from the investment in a second precooling stage. However, the use of further precooling stages pays only in applications where energy costs have a high priority (e.g. plants with high refrigeration capacity, i.e. in the case of He, in the kilowatt region).

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Table D3.0.1. The effects of precooling stages. Boundary conditions: TU = 295 K, TL = 4.4 K, pc = 25 bar.

Figure D3.0.11. A helium refrigerator showing the Claude process and exergy flow diagram.

An example of a He refrigerator with two precooling stages, (expansion machines) is given in figure D3.0.11. The plant delivers 75 W at 4.4 K. This example will be used to describe the design of refrigeration processes in more detail. To determine the characteristic parameters of a process it is divided into different balances. The following equation describes balance B1 of figure D3.0.11

As the plant is operated as a refrigerator, no liquid is drawn off. This means that m5 is equal to m12. To solve the above equation it is necessary to define: (i)

the refrigeration capacity Q as well as the operating temperature and pressure of the refrigerant at this point: normally, these data are specified by the user of the plant

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(ii) pressure and temperature of balance points 5 and 12 to determine the corresponding enthalpies: normally these values are based on experience; if not they have to be found by means of optimization calculations which have to take into consideration physical feasibility. As a result equation (D3.0.22) gives the necessary mass flow rate m5 at the assumed boundary conditions. Balance B2 can be formulated as follows

with m3 = m5 + mx 2 = m12 + mx 2 = m14. The outlet temperature and pressure of the expansion machine correspond to condition 12; to calculate the inlet conditions the efficiency of the expander as well as one state variable at the inlet must be known. This variable is usually the inlet pressure. The enthalpies of points 3 and 14 have to be determined in the same manner as h5 and h12 in balance B1. On this basis mx 2 can be calculated with equation (D3.0.23) leading to the cooling power of expander X2 (last term of equation (D3.0.23)). Balance B3

with m1 = m3 + mx 1 = m14 + mx 1 = m16 allows us to fully determine the process of figure D3.0.11. The conditions that have not been determined until now (i.e. the conditions between the single heat exchangers, points 15, 13, 11, 9, 8, 7 and 6) can be found by setting balances around the single component. Point 10 corresponds to saturated vapour at operational pressure and temperature at the consumer outlet. In connection with the Claude process a state of the art projection of cryogenic processes should be mentioned: the T—s diagram. In a similar way to the exergy flow sheet it can be used to assess the efficiency of a process. A simplified example of the process in figure D3.0.11 is shown in figure D3.0.12. One can see that the following apply. (i)

The gas expansion by means of X1 and X2 leads to an increase of the system’s entropy (and exergy loss). An ideal machine with ηa d = 100% causes no entropy change (ds = 0). A decreasing ηa d , however, results in an increasing ds. The lower the efficiency of the machine, the flatter the curves of X1 and X2. (ii) Work is performed due to expansion in X1 and X2 (h is not a constant); the gas temperature decreases noticeably. (iii) An entropy increase is caused by throttling the gas in valves ZV and V; however no work is performed (h = constant). The temperature decrease due to throttling is significantly lower. D3.0.2.4 The Stirling process By means of the refrigeration processes described so far the refrigerant is cooled down/warmed up within heat exchangers called recuperators. The processes dealt with in the next three sections deal with the use of a regenerative type of heat transfer. The characteristic principle of such processes can be explained with the help of figure D3.0.13. A displacer D divides a cylinder into a volume Vc at low temperature and a volume Vw at a high temperature. The displacer is movable within the cylinder. The two volumes are coupled via the regenerator R, which is a heat accumulator of high heat capacity and exchange area. Moving down the displacer causes a gas flow through the regenerator to volume Vc ; this gas cools down from temperature Tw to Tc . As the system is an isochoric arrangement, i.e. of constant volume, the system pressure decreases. Moving the displacer upwards shifts cold gas through the regenerator into Vw ; this gas is warmed up to Tw and

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Figure D3.0.12. The T—s diagram for the Claude process of figure D3.0.11.

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Figure D3.0.13. The principle of a regenerative heat exchanger.

the system pressure increases. Under ideal conditions, where pressure losses of the regenerator are not considered, the movement of the piston needs no work. When the displacer is driven by means of a crankshaft, the volumes Vc and Vw as well as the pressure alter periodically with time, depending on the rotation angle z . The alteration of pressure p and Vc act in opposite directions

where V0 = Vc + Vw and g = g(Tc , Tw ). At the rotational frequency f the cold produced would be zero in this case because

To achieve a real refrigeration power the alteration of Vc and p in opposite directions has to be changed to an out of phase relationship. In the case of a Stirling cycle, this is achieved by means of a compression piston located within Vw, having a negative phase quadrature. The basic difference between the Stirling and the Claude cycle arises from the integration of a compression and an expansion component within one system. Therefore no valves are necessary. To describe the Stirling process a little more clearly, a stylized course of the real piston movement is assumed (figure D3.0.14). Thereby, the regenerator, compression (and displacer) piston are integrated within one single unit. During phase I, the compression piston K moves up compressing an ideal gas from pressure p0 to p1 ; work is brought into the system. The heat of compression is carried away by means of cooler E2; this is an isothermal reaction. Displacer D thereby remains at the upper dead point of the expansion volume (position 1). In phase II the displacer D moves downwards; at constant volume (isochorically) the gas is pushed from compression volume Vw to expansion volume Vc. On its way through the regenerator the gas cools down (position 2). During phase III displacer D and the compression piston K jointly move downwards (position 3). The gas is expanded isothermally. During expansion, the gas provides cooling within the exchanger E1. The return of the displacer to its starting point is part of phase IV. The compression piston remains at its lower dead point. The gas is isochorically pushed from the expansion volume to the compression volume. On its way through the regenerator it is wanned up again (position 0). In this way each cycle increases the temperature gradient until stationary conditions are reached.

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Figure D3.0.14. A schematic view of the Stirling process. Vc: expansion volume; R: regenerator; E2: cooler; E1: refrigeration exchanger; D: displacer; Vw: compression volume; K: compression piston.

The refrigeration power generated during each period of expansion amounts to

which in the case of an ideal gas, is a lossless and a stylized process. The work to be performed by the compression piston is

During expansion, the compression piston absorbs the work performed by the cycle gas. The quotient of refrigeration power and external work can be expressed as

This term corresponds to the Carnot efficiency. The stylized Stirling process consists of two isothermal and two isochoric lines (figure D3.0.15(a)). The real process shows an elliptical curve (figure D3.0.15(b)). The latter is caused by a harmonic, out-ofphase movement of the compressor piston and the displacer, similar to a sine curve. Cool-down and warm-up follow no isochoric lines. The four phases of the process can no longer be separated clearly: they turn into one another smoothly. So far the work providing expansion has been regarded as an isothermal event. This presumes that the cold thereby produced is continuously consumed. For construction reasons, exchanger E1 is positioned between regenerator R and expansion volume Vc . Therefore the expansion of the amount of gas inside the expansion volume is an adiabatic process whereas the incoming gas is of constant temperature. As a consequence the temperature inside the expansion volume is lower than inside E1; the cold is produced at a lower temperature level than that available for use. As can be seen from equation (D3.0.30), the thermodynamic efficiency of the Stirling process contains a term which consists of the ambient temperature TU and the temperature Tc of the expansion volume. (According to the explanations of the preceding paragraph, Tc should be replaced by T′c the usable temperature within E1.) This term causes very low values of ηt d when Tc is close to TU . The influence of pressure ratios on efficiency is similar to that of the Claude process shown in figure D3.0.10(b): the temperature differences on E1 decrease ηt h more at low pressure ratios than at high pressure ratios.

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Figure D3.0.15. A p—V diagram of the Stirling process: (a) theoretical and (b) experimental.

A variant of the Stirling cycle is represented by Stirling—Philips cryogenerators. Until now both the displacer and the compression piston acted as working pistons; in a Stirling—Philips machine, the compression piston K (see figure D3.0.16) performs both the compression and the work performed during expansion. The displacer only shifts the refrigerant between the warm compression volume Vw and the cold expansion volume Vc —under ideal conditions without friction. Consequently, the compression piston varies the total volume, whereas the displacer is responsible for the partition of this volume into a cold and a warm part. Pressures above and below the displacer are approximately equal; the sealing of the displacer at the cylindrical wall therefore is easy. The loss of cold caused by leakage as well as frictional heat is low. Single-stage Stirling—Philips cryogenerators can reach temperatures down to approximately 40 K. Applications are, for example, in the liquefaction of nitrogen or providing refrigeration power to cool radiation shields of cryogenic components. Plants with two stages can produce approximately 50 K as the lowest temperature at the first stage and approximately 12 K at the second stage. When an He cycle is connected the refrigeration power produced can be transferred, thereby reaching sufficiently low precooling temperatures to liquefy the helium. Three-stage units can even reach temperatures below 8 K.

Figure D3.0.16. The principle of the Stirling—Philips process.

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D3.0.2.5 The Gifford—McMahon process The process of Gifford and McMahon is based on a principle which was discovered by Ernest Solvay in 1886; later on, it was applied with different modifications by different researchers. The difference from the Stirling cycle is that this process uses valves, which alternately connect the low- and high-pressure side of the compressor with the system. Idealized, the Stirling process is an isochoric cycle; essentially the Gifford—McMahon process is an isobaric cycle. A schematic view is shown in figure D3.0.17(a).

Figure D3.0.17. The Gifford—McMahon process: (a) schematic view and (b) T—s diagram.

The process operates as follows. Phase I The system starts from point 0 (low pressure p0 , displacer at lower dead point, low-pressure valve V0 closed). The high-pressure valve V1 opens, the incoming gas is compressed adiabatically—isentropically to pressure p1. The gas flowing in at the beginning finds a high pressure difference; during compression it is warmed up to more than 300 K (line 0–1). Gas coming in later is warmed up less because of the decreasing pressure difference. Finally, the filled (warm) volume is in a condition in between 2 and 3. Phase II Valve V1 is still open; the displacer moves to its upper dead point. Gas flows from the warm volume to the cold one. At the beginning pressure decreases and further gas enters via V1. On its way from the warm to the cold volume the gas cools down isobarically to the regenerator temperature (condition 4). Phase III Valve V1 is closed; the displacer remains in its position. Depending on the pressure ratio, V0 opens uniformly. Gas is flowing through the regenerator with low velocity. This causes an isentropic expansion of gas within Vc . The gas performs work; it pushes the gas streaming out thereby cooling down (condition 5). The equivalent of this work can be removed from the cooling object as thermal energy Qc . Phase IV The displacer returns to its lower dead point. The gas pushed out of Vc warms up within the regenerator (condition 6). The remaining pressure difference degrades within the tubing (condition 7). In the case of a Stirling process, the return gas leaves the regenerator at approximately ambient temperature; the return temperature of a Gifford—McMahon process, however, is significantly higher. The compressed gas entering during phase I therefore is of lower temperature than the gas flowing back during phase IV (compare points 0 and 7 in figure D3.0.17(b)). The resulting amount of heat, which corresponds to the above mentioned thermal energy Qc (first law of thermodynamics), is rejected to the surroundings.

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Assuming a compressor with an efficiency ηK and an ideal gas, the thermodynamic efficiency amounts

As in a Stirling cycle, ηt h , goes down to zero approaching TU . The advantage of a Gifford—McMahon cycle is of a practical nature: a common compressor can be used instead of an integrated one which is not the case in a Stirling cycle. Compared with the Claude cycle, the use of a valveless, force-neutral displacer instead of an expander is advantageous. Similar to Stirling machines, Gifford—McMahon refrigerators can be equipped with one or more stages. Single-stage units, for example, deliver a refrigeration power up to 100 W at approximately LN2 temperature; two-stage types can provide up to 10 W at approximately 20 K and up to 25 W at 80 K. Adding a JT stage at a precooling temperature of approximately 15 K cold can be carried out at the LHe level. With three-stage machines, temperatures below 7 K can be reached. D3.0.2.6 The Vuilleumier process This process, invented by Vuilleumier, was patented in 1918; the broader application, however, started in the 1960s. The principle is shown schematically in figure D3.0.18(a). The Vuilleumier refrigerator consists of two displacer systems which operate within a constant gas volume; consequently it is an isochoric process. The warm piston is placed in a heated cylinder (subscript w); the cold piston (subscript c) represents the refrigerator. In contrast to the Stirling cycle the warm (compression) piston has a phase lead of +90° compared with the cold (expansion) piston. In principle, the Vuilleumier process operates as shown in figure D3.0.18(b), starting at position 0. Phase I The cold piston is close to its right dead point; the cold volume Vc has almost reached its minimum. Because of the phase lead of +90° the warm piston moves to position 1w ; gas flows into Vw and warms up to Tw . This results in a pressure increase within the complete system (p1 ).

Figure D3.0.18. The Vuilleumier process: (a) schematic view and (b) T—s diagram.

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Phase II The cold piston moves to position 2c and the cold volume increases. Gas is transferred from V to Vc . On the way through the regenerator R, the gas temperature decreases to Tc . As a consequence, the pressure of the complete system decreases to p2. In parallel, the warm piston moves from 1w to 2w. During this phase the warm volume absorbs heat. Phase III The warm piston moves to position 3w . Thereby warm gas is displaced from Vw to V. This gas cools down isochorically resulting in a pressure decrease to p3. During this time, the cold volume absorbs the heat Qc . Phase IV Both pistons return to starting position 0. Caused by the displacement of cold gas from Vc to V, this gas is warmed up to temperature T on its way through the regenerator; the pressure increases to p0. It should be pointed out that the expression ‘warm’ is somewhat misleading if used in combination with a Vuilleumier process. If this stands for 300 K, it is valid only for volume V; the temperature Tw of volume Vw is close to 1000 K. This temperature, which drives the process, can be generated by direct use of primary energy (e.g. solar energy, nuclear energy, exothermic chemical processes, fuel gas). In contrast to the other processes, efficiency losses due to conversion of primary energy to electrical energy can be avoided. The principal differences and attributes of the processes described in the previous three chapters are listed in table D3.0.2. Table D3.0.2. A comparison of the different regenerative processes.

All refrigeration processes using regenerators can reach about 10 K as the lowest operational temperature with acceptable effort. With decreasing temperature, the size of the regenerator increases considerably, as the specific heat capacity of the regenerator materials approaches zero. The efficiency of a regenerative process therefore depends strongly on the efficiency of the regenerator.

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Figure D3.0.19 shows the specific power consumption of the three regenerative processes described above as a function of working temperature (Donabedian 1972). For the chosen region (several tens of watts at 77 K, several watts at 10–20 K) the units working with the Stirling (Philips) principle show the best performance: the quotient of energy needed for the cooling power is lower over a wider range of temperatures than in the case of a Gifford—McMahon or a Vuilleumier process. D3.0.3 Harmful influences D3.0.3.1 Compression of gas Compared with an ideal isothermal compression a real compression always causes losses; the effort required depends on many influences e.g. gas and lubricant friction, size of the dead volume, type of bearing, the amount of back-streaming gas already compressed, etc. The ratio of work necessary for lossless compression with heat rejection to the surroundings to the real compression effort describes the efficiency ηK of a compressor. The effort under ideal conditions can be calculated with the equations of section D3.0.1 and corresponds to the exergy difference of the gas before and afte compression. This makes it possible to take into consideration the real suction temperature. Typical efficiencies are in the region of 50–80%. deoendine on the comnressor type.

Figure D3.0.19. Specific energy consumption for regenerative processes.

D3.0.3.2 Work performing gas expansion For piston expanders and expansion turbines working in cryogenic plants the adiabatic efficiency ηa d , which describes the quotient of the real to the isentropic enthalpy drop, in general amounts to approximately 60–85%. Among other reasons, losses arise from • • • • • • •

throttling of gas, e.g. in valves of expansion machines leakages dead volume of expansion machines impulse losses of turbines heat exchange between the warm, high-pressure gas and the cold, low-pressure gas friction losses heat input from the surroundings (isolation losses).

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To calculate the exergy losses of an adiabatic expansion machine, equations (D3.0.20) and (D3.0.21) can be used. D3.0.3.3 Temperature differences at heat exchangers Transferring an amount of heat dq from temperature T1 to temperature T2 = T1– ∆T, the exergy

is lost (exchange loss). As a minimum, this amount is necessary to transfer heat at a temperature difference ∆T. It follows from equation (D3.0.32) that this exergy loss (i) increases proportionally with the transferred heat dq (ii) increases at a fixed ∆T with decreasing temperature by 1/T 2 (T 2 = T1T2 ). But how can the optimum temperature difference ∆T be found for a specific temperature level? From a thermodynamic point of view ∆T = 0 is the optimum, especially at low temperatures (equation (D3.0.32)). This requirement, however, would lead to an infinite size of the heat exchangers as the size of the heat exchange area depends on the mean logarithmic temperature difference of the gas flows. From this point of view, high temperature differences would be desirable. Optimizing these two contrasting requirements it is commonly accepted that the temperature difference should follow

where T represents the temperature at the warm end of a heat exchanger. In some cases, e.g. regarding plants with high refrigeration power, this rule can be slightly modified, especially at higher temperatures. In general, temperature differences smaller than 0.2 K should be avoided; at smaller differences, the accuracy of the gas properties, for example, could affect the calculations. D3.0.3.4 Isolation losses Heat losses Qi s o from the surroundings to components at a temperature T < TU increase with decreasing temperature T. In process calculations they have to be added to the amount of heat entering the equation. The effort required for losses to the surroundings amounts to

To reduce losses due to heat input from the surroundings a high effort for cryogenic plants is required. Temperatures down to LN2 temperature only need some kind of filling material; plants working at LHe temperatures, however, must be equipped with a so-called super-insulation in combination with a vacuum insulation to keep isolation losses within acceptable limits. Accompanying provisions should be: (i)

an arrangement of the single components according to their temperatures, i.e. components at low temperature should not be coupled directly to components at ambient temperature (heat conduction). A better way is to surround low-temperature equipment with components of a temperature T < TU (heat of radiation) (ii) an optimum layout of components, considering the lowest possible surface area at maximum volume.

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D3.0.3.5 Pressure drops It follows from the requirement of low isolation losses as well as of low investment costs that design compact plants should be designed. Especially regarding heat exchangers, high pressure drops of the refrigerant would result in a high power consumption for gas compression (see, for example, the description of figure D3.0.7). The use of compact heat exchangers—components of high transfer area at low volume such as plate fin heat exchangers—lead to an acceptable compromise. Thereby high absolute pressures would of course be advantageous, but this is opposed by the higher energy need for compression as well as potential lower efficiencies of the expanders. It becomes evident from this chapter that exergetic optimization of a process requires the reconciliation of contradictionary influences for the specific application. These requirements must also take into account the wishes of the users such as simple and clear performance, reliability and low investment and operational costs. References Donabedian M 1972 Survey of Cryogenic Cooling Techniques Aerospace Corporation, Vehicle Engineering Division Frey H and Haefer R A 1981 Tieftemperaturtechnologie ed F X Eder (Düsseldorf: VDI)

Gifford W E and McMahon H O 1960 A new low temperature gas expansion cycle, parts 1 & 2 Adv. Cryogen. Eng. 5 354 McCarty R D 1972 Thermophysical properties of helium-4 from 2 to 1500 K with pressures to 1000 atmospheres NBS Technical Note 631 US Department of Commerce Streich M 1977 Thermodynamik der Kryoverfahren (Düsseldorf: VDI Bildungswerk) Walker G 1973 Stirling Cycle Machines (Oxford: Oxford University Press)

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D4 Refrigerators

H Quack

D4.0.1 Connections between application and refrigerator All cryogenic applications need cooling, first for the initial cooldown and later for the compensation of thermal losses during operation. These losses may have their origin in • electrical or mechanical losses in the superconductor • electric current lead losses (conductive and ohmic) • thermal conduction through supports • thermal conduction and radiation across the vacuum insulation • mechanical friction • induction losses • radioactive or particle radiation. The application has to be connected to a machine, where the ‘refrigeration’ is being ‘produced’ (figure D4.0.1). For this machine we generally use the term refrigerator, but also sometimes the term liquefier. Figure D4.0.2 gives an overview of possible connection methods. D4.0.1.1 Open or closed loop cooling (figure D4.0.2) Refrigeration can be stored and transported in the form of a liquid refrigerant such as liquid helium or liquid nitrogen. In an open loop, the refrigerant is brought to the application in a well insulated transport vessel and transferred batch-wise into the cryostat of the application. The boiled off refrigerant is either vented or it is collected and compressed into high-pressure gas bottles, which are transported back to the liquefier. Because the batch-wise handling of the refrigerant is cumbersome, most continuously working applications are directly connected to their refrigerator in a closed loop. Often the refrigerator is mounted directly on the cryostat of the application. D4.0.1.2 Bath, forced and conduction cooling (figure D4.0.3) In the bath cooling arrangement, the application is immersed totally into liquid helium, which boils off due to the thermal losses. This arrangement is quite simple, but it requires two vessels, one to contain the liquid helium and a second one for the vacuum insulation. The two walls of the vessels plus the vacuum

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Figure D4.0.1. A schematic diagram of an application and refrigerator system.

Figure D4.0.2. Open-loop and closed-loop cooling.

Figure D4.0.3. Bath, forced and conduction cooling.

space in between cause a relatively large gap between, for example, a superconducting magnet and the area in which the magnetic field is being used. If forced cooling is being used, only one cryostat wall is needed. The refrigerant flows through channels in the application. Depending on the thermodynamic state of the refrigerant, one can distinguish one-phase and two-phase cooling. One-phase cooling with the refrigerant just above the critical pressure and in the neighbourhood of the critical temperature of the fluid is called supercritical cooling. If the flow of the refrigerant does not get into or around the application, and the refrigerator touches the application just with a ‘cold finger’, the term conduction cooling is used. Some people call it ‘cooling without refrigerant’, because the application does not ‘see’ the refrigerant, but of course the refrigerator itself needs a refrigerant.

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D4.0.1.3 Refrigeration and liquefaction (figure D4.0.4) Applications with large internal losses, and small insulation and leads losses, are well served by a onetemperature-level connection to the refrigerator. On the other hand, applications in which relatively large insulation or conduction losses through supports have to be expected make use of a thermal shield at an intermediate temperature. In this case the refrigerator has to provide refrigeration on two temperature levels. Thermal losses due to current leads and certain kinds of insulation shield are best served by a flow of refrigerant, which is heated up to ambient temperature. The refrigerator supplies liquid refrigerant and receives back ambient temperature gas, i.e. it delivers ‘liquefaction’.

Figure D4.0.4. (a) A one-temperature-level connection; (b) a two-temperature-level connection; (c) ‘liquefaction duty’.

D4.0.1.4 Hermetic and semi-open refrigerators Refrigerators with fluids as refrigerant consist of an ambient temperature section and a low-temperature part. The fluid circulates or oscillates within the operating volume. During steady-state operation the fluid may be in the gaseous or liquid phase. After standstill and warm-up, all the fluid is in the gaseous state at ambient temperature, probably at a high pressure. The equalization pressure depends to a great degree on whether the refrigerator contained liquid refrigerant during steady-state operation. In this case the equalization pressure may be too high either for the pressure rating of the vessels or for the compressor restart. So one needs an ambient temperature buffer volume to reduce the equalization pressure. Dependent on the type of refrigerator, the buffer volume may be open to the suction side of the compressor all the time, or it may be connected to the operating circuit through valves. Some refrigerators contain a fixed amount of refrigerant and are hermetically sealed. They come into contact with the application only through a heat exchanger or a cold finger. Other refrigerators, which one may call ‘semi-open’, have an ambient temperature buffer (B in figure D4.0.5) and a reservoir of liquid refrigerant. They can shift this refrigerant between the buffer and the reservoir dependent on the actual needs of the application and the capacity of the refrigerator. A third possibility is to connect a hermetic refrigerator to a passive thermosiphon loop, which contains a liquid reservoir and an ambient temperature buffer. Which of the above-mentioned connections between application and refrigerator is used depends on temperature levels and refrigeration rates, stability requirements, available space and duty cycle.

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Figure D4.0.5. Hermetic and semi-open refrigerators.

D4.0.2 Other requirements of the refrigerator Besides the refrigeration duty itself, there are other requirements which the refrigerator has to fulfil. Reliability, low maintenance requirements and low price are always of prime importance. For large installations a good thermodynamic efficiency, which leads to low operating cost, is mandatory. As a rule of thumb: for applications with more than 50 W of refrigeration at 4.4 K in continuous operation, the power bill is normally larger than the discounted investment cost. Good thermodynamic efficiency is also a must for mobile refrigerators, because the efficiency determines the size of the compressor, and the compressor is often the heaviest piece of equipment. Besides weight, compactness is also an important requirement in many mobile applications. Applications differ in their sensitivity to vibrations. Superconducting cavities or SQUIDS are very sensitive and require low vibration refrigerators or vibration damping devices. D4.0.3 Review of refrigerators D4.0.3.1 Classification of refrigerators A large number of different refrigeration cycles have been proposed in the past (see also chapter D3) (Hebral 1995). All practical ones in the temperature range between 1.8 and 120 K use fluids as refrigerants. Magnetic refrigerators (chapter D7) and Peltier coolers have been tried, but have not been realized on a practical scale. Cryogenic refrigerators can be classified in different ways. A classification by phenomena was proposed by Walker (1983) and is presented in figure D4.0.6. The main classification is by the type of heat exchanger used. A regenerator (figure D4.0.7) is a device which is alternately passed by warm high-pressure gas in one direction and subsequently by cold low-pressure gas in the other direction. The matrix of the regenerator is used for thermal storage purposes. Refrigerators which use regenerators can be (and must be) very compact, because ‘dead volume’ is detrimental to the process. The ‘cold finger’ is a characteristic for all refrigerators using a regenerator. In a recuperator, the two streams are separated by a wall. Because heat flowing from one stream to the other has to pass through the separation wall, recuperators are more bulky than regenerators. On the

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Figure D4.0.6. Classification of cryogenic refrigerators.

Figure D4.0.7. A regenerator and a recuperator.

other hand, ‘dead volume’ presents no problem to refrigerators using recuperators. So liquid refrigerant storage and extended cooling distribution systems can be directly coupled to their refrigeration loops. D4.0.3.2 Refrigerators for the temperature range 20 to 100 K This temperature range is of importance to applications of high-temperature superconductivity. Figure D4.0.8 shows the types of refrigerator which have found application in this temperature range. (a) Brayton/Claude In the upper capacity range, the preferred solutions are refrigerators based on the Claude and Brayton processes with piston or turbine expanders (Quack 1988, 1989, 1993). In this temperature range they find

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Figure D4.0.8. Refrigerators for the temperature range 20–100 K.

their application in air separation plants and for the liquefaction of oxygen, argon, nitrogen, neon and hydrogen. Used as refrigerators, their lower capacity limit is 100 W at 20 K and 500 W at 80 K. (b) Gifford—McMahon Gifford-McMahon refrigerators are used in single-stage units down to 60 K and in two-stage units down to 10 K (Häfner 1988), and recently to even lower temperatures. The main applications are in cryovacuum pumps, where two-stage units are used: 20 K for the pumping surface and 80 K for the thermal shield. Gifford—McMahon refrigerators are also used for the cooling of the thermal shield of magnetic resonance imaging (MRI) magnets. The largest Gifford—McMahon refrigerators have capacities of 10 W at 20 K or 160 W at 80 K. (c) Stirling There are two size ranges of Stirling refrigerators (Huijgen and Stultiens 1988, Walker 1983). Large units provide capacities between 500 W and 4 kW at 80 K and compete with the smaller Claude units, e.g. for laboratory-scale liquid-nitrogen generators. Small-scale units with a capacity up to 3 W at 80 K are

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used for cooling of infrared sensors. These machines were originally developed for military use, but are nowadays also available for civil applications. Prototype Stirling refrigerators have been developed in the past for the intermediate size range between the two Stirling ‘fields’ shown in figure D4.0.8, but, because of lack of applications, they have not been produced in series. (d) Liquid nitrogen and liquid helium Small-scale refrigerators have to compete with external liquid helium or liquid nitrogen in open-loop configurations, especially in short-term or stationary applications. The refrigeration capacities provided by 1 1 h–1 or 1 litre per day of the two liquid refrigerants are also shown in figure D4.0.8. (e) Pulse tube Recent developments in pulse tube refrigeration have demonstrated that this refrigeration principle is suitable more or less for the same capacity range as the Gifford—McMahon refrigerator and partially the Stirling refrigerator, with the additional advantage that the cold section does not contain any oscillating components (Hofmann and Wild 1994, Radebaugh 1995, Ravex et al 1992, Timmerhaus 1994). (f) Mixed refrigerant Joule—Thomson The classical refrigeration process for the 80 K temperature range used to be the Joule—Thomson cycle, which was used for the first continuous liquefaction of air. Later this process declined in importance, because • • •

it needs a high-pressure compressor it is very sensitive to impurities, which tend to block the throttle valve, therefore nonlubricated compressors had to be used the process has a rather poor thermodynamic efficiency.

However, in the 1960s, Russian scientists found that the pressure ratio could be greatly reduced if a mixture of nitrogen and some low-molecular weight hydrocarbons were used as refrigerant. In the meantime, the largest cryorefrigerators in the world, the baseload liquid natural gas (LNG) production plants, are based on such mixed refrigerant cycles. They are driven by large oil-free turbocompressors (Radebaugh 1995, Timmerhaus 1994). The use of this process at lower capacities was until recently limited by the assumption that oil-free operation was a basic requirement. This changed when it was found that the hydrocarbons in the mixture are able to dissolve some oil and to carry it even through the throttle valve without blocking the cold end of the refrigerator. This opened the door for the use of low-cost, ultrareliable household refrigeration compressors for cryorefrigerators. The first units with a refrigeration capacity of 10 W at 80 K are already being marketed. This refrigerator principle has no real upper or lower capacity limit. D4.0.3.3 Neon as refrigerant With a triple point of 24.55 K and a critical point of 44.44 K, neon is considered to be an ideal refrigerant for many applications of high-temperature superconductivity, especially in bath cooling mode, where temperature stability is provided (Fredrich et al 1995, Richardson and Tavner 1995). Table D4.0.1 compares some properties of neon and helium. Remarkable is the latent heat per volume with a value of 103 kJ l–1, which is 40 times larger than the equivalent value for helium. Neon can be used as secondary refrigerant, where the refrigeration is being produced by a heliumbased Stirling, Gifford—McMahon or pulse tube refrigerator. The liquid neon is then used for temperature stability purposes. Neon could of course also be used as refrigerant in a Claude process.

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Refrigerators Table D4.0.1. Properties of neon and helium.

D4.0.3.4 Refrigerators for the temperature range 1.8–20 K Figure D4.0.9 is the continuation of figure D4.0.8 to lower temperatures. In principle, all refrigerators that can reach the 20 K level can be extended by a helium Joule—Thomson loop down to the 4 K level. Stirling refrigerators have been used in the past as precooling stages for helium liquefiers, but they disappeared from the market, because they were not competitive in terms of price and reliability. The lower limit of two-stage Gifford—McMahon refrigerators used to be 10 K. Recently the use of magnetic regenerator material has extended the temperature range down to 4 K and below with two-stage units. The same development is at present also occurring with pulse tube refrigerators, where temperatures below 10 K with two-stage and below 4 K with three-stage systems have been obtained in laboratory units (Gao and Matsubara 1993). Figure D4.0.9 also shows the lower capacity limit of refrigerators according to the Claude process with piston or turbine expanders. For completeness, the position of the largest 3He/4He dilution refrigerator with a capacity of 1.3 W at 0.2 K is also indicated. An interesting hybrid refrigerator is the Boreas refrigerator, a two-stage Gifford—McMahon refrigerator combined with a wet expander stage (figure D4.0.10). The expander is mechanically a prolongation of the Gifford—McMahon displacer (Gregory et al 1993). In figure D4.0.9 one can recognize a remarkable distance between the smallest Claude refrigerator and the largest Gifford-McMahon unit. The 5–10 K temperature range is of interest to a number of applications using Nb3Sn as superconductor. Multiple Gifford—McMahon units are at present the only cost-effective solution. D4.0.4 Cooldown versus continuous operation Many applications of superconductivity, especially superconducting magnets in persistent current mode, have quite large thermal masses during cool-down, but rather small thermal losses during continuous operation. A refrigerator dimensioned for continuous operation would need weeks to cool down the magnet to its operating temperature. For such systems one foresees for the cool-down the additional use of liquid nitrogen and/or liquid helium. The ‘cooldown refrigerant’ can either be supplied to the refrigerator or to the application itself. The boil-off is normally vented during this procedure. The disadvantage of such an arrangement is that after

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Figure D4.0.9. Refrigerators for the temperature range 1.8–20 K.

each interruption of the operation, e.g. due to a quench, the auxiliary refrigerant has to be shipped in and manual intervention is necessary. An alternative is the use of semi-open systems (figure D4.0.5), where, for example, liquid helium could be liquefied by the refrigerator and stored before the actual cool-down of the magnet begins. D4.0.5 Most common combinations of application/refrigerator All applications, which are being produced in large numbers, have found tailor-made refrigerators over the years. Some examples are given below. D4.0.5.1 Large magnet or cavity systems They are cooled by Claude refrigerators, which use oil-lubricated screw compressors and gas bearing expansion turbines. A number of examples are described in chapter D5.

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Figure D4.0.10. A schematic diagram of the Boreas refrigerator.

D4.0.5.2 Laboratory nitrogen liquefiers Commercial liquid-nitrogen generators in the range between 5 and 50 1 h-1 are available based on Claude or Stirling principles. D4.0.5.3 Laboratory helium liquefiers Commercial helium liquefiers in the range above 5 1 h–1 are available based on the Claude principle. They include a freeze-out purifier and use piston or turbine expanders. D4.0.5.4 Cryopumps with Gifford—McMahon refrigerators More than 10 000 units per year are currently being installed for this application. In particular, modern semiconductor chip manufacturing factories need hundreds of these units. D4.0.5.5 MRI magnets with Gifford—McMahon shield coolers MRI tomographs can nowadays be found in many hospitals. Normally their superconducting magnets are bath cooled. The boil-off is used to cool several layers of shields. The time between refills is in the order

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Reliability

691

of months. This time span can be increased if a Gifford—McMahon refrigerator is used for additional shield cooling. D4.0.5.6 Infrared detectors with miniature Stirling refrigerators This combination was initially developed for military night vision and rocket-guiding facilities, but nowadays it finds new applications in thermography for medical and engineering purposes. D4.0.6 Reliability Cryogenic applications expect reliable performance from their refrigerators. This includes low maintenance requirements. Table D4.0.2 lists the main sources of unreliability, some potential remedies and new problems, which have their origin in the remedy. Table D4.0.2. Sources of unreliability

D4.0.7 Efficiency The minimum power requirement is given by the second law of thermodynamics

where Q• is the refrigeration rate in watts, and Ta m b and T0 are the ambient and operating temperature respectively. The required power can be described by the efficiency η

In figure D4.0.8 typical efficiencies of some refrigerators are indicated: Claude/Brayton 0.15–0.35, large Stirling 0.2–0.4, Gifford—McMahon 0.05–0.1, pulse tube 0.04–0.08 and small Stirling 0.08–0.15.

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The following is an example of an efficiency calculation. For a refrigeration rate of Q• 0 = 10 W at a temperature of T0 = 80 K with an ambient temperature of Ta m b = 300 K, the minimum power requirement is

With an efficiency of η = 0.1, the actual needed power is

For large cryogenic systems efficiency is of importance, because the power bill tends to become larger than the amortization of the investment. The part-load efficiency is also important, since in many instances the refrigerator has to be dimensioned for peak demands, but runs most of the time at reduced capacity. For small systems, efficiency is more a matter of practicability. The compressor duty and size is inversely proportional to the refrigerator efficiency. For various reasons the Gifford—McMahon refrigerator has an efficiency of only 5% of a Carnot refrigerator. Therefore it needs a quite large compressor unit. This may be acceptable for stationary applications, but not normally for mobile units. Overall, efficiency in most applications is a secondary factor far behind reliability. This is not only the case in cryogenic refrigerators, but also in household refrigerators. There the refrigeration system has a relatively low thermodynamic efficiency, but can be reliable and maintenance free for more than 20 years. D4.0.8 Cost/price The cost of a refrigerator to the manufacturer and the price to a buyer depend on a number of parameters. D4.0.8.1 The capacity of the plant One can express the capacity either in terms of refrigeration capacity or in terms of the input power P. These two are (for a plain refrigerator) connected by

Larger plants tend to have a better efficiency η. In a limited capacity range one may assume

So the input power depends on the refrigeration capacity as

From electrical motors and compressors, which form a major part of a refrigerator, it is known that their cost/price is proportional to the input power with an exponent 0.7

This leads to a dependence of the price of a refrigerator on the refrigeration capacity for similar designs and scope of supply

This means that a refrigerator with double capacity is 20.56 = 1.47 times more expensive.

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D4.0.8.2 System boundary A refrigerator including auxiliary items like transfer lines, power supply, cooling water system or (for larger systems) installation costs, may easily be two or three times as expensive as a ‘naked’ refrigerator. D4.0.8.3 Liquid-nitrogen precooling The electric power consumption of a helium liquefier can be reduced by a factor of 0.5 by using liquidnitrogen precooling. For a helium refrigerator the factor would be about 0.8. Applying the above-mentioned dependence of the price of a refrigerator on the power input, the influence of liquid-nitrogen precooling on the first cost could be expressed by a factor of 0.50.7 = 0.62 for a helium liquefier, and 0.80.7 = 0.86 for a helium temperature refrigerator. The influence of nitrogen precooling on the operating costs depends on the relative cost (including handling costs) for electricity and liquid nitrogen. D4.0.8.4 Standard plant versus tailor-made plant The cost and the price depend on the number of units manufactured from a certain design. In many fields of technology an ‘experience formula’ has been found, i.e. the cost depends on the number of units n built Table D4.0.3. A checklist for the specification of a refrigerator.

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since the design was finished

D4.0.8.5 Purchasing process Last but not least, the price depends on currency exchange rates, the competitive situation and the skill of sales persons and purchasers. D4.0.9 Checklist for the specification of a refrigerator See table D4.0.3 for details. References Fredrich O, Haberstroh C and Quack H 1995 Studies on a modified Ericsson cycle with neon as refrigerant Adv. Cryogen. Eng. B 41 1255–63 Gao J L and Matsubara Y 1993 4 K pulse tube refrigeration Proc. 4th Joint Sino-Japanese Seminar on Cryocoolers and Concerned Topics (Beijing, 1993) (Chinese Academy of Sciences) p 69 Giese R F 1994 Refrigeration options for high-temperature-superconducting devices, operating between 20 and 80 K for use in the electric power sector IEA Report Gregory R, Gallagher R and Crunkleton J A 1993 Thermodynamic analysis of the Boreas cryocooler CEC Conf. (Albuquerque, 1993) paper BD7 Häfner H-U 1988 Leistungsstarker einstufiger Kryorefrigerator und seine Applikationsmöglichkeiten im Temperaturbereich ( DKV Tagungsbericht ) vol 1 (Munich: DKV) pp 93–103

LN2-

Hebral B (ed) 1995 Cryogenie, ses Applications en Supraconductivite IIF (chapter on Refrigeration et liquefaction) Hofmann A and Wild S 1994 A model for analysing ideal double inlet pulse tube refrigerators Int. Cryocooler Conf. (Vail, 1994) Huijgen G and Stultiens T 1988 Design approaches to a new generation of Stirling cryocoolers Proc. ICEC 12 (Southampton, 1988) pp 534–8

Quack H 1988 Kälteanlagen für den 80 K Temperaturbereich (DKV Tagungsbericht) vol 1 (Munich: DKV) pp 13–23 Quack H 1989 Kryokälteanlagen für Hoch- und Niedertemperatur-Supraleiter Supraleitung in der Energietechnik (VDIBerichte 733) pp 107–18

Quack H 1993 Maximum efficiency of helium refrigeration cycles using non-ideal components Adv. Cryogen. Eng. B 39 1209–16 Radebaugh R 1995 Recent developments in cryocoolers Proc. 19th Int. Congress of Refrigeration Vol IIIb, The Hague pp 973–89 Ravex A, Rolland P and Liang J 1992 Experimental study and modelisation of a pulse tube refrigerator Cryogenics 32 (ICEC Suppl.) 9–12 Richardson R N and Tavner A C R 1995 Neon liquefaction system for high-Tc experiments Cryogenics 35 195–8 Timmerhaus K D 1994 Cryogenics and its applications IIR-Bulletin 94/95 3–11 Walker G 1983 Cryocoolers (2 vols) (New York: Plenum)

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D5 Examples of cryogenic plants

M Wanner and B Ziegler

D5.0.1 Introduction The low-temperature processes which were described in chapter D3 are technically realized in different ways according to the required temperature levels and multiple operation modes as well as to the general boundary conditions. A survey of the range of actual cooling powers together with the achieved efficiencies of cryogenic plants at the three essential temperature levels (N2 , H2 , He level) are given in figure D5.0.1 (Frey and Haefer 1981). Characteristic of the graphs is the difference in the specific power consumption at the different temperature levels. These changes are a consequence of the decrease of the Carnot efficiency and of increasing influence of thermodynamic losses at low temperatures

where Tu is the ambient temperature and T the operating temperature. Cryogenic plants may be characterized according to their main duty as follows. (i)

In a liquefier the plant continuously takes feed gas at ambient temperature and delivers cold liquid. Hence the plant has to provide refrigeration in the whole temperature range between Tu and T. In order to remove potential gaseous impurities from the product, liquefiers usually incorporate purifiers. (ii) In a refrigerator cooling power is supplied either at a constant temperature level by evaporation of a cryofluid or within a dedicated temperature range by warming up a cold gas flow within the consumer. Typical examples are, for example, the cooling of a superconducting magnet at 4.4 K by boiling helium or the cooling of a cryostat radiation shield between 40 K and 80 K by helium vent gas. In contrast to liquefiers, refrigerators operate in a closed loop under stationary conditions. (iii) Low-temperature purifiers are used to separate gaseous impurities from the feed gas (e.g. N2 , O2 in H2 or He) by liquefaction, adsorption or freeze-out processes. Again refrigeration is required continuously down to the boiling or melting temperature of the impurity. Consequently the liquefaction capacity of the plant reduces with increasing levels of impurities unless additional sources of refrigeration (e.g. liquid nitrogen) are used. (iv) Recondensers are a special form of refrigerators. They may be used in large liquid gas storage tanks (e.g. liquid-nitrogen gas/liquid hydrogen) in order to reduce the evaporation losses. In that case refrigeration is used to remove the latent heat of evaporation for reliquefaction of the vent gas.

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Figure D5.0.1. Specific power and thermodynamic efficiency of cryogenic power plants as a function of cooling power and operating temperature.

In many cases cryogenic plants have to be designed for very different and combined operation modes. A typical specification for a helium plant which is intended to cool a superconducting magnet could be: (i)

refrigeration duty at 70 K < T < 80 K for cooling of radiation shields 800 W—additional capacity at 4.4 K refrigeration power 400 W or liquefaction rate >100 1 h-1

During the cooldown of the magnet this plant works as a refrigerator at decreasing temperatures. When the magnet has reached liquid-helium temperature the plant turns to liquefaction mode by filling the cold volumes with liquid helium. Finally the cryoplant again changes to refrigeration mode, now at a constant temperature of 4.4 K. In response to the individual requirements and cooling capacities different types of cooling process and equipment have been built at the different temperature levels (4.4 K = liquid helium, 20 K = liquid hydrogen, 77 K = liquid nitrogen) (Baldus et al 1983): (a) Liquid-hh elium temperatures In the range of average (approximately 100 W) to large cooling powers (20 kW) the Claude process with two or more precooling stages (e.g. liquid-nitrogen precooling) is exclusively used. At the lower end of these cooling powers reciprocating expanders can be found whereas turboexpanders dominate in high-capacity plants. In analogy, reciprocating compressors are used for medium throughputs whereas oil-lubricated screw compressors have their benefits at large flow rates. Minirefrigerators (<1 W) often make use of two-stage precooling by Joule—Thomson throttling, or by expanders or regenerative cycles in combination with an additional Joule—Thomson stage for the lowest temperatures. Although even helium liquefiers based on the Joule—Thomson process have been realized, using N2 and H2 for precooling, their efficiency ηt d is typically below 1–2% for small cooling powers. (b) Liquid-hh ydrogen level Low cooling powers of 1–10 W are typically realized by multistage Gifford—McMahon processes in competition with high-pressure H2 cycles with liquid-nitrogen precooling. Two-stage Stirling machines can supply several hundred watts of refrigeration. Both the Stirling and the Gifford—McMahon process can provide refrigeration down to approximately 15 K with reasonable efficiency and hence avoid the irreversibilities of the Joule—Thomson expansion. Large cooling powers especially for bulk hydrogen liquefaction are exclusively produced by the Claude process.

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(c) Liquid-nn itrogen level The processes for medium cooling powers correspond to those for the liquid-hydrogen level. Because of the lower insulation losses and the higher regenerator temperatures higher specific efficiencies can be reached. For minirefrigerators a variety of Joule—Thomson coolers offered is characterized by very small cold heads and quick cool-down periods. On the other hand the specific power consumption is relatively high. The upper capacity range is again dominated by single- or two-stage Claude cycles. In that case the efficiency of the cold box is approaching that of the ideal Carnot process. As a consequence the total efficiency of the plant is determined by the compressor. D5.0.2 Technical requirements for cryogenic plants The ideal goal of each operator would certainly be to have a cryogenic plant which is maintenance-free, exhibits optimum thermodynamic efficiency in various operation modes and which has a very comfortable control system. In general, however, these requirements are conflicting from a thermodynamic or mechanical point of view and last but not least have to be realized within a certain budget. Therefore a technical solution has to be found for each application which balances the competing requirements and which can perhaps be based on a standard product of the manufacturer. Thermodynamic efficiency is certainly an essential criterion for a cryogenic system, since the power consumption directly affects the operational costs of the plant. However, the energy costs are only one element of several cost factors. In figure D5.0.2 the specific energy costs of medium-pressure nitrogen liquefiers are compared with the investment costs as a function of the plant capacity. This comparison shows that additional effort to improve the thermodynamic efficiency is only useful from an economic point of view for plants with higher capacity. In addition to investment and power costs all other cost elements, as well as specific conditions on site, have to be considered in a comprehensive economic analysis. The following factors may contribute to such an analysis: • investment costs (including financing, depreciation, etc) • general costs (building, lighting, overheads)

Figure D5.0.2. Costs (arbitrary units) for investment and operation (energy consumption) of a medium-pressure liquid-nitrogen plant.

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maintenance (personnel costs, spare parts) utilities (electricity, cooling water, pressurized air) operator costs (personnel costs) transfer losses for liquefied gases (for liquid helium this may amount to 20% per transfer) gas losses during transportation and storage (these depend on the size of the storage tank) impurities (effort for purification) actual operation of the plant (cool down, warm up, actual demand).

This list shows that, depending on the individual situation, the thermodynamic efficiency may be more or less important. In addition the benefits of technical measures to improve the efficiency have to be balanced against eventual risks or disadvantages. In such a way an improvement of the efficiency by additional precooling stages and, for example, by an increase in the number of expanders might imply a larger number of valves and increased control effort. The power savings of the plants have, however, to be weighed against the increased complexity, the additional effort for control and maintenance as well as the increased risk of failure. Plants can be designed for optimum efficiency only for a well defined operation mode. If, however, the plant has to work often in different operation modes (e.g. liquefaction, refrigeration, part load, overcapacity) both the process design and the choice of the components have to be compromised in order to avoid a significant reduction of the efficiency in individual modes. The cool-down time of the plant may be of importance especially in situations where the plant is operated only intermittently. Cool-down time may be a couple of minutes for minirefrigerators but can reach 5–10 h for very large plants. In some cases, however, the cool-down time may also be dictated by requirements from other parts of the plant, e.g. to reduce thermal induced stresses. Generally the cool-down time of the system is determined by the stored enthalpy of the cold components of the plant (total mass and specific heat of the exchangers, adsorbers, etc) as well as the effective refrigeration capacity of the different cooling stages. The following terms affect the net cooling capacity during cool-down: (i) the cooling power at the design point (ii) reduced insulation losses because of the initially reduced differences to ambient (iii) the reduced heat-exchange losses at the warm end of the plant; since in general the mass flow rate through the plant is reduced during the cool-down period the heat exchangers are effectively overdimensioned (iv) excess capacity of the precooling stages at elevated temperatures. The contributions (i) to (iii) are valid for all cryogenic processes. The reduced insulation losses are especially important for minirefrigerators where Qi s o l usually exceeds by far the net refrigeration Qn e t . The contribution (iv) strongly depends on the applied process. For instance in pure Joule—Thomson processes the cooling power depends only on the difference of the enthalpies at the warm end of the plant; hence the refrigeration power is constant during the whole cool-down phase. A different situation exists for expansion turbines without capacity control (i.e. with fixed nozzles). At an inlet temperature T the flow rate m is roughly proportional to 1/p T, whereas the adiabatic enthalpy head ∆ha d is proportional to T. Consequently the ideal cooling power (m∆ha d ) should vary proportionally to p T. Since on the other hand the efficiency of the turbine decreases for off-design conditions, the actual cooling power during cool-down depends on the specific properties of the type of turbine in use. For H2 or He plants excess refrigeration and hence a significant reduction in cool-down time can be achieved by the use of liquid nitrogen. A different behaviour can be found in plants which are based on the regenerator principle. As can be seen in figure D5.0.3 for a Philips gas refrigerator, considerable excess refrigeration is available at higher temperatures despite reduced efficiency.

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Figure D5.0.3. (a) Thermodynamic efficiency versus condensate temperature for a Philips gas refrigerator, (b) Cooling power Pt h and electrical input power Pm versus condensate temperature for a commercial gas refrigerator.

Among the different operational needs the requirement to minimize the effort for operation and maintenance has a high priority. This requirement is clear especially for industrial plants where the personnel required for operation constitute an essential cost factor. The request for a fully automatic ‘singlebutton plant’ becomes essential in cases where the plant forms auxiliary equipment and therefore the operator does not need to be familiar with the details of the process. A consequence of this requirement may be increased investment costs and, on the other hand, reduced operator control capabilities. Reliability is a key function for all plants which are used in permanent mode for cooling of critical components (e.g. the cooling of a superconducting generator or the cooling of a receiver of a satellite station). In addition military and space-bound cooling systems demand the highest reliability, which is necessary because of the absence of personnel for operation and service. A typical requirement of 8000 h uninterrupted and maintenance-free operation has to be considered in the process design (e.g. by the installation of intermittent operating adsorbers to avoid blockage) as well

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as during the selection of the mechanical components (wear of bearings, piston-rings, sealings). In addition to that, operational restrictions may become important e.g. limited mass and dimensions for space-bound systems, emission of noise or vibrations by compressors, and the adaptation to existing utilities (cooling water, cooling air, diesel drive) or the operation under different environmental constraints (weightlessness, magnetic fields, mobile plants). From the plurality of requirements and constraints it becomes evident that there is no single optimum plant for all purposes. Plants have therefore to be matched to the specific boundary conditions. This explains the broad range of existing cryoplants. As discussed before the different requirements for a cryoplant have to be considered during the selection and design of the individual elements. In addition to reliability and efficiency of the single component economic aspects are also essential. Therefore, whenever possible, proven and economic standard products will be preferred which may have to be matched to the specific requirements of the cryoplant by design modifications. For example, helium compressors were originally developed for the compression of air and refrigerants. In order to achieve the necessary leak tightness for the operation with helium, standard sealings of the casing had to be replaced by O-rings. Special care has to be paid to the choice of suitable low-temperature materials. This is very obvious for cold valves or cold machines (expander). It has to be analysed, however, if ‘warm’ parts (pressure vessels, valves, machinery) may cool down significantly below ambient in special situations (failure mode, maloperation, warm-up of the plant). For safety reasons such parts have to be manufactured from lowtemperature materials like stainless steel, brass, copper or special aluminium alloys. (a) Compressors Until about 1930 cryogenic processes were almost exclusively based on high pressures (50–200 bar, i.e. 50–200 × 105 Pa) (Baldus et al 1983). This was justified for several reasons: (i) improvement of the efficiency at high pressures (ii) reduction of the energy losses at high pressure ratios (iii) reduced volume flow rates leading to smaller components. Compression of the gases was therefore performed by multistage oil-lubricated piston compressors which were able to achieve a high pressure ratio per stage. The development of larger and larger cryoplants pushed the piston compressors to their technical limits and opened the way for medium- and low-pressure processes and the use of rotary machinery. In figure D5.0.4 the range of operation of piston-, screw- and turbocompressors is illustrated for air. Helium plants with small and medium capacity often make use of piston compressors. Although dryrunning compressors have the advantage of delivering absolute oil-free gas they show shorter lifetimes of the piston rings and hence require greater maintenance. Self-lubricating piston rings made from carbon or plastic lead to higher temperatures and may result in increased wear, the debris from which may be deposited in the plant. By injecting small amounts of oil the friction and hence the lifetime of the piston rings could be significantly improved. The small amounts of oil are completely removed by downstream oil adsorbers. A different approach has been realized in labyrinth compressors. In this case a labyrinth in the piston wall with a very small clearance between the piston and the cylinder of typically a fraction of a millimetre provides the necessary sealing (Klein 1978). Because of the pressure difference across the labyrinth there is a small amount of leak gas. This has the disadvantage of raising the average temperature of compression and reducing the efficiency of the compressor somewhat. A further solution for oil-free compression is realized in membrane compressors. In this case an elastic metal membrane is used for compression of the gas. Since the gas space is hermetically separated from the drive, any contamination of the process gas is excluded. Although very high pressures of 200

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Figure D5.0.4. Range of operation of piston-, screw- and turbocompressors for air.

bar and more can be realized the capacity of the membrane compressors is limited to typically less than 30 m3 h-1. In low-temperature technology they are therefore used for the compression of the rare noble gases (Ne, Kr, Xe). In contrast to reciprocating compressors rotary machines have certain advantages. (i)

Due to the rotary movement the emission of noise and vibration is greatly reduced. For that reason compressors of small and medium capacity can be installed without foundation which reduces the investment costs. (ii) The absence of mechanical wear makes them maintenance-free to a large extent. The absence of inlet and outlet valves again reduces the maintenance costs and increases the reliability. (iii) The relation of the size of the machine (and hence its price) to the installed capacity becomes more and more favourable with increasing flow rates. (iv) Capacity control can be realized in a relatively simple and economic way, e.g. by adjustable guide vanes before or after the wheel of a turbocompressor or the use of a slide valve in screw compressors. Screw compressors dominate in small air separation plants and are widely used in helium plants of medium and large capacities. This type of compressor is based on two screw-shaped rotors (a primary rotor with a convex profile and a secondary rotor with a concave profile). As shown in figure D5.0.5 the counter-rotating rotors enclose a gas volume which is progressively compressed towards the discharge side. Oil-flooded screw compressors are mainly used unless special process restrictions require the use of dry-running versions. The oil which is injected after the suction process has two functions. On one hand it acts as a sealant between the primary and the secondary rotor and against the casing which allows larger clearances. On the other hand it absorbs a large amount of the heat of compression since the specific heat of oil is about 2.5 times that of air. In this way typical compression temperatures are limited to about 80–100°C. Table D5.0.1 shows a comparison of the features of dry-running and oil-flooded screw compressors. Again the big advantage of oil-flooded screw compressors becomes evident; however, they require reliable and highly effective oil separation systems. The thermodynamic efficiency of a screw compressor depends on different factors.

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Figure D5.0.5. Cross-section of a screw compressor. Table D5.0.1. A comparison of the features of dry-running and oil-flooded screw compressors.

First the intrinsic volume ratio Vi n /Vo u t , between the inlet and the outlet volume of the screws determines the optimum pressure ratio through the polytropic coefficient n

If the compressor is operated at a pressure ratio smaller than the intrinsic pressure ratio overcompression occurs. If it is operated at a larger pressure ratio than the optimum pressure ratio undercompression happens and high-pressure gas flows back into the discharge volume. In both cases the efficiency decreases. As can be seen from figure D5.0.6 (Klein 1978) there is another optimum with respect to the tip speed of the rotor and the injected oil flow rate. At low circumferential velocities the losses due to gas back-flow across the clearance increase in relation to the flow rate. High rotor speeds on the other hand increase the

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Figure D5.0.6. Specific input power for a screw compressor as a function of the tip speed of the rotor UH L and the injected oil flow rate Vo i l .

losses by gas friction. A higher oil injection rate tends to decrease the compression temperature. However, since this oil has to be pushed again through the screw this will result in increased frictional losses. Helium applications can only make use of oil-lubricated screw compressors since the gas back-flow increases with decreasing molecular weight. Low-temperature plants require very effective oil separation systems since oil impurities even in the parts per million (ppm) range would freeze out and block the heat exchangers and low-temperature valves. Figure D5.0.7 schematically shows a typical multistage oil separation system for a helium screw compressor. After the compressor block the oil—gas mixture first enters a gravity separator (cyclone, demister) where the bulk oil is separated from the helium. The subsequent separator elements retain larger droplets before the gas passes the gas cooler with a purity of approximately 50 ppm by weight of oil. After that two coalescers in series remove residual oil droplets to a level of <0.5 ppm. Finally a charcoal adsorber

Figure D5.0.7. A schematic diagram of a multistage oil separation system for a helium screw compressor.

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removes oil vapour leading to a final oil contamination level of <10−2 ppm. Whereas in helium plants only piston and screw compressors are used, large air separation plants (up to 300 000 N m3 h−1 ) require turbocompressors. In that case the gas usually enters the wheel of the first stage axially. The wheel transmits momentum to the gas whereby only a small fraction is used for pressure increase and most of the energy is converted into speed and hence into the kinetic energy of the gas. In a subsequent diffusor the kinetic energy is again transformed into pressure energy. In a very simplified way and assuming adiabatic conditions (change in entropy dS = 0) the first law of thermodynamics can be applied to a mass element of an ideal gas as follows

i.e.

From this relation it can be deduced that for a given gas velocity υ0 which is determined by the speed of the compressor wheel the achievable pressure ratio p1 /p0 will decrease with decreasing specific heat cp (number of atoms per molecule). In addition the energy transfer to the gas is reduced with lowering density because of the conservation of energy and momentum. From that it becomes obvious that, for example, helium turbocompressors have to run at higher circumferential speeds and need to have multiple stages as would be required for operation with a multiatomic gas of higher density. For heavy gases the achievable pressure head and consequently the speed per stage is limited by the velocity of sound. For light gases, however, the material stresses limit the maximum circumferential speed of the wheel. (b) Expanders The essential task of an expansion machine in low-temperature applications is not the gain of the energy by the expansion of the gas but to achieve cooling by the reduction of the enthalpy. In analogy to the considerations for the compressors the main application of piston expanders is in achieving lower throughputs and higher pressure ratios. On the other hand expansion turbines have their advantages at higher throughputs and lower pressure ratios. At the very beginning of cryogenic technology only oil-lubricated piston expanders were used. These machines had large pistons and oil-lubricated metallic piston rings at ambient temperature. Oil impurities which could be trapped by the gas were a permanent risk to the low-temperature parts of the plant (risk of explosion with O2 blockage). Nowadays mainly dry-running pistons with plastic rings (Teflon with graphite or a glass-fibre compound) are used, which are gently pressed to the cylinder wall by their own stiffness, pressure or by additional springs. An example of a helium expansion machine is shown in figure D5.0.8. This piston expander developed by the Linde AG has piston rings which are at operational temperature (80 K and 13 K) (Kneuer et al 1980). The inlet and outlet of the expander are initiated by cryogenic solenoid valves which are controlled by inductive switches at the crankshaft. The expansion is from about 25 bar to 1.2 bar. The gas throughput can be varied between 20% and 150% of the nominal capacity by electronic control of the opening time of the inlet valves. Adiabatic efficiencies of up to 85% have been reached. The use of expansion turbines started with the manufacture of the first cryogenic air turbine in 1936 by the Maschinenfabrik Sürth (now Atlas Copco). This radial turbine had a speed of 7000 rpm (revolutions per minute) and used a capacity control by switchable nozzles. Since 1945 the use of turbines in low-temperature technology has significantly increased. Almost all radial turbines built were of the reaction type where the gas flows radially inwards (centripetal turbine).

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Figure D5.0.8. Cross-section of a piston expander for helium. Reproduced by permission of Linde AG.

The reason for that is that the centrifugal forces are acting against the flow which supports the energy transfer to the wheel and allows the design of shorter turbine wheels. A simplified enthalpy—entropy diagram of a turbine is schematically shown in figure D5.0.9. In the nozzle ring the gas is throttled from the inlet pressure p0 to the nozzle pressure p1 whereby the enthalpy is converted into kinetic energy. The further expansion to the outlet pressure p2 and conversion of the kinetic energy of the gas into mechanical energy is performed at the wheel. Downstream of the wheel the gas is passing a diffusor where the exit velocity from the wheel is again converted into pressure. The degree of reaction r of a turbine is defined as the ratio of the enthalpy head across the wheel to the total enthalpy head for the case of ideal expansion

Optimum efficiencies are achieved with r between 0.45 and 0.6. In addition the specific speed which is defined as the ratio of the tip speed u to the gas inlet speed c0 should be in the range 0.65–0.75 for centripetal turbines. In order to reach an optimum efficiency several loss factors have to be minimized as far as possible: (i)

gas friction in the nozzle ring

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Figure D5.0.9. A typical enthalpy—entropy diagram of a turbine.

(ii) (iii) (iv) (v) (vi) (vi)

shock losses between the nozzle and the wheel (e.g. by limitation of the exit gas speed from the nozzles) gas friction at the surface of the wheel frictional losses caused by the pumping effect of the backside of the wheel gas leakage between the wheel and the contour losses by incomplete recovery of the kinetic energy of the gas e.g. by whirls in the diffuser insulation losses by heat input along the rotor and the low-temperature housing.

For air separation plants the cooling powers achieved in the turbines are sufficiently high and at the same time the speed of the turbines are sufficiently low (20 000–30 000 rpm) that the recovery of the energy may be justified from an economic point of view. In that case the power gained by the brake is converted into electrical energy in a generator or can be used in a coupled blower to recompress the process gas. For helium expanders the cooling powers are small compared with the installed compressor power and the speeds are generally rather high (80 000–400 000 rpm). Therefore the turbine power is converted into heat in an oil or gas brake and removed from the system through a heat exchanger to the cooling water. With respect to the use of turbines for the light gases helium and hydrogen the same arguments apply as for the turbocompressors. This means that especially at higher temperatures reasonable efficiencies can only be obtained with relatively low pressure ratios. In order to maintain a high efficiency for a given enthalpy head the cross-section of the wheel has to be proportional to the volume flow of the gas. Since there are certain limits for the blade height of the wheel the diameter of the wheel is roughly proportional to the volume flow for small turbines. On the other hand the optimum speed is inversely proportional to the flow. This leads to technical problems with bearings and sealings for turbines with small power and high speeds. In many cases the maximum tolerable tip speed is limited by the stiffness of the expansion or compressor wheel (for Al alloy um a x = 400–500 m s−1, for titanium 600 m s−1). The development of high-speed turbines started in about 1960 with the need for larger cooling capacities and because of their inherent advantages against piston expanders, e.g. compact design, absence of wear and maintenance, high reliability and smooth running. Nowadays standard turbines with gas

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Figure D5.0.10. A high-speed expansion turbine with dynamic gas bearings. Reproduced by permission of Linde Kryotechnik AG.

bearings and speeds up to 300 000 rpm are available. In experimental turbines speeds up to 700 000 rpm have been achieved. An example of a well proven turbine is shown in figure D5.0.10. In general two radial and two axial bearings are required to centre the rotor and to balance the axial forces. There are basically three types of bearing, namely oil, static gas and dynamic gas bearings (figure D5.0.11). Turbines with high capacity and high power in general use oil bearings to control the large forces. In that case the injected oil is used both for lubrication of the bearing and to brake the turbine rotor. The speed control is performed through the flow rate of the injected oil. In order to avoid oil leakage towards the cold process side, a small leak of the process gas towards the bearing side is tolerated. This small amount of leakage reduces the efficiency of the turbine only insignificantly. The leak gas is fed back to the compressor suction side after an appropriate oil purification. In a static gas bearing the bearings are directly pressurized by a small part of the process gas from the cycle compressor. This helps to stabilize the rotor in the radial and axial bearings. In order to avoid leakage of the process gas into the bearing or of the bearing gas to the cold side an active control of the bearing pressure is necessary. A single-stage centrifugal compressor which is mounted on the rotor opposite the turbine wheel acts as a brake. The compressed warm gas is expanded again in a brake valve and recooled in a water cooler. The speed of this centrifugal compressor can be varied within certain limits by adjusting the gas circulation. Turbines with dynamic gas bearings do not require an external gas supply. Their self-acting bearings

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Figure D5.0.11. A schematic diagram of different kinds of bearing for high-speed expansion turbines: (a) oil bearings; (b) static gas bearings; (c) dynamic gas bearings.

are charged by the process gas and through the rotation of the shaft. Because the pressure in the bearing equals the inlet pressure at the wheel, any leakage between the process side and the rotor side is avoided. The pressure for the radial and axial support of the rotor is established by the rotation and special flow elements such as tilted pads, spiral grooves or herringbone grooves. Hence no active control of the bearing is required. The operational range of gas bearings is from about 0.1 kW, limited by gas friction, to approximately 100 kW, limited by the maximum load of the gas bearing. For cooling powers from about 5 kW up to more than 150 kW oil bearings are mainly used. Static oil bearings or static gas bearings which are permanently supported by oil or gas films respectively do not show mechanical contact between the rotor and the bearing even at zero speed. This helps to spin up the rotor. In order to reduce the frictional forces during start-up in dynamic gas bearings auxiliary axial bearings (e.g. a permanent magnet or an auxiliary static gas bearing) may be used. (c) Cold compressors There are several reasons for operating superconducting magnets or radiofrequency (RF) cavities at temperatures lower than the Tλ of helium (2.18 K). Maintaining a helium bath at such temperatures requires its pressure to be kept below 0.050 bar. Typically temperatures between 1.8 K (0.016 bar) and 2.0 K (0.031 bar) are used. For small refrigeration capacities the helium is pumped off the bath by vacuum pumps operated at ambient temperature and the detected heat is partially recovered. With increasing capacities perfect recovery of the refrigeration stored in the boil-off stream becomes more and more relevant. The required heat exchanger, however, is very delicate and expensive to build since volumetric flow is high and the

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Survey on field calculations

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acceptable pressure drop should not exceed a few millibar. Cold vapour compression of the boil-off can help this situation. Radial-flow turbocompressors have been developed for this purpose. They typically are powered by high-frequency electric motors or expansion turbines. Several types of bearing are used ranging from static or dynamic gas bearings, hybrid ball bearings to active magnetic bearings. Isentropic efficiency is in the range of 0.6 to 0.7 depending on size. This type of machine is used either in combination with warm vacuum pumps or for full compression to roughly ambient pressure (Gisteau-Baguer 1996). (d) Heat exchangers Heat exchangers play an essential role in low-temperature technology, since the thermodynamic losses caused by finite temperature differences increase with decreasing temperature of the fluid. The classic heat exchangers were of the tube-in-shell type. They were optimized for operation at high process pressures and reached specific densities of the heating area of approximately 400 m2 m−3. Nowadays vacuum-brazed aluminium-plate fin heat exchangers are the most commonly used type in low-temperature technology. Their advantages are a compact design with up to 1500 m2 heating area per m3 of block volume and an economic manufacturing process, since the blocks are assembled from standard elements. The vacuum-brazing process eliminates contaminations which is a further advantage for cryogenic plants. The construction of an Al heat exchanger is shown in figure D5.0.12. The separating aluminium sheets between the flow passages are connected by corrugated sheets, socalled ‘fins’. These fins act as secondary heating areas and may account for up to 90% of the total heat exchange area. Special distribution fins ensure a uniform flow through the individual flow passages. The

Figure D5.0.12. An aluminum heat exchanger; 1—separation sheet; 2—fins (corrugated sheet); 3—distribution fins; 4—profiled rod; 5—housing; 6—header.

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separating sheets are plated with the solder in such a way that the melting point of the solder is just below that of the base material (approximately 600°C). Brazing of the staggered blocks is performed in vacuum ovens. Finally the ‘headers’ and tube connections are welded to the blocks. The high thermal conductivity of the aluminium ensures a good heat transfer between the gas flows. In cryogenic plants, however, longitudinal heat conduction is undesirable and has to be accounted for by appropriate sizing factors for the block length. Depending on the applied codes, the maximum design pressures are between 60 and 90 bar which does not represent a restriction for modern low-temperature applications. The considerable thermal expansion coefficient of aluminium may represent an operational constraint. Because of the compact design of this type of heat exchanger, excessive material stresses have to be avoided. Therefore the maximum tolerable temperature differences between adjacent flows are in general limited to 50°C. D5.0.3 Examples of cryogenic plants The selection of the following examples of cryogenic plants is neither representative nor complete. The selection is primarily for reasons of method. D5.0.3.1 Multipurpose helium refrigerator of medium capacity (Kneuer et al 1980) This modular-designed helium refrigerator can liquefy approximately 4 1 h–1 liquid helium in a minimal configuration using one air-cooled compressor. Increasing the cycle flow to 12 g s–1 by using more compressors in parallel the liquefaction rate can be increased to 18 1 h–1. This rate can be further increased by liquid-nitrogen precooling to a maximum of 30 1 h–1. As can be seen from figure D5.0.13 the applied Claude process uses two temperature levels (approximately 80 K and 13 K) for precooling which are provided by two piston expanders working in parallel. These expanders are driven between 25 bar and 1.2 bar.

Figure D5.0.13. A schematic diagram of a helium refrigerator. Reproduced by permission of Linde AG.

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If the plant is operated as a refrigerator an intermediate throttling before the coldest heat exchanger results in an increase of the specific heat of the high-pressure flow. By this means a crossover in the Q—T diagram of the last heat exchanger is avoided. The purification of the feed gas is performed in several process steps at a pressure of 20 bar: (i) changeable dryers, which use a molecular sieve and automatically remove moisture and CO2 (ii) gross air impurities (N2 /O2 ) are removed down to a level of 1% by a low-temperature purifier. The air constituents are cooled down and condensed in a heat exchanger and vented to atmosphere. Impurities of up to 20% in the helium gas can. be handled; however, this is at the expense of a reduction of the capacity of the plant (iii) ultrapurification down to <1 ppm N2 /O2 is achieved in automatic pressure swing adsorbers at ambient temperature. These adsorbers are based on the physical principle that the adsorbers can take higher levels of impurities at high pressures than at low pressures. Additional guard adsorbers are integrated into the cooling cycle to remove traces of N2 /O2 as well as Ne/H2 . These adsorbers are most effective if they are operated at about the condensation temperature of the gas to be adsorbed (under these conditions the binding energy of the molecule/atom is approximately equal to the thermal energy). A further element of this cryogenic plant is an oil-lubricated piston compressor with subsequent fine purification of oil. As an alternative, two-stage units with water cooling or three-stage units with air cooling can be used. As a refrigerator the plant can deliver up to 78 W at 4.4 K with a power consumption of 64 kW for the compression of 12 g s–1 to 25 bar. These figures correspond to a thermodynamic efficiency of 8.4%.

Figure D5.0.14. A schematic diagram of a two-stage Stirling gas refrigerator: A-20—refrigerator; E1–E7—heat exchanger; A—adsorption cleaner; K—condenser for air; D—ejector; Dr—throttle valve; DMV—pressure reduction valve; V—valve for transfer line. Reproduced by permission of Phillips.

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D5.0.3.2 A 10 l h-1 helium liquefier Amongst the very different types of helium liquefier for laboratory use the Philips helium liquefier (Haarhuis 1968) (which is no longer manufactured) will serve as an example because of its outstanding process. As shown in figure D5.0.14, two two-staged Stirling gas refrigeration machines are used to provide refrigeration at four temperature levels (87 K, 62.5 K, 20.1 K, 15.1 K) in order to cool the compressed cycle gas at 20 bar. As a consequence the specific power consumption of only 2.8 kW h 1–1 is very considerable for a plant of this capacity without liquid-nitrogen precooling. The lowest refrigeration level of 15.1 K is a compromise. On one hand the efficiency of the Stirling machine rapidly approaches zero at low temperatures (Q = 0 W at T = 12 K) because of the strong decrease of the specific heat of the regenerator material. On the other hand the energy losses of the Joule—Thomson expansion increase with increasing temperature. Another speciality of this plant is the use of an ejector upstream of the usual Joule—Thomson valve. This ejector (figure D5.0.15) works like a water jet pump: the high-pressure cycle gas enters the ejector at a pressure of 20 bar and a temperature of 5.4 K and is expanded in the nozzle to 2.5 bar. The acceleration of the gas causes a local decrease of the pressure which can be used to pump the vent gas from the storage tank at a level of 1 bar and compress it to 2.5 bar. Part of the 2.5 bar helium gas is further expanded in a Joule—Thomson valve to produce liquid and the remainder is fed back to the cycle compressor. The basic advantage of the ejector is on the compressor side. Because of the higher suction pressure

Figure D5.0.15. Cross-section through an ejector.

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(2.5 bar) the compressor needs fewer stages and because of the smaller effective volume flow rates it can be smaller. A special feature of this two-stage compressor is a unique sealing between the piston and the cylinder by a rolling membrane. By this means any oil impurities in the helium gas can be avoided. As can be seen from the flow schematic (figure D5.0.14), the refrigeration cycle is almost completely independent of the liquefaction cycle. Hence the removal of impurities is performed in adsorbers in the feed gas only. The adsorbers for H2O, CO2 , Ne and H2 are designed for a fixed operation period of 100 h. The adsorbers for O2 , N2 and Ar are automatically changed and regenerated at 145 K. D5.0.3.3 A 2 W Gifford—McMahon refrigerator Although the Gifford—McMahon process exhibits slightly poorer efficiencies than the Stirling process it has nevertheless the advantage that the compressor can be installed independently of the cold head which reduces the necessary volume and the noise emission of the cold part. For that reason standard piston compressors working at high speed (2900 rpm) can be combined with low-speed displacers (120 rpm). For special applications where low cooling powers at liquid-helium temperatures are required a two-stage Gifford—McMahon refrigerator was combined with an additional Joule—Thomson cycle (Müller et al 1984) (refer to figure D5.0.16).

Figure D5.0.16. A schematic diagram of a three-stage refrigerator.

Since only part of the refrigeration power of the first stage (40 W) and the second stage (10 W) is required for the cool-down of the Joule—Thomson flow, excess refrigeration is available at 80 K for the first stage and 20 K for the second stage for cooling of the radiation shields. The characteristic figures for this cryogenic plant are shown in table D5.0.2. As a special element a thermal switch between the second and the third stage which uses gaseous H2 is worth mentioning. At temperatures above the inversion temperature of helium the heat conduction of the H2 gas provides the necessary heat transfer between the Joule—Thomson stage and the second stage (in that temperature range the negative Joule—Thomson effect would preclude any cool-down of the heat exchanger). Below the inversion curve the expansion results in further cool-down whereby the hydrogen freezes. As a consequence the heat conductivity of the switch is strongly reduced which leads to thermal insulation between the third stage and the second stage.

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Examples of cryogenic plants Table D5.0.2. Characteristic figures for a cryogenic plant.

D5.0.3.4 A 300 W, 1.8 K refrigerator In general, temperatures around 4.2 K are sufficient for the cooling of superconducting coils. Lower temperatures may be required, for example, if very high current densities have to be achieved in the superconductor. Another low-temperature application is in high-frequency cavities which are used, for example, for the acceleration of elementary particles. The quality of the resonant oscillator and hence the losses of the accelerator depend on the high-frequency impedance of the surfaces of the resonator. As a consequence the inner walls of the resonator are plated with superconducting material. Temperatures around 1.8 K turned out to be optimum for this application. An example of such a low-temperature refrigerator is shown in figure D5.0.17. This plant has been in operation at the Kernforschungszentrum at Karlsruhe since 1971 (Baldus 1970). The process is again based on a two-stage Claude cycle where the low vapour pressure of 16.4 mbar (corresponding to a bath temperature of 1.8 K) is achieved by a vacuum pump of the roots blower type with eight stages and a capacity of 10 m3 s–1. This roots system is installed upstream of the cycle compressor. Special emphasis has to be paid to the design of the low-pressure side of the heat exchanger. Any pressure drop in that heat exchanger would contribute significantly to the energy losses and would increase the size and the investment costs of the vacuum pumps (suction volume). On the other hand a minimum pressure drop is required to achieve sufficient flow velocity and hence have adequate heat transfer at a reasonable size of heat exchanger. In this special case approximately 90 Cu tubes with a diameter of 3 mm have been used and result in a pressure drop of less than 5 mbar at the low-pressure side. In addition to the refrigeration mode at 1.8 K, excess refrigeration can be provided by the plant at 4.4 K to other consumers as shown in table D5.0.3. Other essential components are: (i)

two oil-lubricated expansion turbines working in series at speeds of up to 100 000 rpm and cooling powers between 2 to 6 kW (ii) oil-free compression from 1 to 20 bar by dry-running compressors using plastic piston rings and a suction capacity of 3000 m3 h–1 The plant described above was a milestone in large-capacity refrigeration at temperatures below the lambda point. It was followed by other similar systems: • • • •

a 100 W 1.8 K plant at TU Darmstadt, Germany a 300 W 1.8 K plant at CERN, Geneva, Switzerland a 320 W 1.75 K plant for Tore Supra in Cadarache, France a 4800 W 2.0 K plant for CEBAF, Newport News, VA, USA.

The Tore Supra plant features two cryogenic turbocompressors based on active magnetic bearings and two liquid ring pumps operated at ambient temperature switched in series. The CEBAF plant uses four cold compressors for helium compression from approximately 16 mbar to ambient pressure. They are all based on active magnetic bearings.

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Figure D5.0.17. 350 W, 1.8 K cryogenic power plant at the Kernforschungszentrum Karlsruhe: C—compressor; D—medium pressure buffer; E—heat exchanger; A—adsorber; T1 and T2—expansion turbines; V—evaporator; K—cryostat; T—liquid helium tank; PC—pressure control; LC—regulation of helium level. Reproduced by permission of Linde AG. TableD5.0.3. Refrigerator data.

D5.0.3.5 A 400 W refrigerator The next example is a helium refrigerator which was established for the testing of superconducting prototype magnets of the HERA storage ring at DESY (Hamburg). For this application multiple operation modes had to be considered in the design of the plant: (i) refrigerator mode at 4.4 K (isothermal)—400 W (ii) liquefaction mode—100 1 h–1 (iii) refrigeration mode at 3.8–5 K—150 W including liquid withdrawal, +1.6 g s–1

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(iv) in addition to all operation modes a radiation shield had to be cooled at approximately 70 K, 800 W. The refrigeration power at 3.8 K had to be provided either as a two-phase fluid or as a single-phase fluid with an adjustable pressure between 1.4 and 2 bar. In addition any excess refrigeration of the plant should be stored as liquid helium through a second transfer connection. Figure D5.0.18 schematically shows the selected two-stage Claude process.

Figure D5.0.18. A schematic diagram of a two-stage Claude process plant: 400 W, 100 1 helium per hour. C—compressor; D—medium-pressure buffer; E—heat exchanger; A—adsorber; T1 and T2—expansion turbines. Reproduced by permission of Linde AG.

During stationary operation at 4.4 K an oil-lubricated screw compressor is processing approximately 145 g s–1 helium from 1.05 bar to 13 bar with a power consumption of 380 kW. Temperatures below 4.4 K with a corresponding lower vapour pressure in the subcooler are realized by a two-stage dry-running roots blower with bypass control. The two Joule—Thomson valves operate independently of each other and allow the delivery of the refrigeration through two transfer lines to the magnets and then to a liquid-helium Dewar. Any excess refrigeration from the magnets is expanded into the subcooler and eventually evaporated by a heater. In that way sudden load changes during the magnet tests can be controlled and the turbines can be kept at constant operational conditions. Further features of the plant are: (i) aluminium plate fin heat exchangers (ii) static gas bearing turbines with powers between 2 and 8 kW (iii) a free programmable control system for process control and alarms. In order to be able to match the operation of the plant to the prevailing test conditions at short notice the control system allows manual adjustment through conventional controllers.

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D5.0.4 Minirefrigerators An important driver for the development of minirefrigerators (with typical powers of <10 W at T = 70 K and <1 W at T = 20 K) was the use of infrared detectors for earth observation and for night reconnaissance systems. Such semiconductor detectors are sensitive to wavelengths between 1 and 20 µm where the electrical conductivity of the material is increased by free electrons which are excited by the energy of the impinging photons. Since the energy of these photons is comparable with the thermal energy kT at room temperature, sufficient sensitivity can only be achieved by a reduction of the thermal activation of the free electrons. Therefore these detectors have to be cooled. For most applications temperatures between 20 and 70 K are sufficient. In order to reach the highest sensitivities and at the same time to reduce the thermal noise of the electronic amplifiers, temperatures down to 4 K and below are now required by infrared astronomy. Another market for minirefrigerators at liquid-helium temperatures is for instance in cooling devices for superconducting components e.g. SQUIDs or small magnet systems. A major criterion for the use of reliable minirefrigerators with high efficiency and very low temperatures is their economy when compared with conventional cooling by liquid nitrogen or liquid helium. Therefore their application so far has been rather limited. A major problem for the industrialization of such systems is the high costs for development which are opposed by economy aspects which expect small systems to be cheap. Therefore the development of such systems has been mainly concentrated on military applications (either on ground or in space) where the simple supply of the experiment with the cryo-fluid is excluded and a special development is justified. For earth-bound minirefrigerators the major requirements are small size and high reliability under rough environmental conditions. Special requirements are, however, put on future space-bound closedloop cooling systems: (i) (ii) (iii) (iv) (v)

the highest possible thermodynamic efficiency, since the supply as well as rejection of powers > 1 kW cause considerable problems in space mechanical stability against the large static and dynamic loads during the take off of the launcher low mass compared with the refrigeration power absence of mechanical vibrations (which would disturb, for example, infrared detectors) highest degree of reliability of all single parts.

Minirefrigerators have been built using almost all of the processes described in chapter D3. Although regenerative processes dominate, even Joule—Thomson coolers have been used in situations where the efficiency is not so important. Some recent developments of prototype refrigerators which were intended for space applications will be described in more detail. (a) Magnetically supported linear Stirling refrigerator This type of machine which was developed by Philips USA (Daniels et al 1984) is characterized by permanent magnets integrated into both the piston and the displacer which are again driven by external solenoids at a frequency of around 25 Hz (refer to figure D5.0.19). In that case the regenerator is moved together with the displacer. The bearings use electromagnetic forces and are actively controlled by radial distance sensors and compensation coils. Imbalances during operation are kept below 1 µm. Sealing of the working spaces is performed by very close clearances. Refrigeration values of 5 W have been achieved at a temperature of 64.6 K with a single-stage unit. If one relates this capacity to an electrical power consumption of 220 W this results in a thermodynamic efficiency of only 1.7%. (b) Turborefrigerator For several years the US company AiResearch/Garrett developed a minirefrigerator for space applications which exclusively used turbomachinery (Wapato and Norman 1980). The two-stage Brayton cycle

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Figure D5.0.19. A magnetically supported linear Philips type Stirling refrigerator.

processes a He/Ne mixture which is compressed in a two-stage turbocompressor. The machine can reach temperatures down to 30 K (refer to figure D5.0.20). A particular feature is the two turbines which are located on either end of a common rotor but are thermodynamically working in parallel. The rejection of the expansion energy is performed through a turboalternator. Here magnets which are integrated into the rotor produce induction currents in the surrounding coils. The induction loop is closed and the electrical energy is dissipated through a load resistor to the environment at ambient temperature. The big advantage is that the whole turbine, including the bearing, is located at cryogenic temperatures which minimizes the harmful conductive heat losses through the housing and the rotor. On the other hand the frictional losses of the gas bearings greatly affect the efficiency of the turbine. Foil-type gas bearings were used for the radial and axial bearings of these high-speed rotors (compressor 60 000-120 000 rpm, turbine 200 000 rpm). So far 5–20 W of refrigeration at temperatures of about 30 K have been reported. As a result of the use of turbomachinery, this system is considered promising for long life operation. If, however, only very small cooling powers have to be provided at liquid-helium temperatures these systems are no longer very effective, since the low cycle flow rates imply many-stage compressors with very small wheels and very high speeds. This leads to a rapid degradation of the efficiency especially with the turbines (tolerances and frictional losses become comparable with the cooling power).

Figure D5.0.20. A schematic diagram of a small, two-stage turborefrigerator.

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References

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(c) Rotary reciprocating refrigerator (R 3 ) This system was developed by Arthur D Little Inc. and is characterized by a rotating piston both for the compressor and for the expander. Stirling, Vuilleumier and Brayton cycles can be realized with these components (Breckenridge 1980). An induced rotation which is superimposed on the linear motion of the pistons behaves as a self-acting gas bearing. In this way the harmful friction of the piston rings is avoided, which is a basic requirement for long periods of maintenance-free operation. However, very narrow clearances of only a few micrometres have to be maintained to avoid excessive leakage during compression. The pistons are driven by electromagnetic linear and synchronous motors. In order to balance the reactive forces, two pistons are placed in one housing and operate in counter-phase and with opposite directions of rotation. Since the piston in the compressor is rotating at around 2400 rpm and is oscillating at 1200 rpm it turns out that the buoyancy forces of the piston far exceed those of the compression forces of the gas. The compressed gas volumes act as springs and are used to support the reactive forces by the linear drive. For a combination of two such systems which were combined to realize a two-stage Brayton process the figures given in table D5.0.4 were published. Table D5.0.4. Data for two-stage Brayton process.

In addition to the different ‘classic’ processes which were described above, magnetic refrigeration processes may in future also be applied. A potential application could be in the combination of a Stirling or Gifford—McMahon precooling stage with a magnetic refrigerator. Essential prerequisites for magnetic refrigerators, however, are suitable magnetic materials with a high magnetic moment and a low ordering temperature, a low specific heat and a high heat conductivity as well as reliable mechanical solutions for the movement of the magnetic material with a magnetic field. References Baldus W 1970 Helium refrigerator for 300 W at 1.8 K Adv. Cryogen. Eng. 16 163

Baldus H, Baumgärtner K, Knapp H and Streich M 1983 Chemische Technologie vol 3, 4th edn (Hauser) pp 589–600

Breckenridge R W 1980 Refrigerators for cooling spaceborne sensors Proc. SPIE 245 112 Daniels A, Stolfi F, Sherman A and Gassner M 1984 Magnetically suspended Stirling cryogenic space refrigerator: test results Adv. Cryogen. Eng. 29 639 Frey H and Haefer R A 1981 Tieftemperaturtechnologie (Düsseldorf: Eder VDI) Gisteau-Baguer G 1996 High power refrigeration at temperatures around 2.0 K Proc. ICEC 16 (Kitakiyushu) at press Haarhuis G J 1968 Der Philips Heliumverflüssiger Philips Tech. Rev. 29 202 Klein R 1978 Schraubenverdichter für den Einsatz in Wärmepumpen, Die Kälte und Klimatechnik vol 5 p 209 Kneuer R, Petersen K and Stephan A 1980 Automatische Mehrbereichs—Helium Verflüssigungsanlage Cryogenics 20 132 Müller D, Klein H H, Legrand N and Kiese G 1984 A three-stage refrigerator for temperatures down to 4.2 K Vide special issue April p 93 Streich M 1981 VDI Kryolehrgang BW 2025 (Düsseldorf: VDI Bildungswerk) Wapato P G and Norman R H 1980 Long-duration cryogenic cooling with the reversed Brayton turbo-refrigerator Proc. SPIE 245 120

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D6 Small cryocoolers

A Ravex

D6.0.1 Introduction The development of small mechanical cryocoolers has been strongly stimulated over the past years by the emergence of specific applications requiring low-temperature operation with relatively low cooling power. Today the main applications for cryocoolers are in (i) (ii) (iii) (iv) (v)

cryopumping for high and clean vacuum (semiconductor industry, space simulation chambers, particle accelerators); cooling of detectors (for example infrared detectors for earth observation, night vision and missile guidance, gamma-ray detectors, bolometers for astrophysics); cooling of electronic components (cold amplifiers) or of devices including superconducting materials (SQUIDs, Josephson junctions, high-field magnets); cooling of samples for physicists; cooling of radiation thermal shields and recondensation of boiling off in cryogenic liquid storage tanks or large superconducting magnet cryostats for magnetic resonance imaging (MRI).

Typical cooling powers of the small cryocoolers devoted to these applications range from a few tenths of a watt at 4 K or less to about a few tens of watts at 80 K or more. These heat loads are small compared with the large duties required for industrial gas liquefaction plants which in the past have driven the development of large refrigerators or liquifiers based on the Linde—Hampton or Claude processes. Large cold pistons or centrifugal expanders are commonly used in the Brayton cycles (Claude processes) to achieve a good thermodynamical efficiency by isentropic expansion of the cycle fluid with external work extraction. These expanders undergo a drastic efficiency decrease when miniaturized. Thus the small mechanical cryocoolers are developed on the basis of other cycles such as the Stirling, Ericsson (also known as Gifford—MacMahon), Joule—Thomson and more recently pulse tube cycles which do not involve such isentropic expanders. The thermodynamics of these cycles, their practical operation, their main applications and present and future developments are discussed in the following sections. D6.0.2 Thermodynamic considerations about cryocoolers The purpose of a refrigerator is to extract an amount of heat Qc at a cold temperature Tc . This heat load Qc has two contributions: the intrinsic inefficiency of the refrigerators and either a parasitic heat input (conduction through mechanical structures, convection due to residual gas, thermal radiation from

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surrounding warm surfaces, etc) or a local heat dissipation (electronics, a.c. losses in a superconductor, etc). Thermodynamics tell us that for such an operation, some mechanical work Wc has to be transmitted to a fluid following a closed cycle during which the heat load Qc is removed at the cold-sink temperature Tc and an amount of heat Qa(=Qc+Wc ) is rejected at ambient temperature Ta to he surroundings. The efficiency or the energetical costs for this operation are commonly measured either by the coefficient of performance (COP = Qc /Wc ) or by the specific energy consumption (Wc /Qc ) which depend upon the effective cycle followed by the fluid. D6.0.2.1 Theoretical reversible cycles The maximum value of the COP is obtained in a reversible cycle such as the Carnot cycle which is represented (1 → 2 → 3 → 4) on a temperature—entropy (T—S) diagram in figure D6.0.1.

Figure D6.0.1. A Carnot cycle (T—S diagram).

In this cycle the heat transfers between the cycle fluid and the heat sinks at Tc and Ta are assumed to be reversible and isothermal. The compression and expansion of the fluid are supposed to be reversible and adiabatic (i.e. isentropic) transformations. In the T—S diagram, the reversible specific heat exchanges (q = ∫ TdS) are represented by the area under the lines showing the evolution of the fluid. For a reversible cyclic operation, the specific work w is represented by the area inside the closed loop of the cycle on the T—S diagram. Thus, the maximum COP of a refrigerator operating in a reversible way between heat sinks at temperatures Tc and Ta can be written as

Typical values for the maximum achievable COP are given in table D6.0.1 for cold-sink temperatures corresponding approximately to the liquid-nitrogen, hydrogen or helium boiling temperatures under normal atmospheric pressure (1 atmosphere). For mechanical coolers devoted to operation at a cryogenic temperature (Tc < 100 K) it is practically impossible to operate following a Carnot cycle. In fact the required pressure ratio would widely exceed the present technological and mechanical limitations in compressor technology. As shown in figure D6.0.2, some modifications of the basic Carnot cycle allow us to overcome these limitations and yet keep the COP of the cycle equal to that of a Carnot cycle as shown hereafter.

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Table D6.0.1. Maximum COP for reversible cycles operating between Tc and ambient temperature (Ta = 300 K).

Figure D6.0.2. Carnot, Stirling and Ericsson cycles (T—S diagram).

To bring the high pressure (point 4 in figure D6.0.1 for a Carnot cycle) back to a reasonable value, the isentropic compression and expansion in the Carnot cycle are substituted either by reversible isochoric evolutions (Stirling cycle: 1 → 2 → 3’ → 4’ ) or by reversible isobaric evolutions (Ericsson cycle: 1 → 2 → 3” → 4”). It is obvious, in figure D6.0.2, that these new cycles allow us to extract at the cold sink Tc the same amout of heat (∫12 T ds = area under the 1 → 2 line) as in the original Carnot cycle with also the same mechanical work requirement (the area of the parallelepipeds representing the various cycles remains constant). Thus these new cycles may theoretically allow us to obtain the maximum COP if reversible operation is achieved. However, a new feature characterizes these cycles. During either the isochoric processes ( Stirling ) or the isobaric processes (Ericsson) a heat transfer at variable temperature between the fluid and the surroundings is required. This heat transfer (∫ T ds ) is represented on the T—S diagram in figure D6.0.2 by the area under the lines 2 → 3’ or 3” and 4’ or 4” → 1. It corresponds to the variation of the energy of the fluid when it is transferred back and forth between the heat sinks at the temperatures Ta and Tc where isothermal heat exchange is accomplished. For a perfect gas we have

where u, h, s and υ are respectively the specific or massic internal energy, enthalpy, entropy and volume of the fluid and cp and cυ are the isobaric and isochoric specific heats. Consequently we can write: (i) for an isobaric process (dP = 0) dT/T = dS/cp (ii) for an isochoric process (dυ = 0) dT/T = −dS/cυ . From these relations we deduce that in the T—S diagram the isobaric lines and the isochoric lines for a perfect gas are parallel lines with respective slopes T/cp and T/cυ . It means that the amounts of heat transferred during the 2 → 3’ or 3” and 4’ or 4” → 1 processes, which are represented in the T—S

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diagram by ∫23 or 3” T dS and ∫23or 4” T dS, are equal in magnitude and opposite in sign. In practice, when developing a cooler based on either the Stirling or Ericsson cycle, it will be possible to use either a counterflow heat exchanger (in the case of a continuous flow refrigerator) or a regenerator (in the case of an alternate flow refrigerator) to directly (counterflow heat exchanger) or indirectly (regenerator) exchange the energy between the fluid flowing from ambient temperature to low temperature and the fluid flowing back in the opposite direction. The development of small mechanical cryocoolers has been mainly based on the technology of the regenerative heat exchangers. These regenerators are generally constituted of a porous matrix (metal wire mesh or spheres) which acts like a thermal sponge alternately storing or rejecting heat. (a) Gifford—MacMahon-tt ype cryocooler A Gifford—MacMahon (GM) cryocooler is designed to allow the gas to follow an isobaric/isothermal Ericsson cycle. The high- and low-pressure sides of a compressor, in which the gas undergoes an isothermal compression, are alternately connected via an inlet and an outlet valve to a cylinder in which a displacer containing the regenerator can be moved. The operation of the valves is synchronized with the position of the displacer as shown schematically in figure D6.0.3 and the process theoretically operates as follows: Phase (1): The displacer is at its lowest position, the outlet valve is closed and the inlet valve is opened. The high-pressure gas fills the regenerator and the space above the displacer at room temperature. Phase (2): The inlet valve is still open, the displacer moves to its upper position. The high-pressure gas passes through the regenerator, is cooled down isobarically by the matrix and fills the space below the displacer at low temperature. Phase (3): The displacer is at its upper position, the inlet valve is closed and the outlet valve is opened. The gas in the regenerator and in the cold space undergoes expansion: the cooling effect achieved can be used for refrigeration which is theoretically assumed to occur at constant temperature. Phase (4): The outlet valve is still open, the displacer moves back to its lowest position. The low-pressure gas passes through the regenerator, is warmed up isobarically by the matrix and fills the space above the displacer at room temperature. A heat exchanger at the exhaust of the compressor is used to reject heat at the ambient temperature and to theoretically achieve an isothermal compression. (b) Stirling-tt ype cryocooler A Stirling cryocooler is designed to allow the gas to follow an isochoric/isothermal cycle. A typical arrangement is shown schematically in figure D6.0.4. A cylinder contains two pistons. The volume between the pistons (working volume) is divided into two parts by the regenerator. The compression piston can be moved in the compression volume which is kept at ambient temperature by means of a heat exchanger; the expansion piston can be moved in the expansion volume which remains at the cooling temperature. The ideal Stirling cycle can be described as follows: Phase (1)—isothermal compression: The expansion piston is kept close to the regenerator. The compression piston is moved to compress isothermally the gas in the compression volume. The compression work wc is transmitted to the gas and heat qa is rejected at ambient temperature. Phase (2)—isochoric precooling: Both pistons are now moved simultaneously to transfer the compressed gas at constant volume through the regenerator from the compression volume to the expansion volume. The gas is cooled from the ambient temperature to the cooling temperature thus transferring heat to the regenerator matrix. Phase (3)—isothermal expansion: The compression piston is kept close to the regenerator. The expansion piston is moved to expand the gas in the expansion volume. The expansion work we is extracted

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Figure D6.0.3. A schematic diagram of the operation of a GM cooler.

from the gas. The cooling effect qc theoretically assumed to occur at constant temperature can be used for refrigeration. Phase (4)—isochoric reheating: Both pistons are now moved simultaneously to transfer the expanded gas at constant volume through the regenerator from the expansion volume back to the compression volume. The gas is heated from the cooling temperature to the ambient temperature. The heat transferred from the regenerator matrix to the gas theoretically equals the heat previously transferred from the gas to the regenerator.

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Figure D6.0.4. A schematic diagram of the operation of a Stirling cooler.

D6.0.2.2 Joule—Thomson expansion The expansion of a previously compressed gas through a calibrated orifice or an adjustable valve without heat (q) or work (w) exchange with the surroundings is an isenthalpic process. This results from the first principle of thermodynamics for an open system

Such an isenthalpic expansion process is referred as a Joule—Thomson (JT) expansion. From a basic expression for the enthalpy variation

where Cp is the specific heat at constant pressure of the gas and υ its specific volume, we can determine the ratio of the temperature variation (δT) to the pressure drop (δP) during a JT expansion. This ratio is known as the JT effect coefficient µ

Note that for an ideal gas we have a specific equation of state: Pυ = rT. Thus, we can write

and we get µ = 0.

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There is no temperature variation associated with an isenthalpic expansion for an ideal gas. In contrast for a real gas, the JT coefficient µ can either be negative or positive. This fact is illustrated in figure D6.0.5 in which constant-enthalpy lines for nonideal gas conditions are represented in a temperature versus pressure diagram.

Figure D6.0.5. The JT expansion inversion curve.

The full line, referred to as the inversion curve, corresponds to the condition µ = 0. For an isenthalpic expansion, a cooling effect (µ > 0) is observed inside the inversion curve, a heating (µ < 0) outside. A temperature can also be defined above which no cooling effect can be achieved by any isenthalpic expansion. This temperature, referred as the maximum inversion temperature, is represented by a broken line in figure D6.0.5. Table D6.0.2 gives the maximum inversion temperatures for some cryogenic fluids.

TableD6.0.2. The maximum JT inversion temperature for different cryogenic fluids.

The isenthalpic JT expansion is basically an irreversible process and therefore the associated theoretical COP is poor. Nevertheless it is commonly used for cooling because of its simplicity and its capability for miniaturization due to the fact that it does not require any moving part at low temperature. The representation of a typical refrigeration cycle including a JT expansion process is shown on the T—S diagram for nitrogen in figure D6.0.6. The high-pressure gas is precooled (1 → 2) by the low-pressure gas (4 → 5) in a counterflow heat exchanger. If the counterflow heat exchanger is properly sized, the isenthalpic expansion of the precooled high-pressure gas leads to a diphasic mixture of liquid (3) in equilibrium with its vapour at the low pressure. Thus the cooling temperature Tc is determined by the low-pressure value (boiling temperature of the cycle fluid under the cycle low pressure). The heat load is removed at constant temperature (Tc ) by evaporation of the liquid fraction in the mixture (3 → 4).

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Figure D6.0.6. The JT cycle (T—S diagram for nitrogen).

If the counterflow heat exchanger operation is adiabatic, it can be easily shown that the specific cooling power qc = Q• c/m• (where m• is the fluid mass flow rate and Q• c the cooling power) is given by the specific enthalpy difference between the high- and low-pressure gas at the warm end of the heat exchanger (qc = h1 – h5) For a perfect heat exchanger there is no temperature difference between the high- and low-pressure gas at the warm end (Ta ). Then the maximum specific cooling power is available and is equal to the isothermal enthalpy variation

For a given low pressure (which is often atmospheric pressure), the maximal specific cooling power is a function of the high pressure and there is an optimal value of this high pressure which maximizes the specific cooling power. Figure D6.0.7 shows the variation of the specific cooling power with high pressure for nitrogen and argon which are commonly used for cooling at about 80 K. D6.0.3 Inefficiencies and parasitic losses in actual cryocoolers Real cryocoolers operate in practice in a markedly different way from the ideal description given in the previous section and the resulting performances are strongly degraded. Some of the reasons for inefficiency or parasitic losses are briefly discussed in this section. D6.0.3.1 Piston motion In the ideal Stirling cycle description given in section D6.0.2.1(b), the compression and expansion piston motions are discontinuous to achieve a truly isochoric gas transfer between the compression and the expansion volumes. In practical Stirling cryocoolers these pistons move with continuous quasi-sinusoidal

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Figure D6.0.7. JT specific cooling power for nitrogen and argon. From Buller (1971).

motions, the expansion piston leading the compression piston by a phase angle generally about 90° to approximate the theoretical figure. The resulting overlap in the motion of the compression and expansion pistons induces a deformation of the ideal work diagram and a loss in efficiency. D6.0.3.2 Dead volumes In ideal regenerator cycles the gas is assumed to be totaly expelled from the cold volume and the regenerator when it undergoes compression. In practice, the existing dead volumes waste part of the compression work. The reduction of void volumes is thus important but is in practice restricted since in the heat exchangers and regenerators there is competition between the efficiency of heat transfer and a low-pressure drop constraint. A similar parasitic effect occurs during the expansion process. D6.0.3.3 Pressure drop The pressure drop in the regenerator matrix and other heat exchangers induces a reduction in the amplitude of the pressure variation in the expansion space compared with the pressure variation in the compression space, resulting in a decrease of the specific refrigeration effect and a relative increase in the compression work. D6.0.3.4 Nonisothermal operation In ideal regenerative cycles, reversible isothermal compression and expansion processes are assumed. In real machines large variations of the gas temperature are observed either in the compression or in the

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expansion volume, due to the limited surface area for heat transfer, resulting in a significant efficiency loss. When it is technically possible, heat exchangers are introduced on both sides of the regenerator to improve the heat transfer between the cycle working gas and the ambient or cold heat sinks. D6.0.3.5 Regenerator or counterfow heat exchanger inefficiency In a regenerative cycle, the thermal efficiency of the regenerator is of major importance. This efficiency can be defined in a rough estimate as ε = (Ta — T)/(Ta — Tc ) where Ta and Tc are respectively the ambient and cold-sink temperature and T is the average temperature of the gas after it has passed through the regenerator for precooling before entering the expansion volume. Thermal efficiencies larger than 0.99 are necessary to achieve reasonable overall efficiency of the cryocooler. The actual process of regenerator heating and cooling is very complicated since it involves periodical variation of the gas and matrix temperatures in both space and time. A precise modelling of the operation of the regenerator is nevertheless necessary to obtain a realistic simulation of the operation of the regenerative cryocoolers for sizing and optimization. Extensive effort has been and is always devoted to theoretical analysis and numerical simulation of regenerators. Extensive information on the theoretical analysis and computer simulation status of cryocoolers is given in two recent specialized books (Walker 1983, 1989). Similarly for cycles based on JT expansion, the efficiency of the counterflow heat exchanger is critical. For a miniature JT cryocooler the heat exchanger is usually made of a cupro-nickel capillary tube, with copper fins soldered to its outside, wound around a cylindrical mandrel and inserted in a stainless steel or glass sheath (i.e. internal well of a Dewar). The compressed gas flows inside the coiled capillary tube. The expanded gas flows transversely over the coiled finned tube in the annular gap between the mandrel and the outer sheath. Such heat exchangers are shown in figure D6.0.9 in section D6.0.4. The modelling and heat transfer analysis of this type of heat exchanger has been studied and compared with experiment by Giest and Lashmet (1959). D6.0.3.6 Thermal losses Thermal conduction along the walls of the regenerator sleeve and/or the expansion cylinder and through the porous matrix of the regenerator reduce the net cooling power of any practical cryocooler. To minimize these losses high-strength and low-thermal-conductivity material (such as stainless steel or titanium) is used for the cylinder walls. Epoxy—fibre-glass material or plastic materials with low thermal conductivity are often used for the regenerator sleeve. The regenerator matrix itself is generally made from stacks of metallic wire mesh or balls. The axial heat conduction from one disc or ball to another is negligible compared with the radial heat conduction along the wire or in the ball. Parasitic convection heat transfer is generally eliminated by using cryocoolers in high-vacuum Dewars. The radiative heat transfer from all surrounding surfaces at higher temperature than the cold-sink temperature can be minimized by properly controlling the reflectivity and the emissivity (polishing, gold-or silver-coating) of these surfaces or by incorporating multilayer insulation (MLI: aluminized Mylar or Kapton sheets acting as thermal radiation screens between the cold and the warm surfaces). For Stirling and GM cryocoolers specific thermal losses are linked to their dynamic operation. A temperature gradient exists along the cylinder and the expansion piston (or displacer) walls between the ambient and the cold tip temperature. When the expansion piston (or displacer) is moving in the cylinder, large differences in the temperature of the facing surfaces may occur, resulting in a heat transfer from the ambient side to the cold tip. This process, generally called shuttle heat transfer, can be limited by reducing the stroke of the expansion piston. Another heat loss may occur from the gas mass flow circulating in the annular dead volume between the expansion piston and the cylinder wall.

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D6.0.3.7 Conclusion As a result of various irreversibilities and parasitic losses occurring in real cryocoolers, their actual efficiency is quite far from the theoretical figure. To illustrate this point we report in figure D6.0.8 the result of a comprehensive compilation made at The National Bureau of Standards (Strobridge 1974) about the performances of over 100 cryocoolers. The efficiency, in terms of percentage of Carnot value (actual COP/Carnot COP), is given as a function of the cooling capacity for various types of cryocooler at various cooling temperatures.

Figure D6.0.8. Actual efficiency of cryocoolers. From Strobridge (1974).

For small cryocoolers (cooling capacity <100 W ) we note that the actual COP ranges from 1% to 10% of the theoretical Carnot COP. Another reference is given by the measured performances of the most efficient Stirling cryocoolers which have been recently developed for space applications. Typically, for a cooling power of a few watts at 80 K, the specific energy consumption is about 30 W of electrical input power per watt of cooling power (about 10% of the Carnot COP). D6.0.4 Cryocoolers: applications and state of the art D6.0.4.1 Joule—Thomson expansion cryocoolers Miniature cryocoolers employing JT (i.e. isenthalpic) expansion were developed almost simultaneously by Hymatic Engineering Company in England and Air Products Inc. in the US in the mid-1950s. A well established market for these coolers is in the military infrared (IR) missile guidance systems which require cooling capacities of a few hundred milliwatts at liquid-nitrogen temperature (about 80 K). JT expansion cryocoolers have been preferred for this application because of their capability for miniaturization, their ability to withstand large accelerations (no moving parts) and their ability to provide rapid cool-down times (a few seconds). In these applications they are used in an open-cycle mode. High-pressure gas is supplied from a storage reservoir (rechargeable if necessary if subsequent cool-downs are expected). A typical miniature

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JT expansion cryocooler is represented in figure D6.0.9. A finned-type counterflow heat exchanger wound around an insulating mandrel is introduced in an evacuated Dewar. High-purity compressed gas (air, nitrogen or argon) is required for operation to avoid plugging of the expansion orifice by condensation of contaminants such as water, carbon dioxide or hydrocarbons. A microporous filter is incorporated at the inlet to remove solid particles. After precooling by the low-pressure return flow in the counterflow heat exchanger, the high-pressure gas is expanded isenthalpically in an expansion orifice. The liquid-vapour mixture obtained after expansion is collected in a separation chamber on the bottom of which is attached the IR detector to be cooled. Two different types of expansion device are commonly used. The simplest system is a fixed-area orifice which has the main advantage of being easy and cheap to manufacture. However, a fixed-area expansion orifice has a major disadvantage: the mass flow rate through the orifice in creases continuously as the temperature decreases (since gas density increases). The result is that if the orifice is sized for high mass flow rate at room temperature to ensure a quick cool-down time, the mass flow rate at operating conditions will be greatly excessive. In contrast if the orifice is sized for the proper mass flow rate at nominal operating conditions, the cool-down time will be very long due to a very low mass flow rate at room temperature. To overcome these difficulties, variable area, temperature sensitive, expansion devices have been developed. A schematic diagram of such a system is represented in figure D6.0.9. The area of the expansion orifice is adjusted by moving a needle in or out of it. The motion of the needle is controlled by metallic bellows to which it is attached. The pressure inside the bellows is the JT cooler low pressure. A gas charge in a chamber surrounding the bellows squeezes it. The pressure in this chamber is controlled

Figure D6.0.9. A schematic diagram of a JT cooler.

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by a vapour-pressure thermometer with its bulb immersed in the expanded gas. Thus, as the temperature of the expanded gas decreases, the pressure in the chamber decreases, the force acting on the bellows decreases allowing the needle to enter into the orifice, resulting in a mass flow rate reduction. Thus, an automatic control is achieved, allowing for a maximal mass flow rate during cool-down and an adjusted mass flow rate at nominal operating conditions to compensate strictly for the heat load. Open-cycle systems do not require any power input and generally the low-pressure gas is directly vented to the atmosphere. They are well adapted to restricted space applications. Typical miniature JT systems are shown in figure D6.0.10.

Figure D6.0.10. JT systems. Reproduced by permission of L’Air Liquide.

For some applications such as cooling electronic equipment, continuous operation is required. In this case a closed-cycle-type operation may be achieved by connecting a compressor to the JT expansion unit. These compressors are sophisticated devices since the compression ratios involved are large (200:1 to 400:1). To minimize the work of compression and achieve quasi-isothermal operation, multistaged reciprocating compressors with intercooling are generally used. A major problem is gas cleaning to remove any contaminant after compression. Such a compressor is shown in figure D6.0.11. Microminiature JT cryocoolers of small cooling capacity (less than 100 mW at 80 K), using photolitographic processes to etch on a flat glass plate the counterflow heat exchanger and a fine capillary for gas expansion, have also been developed by Little (1984). Capable of rapid cool-down because of their small mass, they are well adapted for laboratory study of superconductors or semiconductors. D6.0.4.2 Gifford-MacMahon cryocoolers Cryocoolers attempting to follow an Ericsson cycle (isobaric-isothermal) were invented and patented at the end of the 19th century by Solvay (with a fixed regenerator and expansion piston for work extraction) and Postle (displacer with integrated moving regenerator). In the late 1950s Gifford and MacMahon of A D Little, Inc. industrialized and commercialized this type of cryocooler which is nowadays widely produced by several companies in a large range of cooling capacities. Its success comes from a proven high

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Figure D6.0.11. A multistaged compressor for a JT cooler. Reproduced by permission of Cryotechnologies SA.

reliability and a minimal maintenance due to a low operating frequency (1–2 Hz) and a robust technology. GM cryocoolers are mainly integrated into mechanical cryopumps for applications requiring a clean and high vacuum. They are also widely used now by physicists for sample cooling in cryostats, replacing cryogenic fluids (liquid nitrogen or liquid hydrogen). The high- and low-pressure levels required in the Ericsson cycle are provided by helium compressing units. These units incorporate compressors developed for domestic refrigerators thus taking advantage of a large-scale production process and the resulting cost reductions. These units include piston-, rolling-or scroll-type compressors. The cycle gas used in these cryocoolers is helium. As a monatomic gas, its isobaric to isochoric specific heat ratio (γ = Cp /Cυ = 1.67) is larger than those of polyatomic fluids (CFC or hydrocarbons with γ ranging between 1 and 1.3) commonly used in domestic refrigerators. It results in a larger temperature rise during compression. To avoid any damage to the compressor valves or any cracking of the lubricant, oil is injected in the helium at the level of the compressor suction line to act as a thermal moderator during compression. Most of this oil is later removed from the compressed helium by centrifuging and coalescence. The remaining traces (a few parts per million (ppm)) are trapped in an activated charcoal adsorber. The compression unit is connected to the cold ringer by two (high- and low-) pressure lines allowing for a separation of both subsystems without any noticeable efficiency loss. A schematic diagram of a two-stage cryocooler cold finger with two expansion volumes providing cooling at different temperature levels is shown in figure D6.0.12. This is a common feature which does not greatly increase the mechanical complexity of the cooler and improves the efficiency by reducing the temperature difference between cold and hot heat sinks for each stage. For cryopumping applications it provides a first temperature level of about 80 K which is useful for the cooling of baffles for water vapour condensation and thermal shielding of the second stage which is operated at about 15 K for the cooling of activated charcoal allowing for air, hydrogen and helium cryotrapping. Such an arrangement of a mechanical cryopump is shown in figure D6.0.13. To operate and synchronize properly the valves and displacer motions two types of drive mechanism are commonly used. The first mode of operation uses a mechanical drive with a crankshaft and/or eccentric cams kept in rotation by a motor; the drive governs the movement of the displacer and the opening and closing of the

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Figure D6.0.12. A schematic view of a pneumatically driven GM cold head. Reproduced by permission of Edwards.

valves. In an alternative mode, the inlet and exhaust of the cycle gas is controlled through a rotary valve driven by a motor as shown in figure D6.0.12. Then the mechanically driven displacer is replaced by a free displacer moved by differential pressure forces acting on a small-section extension of the displacer. The high and low cycle pressures alternately act on one extremity of this small-section cylinder, the other extremity remaining at the mean cycle pressure, resulting in a reciprocating force on the displacer. The forces required to move the displacer are quite small since the resistant foces acting on it are only due to the pressure drop the through the regenerator and the sliding piston seal fiction. The sealing is not a severe problem since the pressure is almost the same on both sides of the displacer and the frequency of operation is low. This is one reason why maintenance is only required after about 15 000 hours of operation for the GM cryocoolers. The first-stage regenerators (300 K-50 K) are commonly made of metallic wire mesh (stainless steel or bronze) with wire diameters and apertures of about 50–100 µm. Since the heat capacity of common materials decreases at low temperature (below 50 K) whereas the heat capacity of the fluid helium in the

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Figure D6.0.13. A cryopump with a GM cooler. Reproduced by permission of Edwards.

cycle increases, the efficiency of the regenerator is strongly affected. To minimize this effect, lead shot (diameter about 200 µm) is used in the second-stage regenerator since lead exhibits the largest volumic specific heat at low temperature among the common materials used. In practice, the regenerator void volumes and the material specific heat decrease result in a limitation of the useful operating temperature of GM cryocoolers to about 10 K. The ideal cooling power of a cryocooler is given by the integral ∫ υ dP calculated in the expansion volume of the cold finger. For an ideal GM cooler we get

where ∆P is the difference between the high and low pressures delivered by the compressor, Vc is the displacer swept volume and N the operational frequency. The principal drawback of the GM cryocooler is its poor efficiency since no work is actually extracted during expansion. Nevertheless its high reliability largely outweighs this point for practical applications for which size and efficiency are not the main point. D6.0.4.3 Compound Gifford—MacMahon and Joule—Thomson cryocoolers GM cryocoolers are reliable but are limited to ultimate temperatures of about 10 K. Various applications or systems such as niobium—titanium superconducting magnets or bolometers and Josephson junctions for

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detectors still require cooling at the liquid-helium temperature level. The use of pool-boiling cryostfosuch applications may be constraining and expensive for long-term operation. To provide mechanical cooling and avoid any cryogenic fluid refilling and consumption at 4 K, it has been suggested that a GM cooler and a JT expander should be combined. A schematic diagram of such a compound cycle is given in figure D6.0.14.

Figure D6.0.14. A schematic diagram of a compound GM/JT cycle.

The two-staged GM cryocooler is used for the precooling of the high-pressure helium gas in the JT loop. The JT counterflow heat exchangers are inserted between the GM precooling stations. Part of the high-pressure helium mass flow rate from the GM cooler compressor is diverted to the JT loop. An additional compressor is then incorporated to recompress helium from the JT low pressure to the GM low pressure. Several systems based on this technology have been developed at CEA/SBT (Claudet et al 1992, Poncet et al 1994) with cooling powers ranging from 100 mW at 2.8 K up to 5 W at 4.5 K. Specific pneumatic valves have been incorporated in the process for automatic cool-down (bypassing the low-temperature counterflow heat exchanger during initial cooling down to 15 K) and automatic temperature control (see figure D6.0.14). The JT low pressure is used to control the opening of the JT expansion valve resulting in a temperature stability of a few millikelvin. An appropriate control of gas purity to prevent plugging at the JT expansion nozzle allows for long-duration trouble-free operation. A typical system is shown in figure D6.0.15. D6.0.4.4 Stirling cryocoolers In the middle of the 1950s, the first Stirling cryocooler was developed and industrialized by the Philips Company for air liquefaction. This type of engine is still in production today as well as derived products such as two-stage versions for hydrogen liquefaction and multicylinder engines for cooling power of up to a few kilowatts. However, today the main developments and applications of Stirling cryocoolers are related to the cooling of IR detectors and associated cold electronics for night vision and missile guidance and may in the near future lead to the development of applications including high-temperature superconducting materials. For these applications, compactness and efficiency are often the more important criteria: thus high-frequency-operated miniature Stirling coolers have emerged as the best possible compromise. The typical cooling power requirement for Stirling refrigerators is from about a quarter of a watt up to a few watts in the temperature range 50–80 K depending on the type and size of the detector to be cooled. For

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Figure D6.0.15. A GM/JT cooler for superconducting magnet cooling. Reproduced by permission of CEA/SBT.

this level of temperature a single-staged cold finger is sufficient. Most development efforts are aimed at increasing the lifetime and the efficiency of these coolers and lowering their size, their electromagnetic noise and their exported vibrations. A large number of mechanical arrangements have been explored. We shall summarize in this section the main basic configurations. A Stirling cryocooler is made of two specific elements: (i)

a pressure oscillator operating at ambient temperature in which a reciprocating piston transmits mechanical work to the cycle gas and generates a pressure oscillation in the refrigerator cold finger; (ii) a cold finger cylinder in which a displacer containing the regenerator matrix separates two volumes at an ambient and a cold temperature. When reciprocating in the cylinder, ideally without any work, the displacer forces the gas from one volume to the other. An appropriate phasing of the displacer motion versus the pressure oscillations generated by the oscillator provides the cooling effect. The cooling capacity of a Stirling cooler operating with a real gas is given by Qc = N ∫ α Vc dP,

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where α is the expansivity (α =(T/υ)(∂υ/∂T)p), P the pressure in the cooler, Vc the variable expansion space and N the frequency of operation. The development of this expression, taking into account the motion and phase shift of both the compression piston and the displacer as well as other parameters such as the cycle mean pressure and the cold tip temperature, has been discussed by Schmidt and the main results of this analysis have been reported by Walker (1983). In an integral cooler the pressure oscillator and the cold finger are integrated in the same casing and mechanically driven by the same crankshaft. The two cylinders are often arranged on orthogonal axes which simplifies the achievement of the 90° phase angle between compression pistons and expansion displacer required for a maximum cooling effect. Such an arrangement is represented in figure D6.0.16.

Figure D6.0.16. Integral-type Stirling coolers. Reproduced by permission of L’Air Liquide.

The main advantages of the integral arrangement are the mechanical control of both piston and expander stroke and phase and the minimization of the connecting dead volume between the pressure oscillator and the cold finger. In contrast it is very difficult to cancel or at least damp the mechanical vibrations generated by the piston motion and directly transmitted to the tip of the cold finger. In a split arrangement the pressure oscillator and the cold finger are completely independent and connected by a small-internal-diameter tube. Split Stirling coolers are shown in figure D6.0.17. A larger flexibility in the integration of the cooler to the system results from this arrangement. It should not be forgotten that the connexion line acts like a dead volume and may reduce the efficiency of the cooler. Conventional split Stirling pressure oscillators are of rotary design: the rotor of a brushless motor is directly coupled to a crankshaft coupling mechanism connected to the compression piston. To avoid cycle gas contamination by oil lubrication, sealed ball bearings or dry-rubbing materials are used for the shaft mounting. A component of the force acting on the piston pushes the piston against the cylinder wall, resulting in a friction and wear which limits the life of the oscillator (typically a few thousand hours). Linear drive systems are widely developed today. In this new arrangement piston side forces are eliminated and consequently rubbing seals have been replaced by clearance seals, resulting in a significant increase of the mean time before failure. Moreover the use of twin pistons operated in opposition leads to a very significant reduction of induced vibrations. The moving piston is submitted on one side to the cycle pressure oscillations and on its opposite side to the mean cycle pressure resulting in a gas-spring-like force acting on it. To minimize the axial force to be delivered by the linear motor, the pressure oscillators

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Figure D6.0.17. Linear split-type Stirling coolers. Reproduced by permission of Cryotechnologies SA.

are thus operated at the resonance frequency caused by the mutual compensation of the inertia and gas spring (this frequency is typically of a few tens of hertz). In the split arrangement there is no mechanical mechanism to control the stroke of the cold finger displacer and its phase relationship with the oscillator piston. It is obvious, for cost reasons (except in space applications), that a specific motorization of the displacer with phase control cannot be considered. The motion of the displacer is obtained by pneumatic means as shown in the schematic diagram of a cold finger in figure D6.0.18. As a result of the pressure oscillations generated by the pressure oscillator, a periodical force acts on the displacer (mean pressure in the buffer volume and oscillating pressure in the expansion volume) and drives its reciprocating motion. The frictional damping (rubbing or clearance seal) generates the appropriate phase shift for effective cooling. The main problem associated with pneumatically driven displacers is the variation in time of the damping force which produces long-term performance degradations. For space applications (earth observation) specific technologies for long-life Stirling coolers have been developed. To avoid any wear, frictionless operation is achieved by clearance sealing and contactless bearing. Prototypes using active magnetic bearings, hydrodynamic gas bearings (Duband et al 1994) or flexure bearings have been designed and developed. A photograph of the linear Stirling cryocooler developed by Jewell et al (1993) at the Rutherford Appleton Laboratory, and later manufactured and tested in space by Matra Marconi Space (previously BAe), is given in figure D6.0.19. Both the pressure oscillator and the expansion displacer are driven by a loudspeaker-type linear motor. The phase between both piston and displacer motions is measured by position transducers and electronically controlled. In its latest version this cooler is capable of about 2 W of cooling power at 80 K with about 60 W of electrical input power to the pressure oscillator. Double-staged versions of this cooler as well as a compound two-stage Stirling and JT expansion loop systems are under development (Orlowska et al 1994).

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Figure D6.0.18. A schematic view of a pneumatically driven Stirling cold finger.

Figure D6.0.19. Flexure bearings in a Stirling cooler for space applications. Reproduced by permission of DRAL.

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D6.0.5 Future trends in cryocooler development Cryocoolers are nowadays used in appreciable numbers for various applications. To promote a larger diffusion, much development time is currently being spent on improving their reliability, efficiency and capability of producing significant cooling power at low temperature (i.e. liquid helium temperature). Recent results obtained during these developments, which probably prefigure the future trends in cryocooler technology, are reported in the following sections. D6.0.5.1 Magnetic materials for regenerators The ultimate temperature achievable by GM or Stirling cryocoolers is limited to about 10 K by the large inefficiency of the regeneration process when temperature decreases. The main reasons for this efficiency drop are the increasing mass of gas kept and cyclically pressurized and expanded in the void volume of the regenerator porous matrix and the reduction in the volumic specific heat of most of the commonly used metallic materials for regeneration in contrast to the increase in the volumic specific heat of the helium cycle gas. The volumic specific heat of some typical materials used for regenerators is reported in figure D6.0.20. New geometries of regenerators based on the stacking of perforated thermally conductive plates with thermally insulating spacers are under validation. They will allow for low void volume fractions, moderate pressure drops and large heat transfers at low temperature. A way of coping with the enhancement of the specific heat of the helium gas is to use materials exhibiting a comparable specific heat anomaly. This is the case in the temperature range of interest (4–15 K) for several rare-earth-based materials which undergo a magnetic ordering. Several compounds have been suggested and their volumic specific heats have been experimentally determined. Among them, Er3Ni appears to be a good compromise as it has a well adapted specific heat (see in figure D6.0.20), chemical stability, insensitivity to oxidation and can be prepared in the form of spheres.

Figure D6.0.20. Volumic specific heats of regenerator materials.

Results on two-staged GM coolers using Er3Ni spheres in the second regenerator instead of (or in association with) lead shot and achieving ultimate temperatures as low as 3 K with cooling powers at 4.2 K of a few hundred milliwatts have been recently published (Hashimoto et al 1993). Nevertheless, the long-term mechanical integrity of this brittle material remains to be proven. GM coolers including magnetic material in the regenerator will probably compete in the near future with compound GM/JT coolers for 4 K cooling.

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D6.0.5.2 Gas mixtures for Joule—Thomson expansion coolers Technological simplicity, reliability, the capability for miniaturization and temperature stability are some qualities of the JT coolers operated in the open-cycle mode which explain their widespread use in IR detectors (night vision, missile guidance) and cooling of electronic equipment. On the other hand the efficiency of closed-cycle JT systems in the 80 K temperature range is poor due to their low specific cooling power and the high pressure required for optimal operation (several hundreds of atmospheres) with a single gas (nitrogen, air or argon). However, large improvements in specific cooling power and efficiency can be achieved by the use of mixtures of gases as proposed by Alfeev (1972). More recently at MMR Technologies, Inc. and at CEA/SBT these results have been confirmed and analysis of the thermodynamical properties of these mixtures has been carried out to optimize the performance. Suitable mixtures including nitrogen and hydrocarbons provide at 80 K, with pressures of the order of 3 MPa, the same specific cooling power as pure nitrogen under 15 MPa. This reduction in operating pressure allows for large improvements in efficiency and for the use of less sophisticated compressors. Recently a prototype of a JT cooler, operated with a GM-like helium compressor using a gas mixture and capable of 10 W of cooling at 95 K, has been developed by Air Products. Future developments will probably give rise to a new generation of closed-cycle JT coolers using gas mixtures in the 80 K temperature range. D6.0.5.3 Pulse-tube refrigerators Miniature Stirling coolers are widely used for fractional watt cooling when miniature size, weight and high efficiency are required. Nevertheless the technological complexity of the cold finger, including a moving displacer—regenerator with a clearance seal and pneumatic drive, is a limitation to the reliability and ease of integration of these coolers. A new concept of cooler is emerging which allows for the design of a cold finger with no moving part: the pulse-tube cooler. In 1964 Gifford and Longsworth published results concerning a pulse-tube cooler (generally referred to as the basic pulse tube), with no moving parts, in which a cooling effect was achieved by surface heat pumping. This heat transfer mode is restricted to low-frequency operation by the gas thermal diffusivity and results in a poor efficiency, only allowing ultimate temperatures of about 150 K to be achieved. Due to this performance limitation, the basic pulse tube has not been further developed. Later on, Mikulin et al (1984) and Zhu et al (1990) proposed new concepts respectively known as orifice and double-inlet pulse tubes. Ultimate temperatures of about respectively 80 K and 40 K have been achieved by these authors. A schematic diagram of the various arrangements for pulse tubes (basic, orifice, double inlet) is represented in figure D6.0.21. The pulse-tube refrigerator consists of two components. (i)

A subsystem for the pressure wave generation. It can be either a pressure oscillator (as in a Stirling cooler) or a compressor associated with a rotating valve distributor (as in a GM cooler). (ii) A set of two tubes. The first one is a traditional regenerator connected at its cold end to a second tube: a simple hollow tube. At its ambient temperature end this tube is connected through a flow impedance (needle valve, calibrated orifice or capillary) to a buffer volume. An analysis of the heat transfer (enthalpy flow) in the orifice pulse-tube arrangement has been proposed by Radebaugh and Storch (1988). The pulse-tube refrigerator can be compared to a Stirling cooler. The pressure oscillator generates adiabatic oscillations of the gas in the tube. If no disturbance (i.e. turbulence) occurs in the hollow tube, a piston-like motion of the gas is achieved. The enthalpy flow in the tube can be described schematically as follows. From the energy conservation equation in the adiabatic tube we get

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Figure D6.0.21. A schematic diagram of a pulse-tube cooler.

where 〈 Q• 〉 is the cooling power, 〈Q• loss〉 is the thermal losses (regenerator inefficiency and conduction losses), 〈H〉 is the enthalpy flow in the tube and 〈Q• a〉 is the heat rejected at ambient temperature. At any point in the tube we can calculate the enthalpy flow c

where τ is the duration of a pressure oscillation period, m• the mass flow rate and T the temperature. From the conservation of mass (m• = ρAu , with ρ = gas density , A = tube area, u = gas velocity) and the perfect gas equation of state (ρ = P/RT) we get

If we assume sinusoidal variations of the pressure (P = P + ∆Τ sin ωt ) and of the gas velocity (u = u0 sin(ωt — φ ) in the tube, we get

If the proper phasing (φ = 0) is achieved between the pressure wave generated by the oscillator and the gas velocity in the tube, a maximal cooling effect is obtained. In the basic pulse-tube configuration the pressure and the velocity are in quadrature and consequently no enthalpy flow (i.e. cooling power) is obtained. By means of the impedance Vl and of the buffer volume, as represented in figure D6.0.21, the phase φ can be properly adjusted and a cooling effect is obtained. Extensive work is being performed using the pulse-tube concept in several laboratories in the world. At CEA/SBT, for example (Ravex and Rolland 1994), an ultimate temperature of 28 K has been obtained in a single-stage cooler. Developments on low-frequency (using a helium compressor and a rotary valve as in GM coolers) and high-frequency (using a pressure oscillator as in Stirling coolers) systems are in progress. Performances and efficiencies comparable to Stirling (1–10 W at 80 K) or GM (100 W at 80 K) coolers have been demonstrated. Multistage systems with magnetic material regenerators are under development for low-temperature cooling (T < 4 K). The technological simplicity of the pulse-tube cold finger results in a high reliability and ease of integration. Moreover the absence of any moving piston and motorization strongly reduces the vibrations

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Figure D6.0.22. Pulse-tube prototypes. Reproduced by permission of CEA/SBT.

and electromagnetic noise exported to the cold tip. Some pulse-tube cold-finger laboratory prototypes are shown in figure D6.0.22. It is obvious that in the near future pulse coolers will strongly compete with traditional GM, Stirling and JT coolers in a large range of temperatures and cooling powers.

Table D6.0.3. Temperature range, cooling power range and applications of cryocoolers.

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D6.0.6 Typical applications of cryocoolers The temperature range, the cooling power and the applications of cryocoolers are summarized in table D6.0.3. References Alfeev V N 1972 Refrigerant for a cryogenic throttling unit UK Patent 1 336 892 Buller S 1971 A miniature self-regulating rapid cooling Joule-Thomson cryostat Adv. Cryogen. Eng. 16 205–13 Claudet G, Lagnier R and Ravex A 1992 Closed cycle liquid helium refrigerators Cryogenics 32 52–5 (ICEC Suppl.) Duband L, Ravex A and Rolland P 1994 Development of a Stirling cryocooler using hydrodynamic gas bearings SAE Technical Paper Series 941528 Giest J M and Lashmet P K 1959 Miniature Joule Thomson refrigeration systems Adv. Cryogen. Eng 5 324–31 Gifford W E and Longsworth R C 1964 Pulse tube refrigeration Trans. ASME J. Eng. Ind. 63 264 Hashimoto T, Eda T, Yabuki M, Kuriyama T and Nakagome H 1993 Recent progress on application of high entropy magnetic material to the regenerator in helium temperature range 7th Int. Cryocooler Conf. Proc. (Kirtland Air Force Base, NM, 1993) 87117-5776 Jewell C, Bradshaw T, Orlowska A and Jones B 1993 Present life testing status of ‘Oxford type’cryocoolers for space applications 7th Int. Cryocooler Conf. Proc. (Kirtland Air Force Base, NM, 1993) 87117-5776 Little W A 1984 Microminiature refrigeration Rev. Sci. Instrum. 55 661-80 Mikulin E I, Tarasov A A and Shkrebyonock M P 1984 Low temperature expansion pulse tubes Adv. Cryogen. Eng. 12 629 Orlowska A, Bradshaw T and Hieatt J 1994 Development status of a 2.5-4 K closed cycle cooler suitable for space use SAE Technical Paper Series 941280 Poncet J M, Claudet G, Lagnier R and Ravex A 1994 Large cooling power hybrid Gifford MacMahon/Joule Thomson refrigerator and liquefier Cryogenics 34 175-8 (ICEC Suppl.) Radebaugh R and Storch P J 1988 Development and experimental test of an analytical model of the orifice pulse tube refrigerator Adv. Cryogen. Eng. 35 1191 Ravex A and Rolland P 1994 Status of pulse tube development at CEA/SBT SAE Technical Paper Series 941525 Strobridge T R 1974 Cryogenic refrigerators an updated survey NBS Technical Note 655 Walker G 1983 Cryocoolers (Part 1: Fundamentals and Part 2: Applications) (The International Cryogenics Monograph Series) (New York: Plenum) Walker G 1989 Miniature Refrigerators for Cryogenic Sensors and Cold Electronics (Monographs in Cryogenics 6) (Oxford: Clarendon) Zhu S, Wu P and Chen Z 1990 A single stage double inlet pulse tube refrigerator capable of reaching 42 K Cryogenics 30 257-61 (ICEC Suppl.)

Further reading International Cryocoolers Conferences: 1980 NBS Special Publication 607, Boulder, CO 1982 NASA Conference Publication 2287, Greenbelt, MD 1984 NBS Special Publication 698, Boulder, CO 1986 David Taylor Naval Ship Research and Development Center, Annapolis, MD 1988 Naval Postgraduate School, Monterey, CA 1990 David Taylor Research Center 91/002, Bethesda, MD 1992 Phillips Laboratory, PL-CP-93-1001, Kirtland Air Force Base, NM Ross R G Jr 1994 Cryocoolers 8 (Pasadena, CA: Jet Propulsion Laboratory, California Institute of Technology) Cryogenics (Oxford: Elsevier) Proc. Cryogenic Engineering Conf. and Int. Cryogenic Materials Conf. (CEC/ICMC) Adv. Cryogen. Eng. Proc. Int. Cryogenic Engineering Conf. (ICEC) (Guildford: Butterworth)

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D7 Magnetic refrigeration

Peter Seyfert

D7.0.1 Introduction Refrigeration in the cryogenic temperature range from 80 K down to about 1 K has been dominated for a long time by helium gas cycle machines. In many fields, their technology has now reached the maturity required for industrial machines or is approaching this goal ever more closely. In the mid-1970s, the prospect of new types of application (small-scale refrigeration on equipment in remote areas or on board satellites, etc), however, stimulated investigations on whether and how magnetic refrigeration could be used as an alternative. The incentive sprang from the realization that magnetizing and demagnetizing a magnetic material was a more reversible, hence more efficient, and a more reliable way of performing work (in the thermodynamic sense of the word) than compressing and expanding a gas by means of special engines. From the historical point of view, magnetic cooling was originally invented as a method for obtaining temperatures below 1 K. The principal aim it was assigned was to reach the lowest possible ultimate temperature. Almost all devices used ‘single-shot’ mode operation which meant that the cold source had to be warmed up periodically for re-magnetization. The method became widely known under the name of adiabatic demagnetization. At present, the advantageous technique of 3He–4He dilution refrigeration has replaced it nearly everywhere. In contrast with the early objectives, the applications which have recently been considered for magnetic cooling require liquid helium (or liquid hydrogen) temperatures and continuous or near-continuous refrigeration. Unfortunately, the techniques and materials used in adiabatic demagnetization were seldom suited to these tasks and different solutions had to be found. Designs of magnetic cycles for continuous or intermittent refrigeration and working substances suitable for temperatures above 1 K have been studied. It has been reported that at least a dozen experimental prototypes have been built and tested. Although substantial progress has been achieved in various directions, a great deal of research and development remains to be done before a clear picture of the new applications of magnetic refrigeration can be drawn. This chapter discusses this situation. It has been written to meet the needs of engineers and applied physicists who wonder about the state of the art. Emphasis has been placed on those aspects which seem to be well established and which will determine future developments. Readers who are interested in the situation of magnetic cooling in the late 1970s are referred to a review article by Radebaugh (1983).

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SI definitions and units have been adopted throughout this chapter, in particular for the equation linking together the physical quantities of magnetism

where B is the magnetic flux density or magnetic induction, H the magnetic field strength and M the magnetization of the magnetic material, i.e. the magnetic moment per unit volume, µ0 is the permeability of free space. B is expressed in tesla and M in A m−1. The correct unit for H would be A m−1, but this is rarely used. In common practice, the quantity µ0H , expressed in tesla, is preferred instead. It will be denoted by B0 in this chapter. The formulae used in the field of magnetic refrigeration contain a certain number of universal constants. Their values have been listed here for the convenience of the reader who wishes to do quick numerical estimations: permeability of free space, µ0 = 4π × 10−7 V s A−1 m−1 Bohr magneton, µB = 9.27 × 10−24 J T−1 Boltzmann constant, k = 1.38 × 10−23 J K−1 molar gas constant, R = 8.314 J mol−1 K−1 Avogadro’s number, NA = 6.022 × 1023 mol−1 D7.0.2 The physical principles of magnetic refrigeration Magnetic refrigeration relies on a physical effect which is known as the magnetocaloric effect. The basic observation is that applying a magnetic field to a suitable magnetic material at a constant temperature brings about a release of heat in the latter and, inversely, removing a previously established magnetic field causes the material at a constant temperature to absorb heat. Roughly speaking, a suitable material is one that shows a strong paramagnetic behaviour. A more detailed discussion of this point will be given in section D7.0.5. A quantitative description of the magnetocaloric effect is obtained by applying the first and second laws of thermodynamics to a magnetic material. For this purpose an appropriate expression for the performed work is required. The work δ A done on a magnetic material is related to the change in magnetization δ M caused by the externally applied magnetic field B0 and is given by (assuming unit volume of magnetic material)

( The total increase in the energy density of the magnetic field is given by Hδ B0 + B0δ M but the first term which represents the rise in stored energy of the applied field has been omitted from equation (D7.0.2) by convention.) By comparing equation (D7.0.2) with the usual expression for the work done on a compressible gas, —pδ V (p is the pressure and δ V is the change in volume), an interesting and useful analogy may be drawn between the two types of systems: if B0 is substituted for p and δ M for —δ V, the entire hierarchy of thermodynamic formulae involving p, V, T may be rewritten for B0 , M, T. As an example, the two partial derivatives of the entropy may be directly related to the specific heat CH and to the temperature variation of the magnetization (∂M/∂T)H , both at constant field

Obviously, equation (D7.0.3) provides a method for constructing the entropy function of a magnetic material from its basic physical properties, which in turn may be determined experimentally or by calculation from a theoretical model.

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Fairly successful models exist at present. They have mostly been developed during the era of adiabatic demagnetization and deal with the system of elementary dipoles in the magnetic material. In all cases of interest here, these dipoles are magnetic ions in solids. The magnetic properties of such ions derive from their angular momentum which is characterized by its quantum mechanical number J. The often used rare-earth ions gadolinium (Gd) and europium (Eu), for example, have J = -27 . When no external magnetic field is present and the temperature is high enough for these dipoles to overcome all internal restraining forces each will be free to take up any of the 2 J + 1 possible spatial orientations allowed by the rules of quantum mechanics. In this completely disordered state, the net magnetic moment of the system is zero and its magnetic entropy reaches the highest possible value

where Sm is the entropy per unit volume, n is the stoichiometric number of magnetic dipoles in the considered compound, ρ is the mass density of the compound, M is its molecular weight and R is the molar gas constant. At lower temperatures, however, the internal restraining forces are no longer negligible as compared to the thermal energy of the ions. These forces will become effective in progressively limiting the spatial freedom up to a point where the magnetic dipoles are ‘frozen’ in an ordered state. As a result, the magnetic entropy will fall below the limiting value of equation (D7.0.4) with decreasing temperature T and approach vanishingly small values as T tends to zero. When a magnetic field is applied, the different orientations of the elementary dipoles are associated with different values of potential energy and, under the influence of the thermodynamic equilibrium forces, will cease to have equal probabilities. A partial magnetic ordering will ensue and cause the material to be magnetized. To a first approximation and for a given material, the magnetization will depend on magnetic field and on temperature through a single parameter, the ratio x of magnetic to thermal energy

where B is the (magnitude of) magnetic induction, T is the absolute temperature, µB is the Bohr magneton and k is the Boltzmann onstant. Starting from zero at x = 0, the magnetization increases with increasing x and eventually tends to a saturation value. In contrast, the derivative of the magnetization with respect to temperature, and hence the magnetic entropy (the second term in equation (D7.0.3)), will fall upon a rise in x and eventually tend to zero. A typical entropy diagram is displayed in figure D7.0.1. It shows a set of entropy curves as a function of temperature for different values of the applied magnetic field. The numerical values of the diagram refer to a frequently used magnetic refrigerant, gadolinium gallium garnet (Gd3Ga5O12 ), often also dubbed GGG. The magnetically active ion in this substance is Gd3+. The molecular weight of GGG is Mw = 1.012 kg mol−1 and there are 3 Gd3+ ions in the compound. The mass density of the single crystal is ρ = 7140 kg m−3. Representation of the thermodynamic properties of magnetic materials by an entropy diagram is particularly useful, since it allows a direct determination of the quantities of heat transferred to or from the system under a given reversible process. Owing to the well known relationship

the transferred heat ∆Q is obtained by simply integrating the entropy path S(T) of the process over S.

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Figure D7.0.1. Entropy diagram of GGG.

The cycle ABCD drawn in the diagram of figure D7.0.1 is an idealized example of a magnetic refrigeration cycle. It is made up of two types of process which are frequently employed. These are isothermal and adiabatic (i.e. isentropic) magnetizations and demagnetizations. In formal analogy to the theory of gas cycles, cycle ABCD is called a magnetic Carnot cycle. The cycle illustrates refrigeration between a hot source at 4.2 K and a cold source at 1.9 K. The total magnetic field swing has been chosen to be 3 T. The refrigeration capacity of the cycle is graphically represented by the shaded area in the diagram. It amounts to 1.9 × 0.062 = 0.12 J cm−3 of GGG, as may be verified readily. The coefficient of performance (COP) of a refrigeration cycle is usually defined as the ratio of heat absorbed at the cold source, Q r e f , to work done. But when the hot source is at a temperature well below room temperature, which is the case in all applications of magnetic refrigeration considered here, it is probably more meaningful to associate the ‘price to pay’ with the heat rejected, Q r e j , instead of the work done. The following definition will therefore be adopted throughout the whole paper

The COP of an ideal magnetic Carnot cycle is just the same as that of an ideal Carnot cycle performed by a gas, that is to say equal to the temperature of the cold source divided by that of the hot source (following the above definition). The efficiency of a real cycle, which can also be viewed as an entropy efficiency, is often indicated by the ratio of its COP to the COP of an ideal Carnot cycle executed between the same temperature levels

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D7.0.3 Technical aspects of magnetic refrigerator design Before getting down to the description of some practical machines, it will be useful to survey common technical problems in magnetic refrigerator design. The technical realization of any path in a magnetic refrigeration cycle usually requires a combination of two types of process: the application of a rising or falling magnetic field to the working substance and the exchange of heat between the working substance and a reservoir. Some aspects of these processes will be discussed now. D7.0.3.1 Variable magnetic fields In magnetic refrigeration, several aspects of the production of magnetic fields have to be considered. As will be shown in section D7.0.5, the working substances known today require magnetic field variations of at least a few tesla for useful cooling. Fields of such magnitude can be obtained by means of conventional electromagnets with copper windings or by superconducting coils. The latter solution is preferred nowadays, because it is cheaper in capital cost and in running cost. For the time being, niobiumtitanium-based wires and liquid helium cooling techniques are commonly employed. The energy consumption of the magnet system must be taken into account when the overall efficiency of a magnetic refrigeration cycle is evaluated. In the case of superconducting coils it is largely determined by the cost of refrigeration associated with the liquid helium consumption of the cooling system. The advent of liquidnitrogen-cooled coils made from high-Tc superconductors in a more or less distant future will certainly bring substantial improvements in this respect. For obvious reasons, permanent magnets would seem a most attractive option. Unfortunately, however, even the most powerful modern permanent magnet materials (e.g. neodymium-iron-based sintered powder) do not allow field strengths much above 1 T when a significant bore volume is required. Permanent-magnet-based magnet systems have therefore not yet been used in magnetic refrigeration. One method of producing field variations consists in moving the working substance in and out of a steady magnetic field, or vice versa. To reduce the running costs, superconducting coils operated in the persistent mode should be considered in that case. The persistent-current operation is obtained by putting a superconducting link between the coil terminals after the magnet has been charged to its full current. This results in an appreciable saving of energy since the external supply of electrical power can be switched off and disconnected from the ‘cold’ terminals so as to diminish the heat leak into the liquid helium cooling system. The persistent mode will, however, only work as long as the field produced by the magnetization of the moving magnetic material is a negligible contribution to the total field on the winding of the magnet. In the opposite case there will be losses in the superconducting wires due to time-varying fields and the persistent current will decay. The forces exerted on the magnetic working substance are given by the following formula in vectorial notation

where F is the force per unit volume of magnetic material, M is the magnetization, ∇ is the gradient operator and B is the magnetic flux density. Two general results follow from direct inspection of equation (D7.0.9): a homogeneous field will result in a zero net force, whatever the value and orientation of the magnetization are; nonuniform fields will bring about forces which attract the magnetized substance towards the maximum of the field distribution if that substance is of the paramagnetic type (which is the case for all magnetic refrigerants). It may be helpful to establish the order of magnitude of these forces which occur in magnetic refrigeration devices when the working substance is made to enter or to leave the magnetizing external

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field. For that purpose, the following practical example will be considered which is quite typical for the design of medium-sized machines. A block of GGG at 4 K is held in a position where the field strength is 2 T and the gradient is 10 T m−1. To apply equation (D7.0.9) the value of magnetization of GGG is needed. It will be estimated by using the well known Curie law of paramagnetism

where M is the magnetization, N is the number of magnetic ions per unit volume, B0 is the applied magnetic field and T is the temperature. The magnetic properties of the ions are characterized by the so-called g factor and the total angular momentum quantum number J ( g ~ 2 and J = -27 for GGG). µB is the Bohr magneton and k is the Boltzmann constant. The calculation of N is straightforward

where n is the number of magnetic ions in the chemical formula of the working substance, ρ is the mass density, Mw is the formula weight and NA is the Avogadro number. The material properties of GGG have been given in the previous section. Inserting their values and B0 = 2 T, T = 4 K in equations (D7.0.10) and (D7.0.11) yields M = 8.3 × 105 A m−1. Taking 10 T m−1 for the field gradient results in a force of 8.3 × 106 N m−3, i.e. roughly 10 N cm−3. A refrigerant unit of volume ∼ 100 cm3 , which is not unusual, would thus experience quite considerable forces. The method of producing a variable field by relative movements between the working material and the magnet therefore requires much care over the mechanical design. To avoid that kind of problem, especially when larger volumes of working material are contemplated, a stationary configuration may be used where the magnetic field is turned on and off by means of a time-varying supply current in a superconducting coil. This solution is not free of problems however. Time-varying magnetic fields dissipate heat within the superconducting wires of the magnet system, which adds to the refrigeration load. Fortunately enough, these losses can be substantially reduced through the use of specially manufactured low-loss conductors. The technology of low-loss NbTi conductors (filament size of a few micrometres and with a resistive alloy matrix) has made great progress in recent years. The cycle time in magnetic refrigeration is typically of the order of some seconds. As an empirical rule of thumb, the heat losses brought about by oscillating currents in that case are of the same order as the heat leak caused by the current leads which connect the superconducting coil with the electrical power source at room temperature. Another problem may arise from the periodic cycling of the electromagnetic energy, -21LI 2, between the magnet (inductance L and energizing current I) and the external power source. In larger machines where that energy is no longer small compared with the cost of refrigeration caused by the cycle and the magnet losses, means should be sought to store it outside the magnet rather than to degrade it into heat in external dump resistors every time the magnet is switched off. D7.0.3.2 Exchange of heat The exchange of heat between the refrigerant and various heat reservoirs is a common problem of refrigeration cycles of all kinds. Since the flow of heat only occurs under a finite (negative) temperature difference all processes involving exchange of heat inherently add to the irreversibility of the cycle and, as a direct consequence, spoil its thermodynamic efficiency. The relevant figure here is the ratio of the temperature difference associated with the exchange process to the temperature at which it takes place. The value of that ratio should not exceed a few per cent if high efficiency of the cycle is an important

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goal. Thus, the lower the temperature, the smaller the temperature difference a heat exchange will be allowed to take with regard to efficiency. In mechanical refrigerators, the working substance is a flowing fluid which can be made to exchange heat by forced convection, a mode of heat transfer which remains efficient down to low temperatures. In contrast, all magnetic working substances suitable for cryogenic refrigeration are solid materials and transferring heat to and from them confronts the designer with two problems: the transport of heat within the bulk of the working substance and the switching of heat flow between the working substance and a heat reservoir. Heat transport in solids can only be achieved by conduction. If the solid is nonmetallic, which is often the case for magnetic refrigerants, the thermal conductivity at low temperatures (below ∼20 K) falls off as T 3 and its value strongly depends on the crystalline form. Pressed or sintered powders conduct heat 10–100 times less than a single crystal of the same substance. Magnetic refrigerants should therefore always be made from single crystals when they undergo heat exchanging processes below 10 K. The solid magnetic refrigerant must be alternately connected to and isolated from at least two (the cold and the hot) thermal reservoirs. The temperature difference associated with the first process and the heat leak of the second are sources of loss for the thermodynamic efficiency of the cycle. Carefully designed heat-switching methods are therefore a major key to making practical continuously operating magnetic coolers. If the reservoir is a fluid convective heat transfer (natural convection or forced convection) offers itself as the only workable solution, which should be improved by providing the largest possible heat exchange surface. The thermal conductance of a mechanical contact between nonmetallic solid surfaces is usually rather poor (<0.1 W K−1 at 10 K) and decreases rapidly with decreasing temperature. It may be enhanced substantially by the presence of exchange gas at the contact area. A last point to be mentioned here concerns the role played by the fluid contained in the voids of the solid magnetic refrigerant during adiabatic processes. In these instances, the fluid acts as a thermal ballast and reduces the desired change in temperature for a given rise or fall of the magnetic field. Owing to the comparatively small specific heat of solids at low temperatures, this effect is significant even for values of the void fraction below 10% if the considered fluid is a liquid (usually helium). D7.0.4 Practical magnetic refrigerators To a certain extent the classification of refrigeration cycles known from gas systems may also be applied to magnetic refrigeration. Thus the Carnot cycle, the Stirling cycle and the Ericsson cycle can all be considered to have their magnetic equivalent. They are most conveniently represented in the temperature— entropy plane, i.e. in the T—S diagram. The magnetic Carnot cycle has already been mentioned in the previous section. By recalling that the pair of variables magnetic field-magnetization replaces the pair pressure-volume of gas systems, it can be seen that, in the magnetic Stirling cycle, two isotherms are connected by two constant-magnetization paths whereas, in the magnetic Ericsson cycle, two isotherms are connected by two constant-field paths. D7.0.4.1 Carnot-cycle refrigerators Several one-stage machines operating on the Carnot cycle have been built and tested. They were aimed at two applications: production of superfluid helium in the 1.7 K to 2.1 K range with a hot source provided at 4.2 K and liquefaction of normal helium at 4.2 K. In the latter case, the magnetic stage was attached to a mechanical cryocooler operating from 15 K up to 300 K and required a magnetic field swing of 4.5 T (Nakagome et al 1986) or 6 T (Barclay el al 1986). A recently completed prototype operates between

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1.8 K and 10 K with a 5 T field swing (Numazawa et al 1993). Two devices concerning the superfluid helium temperature range will be described here in some detail, because they employ particularly ingenious solutions which may lead the way to further developments. One of them has been developed in the cryogenics laboratory at CEA-Grenoble, France (Lacaze et al 1984). It features a reciprocating, double-acting refrigerant unit and a stationary, constant-current magnet system. A schematic diagram of it is shown in figure D7.0.2. The whole device is immersed in a bath of boiling normal helium (∼4.2 K) which acts as the hot source. The magnetic field, generated by a set of concentric superconducting solenoids, is constant in time but varies in space along the main axis. The field strength is typically 3–4 T at the upper and lower end points of the system, 2 T at the bearings and 0–1 T at the centre position.

Figure D7.0.2. A reciprocating Carnot-cycle device.

A piston rod, carrying two blocks of active magnetic material, moves up and down, guided by two linear bearings (cycle frequency 0.1–1 Hz). A driving force of up to 300 N is needed to move the piston against the magnetic forces. The blocks of active material have been cut from a single crystal of GGG. They are cylindrical in shape, with diameter 24 mm and length 20 mm (volume 8.5 cm3 each). The two blocks alternately undergo magnetization in boiling normal helium and demagnetization in the central chamber which contains pressurized superfluid helium, forming the cold source. The chamber is thermally insulated by a vacuum jacket and by two bearings made from sintered alumina. The latter have the shape of cylindrical sleeves, of length 35 mm. The clearance between the bearings and the sliding rod is 20 µm. The bearings serve a threefold purpose: (i) mechanical guidance for the piston movement (ii) partition between the hot- and cold-source helium baths (iii) coverage of the active material blocks enabling the adiabatic processes.

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The optimal useful cooling power of the device at 1.8 K was 1.35 W and the corresponding Carnot efficiency was 53% (cf equation (D7.0.8)). A prototype refrigerator with no moving parts has been built in a research laboratory at Hitachi in Japan (Hakuraku and Ogata 1986). A schematic diagram of it is shown in figure D7.0.3. A time-varying magnetic field is produced by a superconducting solenoid and a pulsed energizing current (peak field 3 T, cycle frequency 0.1–0.5 Hz). The working substance, a cylindrical block of monocrystalline GGG (diameter 50 mm, height 50 mm), is located inside the bore of the solenoid. It is attached on top of a thermally isolating, pressure-tight container which is filled with gaseous helium at low pressure. The whole device is immersed in a bath of saturated normal helium. As the working substance is magnetized, its temperature rises until it reaches the boiling temperature of the normal helium bath. Further magnetization causes the working substance to release heat to this bath through isothermal boiling heat transfer at its top face. In contrast, demagnetization causes the temperature of the working substance to fall below the boiling temperature, thus interrupting the heat transfer to the normal helium bath. At the same time, condensation heat transfer begins at the bottom face as soon as the dew-point temperature of the low-pressure helium gas is attained (e.g. 1.8 K at 16.4 mbar). The strong orientation dependence of natural convection has been used to build these two automatic heat switches.

Figure D7.0.3. A static Carnot-cycle device.

The optimum useful cooling power at 1.8 K of the apparatus was 0.5 W and the corresponding Carnot efficiency was 24%. These somewhat disappointing results were explained by a high loss rate of the upper heat switch during demagnetization. A substantial improvement was expected by using a mechanical shutter at the top exchange surface of the refrigerant block, but no experimental verification of this has been reported. In conclusion, it appears that practicable designs of magnetic refrigerators based on magnetic Carnot

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cycles exist. Their most useful application is probably in the production of superfluid helium. They are able to span the 1.8–4.2 K temperature range with a 3 T magnetic field swing and provide a few watts of cooling power with a Carnot efficiency better than 50%. D7.0.4.2 Regenerative-cycle refrigerators The inherent weakness of magnetic Carnot cycles is that they require prohibitively high magnetic fields as the temperature span between cold source and hot source increases. For example, if the Carnot cycle drawn in the entropy diagram of figure D7.0.1 were to operate from 20 K instead of 4 K, as shown, a field of roughly 30 T would be necessary! The same problem arises in gas systems where the Carnot cycle has never been used in practice because of the gigantic values of pressure it would require. Just as for gas systems, the solution of that problem lies in the use of cycles with internal heat exchange. In gas systems, internal heat exchange is implemented by means of counterflow heat exchangers (recuperative cycles) and regenerators (regenerative cycles). The design of counterflow heat exchangers for solid working substances and, more specifically, of recuperative magnetic cycles, poses enormous engineering problems. A few attempts have been made to build room-temperature devices (Kirol and Dacus 1988, Steyert 1978). Much more work has been done on regenerative magnetic cycles. One development concerned an Ericsson cycle in the 20–77 K temperature range (Matsumoto et al 1988). Test results near 50 K revealed serious problems due to poor heat exchange between the magnetic refrigerant and the solid regenerator. The project no longer seems to be running. Most of the research effort on regenerative magnetic cycles has, however, focused on the so-called ‘active magnetic regenerator’. Several groups have built and tested devices based on this original design concept. The temperature spans investigated were 4.2–20 K (Cogswell et al 1988, Seyfert et al 1988), 20–80 K (DeGregoria et al 1992) and near room temperature (Green et al 1990). The active magnetic regenerator is similar to a passive regenerator in that warm and cold gas is blown through a porous material alternately and periodically. The regenerator core, however, is made from magnetic material and acts as the working substance of the cycle. The fluid (in practice, helium gas) is merely used for heat transport between adjacent layers of the core. Figure D7.0.4 sketches a tandemtype apparatus which has been developed for the 20–4 K temperature range in the cryogenics laboratory at CEA-Grenoble (Seyfert et al 1988). Two vertical regenerator columns (GGG), each placed in the bore of a superconducting solenoid, are connected to a common cold source at their lower ends. Liquid-hydrogen-filled tubular heat exchangers are mounted on top of each column and play the role of heat sinks. A double-acting displacer closes the gas circuit on the warm-end side. It is used to shuttle the helium back and forth through the two regenerator units which work 180° out of phase. The temperature profile in the regenerator cores smoothly bridges the gap between the hot and cold source. The refrigeration cycle is realized in four discrete steps. First the shuttle mass, initially in one of the two working volumes of the displacer, is driven through the hot-source heat exchanger and the regenerator core towards the cold source. During this flow process (hot blow), the magnetic field is decreased in such a way that the temperature profile of the core matrix is maintained and imposes its shape on the temperature profile in the helium stream. In the second step there is no flow. The core is magnetized so that its temperature profile is shifted upwards by a few kelvin. During the third step (cold blow), the shuttle mass flows from the cold source, where it has absorbed heat and warmed up, to the hot source while the rise in magnetic field continues so as to maintain the temperature profile in the fluid stream close to that of the core. In passing through the hot-source heat exchanger, the fluid is cooled to the initial hot-source temperature. The fourth step completes the cycle by demagnetizing the core while the flow discontinues, returning the temperature profile to its initial value.

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Figure D7.0.4. A tandem-type active magnetic regenerator device.

In an idealized view, each layer of the regenerator core follows a particular Carnot cycle. Just like a cascade, these cycles are continuously distributed over the operating temperature interval. The inherent advantage of the global active regenerator cycle is its ability to span a large temperature range with magnetic fields of moderate strength. In addition, since each core layer operates over a rather limited temperature interval, the regenerator column could be a composite stack of more than one magnetic material, each suited to a particular subrange of the overall temperature range. In practice, the development of the active magnetic regenerator has encountered more problems than originally anticipated. In the 20–4 K temperature range, the results obtained so far, although encouraging (helium liquefaction from a heat sink at 14 K with a field swing of 3 T and a cycle period of 4 s), are still considerably behind the predicted performance. Most of the observed shortcomings seem to result from a bad match of the fluid enthalpy flow and the local entropy variations of the magnetic working substance. It has been shown (Smith et al 1990) that the helium and GGG thermodynamic properties in the considered temperature range would require a magnetic field profile in space and time which is incompatible with the use of a single coil over the entire regenerator core length. Additional problems are caused by the helium contained in the voids of the regenerator core during the ‘no-flow’ processes. Its role of thermal ballast has already been mentioned in section D7.0.3.2. Another detrimental effect is related to the thermal expansion or contraction of the fluid which follows the change in temperature of the core (‘breathing effect’). The resulting displacement of the fluid inevitably disturbs the temperature profile in the core and thus causes unwanted irreversibilities. The future of this kind of active magnetic regenerator is not clear. More research needs to be done to see whether other working substances could be used which would ease the realization of the abovementioned matching conditions, or whether a different conceptual design of the magnet system could solve

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the problem. Other designs of active magnetic regenerators do not use magnetic field variations at all during the flow processes. This simplifies the realization but reduces the thermodynamic efficiency (Matsumoto and Hashimoto 1990). D7.0.5 Working substances for magnetic refrigerators There are several parameters that determine the usefulness of a working substance in magnetic refrigeration cycles. The most important is certainly the magnetocaloric effect which should be as large as possible. This point will be discussed in more detail in a moment. Next comes the thermal conductivity. As already pointed out earlier (section D7.0.3.2), heat transport by conduction plays a major role in magnetic cycles, especially when large blocks of refrigerant are employed. In such cases, high thermal conductivity is crucial for reducing the temperature gradients in the refrigerant since they are the direct cause of irreversibility, i.e. of loss of efficiency. If the refrigerant is nonmetallic and if the operating temperature is below ∼10 K the use of single crystals has proven to be a good solution to that problem. Other material parameters are important, but less specific. They include chemical stability and the possibility to form elements of small size (thin wafers, spheres, etc) from the bulk. An example are refrigerant assemblies which are permeable to gas flow as in the active magnetic regenerator. Commercial availability and market price also have to be considered. As explained earlier, the magnetocaloric behaviour of a material may be entirely predicted from its temperature- and magnetic-field-dependent entropy function S(T, B). In the design of magnetic refrigerators, precise knowledge of this function is essential. Equation (D7.0.3) showed how it can be determined from measurable properties. From the thermodynamic point of view, a material is suitable for magnetic refrigeration if it exhibits a large enough change of entropy upon variation of the applied field. The following criterion for the isothermal entropy change coefficient may serve as a rough indicator

Here, the entropy change per unit volume (and not per unit mass) has been chosen because it is this quantity that determines the bore size of the magnet system. As long as this criterion is fulfilled a magnetic field strength of 3–4 T is usually sufficient to provide the desired temperature span and refrigeration power. An additional indicator of suitability of a magnetic material is the adiabatic temperature change associated with a variation of the applied field. This quantity is related to the isothermal entropy change, introduced above, and to the specific heat at constant field

For a given material the coefficient of entropy change is not a constant, but goes over a maximum as the temperature changes. As a result, every magnetic refrigerant is appropriate within a limited temperature interval only. In the 2–10 K temperature range, the rare-earth compound GGG has been universally adopted as the best refrigerant. Its coefficient of volumetric entropy change (equation (D7.0.12)) goes up to 7 × 104 J m−3 K−1 T−1 and, if required, the material is available in the form of large single crystals. It has a paramagnetic behaviour. The entropy chart and some relevant material properties have already been given in section D7.0.2. The thermal conductivity of the single crystal at 5 K equals that of commercial

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copper. Sintered GGG has been shown to have the same entropy, provided its mass density is not more than 10% below that of the single crystal. The thermal conductivity of sintered GGG is not known, but is expected to be smaller than that of the single crystal by a factor of at least ten. Dysprosium aluminium garnet, Dy3Al5O12 (DAG), has been proposed as a paramagnetic working substance superior to GGG at temperatures above 10–15 K, depending on the strength of the applied field (Li et al 1986). The substance has a mass density of 6200 kg m−3 and can be grown in large single crystals with a thermal conductivity as high as that of GGG. In contrast to GGG, however, the entropy change in DAG is anisotropic. As a result, the advantage of DAG is real only if single crystals are used and if the magnetic field is applied along the crystallographic 〈111〉 axis. When temperatures above 15 K are considered, the entropy change coefficients of all paramagnetic materials progressively fall short of the criterion in equation (D7.0.12). The reason is the increasing thermal energy which tends to disorder the magnetic spin system against the ordering power of the potential energy, created by the applied magnetic field. Fortunately enough, it has appeared that ferromagnetic materials can help to overcome this difficulty to a certain extent (Hashimoto et al 1981). They may be useful refrigerants in a relatively narrow temperature interval around their ordering temperature. In the 10–20 K temperature range, the nonmetallic ferromagnetic compound europium sulphide (EuS) thus appears a most attractive candidate. Its ordering temperature is 16 K. The mass density of a single crystal is 5750 kg m-3 and that of good quality sintered samples is about 10% less. The theoretically determined entropy function has been confirmed experimentally and is shown in figure D7.0.5 (Brédy and Seyfert 1988). The coefficient of volumetric entropy change attains the value of 5 × 104 J m−3 K−1 T−1. Measurements of the thermal conductivity on a sintered powder sample give a value of about 8 W m−1 K− 1 at 10 K. EuS is far superior to the above-mentioned DAG if a large thermal conductivity is not a prerequisite, as, for example, in the active magnetic regenerator with its highly subdivided core matrix.

Figure D7.0.5. Experimentally determined entropy diagram of europium sulphide (sintered material, mass density 5140 kg m−3 ).

Ferromagnetic refrigerants for even higher temperatures have been proposed, e.g. GdPd for 20–40 K and GdNi for 40–80 K (Zimm et al 1992), but they require magnetic field swings of 5–7 T and are only efficient within a temperature interval of 20–40 K. If refrigeration over a larger temperature span is desired a cycle must be contemplated which allows the use of a combination of several ferromagnetic working substances with staged ordering temperatures.

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D7.0.6 Conclusion To date, the most successful developments in magnetic refrigeration appear to be one-stage Carnot-cycle machines for the temperature span of 1.8–4.2 K. It has been demonstrated that reciprocating devices can achieve cooling powers of a few watts with efficiencies of better than 50% of the Carnot efficiency. For cooling powers larger than 10 W a design with stationary refrigerant and pulsed magnetic field seems more appropriate, but no successful prototype has been reported yet. Much effort has also been put into research on systems for the temperature span of 4.2–20 K and quite recently 20–80 K. The key problem here is the development of refrigeration cycles with efficient regenerative heat transfer (active magnetic regenerator). Even though results in these fields have often been encouraging it must be admitted that the systems built so far are not competitive in performance with existing gas-cycle machines. Generally speaking, there are good reasons to believe that magnetic refrigeration remains an attractive approach to cryogenic refrigeration but it is likely that no decisive breakthrough will occur until an interesting (and important) application of this alternative cooling technique has been identified which is out of the reach of gas-cycle refrigeration. References Barclay J A, Stewart W F, Overton W C, Candler R J and Harkleroad O D 1986 Experimental results on a low temperature magnetic refrigerator Advances in Cryogenic Engineering vol 31, ed R W Fast (New York: Plenum) pp 743–52 Brédy P and Seyfert P 1988 Experimental results on magnetic and thermal properties of europium sulfide relevant to magnetic refrigeration Proc. 12th Int. Cryogenic Engineering Conf. (Guildford: Butterworth) pp 602–5 Cogswell F J, Smith J L Jr and Iwasa Y 1988 Regenerative magnetic refrigeration over the temperature range of 4.2 to 15 K Proc. 5th Int. Cryocooler Conf. (Naval Postgraduate School, Monterey, CA, 1988) pp 81–90 DeGregoria A J, Feuling L J, Laatsch J F, Rowe J R, Trueblood J R and Wang A A 1992 Test results of an active magnetic regenerative regenerator Advances in Cryogenic Engineering vol 37 Part B, ed R W Fast (New York: Plenum) pp 875–82 Green G, Chafe J, Stevens J and Humphrey J 1990 A gadolinium—terbium active regenerator Advances in Cryogenic Engineering vol 35 Part B, ed R W Fast (New York: Plenum) pp 1165–74 Hakuraku Y and Ogata H 1986 Thermodynamic analysis of a magnetic refrigerator with static heat switches Cryogenics 26 171–6 Hashimoto T, Numazawa T, Shino M and Okada T 1981 Magnetic refrigeration in the temperature range from 10 K to room temperature: the ferromagnetic refrigerants Cryogenics 21 647–53 Kirol L D and Dacus M W 1988 Rotary recuperative magnetic heat pump Advances in Cryogenic Engineering vol 33, ed R W Fast (New York: Plenum) pp 757–65 Lacaze A F, Béranger R, BonMardion G, Claudet G and Lacaze A A 1984 Double acting reciprocating magnetic refrigerator: recent impovements Advances in Cryogenic Engineering vol 29, ed R W Fast (New York: Plenum) pp 573–9 Li R, Numazawa T, Hashimoto T, Tomokiyo A, Goto T and Todo S 1986 Magnetic and thermal properties of Dy3Al5O12 as magnetic refrigerant Advances in Cryogenic Engineering vol 32, ed R W Fast (New York: Plenum) pp 287–94 Matsumoto K and Hashimoto T 1990 Thermodynamic analysis of magnetically active regenerator from 30 to 70 K with a Brayton-like cycle Cryogenics 30 840–5 Matsumoto K, Ito T and Hashimoto T 1988 An Ericsson magnetic refrigerator for low temperature Advances in Cryogenic Engineering vol 33, ed R W Fast (New York: Plenum) pp 743–50 Nakagome H, Kuriyama T, Ogiwara H, Fujita T, Yazawa T and Hashimoto T 1986 Reciprocating magnetic refrigerator for helium liquefaction Advances in Cryogenic Engineering vol 31, ed R W Fast (New York: Plenum) pp 753–62 Numazawa T, Kimur H, Sato M and Maeda H 1993 Carnot magnetic refrigerator operating between 1.4 and 10 K Cryogenics 33 547–54

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References

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Radebaugh R 1983 Fundamentals of alternate cooling systems Cryocoolers, Part 2 (The International Cryogenics Monograph Series) ed G Walker (New York: Plenum) pp 155–60 Seyfert P, Brédy P and Claudet G 1988 Construction and testing of a magnetic refrigeration device for the temperature range of 5 to 15 K Proc. 12th Int. Cryogenic Engineering Conf. (Guildford: Butterworth) pp 607–11 Smith J L Jr, Iwasa Y and Cogswell F J 1990 Material and cycle considerations for regenerative magnetic refrigeration Advances in Cryogenic Engineering vol 35, Part B, ed R W Fast (New York: Plenum) pp 1157–64 Steyert W A 1978 Stirling-cycle rotating magnetic refrigerators and heat engines for use near room temperature J. Appl. Phys. 49 1216–26 Zimm C B, Ludeman E M, Severson M C and Henning T A 1992 Materials for regenerative magnetic cooling spanning 20 K to 80 K Advances in Cryogenic Engineering vol 37 Part B, ed R W Fast (New York: Plenum) pp 883–90

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D8 Cryostats

N H Balshaw

D8.0.1 Overview of the use of cryostats This chapter illustrates some of the diverse range of systems available, from small continuous flow cryostats weighing about 1 kg to large bath cryostats weighing many tonnes and containing thousands of litres of cryogens. The different types of cryostat that are widely available are described individually in the following sections. Each type of cryostat has advantages and disadvantages compared with the others. Thousands of small research cryostats and hundreds of systems for magnetic resonance imaging (MRI) are built worldwide every year. Comparatively few larger and more specialized systems are built, but their value is a significant proportion of the total spend. ‘Bath’ cryostats are commonly used for large laboratory systems and MRI magnets. They are particularly applicable if they have to operate continuously at a constant temperature close to the normal boiling point of a convenient cryogen. It is usually possible to fit a suitable continuous flow insert to achieve variable temperatures in a region inside the bath cryostat. However, if the cryostat has to fit into a small space or has to be thermally cycled rapidly and often, and the experimental equipment does not need a self-contained reservoir of cryogen, it may be better to feed liquid from a remote storage dewar through a special transfer tube. This is called a ‘continuous flow’ cryostat. Most of the largest cryostats built for special research applications also operate on a continuous flow basis, often fed from a reservoir supplied by a dedicated refrigerator system. Small-scale systems can often be bought as standard products which are suitable for a wide range of applications without modification. Larger scale systems are usually custom designed for the application. D8.0.1.1 Cooling powers of different cryostats Although it is difficult to give general cooling power values for different cryostats without knowing the details of their designs, table D8.0.1 summarizes the likely values and may help you to choose the best system for your application. In general, the cooling power of any cryostat is limited by the available surface area for heat exchange with the cryogens that it contains. However, good design can enhance the performance of a heat exchanger of given dimensions. It is generally true to say that standard commercial systems are not designed to give very high cooling powers because of the limited range of applications and the cost of running them. Many companies will undertake the design of special cryostat systems

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Cryostats Table D8.0.1. A summary of cooling power values for different cryostats.

to meet experimental requirements, and some of these systems are shown in the table for comparison. However, unless the design of the whole system is considered carefully, increasing the cooling power of the cryostat might make little difference to the temperature of the experiment. Good thermal contact between components at low temperature can be very difficult to achieve. D8.0.2 Bath cryostats Bath cryostats contain large enough supplies of cryogens for a convenient period of operation. There is no need to refill the cryostat continuously from a storage dewar. The ‘hold time’ depends on a number of factors (for example, size, experimental heat load and cryogen consumption rate). They are typically designed to give operating periods between 10 h and 18 months. According to Stefan’s law, the amount of heat radiated from a warm body to a cold body varies with the difference between the fourth power of their temperatures. A 300 K surface radiates 230 times more heat to a 4.2 K surface than a 77 K surface would radiate onto the same 4.2 K surface. Therefore liquid-helium reservoirs are always shielded from room-temperature radiation by a cooled shield. In most cryostats, the radiation load is further reduced by the use of ‘multilayer superinsulation’. This consists of many thin layers of low-emissivity material, in the insulating vacuum space. All liquid-helium cryostats are vacuum insulated to prevent the heat load due to conduction and convection. However, the helium reservoir is shielded from the room-temperature radiation heat load in different ways. D8.0.2.1 Liquid-nitrogen-shielded cryostats for liquid helium In this type of cryostat, the liquid-helium reservoir is surrounded either by a reservoir of liquid nitrogen, or by a shield cooled by this reservoir. The liquid-nitrogen vessel is thermally linked to the neck of the liquid-helium vessel to form a barrier to heat conducted down from room temperature. Figure D8.0.1 shows a typical small liquid-nitrogen-shielded cryostat used to cool an infrared detector to 1.5 K.

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Figure D8.0.1. A small low-loss infrared detector cryostat (typically containing between 1 and 10 1 of liquid helium).

The advantages and disadvantages of liquid-nitrogen-shielded cryostats are summarized in table D8.0.2. D8.0.2.2 Vapour-shielded cryostats for liquid helium As an alternative to liquid-nitrogen-cooled shields, it is possible to link several thermal shields to the neck of the liquid-helium vessel. The cold gas that has evaporated from the reservoir is then used to cool these shields. This type of cryostat typically has between two and six shields (depending on the required performance) linked to different points on the neck. The space between the shields is filled with superinsulation. The boil-off rates of the two types of cryostat are similar, providing that there are no dimensional constraints on the system. Table D8.0.3 shows the advantages and disadvantages of vapour-shielded cryostats.

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Cryostats Table D8.0.2. Liquid-nitrogen-shielded cryostats for liquid helium.

Table D8.0.3. Vapour-shielded cryostats for liquid helium.

Many low-loss liquid-nitrogen-shielded cryostats are also fitted with a vapour-cooled shield between the liquid-nitrogen and liquid-helium reservoirs. These shields are usually designed to operate at a temperature close to 40 K, reducing the radiated heat load to an acceptably low level. D8.0.2.3 Shields cooled by closed-cycle coolers Mechanical coolers are sometimes used to cool shields in large cryostats. This technique can help to reduce the cryogen losses from the system considerably, by reducing the radiation heat load. They can also reduce the conducted heat load if the shield is fixed to the neck of the helium vessel. However, these coolers have a high initial cost and the pay-back time (in terms of reduced cryogen costs) may be very long. They need to be serviced regularly (typically every 5000 h). There is also the possibility of introducing unwanted vibration into the experiment if the cooler is not mounted very carefully. D8.0.2.4 Bath cryostats for liquid nitrogen In some ways it is more difficult to make a reliable bath cryostat for liquid nitrogen than for liquid helium. Unlike liquid helium, liquid nitrogen is not cold enough to freeze (or cryopump) all the contaminating gases in the insulating vacuum space. The quality of the vacuum is critical for operation of the cryostat, so a sorption pump (containing activated charcoal or a molecular sieve) is normally fitted to the outside

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of the liquid-nitrogen reservoir to maintain the vacuum. The vessel is usually superinsulated, but other insulation techniques are occasionally used; for example, filling the vacuum space with a low-conductivity material such as a suitable mineral powder. D8.0.2.5 Large-scale bath cryostatsfor magnetic resonance imaging One of the most widespread applications of large scale superconducting magnets is MRI. Cryostats for these magnets have to operate for long periods without servicing, in hospital environments. They typically contain more than 1500 1 of liquid helium in a cryostat weighing more than 5 t, and have evaporation rates less than 0.1 1 h−1, giving a hold time in excess of 18 months. It is not necessary to pump the insulating vacuum space for at least ten years. Some of them are designed to be transportable, so that the imaging system can be taken to the patient. The stray field from these systems also has to be minimized so that people can work in areas close to the magnet without risks associated with high magnetic fields. An active-shield magnet incorporates a special array of coils which significantly reduce the stray magnetic field. The magnet consists of several precision manufactured concentric formers. The inner former contains the windings for the main magnet, whilst the outer formers contain the active shield windings. A superconducting switch is connected across the magnet to permit persistent mode operation, whilst a combination of resistors and diodes connected across the magnet provides full protection in the event of a magnet quench. Additional screening coils are fitted which produce a screening effect in the imaging region from the effect of close moving objects. These coils are wound on top of the main coils, but electrically isolated from them, and are wound in series with their own superconducting switch to form an independent circuit. The design of the magnet is chosen to give a high level of homogeneity over a volume suitable for whole-body imaging (see, for example, figure D8.0.2).

Figure D8.0.2. MRI cryostats at various stages of assembly. Siemens-Oxford Instruments venture.

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The magnet assembly is sealed into an all-welded stainless steel vessel (grade 304) which is filled with liquid helium during operation. Total immersion of the magnet is not necessary since the design ensures that isothermal conditions are maintained from a pool of liquid helium. The magnet vessel is assembled into an all-welded, vacuum-insulated cryostat which is designed to minimize the heat leak to the liquid helium. Radiated heat is the dominant factor and this is minimized by using two concentric aluminium radiation shields between the magnet vessel and cryostat. The shields are cooled by a two-stage Gifford—McMahon refrigerator, which maintains the outer shield at a temperature of approximately 65 K and the inner shield at approximately 20 K, depending on specific system operating conditions. The refrigerator is interfaced through a special assembly which allows it to be removed for servicing without breaking the main cryostat vacuum or wanning up the system. The helium evaporation rate increases for a few days until thermal equilibrium is reached again. Conducted heat loads are minimized primarily by the vacuum insulation, and also by supporting the weight of the magnet vessel and shields on a support structure chosen for its combination of mechanical strength and low heat conduction. Cryogenic and electrical access to the magnet vessel is provided by a single service turret located on the cryostat at a 30° angle to minimize the headroom required for normal operation. The helium and vacuum vessels are fully protected against over-pressure by a quench valve (incorporating a bursting disc) and a drop-off plate respectively. D8.0.3 Continuous-flow cryostats A wide range of continuous-flow (CF) cryostats is available. Some of these are supplied with cryogens from a storage vessel; others are mounted in a bath cryostat which supplies liquid. In most of these systems the cooling power available from a flow of cryogen (liquid nitrogen or liquid helium) is balanced by power supplied electrically to a heater near the sample (usually by a temperature controller). D8.0.3.1 Variable temperature inserts Variable temperature inserts (VTIs) are used in laboratory-scale bath cryostats to adjust the temperature of a sample without affecting the helium reservoir. ‘ Dynamic ’ and ‘ static ’ types of VTI are available, and the advantages and disadvantages of each type are described in section D8.0.3.3. The inner parts of the insert are vacuum insulated from the liquid helium. There may also be a radiation shield between the sample space and the liquid reservoir to reduce the radiated heat load on the reservoir when the sample is at a high temperature. This shield is usually cooled by the cold exhaust gas from the main reservoir. The temperature range of a VTI is typically from 1.5 to 300 K, but in certain circumstances this range may be extended. The sample temperature can be controlled continuously at any point in this range. Lower temperatures can often be achieved in single-shot mode: the sample space is filled with liquid and the needle valve is closed to allow the pump to reduce the vapour pressure above the liquid to the lowest possible level. D8.0.3.2 Independent continuous-flow cryostats The operating principles of CF cryostats are generally the same as those of VTIs. Dynamic and static versions are available as described in section D8.0.3.3. However, they normally have their own independent thermal shielding, and they are supplied with coolant from an independent storage vessel through a ‘ lowloss ’ or ‘ gas-flow-shielded ’ (GFS) transfer tube. These cryostats can sometimes be used with a separate superconducting magnet if it has a room-temperature bore. They can also be used with resistive magnets.

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Figure D8.0.3. Optistat—an optical continuous flow cryostat for laboratory-scale applications. It is approximately 45 cm high. Courtesy of Oxford Instruments (UK) Ltd.

Liquid-helium transfer tubes are always vacuum insulated. In order to reduce the losses in the tube, gas-flow-shielded-type transfer tubes use the enthalpy in the exhaust gas from the CF cryostat to reduce the radiation load on the tube carrying the liquid. The temperature range of CF cryostats is typically <4 to 300 K in continuous mode, with lower temperatures available for limited periods in ‘single-shot mode’. However, the range may be extended to give higher or lower temperatures if necessary. In general it is difficult to achieve temperatures as low as

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Figure D8.0.4. The flow system for a small CF cryostat.

those available in VTIs because of the higher flow rates due to thermal losses in the transfer tubes, but some cryostats are designed to reach 1.6 K continuously. Figure D8.0.3 shows one of them schematically and figure D8.0.4 shows the flow system for a small CF cryostat. D8.0.3.3 Static and dynamic continuous-flow systems Although all CF cryostats work on the principle of balancing the cooling power of a flow of cryogen with electrical power from the temperature controller, there are several distinct types of cryostat: the most important are referred to as ‘dynamic’ or ‘static’. (a) Dynamic systems In a dynamic CF cryostat, the sample is mounted in a flowing gas or in liquid, and its temperature is strongly influenced by the fluid. The temperature of the fluid is controlled by passing it through a heat exchanger (usually placed at the bottom of the sample space). The heat exchanger temperature is controlled by the temperature controller. Providing that the flow of cryogen through the heat exchanger is not too

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high the temperature of the flowing fluid can be controlled quite accurately. The fluid flows past the sample and out of the exhaust port of the insert to the pump. This type of insert is easy to operate and it responds very quickly if the set temperature is changed to a new value. However, the temperature stability is not as high as that of a static insert. It is also possible to block the small capillary that feeds the cryogen to the heat exchanger with frozen water or air during the sample changing operation if care is not taken. (b) Running dynamic systems below 4.2 K If dynamic systems are operated below 4.2 K it is possible to fill the sample space with liquid helium. For many applications this is an advantage as the thermal contact to the experiment may be better than it would be to a flow of cold gas. However, some experiments require that liquid is excluded from the sample space. One recipe to ensure that liquid does not collect is to make sure that the pressure in the sample space is always kept below the vapour pressure of liquid helium at the operating temperature. Any liquid entering the sample space will then evaporate and it will be removed by the pump. (c) Static systems Static systems are also fitted with heat exchangers, and the temperature of the heat exchanger is controlled in a similar way. However, the exhaust gas does not flow over the sample, but it passes out of the cryostat to the pump through a separate pumping line. The heat exchanger usually forms an annulus around the sample space, and thermal contact is made to the sample through exchange gas. The exchange gas conditions can be adjusted to suit the conditions. This can be useful for several reasons: (i) to keep the exchange gas pressure high when high-voltage measurements are being made; (ii) to use a different exchange gas; (iii) to allow the exchange gas to be removed to isolate the sample from the surroundings. The sample temperature follows the temperature of the heat exchanger, but rapid temperature fluctuations tend to be filtered out, and the temperature stability of the sample can be improved considerably. In some cases, a heater is fitted to the sample block for fine control of the temperature or to warm the sample quickly. Static inserts are as easy to operate as the dynamic type, and have the advantage that it is not possible to block the heat exchanger during the sample-changing process. Indeed, quite large amounts of air may be frozen into the sample space without affecting the operating procedure. However, the increased sample temperature stability has to be traded off against the increased time taken to change the sample temperature to a new value. In particular, it is not possible to cool the sample as quickly, and static systems are generally used for small sample spaces. D8.0.3.4 Large-scale continuous-flow cryostats Large-scale CF cryostats are sometimes built for specialized applications. Indeed, most large-scale cryostats use a continuous flow of cryogen fed from a reservoir. One such example is shown in figure D8.0.5. This is a set of six kidney-shaped coils assembled to form the 5.8 m diameter CLAS torus at CEBAF, VA, USA (Street et al 1996). The main cryostat contains a header tank of supercritical helium at a pressure of 2.8 bar (2.8 × 105 Pa), fed directly from a refrigerator. This fluid flows through channels which are in intimate thermal contact with the superconducting coils. The cooling channels are arranged in series, with intercooling heat exchangers between the coils, where the fluid is cooled by a reservoir of liquid helium at 4.5 K. After passing through the last coil the supercritical helium is expanded to fill the reservoir. The radiation shields around the coils are made from aluminium honeycomb bonded to epoxy glass facing sheets. They are cooled to approximately 80 K by a forced two-phase flow of nitrogen. The coil

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Figure D8.0.5. Six kidney-shaped coils assembled to form the 5.8 m diameter CLAS torus in CEBAF, VA, USA. The superconducting coils are cooled by a continuous flow of supercritical helium. Courtesy of Oxford Instruments (UK) Ltd.

cases are machined from solid 6061 aluminium and the vacuum vessels for the coils from solid 304L stainless steel, to meet the required dimensional tolerances. The material for the vacuum seals had to be specially selected for the high-radiation environment. D8.0.3.5 Storage (or transport) dewars Storage (or transport) dewars are generally only suitable for supplying cryogens to the cryostat (whether it is of the bath or CF type). They are designed to be robust and to have a low evaporation rate. They usually have very narrow necks and a large amount of superinsulation. A few liquid-helium storage dewars are fitted with liquid-nitrogen jackets (especially older dewars). However, some small VTI, 3He refrigerators and 3He/4He dilution refrigerators are available to fit into storage dewars, providing that the diameter of the neck is sufficiently large (50 mm). These inserts give temperature ranges from 0.03 to 300 K. D8.0.3.6 ‘Stinger’ systems Some closed-cycle cooler systems are used to recondense helium gas into a bath cryostat continuously. They take the form of a cold finger that fits into the neck of the helium reservoir. They need quite high cooling powers both at the 4.2 K stage and at higher temperatures because they have to provide enough cooling to replace the enthalpy of the exhaust gas, which normally cools the neck of the reservoir. The helium reservoir is normally pressurized slightly so that the gas recondenses effectively and so the liquid helium is held at a temperature close to 4.5 K.

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These systems share most of the advantages and disadvantages of a closed-cycle cooler to cool radiation shields. They are available commercially from several closed-cycle cooler manufacturers (including APD and Sumitomo) but interfacing to an existing system can be quite difficult. D8.0.3.7 Cryogen-free cryostats Modern closed-cycle coolers offer a highly reliable method of achieving low temperatures. They may either be used alone, to cool a sample and radiation shields, or with a bath cryostat to cool one or two radiation shields and thus reduce the evaporation rate of the cryostat. This can considerably extend the hold time of a low-loss cryostat, but it is not usually appropriate if the equipment inside the cryostat has a high consumption rate which has a dominant effect on the hold time. It is now possible to build cryogen-free systems containing superconducting magnets. A typical laboratory scale system is shown in figure D8.0.6.

Figure D8.0.6. A laboratory-scale cryogen-free superconducting magnet system, controlled by computer. Courtesy of Oxford Instruments (UK) Ltd.

These cryostats are similar to conventional cryostats which use liquid cryogens. They typically have the following features: (i) radiation shields are cooled by high-temperature stages of the cooler; (ii) the heat loads on the system have to be minimized; (iii) vibrations from the cooler may affect the equipment inside the cryostat and some vibration isolation precautions usually have to be taken; (iv) consider whether the cryostat can be taken out of operation when the cooler is being serviced (typically twice per year). In purely financial terms the pay-back period may be many years even in remote areas where cryogens are expensive. However, there may be sound technical reasons to use a cryogen-free system even if it is

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slightly more expensive to run. For example, the people who run these systems from day to day do not need to be taught how to handle cryogens safely. D8.0.4 Specialized cryostat accessories D8.0.4.1 Lambda point refrigerators Superconducting magnets are usually operated in liquid helium at about 4.2 K. The performance of the superconductor can often be enhanced by cooling the magnet to lower temperatures. The simplest way to achieve temperatures below 4.2 K is to pump the whole liquid-helium reservoir with a rotary pump, to reduce the vapour pressure above the liquid. If the bath is cooled to 2.2 K in this way, about 35% of the helium is evaporated to cool the remaining liquid. Temperatures below 2.2 K can be achieved, but if the bath is cooled below the lambda point, the liquid-helium consumption increases significantly (both to reach the low temperature and to maintain it). This simple approach has several disadvantages. A large amount of liquid is used to cool the magnet down and, since the reservoir is then below atmospheric pressure, access to the reservoir is difficult and all the fittings on the top plate have to be reliably leak tight. The liquid helium can only be refilled by de-energizing the magnet to its 4.2 K field and filling the reservoir to atmospheric pressure with helium gas, which interrupts the experiment. Lambda-point refrigerators (also known as ‘lambda plates’ or ‘pumped plates’) are used to cool superconducting magnets to about 2.2 K and maintain this temperature continuously (see figure D8.0.7). They consist of a needle valve (to control the flow of liquid helium into the refrigerator) and a tube or chamber with a pumping line. They are normally built into the ‘magnet support system’. The refrigerator is in good thermal contact with the liquid helium just above the magnet. Liquid is continuously fed into the refrigerator and pumped to a low pressure so that it cools. The cooling power is determined by the liquid flow rate and the size of the pump, and it can be adjusted using the needle valve. High flow rates are typically used at high temperatures to cool the system quickly or to obtain high cooling power, but when base temperature is reached, the flow can be reduced to make operation as economical as possible. The density of liquid helium changes rapidly with temperature, so strong convection currents are set up around the magnet. The cold liquid from the refrigerator sinks to the bottom of the reservoir, cooling the magnet and keeping it at about 2.2 K. Meanwhile the warmer liquid above the refrigerator is affected very little. The thermal conductivity of the liquid is so low that the region immediately above the plate has a steep temperature gradient, and the liquid surface remains at 4.2 K and at atmospheric pressure. It is important to make sure that this thermal gradient is maintained, and not short circuited by high-conductivity components. Lambda-point refrigerators have several advantages. In particular: (i)

since only a small proportion of the liquid in the reservoir is cooled by the lambda plate less liquid has to be used, and this reduces the cost of operation; (ii) operation can be automated; (iii) the reservoir can be refilled without stopping operation of the system, as long as the transfer tube does not stir the liquid and upset the temperature gradient above the lambda plate. The performance of these systems is dominated by the amount of liquid helium that has to be cooled. Although the mass of the magnet is much larger than that of the liquid, its heat capacity is very much lower. It is possible to calculate the amount of heat that has to be removed if the magnet and liquid are cooled from 4.2 K to 2.2 K. In a typical system, containing a 50 kg magnet, there may be about 3 1 (0.5 kg) of liquid below the lambda plate. Only 5 J has to be removed from the magnet, but about 3 kJ

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Figure D8.0.7. Lambda-point refrigerator.

has to be removed from the liquid. Therefore it is important to minimize the amount of liquid around the magnet so that it will cool quickly and cheaply. In most systems the magnet can only be cooled to 2.2 K in this way, because liquid helium has a phase change (the lambda point) at this temperature. Below the lambda point, the liquid becomes‘ superfluid ’ and has a very high thermal conductivity, so the phase transition can only occur if the whole reservoir is cooled to the lambda point. The heat from any warmer region in the reservoir would be rapidly conducted to the colder region, keeping its temperature above the critical level. However, in a few specialized applications, the refrigerator is built into the top of a separate chamber around the magnet. It is fed from a 4.2 K liquid reservoir, but thermally isolated from it. The lambda plate then cools the whole of this chamber, and temperatures below the lambda point can be reached and maintained continuously, while the liquid is at atmospheric pressure. The optimum temperature is about 1.8 K, as the superfluid is then able to carry heat away from the magnet most effectively. Design guidelines for these special refrigerators have been given in several papers in the journal Cryogenics (Hakuraku and Ogata 1983, Leupold and Iwasa 1986). More conventional systems running above the lambda point are much simpler to make, but the design considerations are similar.

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D8.0.4.2 Ultra-low temperatures Refrigerators working at temperatures below 1 K are used for a surprisingly diverse range of applications in research establishments. A range of specialized techniques is used to achieve these temperatures. Most of the systems described in the previous sections use liquid helium and liquid nitrogen to reach and maintain low temperatures, but it is difficult to achieve temperatures significantly below 1 K using these cryogens alone. However, most ultra-low-temperature systems are immersed in liquid helium (4He) at 4.2 K, so that the heat load from the surroundings is minimized. It is possible to reach temperatures slightly below 1 K by pumping liquid 4He to a low pressure but very large pumps are required and it is not usually economically viable. 4He may also be used to give very low cooling powers at temperatures down to 0.7 K in ‘ vortex refrigerators ’ which rely on the special properties of superfluid 4He. However, the valuable lighter isotope of helium, 3He, is usually used in refrigerators working below 1 K. Evaporating 3He is used in some systems, and temperatures slightly below 0.3 K can be achieved by reducing its vapour pressure. Temperatures below 0.3 K are usually reached by continuously diluting a flow of 3He in liquid 4He using a 3He/4He dilution refrigerator. Only a few years ago the supply of 3He was limited by the nuclear super powers because of its strategic importance and the price of the gas was very high. It is still much more expensive than 4He but the price now seems to have settled in the range 100–200 US$ per litre of gas at normal temperature and pressure (NTP). As a comparison, 4He typically costs around 5–10 US$ per liquid litre. (a) 3 He refrigerators He refrigerators are usually designed for routine operation in the temperature range 0.3–1.2 K, and they use evaporating 3He as the refrigerant. Their operating range can often be extended to 100 K or higher. Some of these systems can run continuously, returning the liquid 3He to the system to replace the evaporated liquid. Others work in ‘single shot’ mode, by pumping on a small charge of liquid 3He condensed into the system. In an efficient cryostat a 20 cm3 charge of liquid 3He may last for longer than 50 h. Small laboratory refrigerators may give a cooling power of a few milliwatts at 0.5 K, but very large and high-powered machines can give cooling powers of several watts at this temperature. 3

(b) Sorption pumped 3 He systems Sorption pumped 3He systems are usually single-shot refrigerators, capable of high-performance operation for a limited time. Several types of system are available to suit the majority of laboratory requirements. Most of them can be used with high-field superconducting magnets if required. The top loading systems allow the sample to be mounted on a probe which is loaded directly into liquid 3He. They may also be designed to operate in rapidly sweeping magnetic fields, and a wide range of special services may be fitted to make connections to the sample. The maximum temperature limit is typically 100 K. The Heliox insert is a low cost miniature 3He system designed to allow inexperienced users to cool samples to 0.3 K. It is designed for operation in a liquid-helium storage dewar, or with a superconducting magnet system. The sample is mounted in vacuum, and wiring can be connected easily. The whole insert is removed from the cryostat to change the sample, but since it is small, the timescale for sample changing is similar to that on the top loading systems. The Heliox system can be run up to about 200 K if it is used with a superconducting magnet, but higher temperatures (up to 300 K) can be reached if the insert is pulled up into the neck of the cryostat. Figure D8.0.8 shows the working parts of a typical system. Although a top loading insert is shown, the principle of operation is similar for most sorption pumped inserts. The insert has an inner vacuum chamber (IVC) to provide thermal isolation from the main liquid-helium bath. The sorption pump, or sorb, will absorb gas when cooled below 40 K, and the amount of gas that can be adsorbed depends on its temperature. It is cooled by drawing some liquid helium from the main

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Figure D8.0.8. The principle of operation of a typical sorption-pumped 3He system (top-loading type).

bath through a heat exchanger. The flow of helium is promoted by a small diaphragm pump and the rate of flow through the heat exchanger is controlled by a valve in the pumping line. A heater is fitted to the sorb so that its temperature can be controlled. The 1 K pot is used to condense the 3He gas and then to reduce the amount of heat conducted to the sample space. It is fed from the main liquid-helium bath through a needle valve, and it can be filled continuously. During condensation, the sorb is wanned above 40 K. When it is at this temperature it will not adsorb any 3He (see figure D8.0.8). The 3He condenses on the 1 K pot assembly and runs down to cool the sample and 3He pot to the temperature of the 1 K pot. When most of the gas has condensed into the insert, the 1 K pot needle valve is closed completely so that the pot cools to the lowest possible temperature for optimum condensation. At this stage the 3He pot is full of liquid 3He at approximately 1.2 K. The sorb is now cooled, and it begins to reduce the vapour pressure above the liquid 3He (see figure D8.0.8), so the sample temperature drops. As the limiting pressure is approached, the temperature of the liquid 3He can be reduced to below 0.3 K. The temperature of the sample can be controlled by adjusting the temperature of the sorb. If the sorb temperature is set between 10 and 40 K it is possible to control the pressure of the 3He vapour, and thus the temperature of the liquid 3He. However, if the best stability is needed, a temperature controller can be set up to measure the sample temperature and control the power supplied to the sorb heater. No heat is supplied directly to the liquid 3He; this would evaporate it too quickly. The temperature of the sorb is continuously adjusted by the temperature controller, and the temperature of the sample can typically be maintained within 1 mK of the set temperature for the full hold time of the system. These systems have limitations both in their cooling power and base temperature, and if high cooling powers (>5 mW) are required, or operation must be continuous, it may be more appropriate to choose a

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continuously circulating 3He refrigerator. If, however, the base temperature is not low enough, a dilution refrigerator should be chosen. In general it is found that a dilution refrigerator has a better performance below 0.4 to 0.5 K, and a continuous 3He system is better above this temperature. In either case, these refrigerators typically have large room-temperature pumping systems, and they are therefore rather more expensive. (c) Continuously circulating 3 He refrigerators Continuously circulating 3He refrigerators are capable of giving high cooling powers and of operating continuously for a long period. They use an external room-temperature pumping system (including a rotary pump and a booster pump). The 3He gas is injected into the cryostat and it is cooled to approximately 4.2 K by the liquid-helium bath before it enters the IVC. It is then cooled to 1.2 K and condensed by the 1 K pot. The liquid 3He then passes through a special heat exchanger where it is cooled by the outgoing 3He gas. Below this heat exchanger an impedance is used to keep the pressure in the condenser high enough even if the pressure in the 3He pot is very low. On some systems a needle valve is used here as a variable impedance to set the 3He flow rate. Since the liquid has already been cooled to a temperature close to that of the 3He pot in the heat exchanger, only a small fraction of it evaporates as it expands through the needle valve. This ensures that the maximum amount of latent heat is available from a given flow rate of 3He. The liquid and gas then enters the 3He pot, which has a large surface area, to give good thermal contact to the sample. The flow rate determines both the base temperature and the cooling power available from the system. In general, a low flow rate will be required for a good base temperature, and a high flow rate will allow a high cooling power to be achieved. (d) 3 He/ 4 He dilution refrigerators The principle of operation of the dilution refrigerator was originally proposed by H London in a discussion after the presentation of a paper (Pryce 1951), but the first working systems were not built until more than ten years later. Since that time, the performance of these systems has steadily improved, and the physical processes involved have become much better understood. When a mixture of the two stable isotopes of helium is cooled below a critical temperature it separates into two phases. The lighter ‘concentrated phase’ is rich in 3He and the heavier ‘dilute phase’ is rich in 4He. The concentration of 3He in each phase depends upon the temperature. Since the enthalpy of the 3 He in the two phases is different, it is possible to obtain cooling by ‘evaporating’ the 3He from the concentrated phase into the dilute phase. The properties of the liquids in the dilution refrigerator are described by quantum mechanics and the details will not be given here. However, it is helpful to regard the concentrated phase of the mixture as liquid 3He and the dilute phase as 3He gas. The 4He which makes up the majority of the dilute phase is inert, and the 3He ‘gas’ moves through the liquid 4He without interaction. This ‘gas’ is formed in the mixing chamber at the phase boundary. This process continues to work even at the lowest temperatures because the equilibrium concentration of 3He in the dilute phase is still finite, even as the temperature approaches absolute zero. However, the base temperature is limited by other factors, and in particular by the residual heat leak and heat exchanger performance. When the refrigerator is started the 1 K pot is used to condense the 3He/4He mixture into the dilution unit. It is not intended to cool the mixture enough to set up the phase boundary but only to cool it to 1.2 K. In order to get phase separation, the temperature must be reduced to below 0.86 K (the tricritical point). The still is the first part of the refrigerator to cool below 1.2 K. It cools the incoming 3He before it enters the heat exchangers and the mixing chamber, and phase separation typically occurs after a few minutes. Gradually, the rest of the dilution unit is cooled to the point where phase separation occurs.

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It is important for the operation of the refrigerator that the 3He concentration and the volume of the mixture is chosen correctly, so that the phase boundary is inside the mixing chamber and the liquid surface is in the still. The concentration of 3He in the mixture is typically between 10 and 20%. In a continuously operating system, the 3He must be extracted from the dilute phase (to prevent it from saturating) and returned into the concentrated phase keeping the system in a dynamic equilibrium. Figure D8.0.9 shows a schematic diagram of a typical continuously operating dilution refrigerator. The 3 He is pumped away from the liquid surface in the still, which is typically maintained at a temperature of 0.6–0.7 K. At this temperature the vapour pressure of the 3He is about 1000 times higher than that of 4 He, so 3He evaporates preferentially. A small amount of heat is supplied to the still to promote the required flow. The concentration of the 3He in the dilute phase in the still therefore becomes lower than it is in the mixing chamber, and the osmotic pressure difference drives a flow of 3He to the still. The 3He leaving the mixing chamber is used to cool the returning flow of concentrated 3He in a series of heat exchangers. In the region where the temperature is above about 50 mK, a conventional counterflow heat exchanger can be used effectively. However, at lower temperatures than this, the thermal boundary resistance (Kapitza resistance) between the liquid and the solid walls increases with T−3, and so the contact area has to be increased as far as possible. This is often done by using sintered silver heat exchangers, which are very efficient even at the lowest temperatures. The room-temperature vacuum pumping system is used to remove the 3He from the still and compress it to a pressure of a few hundred millibar. The gas is then passed through filters and cold traps to remove impurities and returned to the cryostat, where it is pre-cooled in the main helium bath and condensed on the 1 K pot. The primary impedance is used to maintain a high enough pressure in the 1 K pot region for the gas to condense. The experimental apparatus is mounted on or inside the mixing chamber, ensuring that it is in good thermal contact with the dilute phase. All connections to the room-temperature equipment must be thermally anchored at various points on the refrigerator to reduce the heat load on the mixing chamber and give the lowest possible base temperature. If the experiment is to be carried out at higher temperatures, the mixing chamber can be warmed by applying heat to it directly, and a temperature controller can be used to give good stability. (e) Sorption pumped dilution refrigerators It is possible to build continuous dilution refrigerators which do not have external pumps for the 3He/4He mixture. Instead, two sorption pumps are used to pump the still to a low pressure. A cold valve is fitted between each sorb and the still. While one of the sorbs is pumping, the other is regenerating. The temperatures of the sorbs are adjusted by electrical heaters to control the pumping cycle. A special ‘collector’ is fitted below the 1 K pot to hold the liquid condensed by the pot. The pressure in this collector is controlled by maintaining a constant temperature, so that the flow of 3He to the dilution unit is kept constant even though the flow from the pumps to the condenser is not constant. The advantages of these systems are that the vibration levels can be significantly reduced, and the refrigerator system is compact. Since the 3He/4He mixture remains in the cryostat, it is less likely that air can leak into it and block the system. The systems are controlled by a computer, so they can be automated easily. (f) Nuclear demagnetization systems Temperatures below approximately 4 mK cannot be achieved easily or cheaply. Dilution refrigerators capable of reaching temperatures below 5 mK are available but they are large and expensive. Although temperatures as low as 2 mK have been achieved in this type of system, most experimentalists use other techniques.

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Figure D8.0.9. A schematic diagram of a dilution refrigerator.

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Most experiments carried out below 4 mK rely on adiabatic demagnetization of a nuclear paramagnet. Although this is a single-shot process, very long hold times can be achieved, but the total amount of heat that can be absorbed by the demagnetization stage is limited. The performance of the demagnetization stage depends on the heat load that it experiences, either directly from the sample that it is used to cool or in the form of parasitic heat loads from the surroundings. Demagnetization stages are typically pre-cooled to approximately 10 mK in a magnetic field of 8–10 T by a powerful dilution refrigerator. They are then isolated from the mixing chamber by a superconducting heat switch, and the magnetic field is slowly reduced. Temperatures slightly below 1 mK can be achieved using PrNi5 (an enhanced nuclear paramagnet), but copper can be demagnetized to around 10 mK. These systems are described in detail in the book Matter and Methods at Low Temperatures by Frank Pobell (1992). D8.0.5 Experimental access to cryostats It is often necessary to fit windows in cryostats to allow subatomic particles or electromagnetic radiation to reach equipment or samples held at low temperatures. A wide range of materials is available to suit radiation of different frequencies. In most cases the designer aims to maximize the transmission of the windows but minimize the thermal load due to room-temperature radiation. Therefore most cryostats have one window in the outer vacuum vessel and further cold windows designed to absorb the near infrared thermal radiation from the room-temperature environment. If the experimental radiation has to pass through a large number of windows it is important to consider using anti-reflection coatings on the surfaces to reduce signal losses. Table D8.0.4 lists some of the materials commonly used for cryostat windows. This list is not exhaustive, but it gives an idea of the range of materials available.

Table D8.0.4. Typical window materials used for cryogenic systems.

If you are designing windows for a cryostat you should take account of the variation of transmission characteristics with wavelength. Data are usually available from manufacturers of window materials. Detailed thermal design also icquires that a correction is made for the solid angle subtended by the windows at the sample position and internal reflections in the cryostat. It is difficult to measure or calculate the effects accurately because the emissivity of the materials is not always well characterized. However, sometimes it is important to estimate the heat load through the windows to determine the effect on the cryostat’s evaporation rate or the magnitude of thermal gradients induced by the heat load.

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The light pipes and waveguides used for long-wavelength electromagnetic radiation also guide thermal radiation to some extent. Therefore it is usually necessary to fit well-cooled windows or filters at strategic points to absorb radiation in the near-infrared range. Radiation in the near-infrared part of the spectrum is most likely to affect the performance of a cryostat because thermal radiation is also allowed to pass through the windows. It is possible to buy filters with a specified ‘pass band’, and these could be used to allow specific wavelengths to enter the cryostat while all other thermal radiation is excluded. D8.0.5.1 Electrical access and heat sinking A range of materials is used for wiring in low-temperature experiments and it is worth considering which is the most appropriate for your experiment. Optimized wiring for a cryostat is often the result of a compromise between the thermal and electrical requirements of the system. A few simple techniques can be applied to the majority of situations. This section is intended to introduce you to these techniques. Other books give further details of more specialized techniques (Richardson and Smith 1988). (a) Thermal requirements A limited amount of cooling power is available, and it is important to minimize the heat load on the system. Heat conducted along the wires (to the experiment or thermometers) therefore affects the temperature of the system. Materials with high thermal conductivity clearly affect the temperature of the system more than those with lower conductivity. Unfortunately, most materials with high electrical conductivity also have a high thermal conductivity. This means that wires with low electrical resistance are likely to introduce more heat (and affect the temperature more) than wires with high resistance. However, superconducting materials have no electrical resistance, and if they are significantly below their superconducting transition point they have negligible thermal conductivity. Conventional superconductors can be useful at temperatures below approximately 8 K, and new high-Tc materials are now being used above this temperature in some systems. Multifilamentary superconducting wire with a low conductivity matrix (for example Cu—Ni) is therefore especially useful in ultra-low-temperature systems. Note that if the superconductor is allowed to quench (due to the critical field, temperature or current density being exceeded) it is likely to be highly resistive. If high heat loads from the wiring are inevitable because of the electrical requirements, it is often possible to use the exhaust gas from the cryostat to cool the wires. This is especially important in liquid-helium systems as the enthalpy of the gas is many times higher than the latent heat of evaporation of the liquid. If the exhaust gas is allowed to flow freely over the wires the conducted heat load may be reduced by a factor of 20 or more. The low-temperature ends of the wires usually have to be thermally anchored (or ‘ heat sunk ’) to ensure that they are at the required temperature. Details of the most common heat sinking techniques are given in section D8.0.5.2(d). Effective heat sinking of the wiring reduces the amount of heat conducted into the experiment. If the wires are connected to a thermometer without heat sinking, the heat conducted along the wires will certainly warm the thermometer, so that it indicates an artificially high temperature. Indeed the wires are often used to make good thermal contact to the sensor, since the temperature-sensitive element inside the sensor is not always in intimate thermal contact with the case. This can help the sensor to reach thermal equilibrium with its surroundings more quickly. (b) Electrical requirements If you find that your cryostat does not work properly after you have changed the wiring you may have to search for the cause of the problem and replace your new wiring. For example, if you use thick copper wires for an application which only needs thin constantan wires, you are introducing more heat to the

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Table D8.0.5. Choosing appropriate wiring.

system and affecting its temperature more than necessary. Table D8.0.5 shows a range of possible solutions for common wiring requirements. Use it to choose the best material before you start to wire the cryostat. (c) Twisted pairs Electrical noise is often picked up by an electrical circuit, and if sensitive measurements are being made the noise may make it difficult to detect a signal. The noise can also contribute to radiofrequency heating of the sensor in ultra-low-temperature systems. One of the popular and simple ways of reducing the electrical noise pick up is to arrange the wires in twisted pairs. The wires are twisted together for their whole length, so that the currents induced by flux passing between the wires in each twist is cancelled by that in the next twist. Some experimentalists report that the pitch of the twist is important. Others claim that it is only important to maximize the number of twists per unit length. However, it is often difficult to eliminate interference effectively, and a range of techniques is used to reduce noise pick-up (Richardson and Smith 1988). (d) Heat sinking Effective heat sinking or thermal anchoring is one of the most important features of good cryogenic wiring. A variety of techniques is used to ensure that the wires are fixed at the required temperature. Wires are often heat sunk at several points and each heat sink helps to reduce the amount of heat conducted to lower temperatures. These techniques can be so effective that on some systems it is possible to run wires and coaxial cables from room temperature to an experiment at <10 mK without introducing too much heat. The easiest systems to consider are those where the wires are in gas or liquid. For example, if the experiment is carried out in liquid helium or helium gas in a variable-temperature insert, the gas flows over the wires before it leaves the cryostat. This cools them very effectively, and it is only necessary to

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make sure that sufficient length of wire is in contact with the cold gas. It is best to allow the wires to spiral around some convenient mechanical support, such as a pumping line or support leg. However, in many systems the experiment is carried out in vacuum, so no cooling is available from the gas. The temperature of the wires therefore has to be fixed in some other way. The simplest way is to wrap them around a copper post (sometimes known as a ‘thermal dump’) which is held at a known temperature. General Electric 7031 varnish (GE varnish) is typically used to make sure that they are in good thermal contact with the post. Although its thermal conductivity is only moderately good, it gives a large area of contact (see figure D8.0.10). If it is important that the capacitance between the wires and ground is very low (for example, less than 100 pF), alternative methods of heat sinking have to be considered. One method is to clamp the wires firmly and another is to encapsulate them in epoxy resin.

Figure D8.0.10. Heat sinking wires.

(e) Hermetic feedthroughs A wide range of hermetic feedthroughs is available from the manufacturers of electrical connectors. These are suitable for use at room temperature, but they are not guaranteed for cryogenic temperatures. It is possible that some of them could be used, but before relying on them they should be tested by repeated thermal cycling to the working temperature. Some glass-to-metal seals can be used reliably in situations where their temperature is changed slowly. They can be cooled using exchange gas, but if they are immersed directly in liquid nitrogen the thermal shock sometimes causes them to leak after a few cycles. Special feedthroughs can be made using epoxy resins, but it is important that these are designed correctly (Richardson and Smith 1988). (f) Thermoelectric voltages If two dissimilar metals are joined together they tend to act as a thermocouple, and small voltages (typically microvolts) can be generated. If very low-voltage signals are being measured steps have to be taken to reduce the thermal voltages, so that they do not affect the readings. The best way to do this is to ensure that there are no joints in the wires. If there have to be some joints, it is important to ensure that the joints in both wires are at exactly the same temperature. It is possible to buy special feedthroughs for thermocouples, which allow the wire to pass through a metal tube, and these can be used for other similar applications. (g) Four-w w ire measurements If only two wires are connected to the sensor they must be used to supply the excitation current and measure the voltage across the sensor. This current causes a voltage drop along the wires because of their resistance. This voltage is added to the voltage across the sensor, and although it is possible to estimate the fraction of the voltage caused by the wires, the accuracy is limited.

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However, if four wires are connected to the sensor it is possible to make accurate measurements of electrical resistance even if the resistance of the wires is quite large ( figure D8.0.11 ). Two of them are used to supply the excitation current. The voltage drop across these wires is not measured, so it does not matter how high it is. The other two wires are used to measure the voltage across the sensor, using a highinput-impedance voltmeter. Since there is no current flowing in these wires there is no voltage drop along them and their resistance can also be neglected.

Figure D8.0.11. Four-wire measurements.

This is especially useful to allow resistance thermometers to be measured through thin constantan wires. A sensor with a resistance of a few ohms can be measured accurately through leads with resistances of several hundred ohms. Many four-wire sensors have their terminals labelled (V+, I+, V—, I—), and it is important to make connections to the correct terminals. V+ and I+ may not be interchangeable, because they are connected to the sensor at different positions, so that the contact resistance is not measured. Similar techniques can be used for capacitance measurements. (h) Coaxial cables A range of cryogenic coaxial cables is available, made from stainless steel and/or beryllium copper. If they are sufficiently long, and they are heat sunk effectively, they can be used for systems operating at the lowest temperatures. Some of these cables are magnetic; if this is likely to affect your experiment check them carefully. Lakeshore S1 coaxial cable (a flexible stainless steel coaxial cable) is suitable for signals with frequencies up to a few kilohertz. At higher frequencies the insertion loss of the coaxial cable rises very rapidly. Although this cable has a characteristic impedance of 40 Ω ( rather than the usual 50 Ω) this has little effect on the signal provided that the cable length is much less than one wavelength. At low frequencies this is likely to be true. Semi-rigid cables have much better high-frequency performance. They are suitable for frequencies up to about 20 GHz. They can be heat sunk in several different ways. If the cable is in gas, heat is conducted away from the inner conductor through the dielectric material to the outer, and then from the outer to the gas. It may not be necessary to make any other arrangements for heat sinking. (i) Wiring looms It is worth mentioning that for many years commercial systems have been fitted with complex wiring made up in the form of ribbons of wires, held together by GE varnish. These ‘looms’ can be easily manufactured to suit individual requirements and their flexibility will no doubt continue to be a benefit in the future. However, some manufacturers are now starting to use cables made by weaving the thin wires into a polymer ribbon. The strands of polymer separate the wires to reduce the risk of short circuits between adjacent conductors but they also make the wiring loom very robust. It is possible to encapsulate the connections at the end of the looms with epoxy resin to ensure that they are not accidentally damaged during handling. It is also possible to arrange the weave of the loom so that the cross-talk between adjacent conductors is minimized, approximating to the performance of twisted pairs.

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D8.0.6 Demountable vacuum seals D8.0.6.1 Room-temperature seals Most standard vacuum seals are suitable for cryogenic systems provided that they are not subjected to low temperatures during normal operation. Rubber ‘O’ rings lightly covered with silicone vacuum grease are suitable for most applications. They are usually held captive in one of the flanges so that they can be assembled easily, and they can be re-used. Seals which only have to be opened rarely are usually held together by a ring of bolts. However, a variety of quick-release vacuum flanges is available for joints which are regularly separated. Some of the most common flanges of this type are listed below: (i)

Klein flanges (also known as ISO-KF, DN or NW flanges) up to 50 mm diameter—the flanges are held together by a clamp around their circumference; (ii) ISO-K flanges for flanges between 63 mm and 1 m diameter—the flanges are held together by several claw clamps; (iii) ISO-F flanges for flanges between 63 mm and 1 m diameter—the flanges are similar to the ISO-K flanges but they are held together by bolts; (iv) similar systems are available in the USA, and some of them claim to be compatible with the ISO flanges.

D8.0.6.2 Ultra-high vacuum Systems which are required to have very clean high-vacuum spaces must be fitted with special metal-tometal vacuum seals. They often have to withstand baking to at least 200° during the initial pump-down, to outgas contamination from the surfaces that are exposed to vacuum. The most common standard seal for these applications is the ‘Conflat ’ range (sometimes known as ISO-CF). These flanges are designed to cut into the surface of an annealed copper gasket. They also contain the material of the gasket to prevent it from ‘flowing’ even at high temperatures. The gasket has to be replaced every time the seal is re-made. Conflat seals can be used at low temperatures, even if they are exposed to superfluid liquid helium.

D8.0.6.3 Low-temperature seals Most of the standard vacuum seals used at room temperature and above cannot be used at low temperatures. Rubber ‘ O ’ rings become too hard and brittle at temperatures below approximately 250 K. PTFE is sometimes used to give a rough vacuum seal but it is not easily applicable to high vacuum seals. Conflat seals can be used but they are expensive and bulky, therefore other types of seal are usually used. (a) Indium seals Indium wire is used to make a joint which can be removed and replaced easily. The wire is compressed into intimate contact with the two surfaces that are to be sealed together as shown in figure D8.0.12. Similar joints may be made using lead wire, but for low-temperature operation, it is easier to use indium because it is softer. Indium seals larger than about 50 mm diameter have to be designed carefully. It is important that the flanges are not distorted as the seal is tightened. Steps also have to be taken to relieve thermally induced stresses and to ensure that both flanges in the seal cool down or warm up at similar rates so that the seal is reliable even after repeated thermal cycling.

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Figure D8.0.12. Schematic diagrams of indium seals.

(b) Greased cone seals A reliable low-temperature vacuum seal can be made by making the joint between two components a closely matched taper. A thin layer of vacuum grease between the surfaces makes the seal. These can be used as a quicker alternative to indium seals. Cone seals also occupy less space than indium seals but the vacuum grease spreads to cover everything in the area of the seal and there is some evidence to suggest that they are slightly less reliable than well designed indium seals. (c) Seals made at low temperatures Most vacuum seals can only be made at room temperature, which often means that the whole cryostat has to be warmed to room temperature or all the seals that are regularly changed have to be at room temperature. However, Niinikoski and Rieubland (1982) developed an indium seal at CERN which could be made at low temperatures. A sharp knife edge on one flange is pressed into indium cast into the other flange. The seal can be made several times before the system has to be warmed to room temperature and the indium recast. D8.0.7 Temperature control D8.0.7.1 General The aim of a temperature controller is to maintain the temperature of a system as close as possible to some desired temperature (the ‘set point’) and to minimize the effect of changes in heat load on the system. When a steady state is established, the power supplied by the controller will exactly balance the heat lost by the system to its surroundings. A further function of the controller is to follow any changes in the set point as rapidly as possible. Thus the criteria for good control are: (i)

control accuracy—the mean temperature of the system should be as close as possible to the desired temperature; (ii) control stability—the fluctuations above and below the mean temperature should be small; (iii) control response—the system should follow changes in the set point as rapidly as possible. In the following sections a number of possible control systems of increasing complexity are described, culminating in three-term or PID control. D8.0.7.2 Open-loop operation In an open-loop system, a fixed heater power is applied and the system is allowed to come to equilibrium. There is no control as such, since the heater power can only be changed by the intervention of a human operator. The system takes a long time to reach equilibrium and any changes in the heat loss from the

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system produce corresponding changes in the system temperature. The hot plates on a domestic cooker are an example of this form of ‘control’. D8.0.7.3 On—off control In an on–off (or ‘bang-bang’) control system the heater power is either fully on (if the temperature is below the set point) or off (if it is above the set point). The control accuracy and response can be very good with this form of control and the system can be made largely immune to changes in heat loss. However, the control stability can never be good since the system temperature must always cycle above and below the set point. The magnitude of the temperature fluctuations depends on the thermal properties of the system. For some systems, where temperature fluctuations are not important, this is a perfectly satisfactory and simple system of control (for example the domestic electric oven). D8.0.7.4 Proportional control A proportional control system overcomes the problems of temperature cycling by allowing the heater power to be continuously varied. The heater voltage at any instant is proportional to the error between the measured and desired temperatures. Thus a negative error (temperature below the set point) will produce an increase in heater voltage in order to correct that error. If the output voltage were proportional to the error over the whole range of the instrument, a large error would be required in order to generate the necessary output voltage. For example, an error swing of +50% of span to –50% of span would be required to swing the output from zero to full power. Thus, although the control stability might be good, the accuracy would be very poor. By reducing the proportional band of the controller the output can be made proportional to the error over a small part of the total range of the instrument. Outside this range the output is either fully on or off. The range over which the output is proportional to the input is called the proportional band (typically expressed in degrees or in per cent). The accuracy of the controller may then be improved since a smaller error will then be necessary to produce a given change in output. This would seem to imply that, by sufficiently reducing the proportional band, any required accuracy could be obtained. Unfortunately as the proportional band is progressively reduced, there will come a point at which temperature oscillations appear. (In the limit, a controller with a proportional band of 0 is an on—off controller, as described above.) The reduction in proportional band that can be achieved before the onset of oscillations will depend largely on the design of the system being controlled. In some systems it may be possible to achieve the required control accuracy without oscillations but in most cases this will not be so. D8.0.7.5 Integral action To overcome this problem, integral action is introduced. Consider a system controlled by proportional action as described above, with the proportional band sufficiently large to prevent oscillation. The result will be stable control but with a residual error between the measured and desired temperatures. Suppose this error signal is fed to an integrator, the output of which is added to the existing controller output. The effect of this will be to vary the overall output until control is achieved with no residual error. At this point the input to the integrator will be zero and this will therefore maintain a constant output. Integral action has thus served to remove the residual error. Provided the contribution from the integrator is only allowed to vary slowly, proportional action will prevent the occurrence of oscillations. The response of the integrator is characterized by the ‘integral action time’. This is defined as the time taken for the output to vary from zero to full output, in the presence of a fixed error equal to the proportional band.

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To ensure that the integrator itself does not give rise to oscillations, the integral action time is usually set to at least the response time constant of the system. When a controller is responding to a large change in set point, the integrator will charge whilst the system is approaching the new set point. If left unchecked the integrator is likely to be fully charged by the time the temperature comes within the proportional band. This will inevitably result in a large overshoot. To overcome this an ‘integral desaturator’ is usually used to hold the integrator in its discharged state when the measured temperature is outside the proportional band. This can considerably reduce the amount of overshoot. In some temperature controllers a more sophisticated desaturation algorithm is employed, which deliberately pre-charges the integrator (rather than discharging it) to minimize overshoot further. D8.0.7.6 Derivative action The combination of proportional and integral action is sufficient to ensure that accurate and stable control can be achieved at a fixed temperature. However, when the set point is changed many systems will tend to overshoot the required value. This effect may be reduced or eliminated completely by ‘derivative action’. This monitors the rate at which the measured temperature is changing and modifies the control output such as to reduce this rate of change. Like integral action, derivative action is characterized by an action time. If the measured temperature is changing at a rate of one proportional band per derivative action time, derivative action will contribute a signal sufficient to reduce a full output to zero or vice versa. The use of the integral desaturation algorithm (referred to above) may be sufficient to limit overshoot in many systems. In this case no derivative action will be required. D8.0.7.7 North American terminology In North America, a different terminology exists for three-term control: proportional band is replaced by its reciprocal ‘gain’ ; integral action is replaced by ‘reset’ which may either be specified as a time (as for integral action) or as its reciprocal, ‘repeats per minute’; derivative action is replaced by ‘rate’ which again may be specified as a time or as repeats per minute. D8.0.7.8 Gas-flow control Some temperature controllers can drive a motorized needle valve. The position of the needle valve is controlled automatically to provide variable cooling power in conjunction with the heater for temperature control. The valve position is controlled by an integrator. In other words control operates relative to the current position rather than being an absolute setting. This is necessary as it is not generally practical to calibrate a cryogenic needle valve in absolute terms. The integrator is driven by two error signals, a temperature error (the difference between the set point and the temperature measured by the control sensor) and a heater voltage error (the difference between the expected heater voltage and the actual heater voltage delivered by the heater control algorithm). The error signals are scaled by the error sensitivities and are also compensated for valve nonlinearity (decreasing error sensitivity with increasing temperature and lower flows). The sensitivity to temperature errors is such that the valve will respond to large temperature errors ( for example following a change in set point ) but will be relatively unaffected by small temperature errors when close to the set point, allowing the fine control of temperature to be dominated by the heater. When control is established at the set point, the temperature error is by definition zero, so control of the valve integrator is dominated by the heater voltage error, such that this too tends to zero as the cooling is adjusted

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until the heater voltage reaches its expected target value. Therefore the controller aims to reach a target heater voltage which is a function of the set temperature. At the most simplistic level, a target heater voltage of zero under all conditions would provide minimum cryogen usage and hence the most economic operation. The disadvantage is that whilst heater power could be delivered to correct for rapid falls in temperature it would not be possible to correct for rapid rises in temperature and the result would be poor control stability. D8.0.8 Designing or buying a cryostat Cryostat design is such a large and complex subject that it would be difficult to cover it in one chapter. If you are considering designing and building a cryostat check first that: (i) you have access to the necessary information; (ii) a similar system cannot be bought from a reputable supplier or even secondhand (which would save you much time and effort). Whether you decide to buy the cryostat or build it yourself you may find the following checklists useful. At least consider whether the following points are relevant to you. D8.0.8.1 Where can you buy a cryostat? Throughout the world there are many companies specializing in the manufacture of cryostat systems and it may be difficult to choose the best company for your application. The magazine Superconductor Industry regularly reviews products and suppliers, and in a recent edition the list of suppliers ran to well over 200 companies. All these companies are not the same. The following guidelines may help you to choose between different products. (i) (ii) (iii)

(iv) (v)

Survey the monthly and quarterly magazines on the subject to find out who advertises the products that you are interested in. Contact the companies and ask for quotations, specifying clearly what you want to do with the system and listing any special features that you require. Make sure that you compare like with like when deciding which quotation to accept. The quotations may be worded to mislead you so if you think that a critical specification is not clearly defined contact the company again for clarification. (For example, is the cooling power of the system specified at the sample position where you can use it fully or in the liquid helium where it is inaccessible?) The cheapest system on the quotation is not always the cheapest in the long run. Consider the running costs of the different systems (for example by comparing cryogen consumption or power consumption). Ask your colleagues for their opinions before you place an order.

D8.0.8.2 Defining your cryostat system (a) Basic requirements • • • • • • •

What temperature range is the system required to cover? What cooling power is required? Are liquid cryogens readily available? What hold time is required? Will it be easy (or possible) to cool the system down, and how long will it take? If using a closed-cycle cooler, can it be serviced easily? Is vibration isolation required?

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What pumping systems are available or have to be bought? Have all relevant safety features been incorporated? Will the system be controlled through a computer (to automate its operation)? What are the dimensional limitations for the system? How heavy is the system and what lifting equipment is available (for large systems)? How much can you afford to spend?

(b) Thermometry and temperature control • • •

What types of thermometer are appropriate for the system? (Consider sensitivity, accuracy, repeatability, effect of magnetic fields.) How will the thermometer be fixed to the experiment? What accuracy of temperature control is required?

(c) Sample access • • • •

How will the sample or experimental apparatus be mounted? What will be the sample environment (for example, in liquid, gas, vacuum or UHV)? Do you want to be able to change the sample without warming up the whole system (called top loading)? What services are required in the cryostat (for example, wiring, windows, mechanical drive rods, cryogen level probes, etc)?

(d) Superconducting magnet systems • • • • • •

What field strength, homogeneity and decay rate are required? At what temperature will the magnet run? Does the magnet have to be fitted with cancellation coils, sweep coils, modulation coils or gradient coils? Are there any magnetic materials near the magnet or cryostat? Will mechanical damage caused by eddy currents be induced if the magnet quenches? Is there eddy current heating in conducting components near the magnet?

(e) If you are designing the system yourself Consider the following additional points: • • • • • •

conducted heat loads down neck tubes and through supports and spacers gas cooling of conducted heat loads (where possible) convected heat loads (which are very difficult to calculate) radiated heat loads, the number and type of radiation shields and the amount and type of superinsulation check whether liquid nitrogen can be removed from the system easily before liquid helium is transferred into the system how long will it take you to build the system and how much will it cost?

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D8.0.9 Safety—an overview D8.0.9.1 Common hazards The following list shows the range of hazards that you may encounter when you are using laboratory-scale cryostats containing liquid helium and liquid nitrogen (however, you can protect yourself against all of these hazards by following the correct procedures): • • • • • • • • • •

extreme cold and the risk of cold burns or frostbite asphyxiation (if the atmospheric oxygen is displaced) fire and explosion hazards (through oxygen enrichment) high magnetic fields affecting medical implants high magnetic fields resulting in large attractive forces electrical hazards vacuum hazards high-pressure hazards radioactive sources (sometimes used as thermometers, etc) lifting and moving heavy equipment.

Additional and greater hazards may be present in systems containing other cryogens. Liquid air and liquid oxygen are particularly dangerous. Although they are not flammable many other materials combust spontaneously when they come into contact with these liquids. Other cryogens are flammable, explosive or poisonous, and special training is essential. (a) Signs that a hazard might be developing: • • • •

unusually high (or low) boil-off unusual condensation of atmospheric moisture unexpected patches of frost on the outside of the cryostat faulty valves.

D8.0.9.2 Setting up your laboratory When you are setting up your laboratory you should: • • • • • •

•

• •

design the laboratory with safety in mind consult an expert who has experience of setting up other similar laboratories set up a procedure to be followed by anyone using the equipment make sure that the correct procedures and local regulations are always followed train all personnel and supervise them properly display clear notices to warn people that they are entering a potentially hazardous area—remember that even if the door is locked, some other people have keys; for example, cleaners and security staff are often working when there is no one else around and they are at risk too tell the local safety officers about your system, and ask them to make local emergency services aware of the hazards, as this may affect the procedures they follow when they are dealing with fires or other incidents consider carefully whether the floor in your laboratory is strong enough to take the weight of the system—seek professional advice if necessary if you are using a superconducting magnet system, consider whether there are any large magnetic items close to the system (such as magnetic beams in the floor) and check whether the stray field will affect other equipment in your laboratory or in other rooms nearby (even on other floors)

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consult your local fire authority about the equipment you should install in case of a fire—they may require that portable fire fighting equipment is non-magnetic; ask them to check whether your smoke detectors will be set off by helium gas (as some are) install an overhead crane (or other lifting equipment) capable of lifting heavy equipment safely make sure that the laboratory is sufficiently well ventilated—if there is any doubt, install sensors which will warn you if the oxygen level becomes dangerously low refer to relevant local health and safety publications.

References Hakuraku Y and Ogata H 1983 Thermal design and test of a subcooled superfluid helium refrigerator Cryogenics 23 291 Leupold M J and Iwasa Y 1986 Subcooled superfluid helium cryostat for a hybrid magnet system Cryogenics 26 579 Niinikoski T O 1976 Dilution refrigeration: new concepts Proc. 6th Int. Cryogenic Engineering Conf. ed K Mendelssohn (Guildford: IPC Science and Technology) pp 102–11 (This paper describes the mathematical analysis of the main heat exchanger which allows its design to be optimized and tailored to reach a given cooling power at the desired temperature.) Niinikoski T O 1982 Dilution refrigerator for a two-litre polarized target Nucl. Instrum. Methods 192 151–6 (This paper describes the 2 W dilution refrigerator for the so-called 2 1 polarized target.) Niinikoski T O and Rieubland J-M 1982 Large dilution refrigerators Proc. 9th Int. Cryogenic Engineering Conf. ed K Yasukochi and H Nagano (Guildford: Butterworth) pp 580–5 (This paper describes some further improvements to a refrigerator to reach the goal of 2 W, together with the description of a smaller and simpler machine.) Pobell F 1992 Matter and Methods at Low Temperatures (Berlin: Springer) Pryce M H L 1951 Oriented Nuclei, Proc. Int. Conf. on Low Temperature Physics (Oxford) ed R Bowers (Amsterdam: North-Holland) Richardson R C and Smith E N 1988 Experimental Techniques in Condensed Matter Physics at Low Temperatures (Reading, MA: Addison-Wesley) Street A J, Ross J S H, Harrison S M, Jenkins D M, Mason M F, Riggs R J, Smith K D, Wiatrzyk J M, O’Meara J E, Tuzel W and Tilles D 1996 Final site assembly and testing of the superconducting toroidal magnet for the CEBAF Large Acceptance Spectrometer (CLAS) IEEE Trans. Magn. MAG-22 2074

Further reading The following books contain useful background information about the fundamentals of cryogenic practice and the basics of the design of cryostats. Using these books you would probably be able to design a working cryostat but it is often better to buy a well proven product. In this way you gain the benefit of many years of cryogenic experience, with the guarantee that you will be able to concentrate more on designing your experiment. Balshaw N H 1996 Practical Cryogenics—an Introduction to Laboratory Scale Cryogenics (Oxford: Oxford Instruments) British Cryogenics Council 1982 Cryogenics Safety Manual—a Guide to Good Practice (London: Mechanical Engineering) Lounasmaa O V 1974 Experimental Principles and Methods Below 1 K (New York: Academic) Rose-Innes A C 1973 Low Temperature Laboratory Techniques (English Universities Press) (Probably out of print, but worth looking in the library.)

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D9 Bath cryostats for superfluid helium cooling

Gérard Claudet

After its discovery in 1939, superfluid helium (HeII) was mainly considered as a curiosity by physicists carrying out low-temperature research who were trying to understand its very surprising behaviour with respect to standard fluids. Its ability to climb the walls of a reservoir or to flow apparently without any resistance through very narrow leaks was quite fascinating but on the other hand such properties made people reluctant to use such a fluid for practical application. For a long time, superfluid helium was exclusively used as a means to produce temperatures as low as 1 K by reducing the vapour pressure over a liquid bath and was studied to illustrate the new theory of quantum mechanics. With the availability of industrial superconductors for which the critical temperature is around 10 K, their possible operation in superfluid helium near 2 K was soon considered as it would significantly improve their performance in comparison with what could be obtained in normal helium near 4.5 K. Unfortunately, the first experiments using superconducting coils in a helium bath under reduced pressure were generally very disappointing due to the very poor dielectric properties of the low-pressure gaseous helium, inducing electrical arcing in the case of a quench. A completely new situation was introduced when it became possible to use superfluid helium at atmospheric pressure at the λ transition, opening the way to further development of a pressurized superfluid helium bath at any desired temperature and pressure. This new technique pioneered in France (Roubeau 1971) was tested and confirmed by large-scale projects (Claudet and Aymar 1990). Pressurized superfluid helium can be now considered as a realistic alternative for low-temperature cooling in advanced projects. D9.0.1 Boiling superfluid helium By lowering the pressure of the vapour when pumping over a liquid helium bath, one can reduce the boiling temperature. The liquid—vapour equilibrium data are given in table D9.0.1 (Arp and McCarty 1989). The same table gives latent heat for saturated liquid helium evaporation. From these data two important practical points arise. First, implementation of boiling helium II baths by pumping from HeI at initial conditions of 4.2 K and 0.1 MPa is responsible for a large volume reduction, as given in table D9.0.2. In practice 50% of the initial liquid quantity has to be evaporated to produce superfluid helium. Another important parameter to be underlined is the low-pressure level corresponding to boiling Hell in the range of 10–30 × 102 Pa. Boiling at such a low pressure will produce a large volume of gas which can completely cover the heated surface of the experiment to be cooled and suddenly cut off the heat

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Bath cryostats for superfluid helium cooling Table D9.0.1. Liquid helium vapour pressure equilibrium and latent heat versus temperature.

Table D9.0.2. Remaining Liquid volume after pumping from atmosphere pressure.

transfer. However, as the dielectric behaviour of gas is pressure dependent (Hara et al 1990), it would be very dangerous to use a saturated Hell bath when a high electrical voltage could appear. Finally, the implementation of a low-pressure boiling bath does require very high standards of vacuum integrity in order to avoid the risk of air leakage into the experiment and the vacuum circuit.

D9.0.2 Tλ bath (Roubeau’s type) The use of superfluid helium outside liquid-vapour equilibrium was first carried out by Roubeau using the solution represented in figure D9.0.1 (Roubeau 1971).

Figure D9.01. Tλ bath parameters.

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A liquid helium bath maintained at the required pressure (e.g. atmospheric) is indirectly cooled by an immersed cold source which can itself be produced by low-pressure boiling of helium. Taking into account the increasing liquid density when decreasing the temperature, a vertical stratification appears in the Dewar. The free liquid—vapour interface is kept at the equilibrium temperature while the bulk liquid in the Dewar progressively cools. From the example illustrated in figure D9.0.1, equilibrium temperatures and thermal characteristics can be studied by successively considering three different zones. (i)

The upper part is fed with liquid helium to maintain a constant level. Various heat loads can be identified such as those resulting from current leads or thermal shielding. A permament liquid now can be maintained to feed the auxiliary refrigeration loop. As a consequence, this upper part can be considered as isothermal at the temperature Ts a t . (ii) The middle part between Ts a t and Tλ transmits the flux Wc o n d due to the liquid thermal conductivity. The thermal radiation WR goes through the liquid with no heat dissipation. The heat load from the wall can generally be neglected. In steady-state conditions the thickness of this intermediate zone is mainly determined by the flux Wc o n d . The thermal conductivity of liquid helium is so low that the thermal gradient is generally concentrated over a thin layer typically of the order of 1 cm. (iii) A lower part in which heat loads appear from thermal conductivity or radiation through the liquid or from the wall of the cryostat. The main load is Wu coming from the considered experiment and the cooling power Wr e f is removed by the auxiliary refrigerator. In a steady-state condition, the balance can be written

As a result, the value Wc o n d can be deduced if all the other terms are known, and then the thickness of the intermediate zone in which the thermal gradient is concentrated can be estimated. In the lower part, the temperature denoted Tλ– ε in figure D9.0.1 results in a heat flux through the liquid from Tλ as a boundary condition. Hell heat transport properties are well defined and reported by Bon Mardion et al (1979). In the more common case, for laboratory cryostats, heat flux in the range of a few hundred milliwatts through surfaces of several hundred square centimetres are responsible in the superfluid zone for the temperature difference of 10−4–10−3 K at the Tλ interface. As Tλ is itself pressure dependent, the Roubeau stratified bath can be used as a practical temperature reference easily adjustable in the 2.16 to 2.176 K range by pressure adjustment. On the other hand, such a bath provides a useful means to improve the available maximum field by about 25% with impregnated NbTi superconducting coils. For such coils a working temperature 2 K lower than with normal boiling helium is very helpful even when resin impregnation impedes heat transfer between coil and liquid helium. D9.0.3 Baths for any needed temperature (Claudet type baths) As illustrated in figure D9.0.2, if the lower part of the previously considered cryostat is filled with any insulating material devoted to reducing the liquid cross-section: the same heat flux Wc o n d will be concentrated in a smaller cross-section. Increasing the heat flux density will result in a higher temperature difference Tλ — TH e I I . The HeII heat transport properties (Bon Mardion et al 1979) govern the relation between the heat flux density Wc o n d /S and the temperature difference Tλ — TH e I I for a given channel length. By this method (Claudet et al 1974), it is easy to get a HeII bath at any desired temperature at the cost of an admissible parasitic heat load Wc o n d if one can reduce the liquid cross-section to a few square millimetres. This can be done by connecting two different vessels through a small-diameter tube. Another

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Figure D9.0.2. Double-bath parameters.

way, illustrated in figure D9.0.2, is to introduce an insulating plate providing a small clearance with respect to the wall of the cryostat. A few square millimetres as a residual cross-section gives a lot of freedom for the design and implementation of connecting elements such as electrical feedthrough or emergency valves for the quick release of the stored liquid. The double-bath cryostat fulfils all the conditions associated with the practical use of superfluid helium: (i)

it can be operated at the required pressure, e.g. above atmospheric, to avoid air ingress and to ensure a good dielectric behaviour (ii) liquid helium replenishment can be performed in the same way as for a standard helium cryostat (iii) current leads and all other thermal loads coming from outside can be intercepted at the HeI level.

For the user, the upper part of the cryostat can be considered and operated as a standard helium cryostat. The lower part, a pressurized superfluid bath, easy to access, can be adjusted to the required temperature, thus providing a pure liquid medium with very high thermal capacity and very high heat diffusivity. Such a bath would be very attractive for applications in which heat transfer is the main design parameter, as is the case when superconductors are used, to remove dissipated heat in small temperature increments (Van Sciver 1986, Wilson 1983). D9.0.4 Control and operation of stratified HeII baths To obtain the previously discussed thermal load characteristics, needed for an auxiliary refrigerator, entails removing a cooling power Wr e f at a temperature Tr e f lower than the required value for TH e I I . The difference TH e I I –Tr e f resulting from the liquid—liquid heat exchanger design has to be minimized to save refrigeration costs. In the more common case, the cold source is produced by a helium Joule—Thomson loop as illustrated in figure D9.0.3.

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Figure D9.0.3. The cooling circuit for a pressurized HeII bath.

A counterflow heat exchanger fed by liquid helium is cooled by low-pressure gaseous helium. As a result of the very large density difference between these two fluids, the design of such a heat exchanger is not trivial. A useful solution is described elsewhere (Bon Mardion and Claudet 1979). The precooling heat exchanger can be avoided at the cost of a higher liquid helium consumption, resulting in an over-designed pumping system to obtain the required pressure of around 1 kPa. The following orders of magnitude should be kept in mind: • •

with a heat exchanger–0.9 mW for a gas flow rate of 11 h-1 at normal temperature and pressure and a pumping unit of 100 1 h-1 without a heat exchanger and for the same conditions, only 0.6 mW is produced.

The fluid expansion is performed by an isenthalpic process in a throttle valve to feed the cold source with a liquid—gas mixture. Only the liquid phase is helpful when trying to remove heat by vaporization. In the low-pressure vessel, the wetted surface in contact with pressurized liquid on one face and with saturated liquid on the other acts as a coupling heat exchanger with a useful area of S cm2. The design of the cold box and the liquid level control should be made to increase the heat exchange area as much as possible in order to decrease the resulting temperature difference caused mainly by the Kapitza resistance at the solid—liquid boundaries (Seyfert 1982). At this level, an order of magnitude can be given. Using a 1 mm thick tube made of the more common phosphorus copper yields the following results. Near 1.8 K a heat flux of 27 mW cm−2 induces a 0.05 K temperature difference TH e I I - Tr e f (0.02 K at each boundary and 0.01 K through the copper wall). To take into account the thermal load evolution due to the cooling down period or related to the performed experiments it is recommended that the throttle valve aperture is controlled in order to get a constant liquid level in the low-pressure box. In such a way, the heat exchange area will be kept constant and the flow rate in the Joule-Thomson loop will be adapted to the dissipated power with the relationship W = m L where m is the mass flow rate and L the latent

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heat of the liquid helium at the considered pressure level. Taking into account the working characteristic of the pumping system the result will be a varying temperature versus flow or dissipated power quantity. Another simple solution, to avoid level control, is to keep a constant flow rate by using a calibrated orifice to perform the Joule—Thomson expansion. Such a design is very sensitive to circuit pollution by solid impurities and is not relevant for applications in which the dissipated power is not well known. if the imposed flow rate is not well adapted to the dissipated power, the risk is that the cold box will be submersed with a subsequent liquid propagation to the pumping line, inducing flow, pressure and temperature instabilities. In fact, operation at constant flow with a calibrated orifice has to be restricted to use in a Roubeau bath near Tλ. In this particular case, the range of stability is enlarged by the high sensitivity of the parasitic heat flux Wc o n d versus the very small temperature difference TH e I I –Tλ = ε as given by Bon Mardion et al (1979). A large heat load variation will induce a small variation in the ε value in such a way that it automatically adjusts the heat flux coming from HeI through the Tλ interface. References Arp V D and McCarty R D 1989 Thermophysical properties of helium 4 from 0.8 to 1500 K with pressures to 2000 MPa NIST Technical Note 1334 US Department of Commerce, NIST, Boulder, 80303-3328, CO Bon Mardion G and Claudet G 1979 A counterflow gas—liquid heat exchanger with copper grid Cryogenics 29 552 Bon Mardion G, Claudet G and Seyfert P 1979 Practical data on steady state heat transport in superfluid helium at atmospheric pressure Cryogenics 29 45–7 Claudet G and Aymar R 1990 Tore Supra and HeII cooling of large high field magnets Advances in Cryogenic Engineering vol 35 (New York: Plenum) pp 55–67 Claudet G, Lacaze A, Roubeau P and Verdier J 1974 The design and operation of a refrigerator system using superfluid helium Proc. 5th Int. Cryogenic Engineering Conf (Kyoto 1974) (London: IPC Science and Technology) pp 265–7 Hara M, Suehiro J and Matsumoto H 1990 Breakdown characteristics of cryogenic gaseous helium in uniform electric field and space charge modified non uniform field Cryogenics 30 787–94 Roubeau P 1971 Bain cryogénique d’hélium liquide à gradient de température avec coexistence des phases superfluide et normale C. R. Acad. Sci., Paris B 273-14B 581–3 Seyfert P 1982 Results on heat transfer to Hell for use in superconducting magnet technology Proc. 9th Int. Cryogenic Engineering Conf. (London: Butterworth) pp 263–8 Van Sciver S W 1986 Helium Cryogenics (New York: Plenum) Wilson M N 1983 Superconducting Magnets (Oxford: Clarendon)

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D10 Current leads

P F Herrmann

D10.0.1 Introduction Electrical machines for power generation, as used today by power utilities all over the world, operate at room temperature or above. Likewise, power electronic systems which are used in common power supplies for magnets also operate at ambient temperatures. However, most of today’s superconducting magnets or superconducting a.c. power devices must be operated at liquid-helium temperatures, making current links between low and ambient temperature necessary. Even future applications of high-temperature superconductors will not change this situation completely, as they will still require current links to the liquid-nitrogen temperature range. Unless room-temperature superconductors are found, the use of superconducting machinery will always require such current leads. For many years, current leads were almost exclusively d.c. vapour-cooled (VC) all-metal current leads for low-Tc , applications at liquid-helium temperatures. The operating conditions of all-metalcurrent leads are well understood and the heat dissipation to the low-temperature level (low- temperature heat load) is approaching the theoretical limit of performance. Despite this, current leads are often the main heat link to the low-temperature level of cryogenic systems and cause the main part of the boil-off of the cryogenic liquids. For the condensation of these vapours back to the liquid phase, an appreciable power consumption is needed. For most applications, these cooling costs determine the major part of the running cost. Apart from applications in the field of electronics, today’s most common use of superconductivity is limited to superconducting d.c. magnets. The current requirements for these devices depend on the size of the magnet and vary from 30 A for a small research magnet up to currents in the 20 kA range for magnets such as those projected for the Large H adron Collider (LHC) accelerator in Geneva. Large magnets for fusion (for example the International Tokamak Experimental Reactor (ITER)) are projected with operating currents which will reach 50 kA. Charging voltages of 10 V or less are usually sufficient to energize d.c. magnets. Future a.c. applications will of course require higher voltages (some 100 V up to several tens of kilovolts), and currents in the kiloampere range. Indeed, research on devices for 50/60 Hz applications such as superconducting fault current limiters, transformers, motors and generators, is progressing worldwide. This will generate a demand for a new class of current leads adapted for operation under a.c. conditions. Since the discovery of the high-temperature superconductors, the current-carrying capacity of these conductors has been continuously increasing. Several demonstration or prototype current leads with a very low level of low-temperature heat loads have been realized (Grivon et al 1991, Herrmann et al 1994, 1996, Hull 1992, Watanabe et al 1993, Wu et al 1991, Yamada et al (1992) clearly demonstrating the

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interest in this new concept. It is probable that during the coming years the lead technology (at least for large currents) will be dominated by the new high-temperature superconductor design. This contribution is divided into six sections and is structured as follows: the cooling of current leads and some important materials parameters are discussed in section D10.0.2. In section D10.0.3 the design concepts for all-metal current leads, their cryogenic heat load and stability aspects are evaluated. These calculations are compared with the experimental results of metallic current leads. A brief discussion of the design criteria for metallic a.c. current leads is also given. Section D10.0.4 is devoted to current leads using high-temperature superconductors and the design concepts of such hybrid metallic—high-temperature superconductor current leads for d.c. and for 50/60 Hz a.c. applications will be addressed. Their cryogenic heat load and stability aspects are discussed. In section D10.0.5 the test results of hybrid metal—hightemperature superconductor leads for d.c. and a.c. applications are compared with the performance of classical all-metal current leads. In section D10.0.6 a short outlook on the perspectives of high-temperature superconductor current leads is given. Finally the reader is referred to a list of the publications which have been referred to in the preparation of this text. D10.0.2 Refrigeration and materials properties D10.0.2.1 Cooling modes The electric power consumption of a cryogenic refrigerator is a combination of the efficiency of the machine and the Carnot ratio (TR — TL)/TL between room temperature TR and the cryogenic temperature TL . Although the Carnot ratio depends only on the temperature, the efficiency (lying in the range e = 0.05–0.3) also depends on the size of the cryogenic machine (Gistau 1990). The conversion factors of cryocoolers working at 4 K or 77 K, which are the ratios of the cooling power at low temperature, Pc , to the electric power consumption, Pe , are given in table D10.0.1. It shows the enormous benefit that can be achieved by the use of high-Tc superconductors operating at liquid-nitrogen temperatures. For the current lead application it shows why it would be interesting to remove incoming heat in the 77 K range. The highest refrigeration efficiency e = 0.3 is achieved in very large cryorefrigerators (LHC type) where the following conversion factors are achieved (Gistau 1994): for liquefaction of warm He vapour Pe /Pc = 1300; for liquefaction of cold He vapour Pe /Pc = 250. Throughout this work, except for large-scale applications, slightly higher values will be used: for liquefaction of cold He vapour Pe /Pc = 350; for liquefaction of warm He vapour Pe /Pc = 2000; for liquefaction of cold N2 vapour Pe /Pc = 10 and for liquefaction of warm N2 vapour Pe /Pc = 15.

Table D10.0.1. Power conversion factors for different cryogenic loads and for different liquiefaction modes.

D10.0.2.2 Cooling of current leads The heat of evaporation Qe υ and the heat capacity from the boiling point to room temperature QB −R = Cp(T) dT for helium and nitrogen are indicated in table D10.0.2. The available cooling power at liquid-

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Table D10.0.2. For helium and nitrogen: the specific heat (Cp), the evaporation heat (Qev) and the heat which is necessary to warm up to the vapours from the boiling point at atomosphere pressure to room temperature (QB-R).

helium and liquid-nitrogen temperatures and the heat capacities of the gases have an important influence on the optimum design of different current lead types. (a) All-m m etal current leads for 4.2 K operation The most striking aspect about the values in table D10.0.2 is the small evaporation heat of He and its huge heat capacity between liquid-helium temperature and room temperature. In fact, the gas evaporated by one watt at 4.2 K makes available another 70 W of cooling power when it is heated to room temperature. Classical all-metal concepts make use of this important heat capacity of helium vapour to cool the current lead and reduce the heat flow to 4.2 K. (b) All-m m etal current leads for 77 K operation The current lead operating at liquid-nitrogen temperatures can also make use of the heat capacity of cold nitrogen vapour but the available heat capacity of these vapours is only of the same order as the heat of evaporation of liquid nitrogen. Due to this comparatively high evaporation heat, the conduction-cooled (CC) current lead in combination with closed-cycle refrigeration appears as an interesting alternative to the VC case. (c) Hybrid metal—high-tt emperature superconductor current leads for 4 K operation These are built from a normal metal current lead for operation between room temperature and an intermediate temperature ( in the range 60–80 K ) and from a high-temperature superconductor for the electrical connection down to liquid—helium temperature. They can operate with one or two stable temperatures, one at liquid-helium temperature and the other at an intermediate temperature which is often provided by liquid nitrogen. The electrical connection from the intermediate temperature to the low temperature is realized with the high-temperature superconductor and operates therefore almost without dissipation of heat. At the same time the low thermal conductivity of the high-temperature superconductor reduces substantially the heat conduction to the low-temperature level. D10.0.2.3 Conductor materials Conductor materials that are used for current leads are mainly metals, but since the discovery of the high-temperature superconductors some hybrid metal—ceramic superconductor prototype current leads have been realized (Herrmann et al 1994, 1996, Watanabe et al 1995, Wu et al 1991, Yamada et al 1992). The material properties which determine the performances of current leads are thermal conductivity and electrical resistivity. Metals, for not too low temperatures, follow the WiedemannFranz law (WFL) which relates the thermal conductivity k(T) and the electrical resistivity ρ (T) by the expression k(T) ρ (T) = LLT, where LL = 2.45 × 10−8 W Ω K−2 is the Lorentz number. The WFL

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Figure D10.0.1. Temperature-dependent thermal conductivity and electrical resistivity for different conductor materials. It can be seen that the WFL (see text) is valid for the metals and for the ceramic superconductors above the superconducting transition. Only below the superconducting transition do the high-temperature superconductors escape from the WFL showing simultaneously zero resistivity and a very low thermal conductivity.

indicates that a good electrical conductor is also a good thermal conductor. This has the very important implication that the performance of optimized metallic current leads is quite independent of the metal which is used. The temperature-dependent thermal conductivity of some relevant materials is given in figure D10.0.1(a). The difference in the room-temperature value between copper and a ceramic conductor is about two orders of magnitude. This difference is much more pronounced at low temperatures where this ratio reaches 105 for high-purity copper (not shown in figure D10.0.1(a)). A very complete collection of k(T) data for pure metals and alloys is given by Touloukian et al (1970) and a choice for some materials of relevance for cryogenic engineering is given by Reed and Clark (1983), Conte (1970) and in chapter F4. Figure D10.0.1(b) shows the temperature-dependent resistivity for some relevant materials. Only the ceramic superconductors differ significantly from the WFL combining the low thermal conductivity with a vanishing resistivity below the critical temperature Tc . This clearly shows the reason for the interest in the use of these materials for current leads. The ρ (T) curves from other metals and alloys are given by Fickett (1982) and Reed and Clark (1983) and chapter F3. (a) Critical current in some superconducting materials The development of different bulk conductors and superconducting tapes has progressed significantly during recent years. Today large tubes and rods of melt-cast-processed ( MCP ) Bi2212 tubes are available with transport currents reaching 8 kA at 77 K and even 14 kA in cooled nitrogen at 66 K. This and other bulk conductors (sintered Bi2223, sintered and melt-textured Y123) have been developed in two European projects on d.c. (Albrecht et al 1994) and a.c. (Albrecht et al 1996) current leads for magnet and power applications at liquid-helium temperatures. A comparison of the performance of these materials for current leads is given in section D10.0.4.2. Other conductor options like Bi2223 tape conductors with a silver—gold matrix material are considered for current leads. In the opinion of the author, this very expensive material

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has the disadvantage that very long leads are needed to reach cryogenic performance comparable to that of a short MCP conductor of about 100 mm long. For this reason, this work is limited to different types of bulk conductor material.

D10.0.3 Metallic current leads D10.0.3.1 Computer calculations Current leads operate as the link for the operating current from the power supply system, which is working at room temperature, to the low-temperature device. Today’s current leads operating from room temperature TR to a low-temperature level TL which is usually liquid-helium temperature are mostly made of copper or brass. Such all-metal current leads for d.c. applications are very reliable subsystems and their performance is well understood. The low-temperature heat load is mainly determined by the solid-state heat conduction of the conductor, by the Joule heating due to its electrical resistance and by its cooling mode. Analytical calculations of the low-temperature head load Q• L based on the WFL show that the result for optimized leads is independent of the metal that is used. The Q• L value for an optimized metallic 1 kA current lead is determined by Wilson’s value of 1.04 W kA−1 (Wilson 1983). From an engineer’s viewpoint, analytical calculations do not appear to be very practical. Numerical calculations provide the possibility to modify independently the functions k (T), ρ (T). They further allow simple adaptation for different cooling conditions, to include dissipation in contacts, and they can equally be used for hybrid high-Tc superconductor—metal current leads. Low-temperature heat loads and steady-state temperature profiles are calculated by the following computer calculation. As may be seen in figure D10.0.2 the current lead is divided into segments. For steady-state operation, the heat conduction, the dissipative heat generation and the heat exchange with the cooling medium must be in equilibrium for each segment. This condition allows the calculation of the equilibrium temperature profile in the current lead by an iterative calculation.

Figure D10.0.2. The sum of the heat generation and heat exchanges must vanish for each segment of the current lead. The low-temperature heat load Q• L determines the cooling power which is necessary for steady-state operation.

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The conduction heat flow Q•c o n d (in a conductor of cross-section A) which is flowing in the direction opposite to the thermal gradient is described by the Fourier law

In a metallic current lead the dissipative heat generated by the transport current I flowing through the lead is of course given by the Joule heating. Its value is

where J represents the local current density which is given for low frequencies in normal metals by J = I/A. The cold vapours resulting from the boil-off from the refrigerating liquid can be used to remove incoming heat from the current lead. The evaporation of a cryogenic liquid by the low-temperature heat load of the current lead Q•L is given by the gas flow m• = Q• L /Qe υ . The heat which can be removed by the vapours for each segment is then given by

where Cp is the specific heat of the gas and Tυ a p the local temperature of the cryogenic vapours which is usually different from the current-lead temperature. The heat transfer from the current lead to the gas is proportional to the temperature difference between the current lead and the vapours. It further depends on the heat exchange coefficient u which is related to the geometry of the channels for the cryogenic vapours in the current lead

Optimized all-metal current leads are made like heat exchangers where the cryogenic vapours are constrained to flow past the conductors. For conductors with a high aspect ratio (exchange surface/conductor volume) high u values can be realized. For most of the following considerations we shall therefore assume a good heat transfer making the temperature difference (T — Tυ a p ) very small. The main steps in the calculation of the low-temperature heat load of the current lead and its temperature distribution are indicated in figure D10.0.3. The first step introduces material properties k (T) and ρ (T), the cooling mode and the properties of the cryogenic refrigerant, the geometry and the u value of the current lead. Furthermore, operating conditions such as the transport current and the temperature of the lower end of the lead TL are fixed. The temperature of the warm end of the lead is a result of the heat exchange with the surrounding room-temperature environment which can be taken into account through a heat exchange coefficient (different from u). However, for theoretical consideration, when the geometry of the room-temperature end of the lead is not specified, TR = 300 K will simply be assumed. The temperature on the top side of each segment is calculated using equation ( D10.0.1)

and the heat flow entering the segment is calculated according to the condition

Tk and Q•c o n d are calculated for all segments until the upper end of the current lead is reached. Generally the calculated temperature there (Tk) differs from the actual temperature TR suggesting a modification

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Figure D10.0.3. The main steps for the iterative calculations of the low-temperature heat load of the current lead and its temperature distribution.

of the initial Q• L value. The whole calculation is then repeated until TR is obtained within the required accuracy. The last Q• L value represents the low-temperature heat load of the current lead operating at the current I, which is being looked for. The plot of the Tk values as a function of the coordinate z illustrates the temperature profile in the current lead. Such calculations can easily be generalized for current leads (including contacts) made up from different conductor materials of variable cross-section. We will now illustrate how the computer calculation is exploited to approach the optimum current lead design for the following options of practical interest. D10.0.3.2 A vapour-cooled current lead for liquid-helium applications For this calculation we assume a good heat exchange between the lead and the vapours, which is in practice often realized by current leads of at least 1 m in length. There exist of course designs where this heat exchange is maximized, forcing the vapours to flow near the conductor and to cover a large surface thus realizing a good heat exchange even for shorter leads. The simplest case is a current lead with a constant cross-section A. When the nominal current I and the conductor length L are determined, the A value is the only free parameter for optimization. The first step of the calculation is to make a guess of the values of I, L and A. In practice the parameter IL/A turns out to be a good variable for the problem if the restrictions on L for a good heat exchange are respected. The computer calculation determines for a given IL/A value the low-temperature heat load Q• L . As is shown in figure D10.0.4, the optimum IL/A value and thus the optimum cross-section Ao is found by searching for the minimum of the function Q• L(IL/A). For this current lead dimension, the pt current I is at the same time the optimum current Iopt. For the optimum current value Io p t the Q• L values of the copper lead and the brass lead are determined as 1.07 W kA−1 and 1.08 W kA−1 respectively. It is interesting to note that the minimum of the lowtemperature heat load is almost independent of the conductor material as would be expected from the

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Figure D10.04. The heliurm-temperature heat load VC all-metal current leads of copper or brass.

Figure D10.0.5. Temperature profiles in current leads for 4 K devices made from copper.

Figure D10.0.6. Temperature profiles in current leads for 4 K devices made from brass.

WFL. The slight difference from Wilson’s standard value of 1.04 W kA−1 is due to small deviations of the material properties from the WFL. For currents exceeding the Io p t value, a hot domain in the current lead is generated and the current lead is destroyed if the maximum temperature reaches the melting point of the conductor material. The destruction value of IL/A is also indicated in figure D10.0.4. The results of the temperature profile calculation for a residual resistivity ratio (RRR) 100 copper and a brass current lead for 4 K applications are shown in figures D10.0.5 and D10.0.6. Note that for the optimum curve the thermal gradient on the room-temperature side is zero (horizontal slope). Although the low-temperature heat loads of the copper and the brass current leads are almost the same, the stability to over-currents (I > Io p t) is significantly different. For the copper lead, an over-current of a few per cent is sufficient to create a hot spot of 600 K. In the brass lead the same peak temperature is only reached when the optimum current is exceeded by a factor of two. This shows that high-purity low-resistivity materials are not suitable for the design of high-stability current leads. Figure D10.0.7 shows the current dependence of the maximum temperature of the lead. For an all-metal current lead operating in the conduction-cooled mode (no exchange with the helium vapours) the heat load has been evaluated at 42 W kA–1. This is slightly lower than the corresponding

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Figure D10.0.7. Sensitivity to over-currents of current leads made of copper or brass for 4 K applications.

value 47 W kA−1 from Wilson (1983). These results show that for operation at liquid-helium temperatures the cooling by the helium vapours must be considered if economical operation is required. (a) Low-tt emperature superconductors for the low-tt emperature end of a 4 K lead It is seen from the temperature profiles (figures D10.0.5 and D10.0.6) that a part of the lead is colder than the critical temperature of common low-temperature superconductors such as Nb—Ti (Tc = 9.7 K) or even better Nb3Sn (Tc = 18.3 K). It is further seen, from the same figures, that these curves are very flat on the cold end for a copper lead with an RRR of 100 and approximately linear for I = Io p t for a brass lead. In the part of the lead where the temperature remains lower than the critical temperature of the superconductor, the Joule heating can be suppressed by adding a low-temperature superconductor in parallel to the metal of this part of the lead. This allows an escape from the WFL in this part of the lead, and a slight improvement of performance of an all-metal lead can be achieved. For the example of the copper lead, the long length for which T < Tc is verified is at the same time also the domain where the resistance of the copper is very low. Therefore, the achieved gain is comparable to that for the brass lead, where the length of T < Tc is shorter and the change in resistivity between 300 K and 4.2 K is lower. An estimate of the gain in performance is found if a constant resistivity is assumed: the Joule dissipation is then reduced by the proportion of the lead where T < Tc which is about 5% for the Nb3Sn case. For a more precise estimation, the modification of the heat exchange with the helium vapours must be taken into account which will slightly improve this gain in performance. The concept is interesting for applications where the length of the lead changes due to the variation of the liquid-helium level. For stability reasons, the optimum length according to figure D10.0.4 must then be verified for the lowest helium level for which the lead is operated. This causes higher boil-off rates when the helium level is high and the IL/A value is smaller than optimum. Optimum conditions can easily be maintained for varying helium level by dissociating the current conduction in the thermal gradient part and the isothermal conduction of current: the lead is then optimized for the upper helium level and is connected to the low-temperature application through a bus bar of large cross-section, doubled by a low-temperature superconductor to avoid Joule heating. A significant improvement of the performances of current leads for 4 K applications can only be obtained using high-temperature superconducting current leads where a tenfold reduction of the 4 K heat load can be achieved as will be discussed in section D10.0.4.

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(b) Ice formation at the room-tt emperature end Ice formation at the room-temperature end is a problem which is frequent in current leads when the lead is rated below the optimum current. For operation at the optimum current this problem is avoided because the temperature gradient at the upper end is vanishing. Current leads should therefore be optimized as close as possible to this current and should not be over-dimensioned. Even for perfectly optimized leads, the upper end is cold when the low-temperature device is cold but not energized (no current is supplied by the lead). The heat exchanger at the room-temperature end must therefore be sufficiently large so that ice formation can be avoided. In large current leads a valve regulates the vapour coolant flow through the lead avoiding excessive cooling and ice formation at the room-temperature end of the lead (Blessing and Lebrun 1983). D10.0.3.3 Current leads for liquid-nitrogen applications As stated above, the advantages of the use of cryogenic vapours for the 77 K application are not as evident as for the 4 K application. A CC current lead in combination with a closed-cycle refrigerator represents in fact an attractive alternative. In the following we shall therefore compare both cases. Figure D10.0.8 shows the 77 K cryogenic loads Q•L(IL/A) for nitrogen VC and also for CC current leads. The calculations are realized for copper, brass and for stainless steel exactly in the same way as the calculation leading to the results in figure D10.0.4. Steel is actually not used as a conductor for current leads; the calculation indicates that for the CC case very short leads of a resistive material can be considered. Although the difference of the resistivity (and the thermal conductivity) between copper and steel changes by about two orders of magnitude, the minimum Q• L values of the optimized current lead vary only by a few percentage points (for both cooling modes). This demonstrates nicely that the performance of metallic current leads is almost independent of the conductor material, a result which is expected from the WFL.

Figure D10.0.8. Liquid-nitrogen temperature heat load for nitrogen VC and CC all-metal current leads of copper, brass or stainless steel.

The last point (maximum IL/A value) of each curve in figure D10.0.8 represents the destruction limit where the melting temperature of the conductor material is reached. At this limit there exists only a small difference in the IL/A value between the CC and the VC case. In contrast to this, there is an important difference between the 77 K heat load for the two refrigeration modes reaching almost one order of magnitude for the stainless steel case.

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Table D10.0.3. Optimum design for nitrogens VC current leads using copper, brass or stainless steel (SS) conductors. The low-temperature heat load Q• L from the optimum I L / A value and the corresponding value of the 300 K power consumption Pe of the refrigerator is given. According to table D10.0.2, for liquefaction of warm 300 K vapours a conversion factor of 15 is used for liquefaction of warm N2 vapours and a factor of ten is used for liquefaction of cold N2 vapours.

As for the 4 K case, the low-temperature heat load of an optimized VC current lead Q• L ≈ 25 W kA−1 (see table D10.0.3) depends only very slightly on the conductor material, but these values are more than 20 times higher than the value of an equivalent VC current lead for a 4 K operation, demonstrating the importance of the enthalpy of the vapours for the 4 K case. In contrast to this, the electric power consumption for the liquefaction of the nitrogen vapours requires about Pe = 350 W kA−1 (see table D10.0.3) which is small compared to the 2 kW kA−1 for the 4 K current lead. For the CC current leads, heat load values of about 43 W kA−1 are found. These values are of course significantly higher than those of the VC case, but this disadvantage is partially compensated for by the lower liquefaction costs of the cold nitrogen vapours. The electric power consumption for this case is evaluated at Pe ≈ 450 W kA−1. This value is still slightly higher than for the VC current lead. However, the simplicity of the concept of a CC current lead and the possibility of realizing very short leads for 77 K applications justifies for some applications a slightly higher boil-off rate of the nitrogen. The temperature profiles for a brass, VC current lead for operation at liquid-nitrogen temperatures are shown in figure D10.0.9.

Figure D10.0.9. Temperature profiles in a VC current lead for 77 K devices made of brass.

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Figure D10.0.10. The sensitivity to over-currents of current leads for 77 K applications made of copper, brass or stainless steel.

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For all conductor materials and cooling modes, the current leads for 77 K applications are much less sensitive to over-currents. The Tm a x value as a function of the over-current is shown in figure D10.0.10. A copper current lead for 77 K applications can sustain over-currents of twice the optimum current which is an important difference with the 4 K case where over-currents of 4% lead to a comparable Tm a x value. Also for a 77 K application the sensitivity to over-currents is a decreasing function with increasing resistivity of the conductor material. D10.0.3.4 Examples of all-metal d.c. leads The optimum dimensions for current leads can be directly determined from the results shown in figures D10.0.4 and D10.0.8 and from table D10.0.3. Some examples are shown in table D10.0.4. For example, a 1000 A VC copper lead for a 4 K application of length 1 m would have an optimum cross-section of Ao p t = 27 mm2. In such a thin current lead the current densities and the heat generation per unit volume are high. This heat is of course balanced by the high thermal conductivity and by the cold helium vapours, but small fluctuations of the cooling, over-currents, or other perturbations would lead rapidly to a burn-out of the lead. For cases where a small cross-section is required and high-purity copper is used, special attention must be devoted to safety aspects to avoid the bum-out of the lead. An equivalent brass current lead requires a cross-section of 670 mm2 leading to a very stable but also rather heavy solution. As high-purity copper leads are unstable and brass leads are quite heavy, low-purity copper with an intermediate RRR value is often used for 4 K current leads. Table D10.0.4. Optimum conductor section for 100 A current lead for different conductor materials. (1) VC for a 4 K application (stainless stell (SS) is not applicable (NA) in this case), (2) VC for 77 K applications and (3) CC for 77 K applications.

A 77 K VC 1000 A copper lead of length 1 m would have an optimum cross-section of Ao p t = 200 mm2. The same current lead for the CC case requires a cross-section of 290 mm2. The cross- section of an equivalent brass current lead is found at 1540 mm2 requiring 13 kg of conductor material. For short leads, for example a CC lead of length 0.1 m, the steel lead would require a cross-section of 1000 mm2, the brass lead requires 150 mm2 and the copper lead requires a cross-section of only 29 mm2. According to figure D10.0.10, the current lead for 77 K applications is less sensitive to over-currents so that such short and compact current leads are feasible. Figure D10.0.11 shows a photograph of a 19 000 A current lead for the magnets of the LHC which is a particle accelerator projected for completion by the beginning of the next century at CERN in Geneva. This lead has an overall length of 1.71 m. The oxygen-free high-conductivity copper (OFHC) conductor length is 1.04 m and its cross-section is 475 mm2. The cross-section for 1000 A is therefore 25 mm2 confirming the theoretical considerations. Small deviations are due to variations of the material properties. The heat load on the liquid helium at its nominal current for this VC current lead was measured at

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Figure D10.0.11. Photograph of the CERN 19 kA current lead for the LHC accelerator. This VC current lead was built by Oxford Instruments using CERN specifications. Courtesy of Mr H Blessing from CERN.

• QL

= 23 W (Blessing and Lebrun 1983). This value corresponds to 1.2 W kA−1, a value which is close to the theoretical optimum values. The current lead is equipped with a coolant flow regulation valve which avoids the formation of ice on the room-temperature side when the current in the lead is operated below the nominal current in the lead. At zero current and at a coolant flow which is reduced to 30% of its 19 kA value (less than the self-evaporation rate), the low-temperature heat load value is measured at Q• L = 23 W. This is even higher than the value at the design current, showing that for some applications, easy handling aspects (no ice) can become more important than the reduction of the losses to the smallest possible value. D10.0.3.5 All-metal a.c. current leads

Low-loss superconducting wires for a.c. applications became available a few years ago (Février et al 1989, Lacaze et al 1991) and superconducting power applications are now being developed in Europe, the US and in Japan. It is expected that superconducting devices for power applications will be commercialized in the near future. For the adaptation of the calculations to the a.c. case, the additional Joule losses due to the skin effect and to eddy currents must be accounted for. The skin effect is related to the a.c. self-field of the conductor and eddy current losses are related to the screening currents due to an external a.c. field. The current

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distribution is in both cases governed by the skin depth which is defined by

The additional Joule losses are due to the inhomogenous distribution of the transport current in the conductor and can in principle be evaluated by integrating over the conductor cross-section.

The evaluation of J(A) depends on the geometry of the conductor; however, the determination of the current distribution in the conductor can be rather complicated. For a cylindrical conductor, the two Maxwell equations rot(E) = −B and rot(H ) = J together with Ohm’s law E = ρ (z) J lead to Bessel-type differential equations for the current density. A simplified solution (Grellet 1989) can be obtained for the effective resistivity of the conductor

Using these equations the Joule heating can be calculated using the average current density in the conductor

It can be seen from these equations that the a.c. Joule losses are linear with the radius R of the current lead if the condition R > 2δ is true. On the other hand, the a.c. self-field losses will approach the d.c. limit if R is small in comparison to δ. This leads to the practical condition that current leads for a.c. operation must use assembled conductors with transverse dimensions which are small compared with, or of the same size as, the δ value. For example, in copper at liquid-nitrogen temperatures the δ value equals 3 mm and at liquid-helium temperatures this value is only 1 mm. In more resistive metals, for example in brass, the δ value of 12 mm is less critical, and such a.c. leads can be built from conductors of a larger cross-section. Eddy currents are induced in conductors which operate in a.c. stray fields. The corresponding losses can be the main heat generation in the current lead if the field amplitude is larger than the self-field of the current lead. It is therefore also important in this case that transverse dimensions of all conducting parts of the current lead (including contacts) are made as small as possible. Both types of a.c. loss depend also on the size of the lead and are approximately proportional to the length of the lead. A long lead containing a large proportion of copper will therefore generate much more loss than a short and compact a.c. lead. An original approach for a.c. leads was realized based on experience from the development of Nb— Ti a.c. conductors. In these conductors the very thin (0.13 µm) superconducting filaments are de-coupled by a resistive Cu—Ni layer. A composite conductor with thin copper filaments also separated by a Cu—Ni layer (see figure D10.0.12) has been realized. The thickness of the 421 copper filaments is 100 µm and the thickness of the insulating Cu—Ni layer is 16 µm. This design is an example of how the transverse conductor dimensions can be kept small compared with the penetration depth at all relevant temperatures and how a drastic reduction of the a.c. losses under transverse a.c. fields is obtained.

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Figure D10.0.12. A cross-sectional view of a composite copper—Cu—Ni all-metal conductor for a.c. current leads.

An example of the realization of an all-metal a.c. lead using these conductors is shown in figure D10.0.13. The lead was realized for the tests of a demonstration fault current limiter (Verhaege et al 1994) with a nominal root mean square (rms) current of 1.25 kA and a limited short-circuit rms current in the 5 kA range.

D10.0.4 Concepts for hybrid metal-high-temperature superconductor current leads D10.0.4.1 The operation principle of hybrid metallic—high-temperature superconductor current leads Since the discovery of the high-temperature superconductors in 1986, considerable effort has been expended in the development of high-temperature superconductor current leads (Grivon 1991, Herrmann et al 1994, Wu et al 1991, Yamada et al 1992). Indeed the use of high-temperature superconductor materials is very attractive as can be seen from figure D10.0.1. According to the WFL all metals and also superconductors above Tc cannot be at the same time good electrical conductors and thermal insulators, thus leading to the limitation of current lead performance discussed above. This situation is different for superconducting materials below Tc which combine the advantages of zero resistivity with a low thermal conductivity, representing almost ideal properties for current leads. As the use of high-temperature conductors is actually limited to liquid-nitrogen temperatures and below, a normal-conducting metal current lead (section D10.0.3) is used for the connection between 300 K and the liquid-nitrogen level. For the case of the combination of a normal metal and a high-Tc material in the low-temperature part of the current lead (see figure D10.0.14) the 4 K heat load can be reduced considerably. In such a hybrid current lead the heat input varies significantly over the length: Joule heating and high thermal conduction in the upper metallic part, which must be compared with the low thermal conduction in the lower high-Tc part. Furthermore, this heat input of the upper part changes considerably when the current in the lead is varied. The efficient use of the heat capacity of the He vapours is therefore difficult. In contrast to the allmetal current lead for 4 K applications, CC hybrid metal—high-temperature superconductor current leads, in combination with a cryocooler which directly refrigerates the cold helium vapours (see table D10.0.1), is an attractive alternative.

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Figure D10.0.13. An a.c. lead structure built from composite all-metal conductors manufactured by GEC—Alsthom for high-current testing at liquid-helium temperatures. Courtesy of Alcatel Alsthom Recherche.

Figure D10.0.14. A comparison of an all-metal current lead and a hybrid current lead. For both cases high refrigeration efficiency such as found in very large refrigeration systems (for example in the LHC for CERN) has been taken into account.

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D10.0.4.2 High-temperature superconductor current leads without heat generation In the high-temperature superconductor part of a hybrid d.c. lead there is no appreciable heat generation as long as the transport current stays below the critical current. If at the same time dissipation in the contacts is neglected, Q• diss is zero and the low-temperature heat load is determined by heat conduction only. For the CC case Q• vap, = 0, a good estimation of the performance of different conductor materials can be gained without numerical calculation using the integral of the thermal conductivity, which for a conductor of constant cross-section A and length L is determined by

The cross-section A necessary for the current which must be supplied to an application is determined by the current density J which can flow in the conductor. It is therefore convenient to define the low-temperature conduction heat load per ampere as

For the evaluation of Q• cond for the high-temperature superconductor part, we use T2 = 77 K, T1 = 4 K, k(T) of the high-temperature superconductor material (Herrmann et al 1993a) and the operating current density J smaller or equal to the critical current density of the material. For the case of negligible losses in the 4 K contacts Q• cond represents at the same time the total 4 K heat load (Q• L). For the evaluation of the theoretical performance of the hybrid current lead the Q• cond value is converted into the electrical power consumption of the refrigerator using table D10.0.1. This value is then added to the refrigeration load of the normal part according to table D10.0.3. For the case of nonvanishing losses in the 77 K contacts these must also be added by using the appropriate refrigeration mode of table D10.0.1. The d.c. lead development was carried out in a European cooperation (Albrecht et al 1994) with different conductor options including sintered and melt-textured grown (MTG) Y123 conductors, MCP Bi2212 and sintered Bi2223 conductors which have been in strong competition. For a current lead operating between 77 K and 4 K the integral of the thermal conductivity is indicated in table D10.0.5 for the relevant conductor length. The maximum achieved conductor length, the critical transport current, the critical current density and the self-field are also given. These indications allow the determination of the low-temperature conduction heat load (Q• cond/I) of equation (D10.0.12) of the current lead. To avoid an arbitrary choice of the transport current in the lead the transport current is chosen to be equal to the critical current according to the 100 µV m−1 criterion. These values are compared to the threshold value for current leads (performance necessary for achieving a heat load reduction of a factor of three). This shows clearly that both Bi conductors can reach sufficiently small (Q• cond/I) values, while the (Q• cond/I) values of the Y123 conductors still remain too high for current leads. This is due to the higher thermal conductivity and also to the shorter length which could be realized from these conductor options. A comparison of the achievable refrigeration load reduction with the material performance values shown in table D10.0.5 is shown in figure D10.0.15. The refrigeration load reduction factor (a comparison of a complete hybrid metallic-high-temperature superconductor current lead with a conventional, optimized, all-metal current lead) is represented as a function of the conductor length. Each curve corre77 sponds to a hypothetical material with the indicated value of (1/Jc ) ∫ 4.2 k dT The length dependence of the refrigeration load reduction for real materials can be found by interpolation between these curves. The dissipation in the contacts is not included in the calculation and must be added to the heat load of the system. For perfect contacts, the break-even point ( load reduction = 1) is obtained for a modest (1/Jc ) (1/Jc ) ∫ 4.277 kk dT value of 2 × 10−4 W m A−1 and a short conductor (50 mm). For very good materials (e.g. Bi2212)

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Table D10.0.5. Comparison of the required and obtained performance of ceramic high-temperature superconductors from Albrecht et al (1994).

Figure D10.0.15. The refrigeration load reduction factor as a function of the conductor length for different 77 k (1/JC)) ∫ 4.2 dT values. The values corresponding to the k different material options from table D10.0.5 are indicated as examples.

Figure D10.0.16. The field dependence of conductors at 77 K (from table D10.0.5) in a parallel field except where indicated.

and/or for long samples the curves converge to a loss reduction factor of 6.4. This corresponds to the heat load of the normal part which remains when the 4 K heat load of the high-temperature superconductor is vanishing. MTG materials have since been realized up to 10 cm in length which makes a refrigeration load reduction of 4.6 possible. These materials are still very brittle but are by far less sensitive to external fields than the other conductor materials. The above results apply to cases where the stray field does not exceed the self-field of the conductor. In magnetic fields the transport currents in the different conductors are more or less reduced (figure D10.0.16).

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Except for the melt-textured conductor all the other materials have already lost more than 23 of the current transport capacity at a temperature of 77 K and a field of 0.1 T. This can be improved by using the conductors in the flux-flow regime (see next section) or by lowering the intermediate temperature level by a few degrees. In particular, the performance of the Bi2212 conductor is significantly improved at a lower operating temperature (Bock et al 1993). The high performance of the Bi materials is mainly due to the low thermal conductivity in these materials and due to the fact that samples with useful dimensions could be realized with a sufficiently good critical current density. Both Y123 material options have been abandoned during the European project. The reason becomes clear from the comparison of the 4.2 K heat load values in table D10.0.5 and figure D10.0.15. Larger sample dimensions are required for the Y123 melt-textured conductor to make it useful for current leads. This conductor which is still being developed in different laboratories (Bornemann et al 1996) could be of interest especially for applications in high magnetic fields. The performance of the sintered Y123 conductor is inferior to the Bi conductors in almost all aspects and, unless the critical current can be enhanced substantially, this conductor is not worth consideration for current leads. For the calculation of the temperature profile in the superconductor without heat generation the computer calculation code discussed in section D10.0.3.1 is used. Equation (D10.0.6) is used with Q• diss and Q• vap equal to zero. These calculations have been limited to two conductor candidates: MCP Bi2212 tubes (Bock et al 1991, 1993) with an outer/inner diameter of 35/29 mm and a Ic value of 1280 A and sintered Y123 tubes (Herrmann et al 1993c) with an outer/inner diameter of 20/15 mm and an Ic value of 450 A. The result of these calculations is shown in figure D10.0.17. There is only a small difference between the two curves. The steep rise of the temperature on the low-temperature side, which is less pronounced for the Bi2212 material, is related to the decreasing heat conduction at low temperatures. This is naturally related to the thermal conductivity curve which is flatter for the Bi2212 material. The critical current density is temperature dependent and is much higher at 4 K than at 77 K. The cross-section of cylindrical conductors must be sufficient to transport the total current over the whole temperature range. This leads to conductors with a cross-section that is determined by the 77 K properties while the low-temperature end of the conductor is over-dimensioned. The heat leak to the 4 K level can therefore be further reduced by the use of conductors with variable cross-sections. The gain in performance through the use of conical samples has been calculated and is shown in figure D10.0.18. For a cross-section ratio of five, the 4 K load may be reduced to half of the value required for a conductor of cylindrical cross-section.

Figure D10.0.17. Calculated temperature profiles in cylindrical Bi2212 and Y123 tubes.

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Figure D10.0.18. Normalized 4 K conduction heat load:Q•conic/Q•cylin for conical Bi2212 conductors.

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However, this concept of variable cross-section reduces the stability margin of the current lead and it is less efficient for leads which are operating with heat generation in the superconducting part by flux flow or by a.c. losses. For such operating conditions the heat which is generated is evacuated to the 4 K temperature level and the thermal resistance should not be too high to avoid thermal instabilities. Thus a lead with constant cross-section is considered for the following section. D10.0.4.3 High-temperature superconductor current leads with heat generation Heat dissipation in a high-temperature current lead occurs in the superconductor itself either when the lead is operated at currents which are higher than the critical current (flux flow losses) or when the lead is operated under a.c. conditions and the periodical motion of flux lines results in a similar kind of heat generation. The principle that high-temperature current leads can be operated at a current higher than the conventional critical current according to the 100 µV m−1 criterion has been the subject of a publication (Herrmann et al 1993c) which is summarized in section D10.0.4.3(a). This consideration of high currents is important because it gives an estimation of the stability of high-temperature current leads. This allows a further reduction of the 4 K heat load per ampere of the current lead and represents the starting point for the incorporation of a.c. losses in the thermal balance of high-Tc leads which are discussed in section D10.0.4.3(b). Also the mechanisms for dissipation (flux motion for both cases) are very similar and the temperature-dependent losses have been calculated using a model with a linear temperature dependence of Jc for the flux-flow case and the more general E(J) model for the a.c. loss case. However, dissipation may also occur in the contacts which join the metal part and the high-temperature superconductor lead part at the liquid-nitrogen-temperature side and the contacts which join the hightemperature superconductor part to the low-temperature application. The acceptable contact losses and the influence of contact losses on the stability of the current lead are discussed in section D10.0.4.3(c). For the calculation of the low-temperature heat load and the corresponding temperature profiles, these loss contributions are introduced as Q• diss in equation (D10.0.6) of the computer calculation code of section D10.0.3.1. (a) Flux flow losses Operation principle In the previous case, the 4 K losses for a lead are conduction losses due to the heat flow Q• cond from the 77 K stage to the 4 K level. This heat flow also cools the liquid-nitrogen stage, and is negligible in comparison with the heat input from the metallic part of the current lead and does not modify significantly the thermal balance of the liquid-nitrogen stage. It will be shown that the operation of the current lead at currents higher than the 77 K critical current will reduce the cooling of the nitrogen stage, but will not modify significantly the heat flow to the liquid helium. At much higher currents the lead will of course become unstable and thermal runaway occurs. The determination of the highest stable transport current in the current lead is the subject of the following considerations. The field dependence of the U(I) transition curves at 77 K for the Bi2212 conductor which shows rather low n values (inset) is represented in figure D10.0.19. This confirms that considerable gain of the current-carrying capacity of the conductors can be achieved if voltage drops higher than the 100 µV m−1 criterion are tolerated. Heat will then be generated in the ‘warm’ part (near the 77 K stage) of the superconductor. This heat will partially replace the heat flow Q• cond from the liquid-nitrogen stage but, as the calculation will show, does not modify the 4 K heat load significantly if the dissipation remains smaller than Q•cond of the lead when no dissipation occurs. As this dissipation mechanism is related to the motion of magnetic flux lines in the superconductor, this approach is called the ‘current lead in flux flow concept’.

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Figure D10.0.19. The magnetic field dependence of U(I) curves of a 35 mm diameter Bi2212 conductor measured at 77 K up to a voltage drop of 10 mV m−1 (100 µV cm−1 ). The inset shows the magnetic field dependence of the n value.

Flux flow losses An estimate of the temperature-dependent flux-flow losses based on experimental data can be found using the U(I) curves of the conductor measured at 77 K up to relatively important voltage values. The U(I) curves are then fitted to the empirical equation

where E0 is the field value of 100 µV m−1, J0 the current density at this value (J0 = Jc ) and n is a fitting parameter quantifying the homogeneity of the superconductor (see also B7.3). A linear temperature dependence (Wilson 1983) of the current I0(T) = I0(T )(Tc —T)/(Tc —T0 ) is used assuming homogeneous properties in the conductor (I0 = J0 A). As the flux flow occurs only in a small temperature interval, this linear dependence appears to be a good approximation. The fitting parameter, the n value, is assumed to be constant over this temperature interval. The current- and temperature-dependent power dissipation per unit length is then given by

Later measurements of the n value have allowed a more precise determination of its temperature dependence (see figure D10.0.25 later). Temperature profiles Equation (D10.0.14) is introduced into the computer calculation leading to appreciable modifications of the temperature profiles (≈3 K) on the 77 K side of the current lead as is shown in figure D10.0.20. On the low-temperature side the temperature is only modified when the current approaches the thermal runaway current IT R which is the smallest current value for which no stable solution can be found. This shows that the current can rise to almost twice the 100 µV m−1critical current which is 1280 A in this tube before the current lead becomes unstable. At IT R however, the temperature of the current lead rises rapidly and the operating current of the lead must stay below this value to avoid the destruction of the lead. The appropriate safety margin for the current lead must be determined by taking into account the specific application and, of course, the type of superconducting material used.

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Figure D10.0.20. Current-dependent temperature profiles in a Bi2212 tube of length 100 mm at the 77 K end.

Current leads

Figure D10.0.21. Current-dependent temperature profiles in a Bi2212 tube of length 100 mm at the 4 K end.

As thermal dissipation takes place in the upper part of the lead, it is not surprising that the heat profile is strongly influenced on the 77 K temperature side but remains almost unchanged on the 4.2 K side (figure D10.0.21). The length where the lead is in the flux-flow state (the 100 µV m−1 criterion is exceeded) has been calculated. The comparison of the curves from figure D10.0.20 allows the estimation of the size of the temperature at the limit of the dissipating zone which is larger than 70 K. This gives the justification for the linear extrapolation of Jc (small temperature interval) used for the calculation. This result also suggests that there is an important potential for increasing the current if the temperature of the liquid-nitrogen stage is lowered. As an example, a current of about 2400 A can flow without appreciable flux-flow dissipation in the same Bi conductor if the intermediate temperature is lowered to 70 K. At this temperature IT R reaches a value as high as 3200 A. We can take advantage of this property in closed-cycle refrigeration systems where any temperature in the range of 30–80 K can be achieved as the operating point at the intermediate temperature.

Figure D10.0.22. Integrated flux-flow losses and 4 K heat load in a Bi2212 tube of length 100 mm.

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Figure D10.0.23. The current-dependent 4 K heat load for Bi2212 tubes of different length up to the theoretical value of thermal runaway current (IT R ).

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4 K heat load The very small change of the temperature profile on the low-temperature side suggests that the 4 K heat load is not influenced dramatically when the current rises above Ic The total (integrated) flux-flow losses and the 4 K heat load are shown in figure D10.0.22 for a conductor of 100 mm length. Indeed, as long as the flux-flow losses remain small in comparison with Q• cond the 4 K heat load remains constant. When the value of Q• cond is reached, then the 4 K heat load increases and the current lead becomes unstable. The length dependence of the 4 K heat load is shown in figure D10.0.23. Its value for low currents is of course exclusively determined by the conduction heat flow which is proportional to 1/L. Its value remains almost constant until the thermal runaway conditions are approached. The numerical values at IT are about the same as for the flux-flow losses at the same current (see figure D10.0.22). The strong variR ation of IT R when the sample length is changed has an important consequence for the applicability of short tubes as we show by the example of the 50 mm tube: the 4 K heat load at Ic is 0.62 W (0.48 W kA− 1 ). This is about half of the 4 K heat load of an all-metal current lead. This value can be greatly reduced when the lead is operated at a current of 2600 A for example. The heat load is then reduced to 0.24 W kA−1 which is an acceptable value. This creates the possibility for a very short current lead design. For a conductor 200 mm in length, which can be operated at 2000 A, the 4 K heat loads are evaluated at a value of 85 mW kA−1 showing that operation at currents > Ic leads to a very low-loss design. (b) A.c. losses Unlike the previous d.c. case, an optimum length exists for an a.c. lead. In the superconducting part of a d.c. current lead ( I ≤ Ic(T)) the solid-state heat conduction through the conductor is proportional to 1/L and is the only mechanism contributing to the low-temperature heat load. The hysteretic a.c. losses (at I ≤ Ic ), which must be added for the case of a tube or a cylindrical rod, are proportional to L. According to figure D10.0.24, the optimum length of the lead conductor corresponds therefore to the crossover point of both interdependent loss curves.

Figure D10.0.24. The determination of the optimum length of an a.c. lead using a high-temperature superconductor.

Self-field a.c. loss calculations based on the Bean model (Beghin et al 1995, Wilson 1983) have been used successfully for the evaluation of dissipation in low- and high-temperature superconductors. An interesting study using the Bean model for a.c. lead calculations has been realized (Krämer 1995). The Bean model works particularly well for materials with a high n value transition curve resulting in a relatively well defined value for the critical current density Jc . In such materials, the Bean assumption that the current density in the sample is either ±Jc( B ) or 0 represents a good approximation. However, as can be seen from figures D10.0.19 and D10.0.25, the n value for the MCP Bi2212 conductor, which is today the best performing conductor for high-temperature superconductor current leads, does not exceed a value of ten. This n value decreases further for increasing temperatures and it is also reduced when a magnetic

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Current leads

Figure D10.0.25. The field- and temperature-dependent critical current density measured on a small sample (measurements courtesy of S Elschner, Hoechst). The inset shows the corresponding n values.

field is applied. Attempts to use the Bean model have not been successful: Bean-model-based calculations result in a very strong increase of the a.c. losses when Jc (T) is reached, resulting in an unrealistic short optimum length and in a nonphysical diverging temperature when this length is exceeded. It is seen that this instability is an artefact and the computer calculation would be more accurate if the more correct material parameters based on the experimental E(J) curves were used. The calculation of the temperature-dependent a.c. loss contribution Pa.c.(T) was realized by Herrmann et al (1995) and a summary of the optimization work is given in this section. It relies on experimental work realized in a European project (Albrecht et al 1996, Herrmann et al 1996) on a 5 kA rms, 50 kV rms a.c. lead for operation at frequencies of 50/60 Hz. Superconductors have been improved since the work on the d.c. leads which may explain some of the differences in material parameters in comparison with the previous section. Before the a.c. losses are evaluated using the E(J) material properties, a simple Bean-model-based consideration can show that the most important part of the a.c. losses occurs in the warm part of the lead. According to Wilson (1983), the self-field a.c. losses in a cylindrical superconductor are given by

where Bm s is the peak-to-peak amplitude of the oscillating field, µ0 the permeability of free space and Γ(β) ≤ 1 a monotonicaly increasing function of β(T) = I/Ic (T) which depends on temperature through Ic (T). The current I is close to Ic only in the warm section of the lead leading to a similar situation as for the flux-flow case where the losses are much more important in the warm part (77 K) of the lead. As for the flux-flow case, Pa.c. will progressively reduce the heat flow Q• cond from the liquid nitrogen level into the superconductor which will reach thermal runaway conditions when Pa.c. > Q•cond is reached. General E(J) model This approach is very general and the model can be used for the calculation of different cases. Ohm’s law—under d.c. conditions (ρ(T) is independent of the current density) Ohm’s law is used in section D10.0.3 for the calculation of the normal part of a d.c. all-metal current lead. (ii) Under a.c. conditions, also with (ρ(T) independent of the current density, the computer code calculates the current distribution in the conductor. Before applying the computer code to the case of (i)

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superconductors, the skin depth (equation (D10.0.9)) has been calculated numerically for different temperatures. (iii) The early results concerning d.c. current leads under flux-flow conditions from the previous section could be calculated more precisely using the results from figure D10.0.25 which would avoid the approximation of a linear temperature dependence of J0 used in equation (D10.0.14). (iv) Finally, by introducing the current-, temperature- and field-dependent resistivity of a superconducting material, the a.c. losses are calculated. The disadvantage of this method is of course that very precise measurements of the E(J) curves at different fields and temperatures are needed up to several millivolts per metre. To keep self-field effects small in these measurements, care must be taken that sufficiently small samples are used. The results of measurements on a small rod of cross-section 19 mm2 and length 100 mm (length between the voltage taps 50 mm) are shown in figure D10.0.25. The E(J) curves are measured in the temperature range T = 64–84 K in varying fields up to 450 mT. These curves are fitted to the empirical equation

where J0(T) is determined by extrapolation to zero field of Jc according to the E0 = 100 µV m−1 criterion. Typical material parameters for example at T = 77 K are J0 = 22 A mm−2 and n = 10. The main steps of the computer calculation are indicated in the following. • •

The tube is divided into concentric cylinders of thickness d (the result is independent of d if d ≤ 10 µm). The time-dependent current I(t) which is supplied to the conductor is given by

•

The magnetic induction (Be x t ) at the surface of the tube (Re x t ) is calculated using

•

The electric field (E) is evaluated using the Maxwell equation

•

The critical current density (and the transport current Ie x t ) in the outermost cylinder is calculated using (D10.0.1). The magnetic induction at the surface of the second cylinder is calculated using

•

• •

The calculation is repeated until B = 0 is reached. The product of E and J represents the local power dissipation which is integrated over time and the radius of the tube

This result represents the average dissipation for a given temperature in the conductor.

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Current leads

For the current lead these losses must be balanced by the solid-state heat conduction over the whole length of the lead. These calculations lead to time-dependent J(t , B) patterns which enter the superconducting tube from the outer surface (Herrmann et al 1995). Similar to Bean-model-based calculations, the current may flow in both directions parallel to the axis at different radii of the tube at the same time. A.c. lead optimization The result from equation (D10.0.21) is introduced in the computer code from section D10.0.3.1. The results for 70 mm diameter tubes of the calculation of the minimum 4 K heat load and the corresponding optimum length for different J0 values are shown in figure D10.0.26. The results are compared with the CC case of the lead when the transport current vanishes (1/L law). There exists for each J0 value a minimum 4 K heat load. Left from this minimum the behaviour is dominated by the 1/L law from the heat conduction and on the right-hand side the a.c. loss contribution causes thermal instability for increasing length. These calculations are based on a J0 value which was determined on a small sample which is believed to be representative. However, a variation of J0 cannot be excluded and the influence of a reduced J0 value on the conductor dimensions must be considered. It is seen that a lead with J0 = 22 A mm−2 results in an optimum length of 122 mm and for one half of J0 an optimum length of 92 mm is found. Based on these considerations, a length of 100 mm was chosen for the a.c. demonstration lead. For this design, a 4 K heat load of approximately 0.25 W kA rms is expected. For this dimension, the safety margin with respect to J0 variations represents 40%.

Figure D10.0.26. The length-dependent conduction heat load of a 70 mm diameter tube of cross-section 1385 mm2. The deviations from the d.c. case (lower curve) are shown for conductors with different J0 values at 5 kA rms. The contribution of the conduction heat load and a.c. losses is shown for a 100 mm conductor at 0.5 J0.

(c) Losses in contacts Contacts generate losses which can limit the performance of current leads, but the heat generation in contacts can also influence the stability of current leads. A natural limit for the acceptable RT value arises from the requirement that the contact losses should not represent a significant proportion of total cryogenic loads of the current lead. For a 1 kA lead, the dissipation in the contacts at 77 K must therefore be small compared to 25–40 W kA−1, the dissipation of a current lead for 77 K application. A heat load of 1 W kA−1 appears an acceptable value which determines an RT value of 1 µΩ as an upper limit for a 1 kA lead. On the 4 K side of the current lead the cryogenic heat load is about 100 mW kA−1 leading to the requirement that the RT value must be smaller than 0.1 µΩ for a 1 kA lead.

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Table D10.0.6. A comparison of the required and obtained contact performance of ceramic high-temperature superconductors.

It has been shown, by experiment on different high-temperature superconductor materials, that RT depends on the electrical current and on the magnetic field in the contacts (Herrmann et al 1993b). For the following we shall not consider these variations and we will use constant transfer resistance. Table D10.0.6 shows experimental RT values of different conductors with a transport current between 1 and 2 kA. For the Bi2212 conductor at 77 K the RT values are found in the range of 0.3 µΩ and at 4 K the RT value is more than one order of magnitude below the 77 K value and smaller than 0.03 µΩ. Except for the sintered Y123 conductor all contact resistance values are sufficiently low up to the critical current of the conductor. Extremely low loss contacts could be realized with the melt textured Y123 conductor. Even when heating in the contacts is small compared with the incoming heat due to thermal conduction of the normal or the superconducting part, it can influence the stability of the current lead. For the contacts on the 4 K side the heat is simply dissipated to the liquid helium and a slight modification of the temperature will not have dramatic consequences for the stability of the current lead. Dissipation in the contacts at

Table D10.0.7. Calculated cryogenic loads per pole for metal and high-temperature superconductor current lead components (a)-(d) and assembled hybrid current leads (including 0.0.3 W kA-1 losses in the 4 K contacts). Pe is calculated for large-scale refrigerators using the following conversion factors: 10 for refrigeration of colda N2 vapurs (at 77 K); 15 for liquefaction of warmb 300 K N2 vapours; 250 for refrigeration of colda He vapours (at 4 K); 1300 for liquefaction of warmb 300 K He vapours (We — ‘electrical watts’ and Wc — ‘cold watts’). The experimental standard value of 1.2 W kA-1 is used for all-metal leads.

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Current leads

77 K is much more critical because a small change in temperature may decrease the critical current in the superconductor. Furthermore, a part of the heat may not be absorbed by the liquid nitrogen and may be conducted through the high-Tc conductor to the 4.2 K stage. This will lower the tolerable amount of heat which can be generated by flux flow or by a.c. losses in the current lead and the value of the maximum stable current will be reduced. A numerical estimate of the heat load can be found from experimental RT values of 0.3 µΩ at 1000 A and 77 K in a self-field. The corresponding losses are 0.3 µΩ kA−1 which is just about the value of the heat conduction for the 100 mm Bi tube (figure D10.0.22 or figure D10.0.26). For approaching optimum conditions for operation under a.c. or d.c. flux-flow dissipation, it is therefore of great importance that this heat must be removed by the liquid-nitrogen stage and not conducted to the 4 K level. In order to push the conductor towards its a.c. or d.c. flux flow limit , the design of the contacts must take into account that the heat transfer to the 77 K stage must be maximized. D10.0.5 Examples and test results of hybrid d.c. and a.c. leads Available values for cryogenic efficiency depend strongly on the size of the cryogenic refrigerator. Although in early publications on current leads conversion factors of up to 2500 for liquefaction of roomtemperature helium vapours have been used (see table D10.0.2), in this section the conversion factors from large-scale refrigerators (LHC type) are used. This is necessary to compare the performance of the different current lead concepts and realizations which are discussed in this section. D10.0.5.1 Examples of hybrid d.c. leads with a liquid-nitrogen heat sink (a) Expected performance for different design options For a liquid-nitrogen-cooled hybrid d.c. lead which operates below the critical current the global performance is determined using the results from the metallic current leads given in sections D10.0.3.2 and D10.0.3.3 and the results from the high-temperature superconductor lead part without heat generation given in section D10.0.4.2. These results are summarized in table D10.0.7 where the theoretically achievable performance for metal and high-temperature superconductor current lead components (a, b, c, d) and assembled current leads (e, f, g) is given. The low-temperature heat load (Q• L) is indicated for different liquefaction modes and the corresponding 300 K electric power consumption of the refrigerator is labelled Pe . The values for the metal lead parts are directly taken from sections D10.0.3.2 and D10.0.3.2 while the result for the superconducting lead part is found using equation (D10.0.14) and material performances for the Bi2212 material with ∫ 77k k dT = 65 W m−1 and an operating current density of 4 A mm− 4k 2 . The contact losses of 0.03 W kA−1 must be added to the conduction heat load of 0.08 W kA−1 of a 200 mm high-temperature superconductor lead leading to a 4 K heat load value of 0.11 W kA−1. These values do not take into account the possibility of using the conductors in the flux-flow mode which allows a further reduction of the 4 K heat load especially for the 50 mm tube to 0.24 W kA−1 (Albrecht et al 1994). The use of short high-temperature superconductor lead parts together with a short CC metallic current lead for the 77 K operation allows the realization of very short and compact current leads for the 4 K systems. The 77 K heat load of such a lead system is Q•L ≈ 43 W kA−1 and the contact losses of 0.3 W kA-1 from the liquid-nitrogen connection can be neglected. Taking into account the length necessary for both parts it appears possible to realize hybrid current leads of length 0.3 m requiring an electric power consumption of approximately 520 W kA-1. Such leads may modify the design of cryostats which are usually more than 1 m in height because of the requirement for a high heat exchange coefficient. The use of new compact leads, which are built right through the vacuum insulation of the cryostat, can make them smaller and more compact at the same time.

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(b) D.c. lead design A schematic view of the Alcatel Alsthom current lead is shown in figure D10.0.27. The figure shows the nitrogen VC metallic 300–77 K lead and the CC Bi2212 conductor which is situated in an insulating vacuum. The junction between these two parts is stabilized by a liquid-nitrogen stage. For the compensation of differential thermal expansion, a flexible connection is necessary which is situated on the 4 K side of the high-temperature superconductor in the insulating vacuum. The position of the high-voltage Al2O3 insulation is also shown. The lead is built into a conventional cryostat and was dimensioned for a current of 1000 A and an insulating voltage of 20 kV.

Figure D10.0.27. A schematic view from one of two poles of the Alcatel Alsthom d.c. lead.

An example of a realization of a 1000 A–20 kV hybrid current lead (Albrecht et al 1994, Herrmann et al 1994) is given in figure D10.0.28. The figure shows the lower part of the lead structure which is connected to a 1000 A test coil. During operation the coil and the lead are immersed into liquid helium and the level of the high-voltage Al2O3 ceramic insulation in the upper part of the photograph is covered by the cryogenic liquid. The high-temperature superconductor tube and the liquid-nitrogen tank are built inside the stainless steel vessel and the thermal insulation is realized by a vacuum insulation. The liquid-nitrogen tank is located in the upper, larger part and the Bi2212 tube is located in the lower thinner part of the stainless steel vessel. The 77 K metal—high-temperature superconductor connection is located approximately at the junction of both parts. The 200 mm Bi2212 tube is conduction cooled and the metal part of length ≈1 m is vapour cooled and reaches the cover of the cryostat located 1 m above. This hybrid lead corresponds to the cases (a) and (c) = (f) in table D10.0.7. (c)Test results of the d.c. lead This lead was successfully tested with the nominal current being reached many times and more than ten thermal cycles did not modify the current-carrying capacity of the system. The field in the centre of the Nb—Ti test coil was 1.15 T at a current of 1 kA.

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Current leads

Figure D10.0.28. Lower part of a 1 kA–20 kV hybrid high-temperature superconductor-metal current lead connected to a 1 kA superconducting (Nb—Ti) test coil. Courtesy of Alcatel Alsthom Recherche.

A summary of the cryogenic test results is given in table D10.0.8. The 77 K heat load value of 26 W kA−1 comes very close to the theoretical 77 K heat load value of 25 W kA−1 for VC leads (table D10.0.7 or section D10.0.2.3). The conversion of the experimental 77 K heat load results in an electrical power consumption value of about 400 We . This part was dimensioned for the nominal current of 1000 A therefore heat-load measurements at 2000 A have not been performed. Metal lead parts for 2000 A with the performance of the 1000 A part (in W kA−1 ) can of course be realized without difficulties. The critical current of these MCP Bi2212 tubes exceeds 2000 A and the performance is therefore compared with the theoretical performance of a high-temperature superconductor lead without heat generation. The total 4 K heat load is indicated as the sum of the heat conduction of the Bi2212 tube and the stainless steel vessel and the dissipation in the 4 K short-circuit connection. At 1000 A this value is determined as 320 mW and it is converted according to table D10.0.2 for large-scale cryogenic refrigerators into the electrical power consumption (80 We kA−1 ). Adding this to the value of the refrigeration load of the metallic part the total power consumption for the cooling of the

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Table D10.0.8. Test results of the d.c. Alcatel Alsthom lead. Conversion factors for mid-sized cryorefrigeratiors between the low-temperature heat loads of the electrical power consumption (referigation load) of the refrigerator: for liquefaction of cold He vapour Pe / Pc = 250; for liquefaction of warma N2 vapour Pe / Pc = 15 (We - ‘electrical watts’ and Wc - ‘cold watts’). The heat load of the short-circuit connection is subtracted for the determination of the net heat load. b Values from the 1000 A test.

1 kA hybrid current lead is determined as 470 We kA−1 which is 20% higher than the theoretical value. At 2000 A the total 4 K heat load is determined as 500 mW ( 25 mW kA−1 ); converting this into electrical power consumption a value of 60 We kA−1 is obtained. Adding this to the value of the refrigeration load of an optimized 2000 A metal 300–77 K lead, which is also 390 We kA−1, a total power consumption of 450 We kA−1 is found for the cooling of the hybrid current lead. This value is more than four times lower than the refrigeration load of an all-metal current lead, demonstrating very clearly the advantage of the hybrid current lead concept. These results are not the ultimate limit of performance for hybrid current leads. Without improvement of the high-temperature superconductor material properties the heat load can be further reduced by avoiding the conduction heat load in the vacuum vessel and the heat load in the flexible junction ( e.g. Nb-Ti in parallel ). The net 4 K heat load, 115 mW kA−1 for the 2000 A case, indicates the minimum possible value; this result is based only on the conduction heat load of the Bi2212 conductor and the heat load of the Bi2212–copper contact and comes very close to the theoretical value of 110 mW kA− 1 ((c) in table D10.0.7). Thus the 4.2 K heat load of the current lead is reduced to 101 of the value of an all-metal lead. The comparison of these results with the results of an all-metal lead ((e) in table D10.0.7) confirms that a refrigeration cost reduction of a factor of five is achievable with hybrid metal—high-temperature superconductor leads. Very short and compact (<0.3 m) hybrid current lead designs can be realized if an increase in the running costs of 30% can be tolerated (the refrigeration cost reduction factor is about four). It is probable that such compact current leads will find their application in systems where small-size and low-weight solutions are required. D10.0.5.2 Example of a hybrid d.c. lead with a 50 K helium-vapour heat sink The cooling of the metal part by helium vapours is advantageous for application cases where a multistage helium refrigerator is used and helium vapours at intermediate temperatures Ti n t are available. The metal part of the hybrid current lead can then be cooled using for example helium vapours at 50 K. This concept has the advantage of using only one cryogenic liquid and avoids the handling of liquid nitrogen. Also the

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Current leads

Figure D10.0.29. Hybrid superconducting current leads with a 50 K helium-gas-cooled normal part placed in the LHC cryostat. The main parts are indicated by numbers and their significance is indicated in the text.

temperature of the intermediate temperature level becomes a free parameter which can be adjusted for the optimization of the lead system. An example of a design of a hybrid current lead for magnets for the LHC with an intermediate heat sink cooled by helium vapours is shown in figure D10.0.29. It is composed of the following parts: (1) Nb-Ti superconductor; (2) copper junction (Bi2212-Nb-Ti); (3) 200 mm Bi2212 conductor; (4) 50 K, 15

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bar (15 × 105 Pa) helium gas supply; (5) needle valve for the helium flow regulation; (6) heat exchanger; (7) temperature sensor; (8) helium VC copper tubes; (9) helium flow regulation; (10) motor for helium flow regulation; (11) temperature sensor; (12) regulation of the heater; (13) heater avoiding ice formation; (14) current terminal; (15) over-pressure security valve; (16) 300 K, 1 bar (105 Pa) helium recovery line. The stabilization of the intermediate temperature (Ti n t ) at 70 K is realized using helium vapour at 50 K and at a pressure of 15 bar (15 × 105 Pa). The calculation of the helium mass flow for the cooling of the normal part and the heat exchanger is realized using the computer code from section D10.0.3.1. The helium mass flow m• for this case is an adjustable parameter and is not determined by the self-evaporation given by equation (D10.0.3). The optimum helium gas flow m• is indicated in table D10.0.9 for the optimized metal part at nominal current and without current for two different standby temperatures 70 K and 90 K of the heat exchanger. The regulation of the helium flow through the heat exchanger is realized by a feedback system using a motor-driven needle valve. The helium leaving the heat exchanger is used for the cooling of the metallic lead part and is recovered in the 300 K, 1 kbar (105 Pa) recovery line of the refrigeration system. The conversion to the electrical power consumption is realized by

For the efficiency of the cryogenic refrigerator a value of e = 0.3 for large-scale refrigeration is chosen. A heater avoids ice formation on the room-temperature side of the lead and the whole system is protected against over-pressure by a security valve. In table D10.0.9 the low-temperature heat load and the corresponding refrigeration loads are indicated. The refrigeration load of the high-temperature superconductor part ≈ 28 We kA−1 is small in comparison with the refrigeration load of the metal part 275 We kA−1. The total refrigeration load is found to be approximately 300 We kA−1. This last value is to be compared with 400 We kA−1 for a high-temperature superconductor current lead with a nitrogen-stabilized intermediate temperature and with 1560 We kA−1 for an all-metal lead. The refrigeration load is therefore evaluated to be reduced by a factor of five. TableD10.0.9. Calculated cryogenic loads per ple and per kA for the 50 K He VC meta l part, the CC high-temperature superconductor part and the assembled hybrid current lead. Pe is calculated for a large-scale refrigerator using the following conversion factors: 250 for refrigeration of colda He vapours (at 4 K); for the refrigeration of the 300 K vapours to 50 K an efficiency value of e = 0.3 is used.

For the standby case, two examples Ti n t = 70 K and Ti n t = 90 K have been indicated, showing that a small change of Ti n t has a quite important influence on the refrigeration heat load. A Ti n t value larger than 70 K will of course cause a cool-down phase before the lead can be energized. The acceptable delay before the current supply to the magnets and the optimum loss reduction will determine the intermediate temperature Ti n t . The minimum standby refrigeration heat load has been evaluated in figure D10.0.30. It is reached at Ti n t = 125 K and corresponds to the crossover point where the refrigeration load of the metal part becomes smaller than the refrigeration heat load of the high-temperature superconductor part.

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Figure D10.0.30. Determination of the optimum standby temperature To p t = 125 K of the intermediate temperature level and the corresponding minimum refrigeration load Pe = 110 We kA−1.

It turns out that the cooling with gaseous helium offers an interesting solution where the overall refrigeration load can be slightly reduced compared with the liquid-nitrogen cooling of the intermediate temperature level; compared with an all-metal lead, a heat load reduction factor of about five is achieved at nominal current. During standby when the magnets are cold but not energized the refrigeration load reduction reaches a factor of seven to eight compared with an all-metal lead under standby conditions. D10.0.5.3 An example of a hybrid metallic-high-temperature superconductor a.c. lead (a) Expected performance The expected performance of the superconducting a.c. lead part operating at a frequency of 50 Hz was calculated (figure D10.0.26) to be Q•L = 0.25 W kA−1 rms. The theoretical performance of the CC metal part under d.c. conditions has been evaluated in section D10.0.3.3 as 43 W kA−1. This value is increased for a.c. application to an unknown value. It was discussed in section D10.0.3.5 that the a.c. losses can be minimized using short leads assembled from conductors with transverse dimensions which are smaller than the skin depth δ. For a 77 K copper lead the stability margin (figure D10.0.10) is much more important than the stability margin for a copper 4 K lead (figure D10.0.7), thereby allowing short leads to be realized for 77 K application. Long all-metal a.c. leads for 4 K application have been realized for the development work of the Alcatel Alsthom Fault Current Limiter (Verhaege et al 1994) and also for the Brookhaven demonstration cable (Forsyth and Thomas 1986). The exact cryogenic performances of these realizations are not known, either because they were not measured separately or the results are not published. The information which has been gathered during the European a.c. lead project (Albrecht et al 1996) has allowed us to fix the achievable 4 K heat load value of an all-metal a.c. lead for 4 K application at about 1.6 W kA-1 rms. This value is used in the following for the comparison between the all-metal and the hybrid metal-high-temperature superconductor a.c. lead. (b) A.c. lead design The coaxial Alcatel Alsthom lead concept (Herrmann et al 1996) is shown in figure D10.0.31. The design is based on the short lead concept which is independent of the orientation of the assembly. This concept allows us to approach the optimum conditions needed for a feasibility demonstration in low stray fields such as found in a current limiter. For this application, the limiter would be located in the 4 K heat exchanger labelled (4) in figure D10.0.31, which would be of course much larger than in this demonstrator. The key points which have determined the lead design are given in the following. Precise a.c. loss calculations (Herrmann et al 1995) of the lead components can be carried out for the coaxial lead geometry (figure D10.0.32). In this design, the magnetic field has a cylindrical symmetry and

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Figure D10.0.31. The Alcatel Alsthom concept of the 5 kA rms, 50 kV rms a.c. lead.

Figure D10.0.32. The current axis and the cryogenic connections inside the grounded A1 screen. Courtesy of Alcatel Alsthom Recherche.

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mutual losses induced by the self-field from one lead in the other lead are avoided. The current distribution can therefore be considered as being symmetric around the axis of symmetry of the system. This justifies on one hand the approximation of the a.c. loss by considering self-induction only (the skin effect) and neglecting eddy current losses. On the other hand it motivates the approach for a.c. loss calculations in superconducting tubes (diameter 70 mm) where an optimum length of about L = 100 mm is found (see section D10.0.4.3(b)). Both current lead parts marked (5) and (6) in figure D10.0.31 are conduction cooled avoiding the need of heat exchange with the cryogenic vapours and thus allowing the installation of the lead from the bottom, from the top or from the side of the cryogenic system. The normal-metal lead part (5) was optimized for 77 K operation according to the criteria developed in sections D10.0.2.3 and D10.0.3.5. These calculations have shown that such a d.c. copper lead for liquid-nitrogen temperatures can accept over-currents of up to 100% before reaching the instability limit. Although these design criteria were established for d.c. leads they are applicable to a coaxial a.c. lead if the effective resistivity change due to the skin effect along the lead is taken into account. Twelve lead parts are arranged such that the tubular geometry of the coaxial structure is respected and the apparent wall thickness is everywhere small in comparison with the temperature-dependent skin depth (in copper: δ ≈ 9 mm at 300 K and δ ≈ 3 mm at 77 K). The only deviation from perfect coaxial geometry is the vacuum feed-through, which is realized by 12 rods of diameter 6 mm arranged on a circle of diameter 70 mm. The main dielectric medium of the test structure is a vacuum (below 10−6 mbar i.e. 10−4 Pa). The coaxial structure allows the realization of a concept with a simple cylindrical geometry where the high-field domain is confined inside the grounded screen labelled (3) in figure D10.0.31 (radiation shield). The only low-temperature insulating connections which remain necessary are the ceramic Al2O3 tubes for the supply of the cryogenic liquids in the heat exchangers labelled (1) and (2) in figure D10.0.31. The room-temperature insulator is realized from a special epoxy part of inner/outer diameter 130/340 mm labelled (8) in figure D10.0.31 and an Al2O3 disc of diameter 130 mm which houses 12 copper current leads through flexible connections. High-voltage calculations by the finite- element method have been performed for different designs for both insulators. The electrical field calculations have shown that the electrical field of 2.8 × 106 V m−1 is not exceeded in the vacuum (and 2 × 106 V m−1 is not exceeded in the insulators); these values are considered to be sufficiently low. The mechanical requirements are mainly concerned with the thermal expansion which causes an overall length reduction of the lead axis of a few millimetres during the cool-down.Flexible connections are therefore incorporated at all sites where dangerous mechanical stresses are possible. For example, each of the 12 rods from the vacuum feed-through is connected flexibly to the Al2O3 insulator by a short stainless steel corrugated tube. A photograph of the open system is shown in figure D10.0.33. One half of the grounded Al screen is taken away which allows us to show the coaxial current axis from left to right: the room-temperature terminal; the ceramic insulator; the 300–77 K copper lead; the 77 K heat exchanger with the Al2O3 ceramic tube for the liquid N2 supply; the Bi2212 lead for the electrical connection of the 77–4 K level; the 4 K Cu-Ni heat exchanger with 60 GEC-Alsthom Nb-Ti a.c. wires soldered in parallel. Symmetrically opposite to the 4 K heat exchanger are: the second Bi2212 tube; the second 77 K heat exchanger; another copper lead. The three heat exchangers are connected through the insulating tubes to the liquid-nitrogen and liquid-helium vessels which are shown in the upper part of the photograph. All the cryogenic elements and the electrical connections are fixed on the right-hand cover which is mounted on a rail system allowing the introduction of whole system into the cryostat on the left side of the photograph. (c) Material options for the 5 kA rms a.c. lead From the four material options which were investigated at the beginning of the d.c. lead project, by the end of the project both the Y123 conductor options (sintered and MTG) had been abandoned. From the

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Figure D10.0.33. The open system where one half of the grounded Al screen is taken away to show the coaxial current axis (see text). Courtesy of Alcatel Alsthom Recherche.

Figure D10.0.34. Large MCP Bi2212 tubes for a.c. leads of diameter 70 mm and length up to 200 mm. Courtesy of J Bock, Hoechst.

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two remaining Bi material options the sintered Bi2223 did not reach the necessary high transport currents. Finally only the MCP Bi2212 conductor was successful and could reach the high peak currents of 7100 A corresponding to 5 kA rms. By the end of 1995, tubes with Ic values of up to 8 kA could be achieved. It is a characteristic of the MCP Bi2212 material that Jc increases significantly with falling temperature. In a tube with a comparable thin wall thickness, as shown in figure D10.0.34, the critical current at 5.3 kA (at 100 µ V m−1 ) and 77 K is increased to 14 kA d.c. at 65.5 K (at only 10 µ V m−1 ). These current capabilities are outstanding for high-temperature superconductor bulk parts and also for high-temperature superconductors in general. The MCP tubes were an important prerequisite for the successful realization of 5 kA rms a.c. leads. (d) Test results of a.c. leads The test results of the coaxial Alcatel Alsthom demonstration a.c. lead are summarized in table D10.0.10 which gives a comparison of the performance of different leads or lead parts. The conversion factors from the LHC have been chosen for the conversion of the low-temperature heat load to the power consumption of the refrigerator. First the minimum achievable refrigeration load (electrical power consumption of the refrigerator) of a classical all-metal lead is given. This must be compared to the sum of the refrigeration loads of the metal and the high-Tc part of the a.c. lead which is indicated as the ‘total measured refrigeration load’. It is seen that for d.c. operation a total refrigeration load of 470 We kA−1 is obtained, which comes close to the results of the 2 kA d.c. lead. For a.c. operation a refrigeration load value of 615 We kA1 is reached which is significantly lower (by a factor of 3.4) than the corresponding value of the a.c. allmetal reference lead. This value does not represent, however, the achievable minimum refrigeration load of the lead which comes close to a factor of four for the actual material performance. The testing was carried out through the following steps. (i)

First, high voltage tests were performed on the individual insulators. Then tests of the structure were

Table D10.0.10. Test results of the Alcatel Alsthom lead. Conversion factors for large-scale cryorefrigerators (e.g. the LHC type; Gistau 1994) of the low-temperature heat load and the electrical power consumption (refrigeration load) of the refrigerator: liquefaction of warma He vapour Pe / Pc = 1300; liquefaction of coldb He vapour Pe / Pc = 250; liquefaction of coldb N2 vapour Pe / Pc = 10. The heat load of the short-circuit connection is subtracted for the determination of the net heat loads.

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carried out at low temperatures with liquid helium and liquid nitrogen in the heat exchangers. An alternating voltage of 50 kV rms (71 kV peak value; 50 Hz) was applied for one minute without dielectric breakdown and it is concluded that the coaxial high-voltage concept is valuable for highvoltage applications. (ii) The lead was tested under direct and alternating 50 Hz currents up to 5 kA rms. The a.c. measurements were performed by applying an increasing current amplitude up to 5 kA rms. The low-temperature heat load tests have shown that the global refrigeration load of the a.c. lead at nominal current is reduced by a factor of 3.4 compared with a classical all-metal a.c. lead. (iii) The high-temperature superconductor part of the a.c. lead contributes only 24% of the total refrigeration load (615 We kA−1) of the system at nominal current. This means that the performance is mainly determined by the normal-metal part. The predominance of the normal lead part was also found during tests on the d.c. lead (see section D10.0.5.1) where a higher refrigeration reduction factor of five was achieved, mainly due to vapour cooling of the normal lead part. (iv) The 4 K heat load of 0.58 W kA−1 is affected by eddy current losses in the Nb-Ti/Cu-Ni shortcircuit connection between both leads, and the net 4 K heat load of the lead is determined as only 0.3 W kA−1, showing that a reduction of the 4 K heat load by a factor of seven compared with the

Figure D10.0.35. The Siemens functional lead model: overall view. Courtesy of C Albrecht, Siemens.

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all-metal a.c. lead is possible. However, this important potential for 4 K heat load reduction has only a limited impact on the overall efficiency of the system: the loss reduction factor is indeed only increased to a value of four. (v) The choice of short lead parts is an imperative condition for low-loss a.c. leads. The high-temperature superconductor part is simply not stable above a critical length and the performance of the metal part becomes worse if long conductors are used. The CC normal part has approached the theoretical performance for current leads for d.c. operation and it was further shown that the 77 K heat load at a nominal a.c. current of 5 kA rms is only increased by 10% which is much better than what is achievable in all-metal a.c. leads for 4 K operation. Similar test results have also been achieved (Albrecht et al 1996) in the vertical current lead pair from Siemens. The top-loading lead structure from Siemens is characterized by two parallel, vertical lead poles mounted on the horizontal cover of a classical stainless steel cryostat. The respective lengths of the normal/superconducting lead parts are 400/100 mm. A 77 K interface is directly cooled by liquid nitrogen. With their lower ends, the two poles lead to a liquid-helium chamber where they are connected to each other. High-voltage insulation is accomplished by Teflon and epoxy elements, ceramic potential separators and a vacuum. The Siemens lead is shown in figures D10.0.35 and D10.0.36.

Figure D10.0.36. The Siemens functional lead model: high-temperature superconductor module design area. Courtesy of C Albrecht, Siemens

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D10.0.6 Perspectives for hybrid metal—high-temperature superconductor current leads Hybrid metal-high-temperature superconductor demonstration current leads have been produced for application under d.c. and under a.c. conditions and their principal advantages can be summarized as follows (see section D10.0.5.1): a hybrid d.c. lead can reduce the 4 K heat load value by a factor of ten and the global need for electrical power (room-temperature refrigeration heat load) of the cryorefrigerator can be reduced by a factor of five compared with a classical all-metal lead. For hybrid a.c. leads for power applications (50 Hz) (see section D10.0.5.3) the 4 K heat load is reduced by a factor of seven and the room-temperature heat load is reduced by a factor of three to four compared with a classical all-metal a.c. lead. D.c. currents in the 10 kA range and a.c. currents of 5 kA rms have been reached in MCP Bi2212 conductors. Electrical insulation at cryogenic temperatures has been realized at a high voltage level of 50 kV rms. The concept of CC hybrid leads allows the installation of the lead independently of the orientation, and the total length from the room-temperature terminal to the 4 K region can be reduced to 300 mm. The choice of short lead parts is an imperative condition for low-loss a.c. leads. The reliability of all-metal d.c. leads has been proven through experience gained in magnet technology over many years. This is, of course, not yet the case for the new hybrid current leads which have only been tested for reliability for about one year. For the use of such current leads in magnets which will be operated for tens of years, the use of a high-temperature superconductor solution therefore still represents a slight uncertainty which will be sorted out through increasing experience during the next few years. An estimation of the economic impact of hybrid metal-high-temperature superconductor current leads can be found using the example of the LHC from CERN at Geneva. This machine relies on a great number of different magnets which are powered using more than 1000 current lead pairs. Some of these leads will carry a d.c. current of 12 kA, but other leads for currents of 8 kA, 5 kA, 1.6 kA and smaller leads for currents down to 25 A are also required. Altogether the current leads of the LHC supply approximately 1700 kA to the different magnets. The intermediate temperature level of the LHC leads will not be cooled by liquid nitrogen but by gaseous helium. As pointed out in section D10.0.5.2, this allows slightly

TableD10.0.11. A comparison of the energy consumption and operating costs in the LHC resulting from the use of hybrid superconducting current leads or the use of classical all-metal current leads. The calculations are based on the following numerical values: for refrigeration at 4.2 K the refrigeration conversion factor is Pe / Pf = 250 and the liquefaction conversion factor is Pe / Pf = 1300, and at 70 K an efficiency of the cryorefrigerator of 0.3 is assumed; price of power is 0.05 EURO/kW h.

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better performance than for the liquid-nitrogen case. Under the assumption that all current leads for the LHC accelerator will be produced using hybrid lead technology, an important reduction of the power consumption and of the operating costs will be realized. The accelerator will operate every year for 24 weeks and the magnets will be cold for another 15 weeks (standby conditions). The refrigeration loads of all these lead pairs at nominal current and under standby conditions have been calculated for a hybrid (section D10.0.5.2) and for an all-metal design. The power and money savings assuming a price of 0.05 EURO/kW h are shown in table D10.0.11. It can be concluded that important energy and money savings can be realized if hybrid superconducting ceramic current leads are used for the current supply of the magnets for accelerators such as the LHC and for future a.c. power applications.

References Albrecht C, Bock J and Herrmann P F 1996 HTS current leads for power devices Synthesis Report From BRITE/EURAM Project 7856 contract BRE2 CT 93 0589 Albrecht C, Bock J, Herrmann P F and Tourre J M 1994 European development on superconducting oxide-based 1 kA (2 kA) d.c. current leads Synthesis Report From BRITE/EURAM Project 4071 contract BREU CT 91 0460 Beghin E, Bock J, Duperray G, Legat D and Herrmann P F 1995 A.c. loss measurements in high-Tc superconductors Appl. Supercond. 3 339–49 Blessing H private communication Blessing H and Lebrun P 1983 Development of modular thermostatic vapour-cooled current leads for cryogenic service CERN-Service d’Information Scientifique-RD/613–1800 Bock J, Bestgen H, Elschner S and Preisler E 1993 Large shaped parts of melt cast BSCCO for applications in electrical engineering IEEE Trans. Appl. Supercond. AS-3 1659–62 Bock J, Elschner S and Preisler E 1991 The impact of oxygen on melt processing of Bi-HT superconductors Adv. Supercond. III (Proc. of ISS’90) p 797 Bornemann H J, Burghardt T, Hennig W and Kaiser A 1996 Semi-finished parts and products from melt textured bulk YBCO materials for superconducting magnetic bearings Proc. Appl. Supercond. Conf. (Pittsburgh, PA, 1996) Conte R R 1970 Elements de Cryogenie (Paris: Masson) pp 24–30 Février A, Gueraud A, Tavergnier J P, Lacazc A and Laumond Y 1989 Thermo-electromagnetic stability of ultrafine multifilamentary superconducting wires for 50–60 Hz use IEEE Trans. Magn. MAG-25 1496–9 Fickett F R 1982 Technical Note 1053 (Gaithersburg, MD: National Bureau of Standards) Forsyth E B and Thomas R A 1986 Performance summary of the Brookhaven superconducting power transmission system Cryogenics 26 599–613 Gistau G 1990 Elements de cryogénie pratique á 100 K L’Air Liquide Gistau G 1994 private communication Grellet G 1989 Pertes dans les machines tournantes Techniques de l'Ingenieur vol 8 (Paris: Place de l’Odeon) Grivon F, Leriche A, Cottevieille C, Kermarrec J C, Petitbon A and Fevrier A 1991 YBaCuO current lead for liquid helium temperature applications IEEE Trans. Magn. MAG-27 1866–9 Herrmann P F, Albrecht C, Bock J, Cottevieille C, Elschner S, Herkert W, Lafon M-O, Lauvray H, Leriche A, Nick W, Preisler E, Salzburger H, Tourre J-M and Verhaege T 1993a European project for the development of high Tc current leads IEEE Trans. Appl. Supercond. AS-3 876–80 Herrmann P F, Beghin E, Bottini G, Cottevieille C, Leriche A, Verhaege T and Bock J 1994 Test results of a 1 kA (2 kA)-20 kV HTS current lead model Cryogenics 34 543–8 Herrmann P F, Béghin E, Bouthegourd J, Cottevieille C, Duperray G, Grivon F, Leriche A, Winter V and Verhaege T 1993b Current transfer condition from a metallic conductor to a high Tc superconductor Cryogenics 33 296–301 Herrmann P F, Cottevieille C, Duperray G, Leriche A, Quemener M and Bock J 1996 Test results of the 5 kAr m s –50 kVr m s HTS a.c. lead Proc. Appl. Supercond. Conf. (ASC 96) (Pittsburgh, PA, 1996) Herrmann P F, Cottevieille C, Duperray G, Leriche A, Verhaege T, Albrecht C and Bock J 1993 Cryogenic load calculation of high Tc current lead Cryogenics 33 555–61

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Herrmann P F, Cottevieille C, Elschner S and Leriche A 1995 Refrigeration load calculation of a HTS current lead under a.c. conditions Conf. Proc. MT-14 (Tampere, 1995) Hull J R 1992 High-temperature superconducting current leads IEEE Trans. Appl. Supercond. AS-3 869–75 Krämer H P 1995 A.c. loss calculations for HTS current lead modules Adv. Cryogenic Eng. 41 Lacaze A, Laumond Y, Bonnet P, Février A, Verhaege T and Ansart A 1992 Coil performances of superconducting cables for a.c. applications IEEE Trans. Magn. MAG-127 2178–81 Reed R P and Clark A F 1983 Materials at Low Temperatures (Metals Park, OH: American Society for Metals) pp 131–96 Touloukian Y S, Powell R W, Ho C Y and Klemens P G 1970 Thermophysical properties of matter TPRC Data Series vol 1–3 (New York: Plenum) Verhaege T, Cottevieille C, Weber W, Thomas P, Thérond P G, Laumond Y, Bekhaled M and Pham V D 1994 Progress on superconducting current limitation project for the French electrical grid IEEE Trans. Magn. MAG-30 Watanabe K, Yamada Y, Sakuraba J, Hata F, Chong C K, Hasebe T and Ishihara H 1993 (Nb,Ti)3Sn superconducting magnet operated at 11 K in vacuum using high-Tc (Bi, Pb)2Sr2Ca2Cu3O10 current leads Japan. J. Appl. Phys. 32 L448–90 Wilson M N 1983 Superconducting Magnets (Oxford: Clarendon) pp 58–78, 194–7, 256–71 Wu J L, Dederer J T, Eckels P W, Singh S K, Hull J R, Poeppel R B, Youngdahl C A, Singh J P, Lanagan M T and Balachandran U 1991 Design and testing of a high temperature superconducting current lead IEEE Trans. Magn. MAG-27 1861–5 Yamada Y, Yanagiya T, Jikihara T, Ishizuk A M, Yasuhara S and Ishihara M 1992 Superconducting current leads of Bi-based oxide IEEE Trans. Appl. Supercond. AS-3 923–6

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D11.1 Forced-flow cooling of superconducting magnets

I L Horvath

D11.1.1 Introduction To circulate liquid, vapour or a mixture of these two fluids in a loop, a differential pressure between inlet and outlet has to be applied. This pressure difference is generated by the compressor of the refrigerator, by a cryogenic pump or by density differences between downstream and upstream in the case of thermosiphon cooling. The cooling method in which an inlet-to-outlet pressure difference forces the coolant through cooling channels is called forced-flow cooling. Up to the late 1960s, the usual method for cooling superconducting magnets was to immerse them in liquid helium. This practice was used for small laboratory magnets as well as for large bubble chambers, like the Big European Bubble Chamber (BEBC) at the Centre Européen pour la Recherche Nucléaire (CERN) in Geneva. In these bath-cooled coils local thermal stability is performed by a large amount of stabilizer as well as by a maximum wetted perimeter along the whole conductor. The necessity of cryogenically well stabilized magnets demanded an open construction with space holders between turns and layers. The excellent cooling capacity suppressed the normal zone propagation, which sometimes entailed a burnout in the hot spot, before the normal zone had been detected. The open ducts in the winding reduced the mechanical stability of the coil, which led to many training steps while energizing the magnet. The application of forced-flow cooling has provided a series of advantages for the design and construction of superconducting magnets. (i) (ii) (iii) (iv) (v)

Because the cooling fluid is inside a tube or the hollow conductor, the magnet can be constructed without a huge liquid-helium Dewar vessel. The liquid-helium requirement within the magnet regime is reduced at least by an order of magnitude. The fully impregnated coil sustains higher magnetic forces because it has an improved mechanical stability. With the reliable operation of refrigerators the cooling process became a viable industrial process. Forced-flow-cooled magnets can be built in either orientation. In particular for internally cooled conductors which are used for thermonuclear fusion reactors the forced-flow cooling guarantees high current density, excellent transient stability and mechanical stability of the magnet structure.

The major advantages of forced-flow cooling with a single-phase fluid are the small volume of helium required to maintain the magnet operation and the simplicity of magnet operation, especially during cooling down and warming up of the magnet. The mass flow distribution is not affected by local heat input differences. The absence of vapour prevents flow being hampered by vapour locking, by the so-called ‘garden hose effect’. The cooling tubes can be wound even around solenoid magnets or used as helical

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heat exchangers. Pressure drop and heat transfer are analytically predictable. A forced-flow system has a high degree of confidence. Often the need for a circulating device is mentioned as a disadvantage of the forced-flow system, although the refrigerator can be connected directly to the users and operates in a closed circuit; the refrigerator outlet pressure forces the coolant through the narrow cooling tubes of the magnet. In an economically well designed system the evaporation enthalpy is used in a bath-cooled heat exchanger, which is fed by the forced-flow circuit, whilst the mass flow is expanded through a Joule-Thomson valve into the bath heat exchanger. In the case of forced-flow cooling, however, the cooling medium is ‘forced’ through cooling tubes, narrow gaps or fully transposed superconducting strands, as in the case of cable-in-conduit conductors (CICCs). The flowing fluid in narrow channels produces a pressure drop between the inlet and outlet of the system. This pressure drop must be compensated for by higher inlet pressure in advance. To overcome pressure drop, subcooled liquid can be circulated by a piston or rotating pumps, supercritical fluid can be circulated by the higher outlet pressure of the refrigerator or the fluid is even forced around by small differences in density in the downstream and upstream of a closed circuit, as in the case of thermal siphon cooling (Baze et al 1988, Lottin and Duthil 1988). To provide easy use of formulae for pressure drop and heat transfer calculations is the main purpose of this chapter. In this chapter the expression ‘fluid’ will be used for gases and vapour with greatly changing density as well as for incompressible liquids. However, in formulas which contain physical properties of the fluid, as in the case of gases and vapour, these properties have to be calculated at average temperature and pressure. Advances in superconducting technology and improved impregnation techniques on one side and new requirements on the other encouraged scientists and engineers to study new possibilities for magnet cooling. Already in 1966 Kolm had the idea of heat transfer improvement by the circulation of supercritical helium (Kolm et al 1966). In the late 1960s CERN set up a research and development programme with the purpose of carrying out experimental investigations of problems related to forced-flow cooling of superconducting magnets. At first a small coil and a circulating apparatus were built and tested (Morpugo 1970a). As a result of this development an initial large magnet with forced-flow cooling was ordered by CERN. This was the spark chamber background field magnet Omega (Morpugo 1970b), which is shown in figure D11.1.1. Parallel to the engineering progress in the design and construction of forced-flow-cooled superconducting magnets, the scientific background for forced-flow cooling was developed by Arp and others (Arp 1972, 1975, Giarratano et al 1971) in the early 1970s. In 1977 under the responsibility of the International Energy Agency (IEA) the construction of the Large Coil Task (LCT) (Conn et al 1988) facility at Oak Ridge National Laboratory (ORNL), USA was started. By the time the six teams (three from USA, one from Japan, one from Euratom and one from Switzerland) began their work, magnet technology offered several promising options for meeting the special requirements of huge D-shaped coils for Tokomak magnets. On the choice of cooling method, the teams were evenly divided, three choosing pool-boiling cooling and three others forced-flow cooling. This decision has given an enormous impetus to forced-flow cooling. Basic discussions about the stability of forced-flow-cooled superconductors were going on (Dresner 1980, 1984) and new solutions for internally cooled cable superconductors were also proposed for other applications such as superconducting magnet energy storage (SMES) (Hoenig 1980). Further to the successful operation of the Omega magnet and the decision in favour of the LCT, in the 1970s several superconducting magnets with forced-flow cooling were built around the world. At the Kurchatov Institute, Moscow, Russia for the T-7 Tokomak magnet a hollow conductor with nine parallel channels was built. Supercritical helium as well as two-phase helium were used (Ivanov et al 1979). Parallel to the developments in Europe and in the USA, in 1985 the Demo Poloidal Coil (DPC) Project at

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Figure D11.1.1. The Omega magnet at CERN.

the Japan Atomic Energy Research Institute (JAERI) started. The aim of this project is the development of large, forced-flow-cooled superconducting poloidal and toroidal coils for the next generation of fusion reactors, such as the Fusion Experimental Reactor (FER) and the International Thermonuclear Experimental Reactor (ITER) (Tsuji et al 1990). In Europe, besides the development at CERN and other national laboratories, at the Paul Scherrer Institute (PSI, formerly SIN) in Villigen, Switzerland (Vecsey et al 1975) a forced-flow superconducting solenoid for the 8 m long muon channel was installed in 1975. As an example of the cooling systems used in many forced-flow-cooled superconducting magnets, the cooling system of the Piotron magnet at PSI will be mentioned here. This medical facility for cancer radiation therapy was built in the late 1970s, and it was used for over ten years to treat more than a thousand patients (Horvath et al 1981). Negative π-mesons are produced by a 20 µA proton beam on a small carbon or molybdenum target. The magnet system of this facility, which consists of two torii with 60 single superconducting pancakes each, aims to bend and focus the 60 beams onto the patient, who is placed at the focus point of the second toroidal magnet. For this method only a superconducting magnet solution was applicable, because a room-temperature magnet would completely close the beam line and no pions would have been forwarded to the patient. In figure D11.1.2 the cooling system of one of the torii is shown. Each toroidal magnet represents an independent heat load to the refrigerator. In the toroidal magnet around the target, in addition to the heat radiation load, a thermal neutron radiation has to be added and cooled by the forcedflow cooling. The cooling circuit is subdivided into 12 sections. Each section is provided with its own heat exchanger, which is cooled by a helium bath fed by the helium return. During cooling down four sets of the 60 pancakes are always cooled in parallel mode. Figure D11.1.3 shows one of the torii with the 60 pancakes fixed onto a support structure.

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Figure D11.1.2. The flow scheme of the Piotron magnet cooling system.

D11.1.2 Subcooled liquids and supercritical fluids In this chapter one-phase fluids, such as subcooled liquids and supercritical gases, are considered. In the case where a liquefied gas is additionally cooled below its saturation temperature at constant pressure, it is ‘subcooled’ in comparison with a boiling liquid. On the other hand, if the system pressure over a boiling liquid is increased, the level of the liquid disappears and such a fluid is called ‘supercritical’. Its temperature and density are about the same as those of the liquid, but its behaviour is like a gas.

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Figure D11.1.3. One of the Piotron toroidal magnets at PSI. Courtesy of the Paul Scherrer Institute.

D11.1.2.1 Pressure drop For a sufficient heat transfer between wall and fluid, either a high heat transfer coefficient or a large heat transfer area or a large differential temperature between wall and fluid is necessary. Large area reduces the overall current density and increases the pressure drop of the fluid; an increasing heat transfer coefficient demands a higher flow velocity, which results in a higher pressure drop in the cooling channels. A higher operation temperature reduces the safety margin of the superconductor. Concerning all these aspects the pressure-drop calculation of a system is extremely important and needs empirical confirmation. In general the pressure drop of a flowing fluid is given by Bernoulli’s equation

where ∆p is the differential pressure between the inlet and outlet of tubes or fittings. The friction factor ξ is a nondimensional number, which is literally the ratio of the wall shear stress, τ ≈ η ∂u/∂y, divided by the kinetic energy density, and it is concluded from the Hagen—Poiseuille law. It is a function of the Reynolds number and the pipe surface conditions. For laminar flow the friction factor ξ is well defined with ξ = 64/Re. Authors from the USA use the so-called Fanning friction factor Cf or f , which is defined with Cf = 16/Re, i.e. it is four times smaller than ξ (Van Scriver 1986). In this chapter the friction factor ξ will be used systematically as it is used in technical references in Europe. The definition of the Reynolds number is given in equation (D11.1.3). The body factor af = 1, if the pressure drop is calculated inside valves and fittings, and af = l/di , if it is calculated inside tubes. Here l is the overall length of the tube and di the inside diameter. Moreover, in the above equation ρ is the averaged density and w is the averaged velocity of the fluid. In cryogenics, non-Newtonian fluids, i.e. flowing substances with a viscosity which is shear-stress dependent, like flowing plastics, are not significant, so this chapter deals only with Newtonian fluids. Also for rarefied gases, as in the case of the vacuum technique, the following equations may not be used. Furthermore, in cryogenics only flow inside straight or curved pipes is used, so this chapter deals only with pressure-drop calculations inside closed cross-section.

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Forced cooling The pressure drop inside a tube is calculated by the formula

As mentioned above, the friction factor ξ is a function of the nondimensional Reynolds number and the pipe surface conditions. The Reynolds number is defined by the following equation

In this equation m• is the mass flow, U is the wetted perimeter and η the dynamic viscosity. In calculations material properties are always taken at the average temperature and pressure, i.e. at p 12 (Pi n l e t + Po u t l e t ) and t = 12 (ti n l e t + to u t l e t ) respectively. In the case of large pressure or temperature differences the calculation must be carried out in several steps. Below a Reynolds number of 2320 the flow is always laminar; above Re = 8000 it is always turbulent. Between these two limits depending on surface roughness, on curved or straight pipes, on hydrodynamically well formed inlet and outlet, either laminar or turbulent flow can appear. (a) Laminar flow For laminar flow in smoothly extruded copper, brass or aluminium tubes measured values of the friction factor fit very well to the Hagen—Poiseuille law

(b) Turbulent flow In the turbulent flow range the surface roughness has a significant impact on the pressure drop. The pressure drop can only be calculated for smoothly extruded copper, aluminium or brass tubes. In accordance with Blasius’ equation, in a Reynolds number range between 3000 and 105 the friction factor can be calculated by the simple formula

In the case of higher Reynolds numbers between Re = 2 × 104 and Re = 2 × 106 the equation given by Hermann delivers good results

For Reynolds numbers higher than 106 the friction factor ξ can be calculated in accordance with the Prandtl-von Kámán (Eck 1978) equation

(c) Impact of the surface roughness With increasing surface roughness the pressure drop is also increasing. In areas with high roughness the pressure drop is only a function of the roughness. Inside a width range the pressure drop depends on the Reynolds number as well as on the surface roughness. In general two types of roughness exist, a long wavy roughness on the one side and rather short peaks on the other. Irrespective of a wide variety of roughness forms and their distribution over the surface, for pressuredrop calculations a ‘relative roughness’ is assumed in accordance with the following formula

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TableD11.1.1. Empirical values for the relative roughtness K.

where K is the average height of all elevations of the surface. Table D11.1.1 shows empirical values for the relative roughness K of pipe inside surfaces usually used in cryogenic plants. The hydraulic diameter dh is an equivalent diameter of the cross-section. In the case of cylindrical channel cross-section, dh = di . If the cross-section is noncylindrical, than dh is defined as

where A is the channel free cross-section and U is the wetted perimeter of the channel. As mentioned before, in a fully developed turbulent flow the friction factor depends only on the surface roughness: accordingly in this area the quadratic resistive law is valid and the friction factor can be calculated in accordance with the Prandtl—von Kármán equation

Within the transition area, where the wall roughness as well as the Reynolds number has an impact on the pressure drop, Colebrook and White developed a formula which considers both facts

In figure D11.1.4 the impact of roughness and Reynolds number on the friction factor is shown. The formulae for pressure-drop calculation and figure D11.1.4 are taken from Schlünder (1988). (d) Pressure drop in curved pipes In the case of solenoidal magnets or helical heat exchangers, a tube is wound around a cylinder. Due to the bending radius of the tube in a flow, centrifugal forces become active which initialize a superposed stream and through it an increase in the pressure drop. In figure D11.1.5 the superposed flow is shown schematically. The geometry of a helix is described by the tube inside diameter di , by the average winding diameter Dw and the gradient h, i.e. the distance between the centre lines of neighbouring windings, of the winding. Although the real diameter of the helix curvature D has to be calculated according to the formula

in the design of solenoids or helical heat exchangers h << Dw and therefore for the calculation of the additional pressure drop D can be set to Dw . In general the pressure drop is calculated in the same way as for straight tubes; however, the friction factor is increased by a factor for the curved pipe in accordance with formulae from Mishra and Gupta (1979). For laminar flow the factor is calculated from the equation

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Figure D11.1.4. Friction factor as a function of Reynolds number and tube surface roughness.

Figure D11.1.5. Flow in curved pipes.

For a turbulent flow in a curved tube one can write for the correction factor

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Applying the above formulae it is important to know where the laminar and turbulent flow areas are. Due to the superposed circumferential flow, energy is lost in the flow and this fact increases the laminar flow area to a higher Reynolds number. In a curved pipe the critical Reynolds number can be calculated in accordance with the formula of Schmidt (1967)

(e) Pressure drop in manifolds, valves and fittings As was shown, in equation (D11.1.1) for manifolds, valves and fittings the body factor af = 1, i.e. the friction factor x includes the form and size of the armatures and the dependence of the Reynolds number is negligible if the friction factor estimation is used for turbulent flow only. There is a wide variety of measured values (Schlünder 1988); however, in the frame of this chapter only a few values are given. These values show the order of magnitude and the impact of the form and size of the armatures. The friction factors collected by Schlünder (1988) are given in tables or charts. Below, for several types of armature the measured values are fitted with a simple equation. Narrowed tubes. In tubes where the cross-section changes from a larger diameter dt to a smaller one (ds ) the friction factor can be estimated from the following formula

When the contraction of the tube is carried out with a conical piece, of which the acceptance angle is at least 40°, the friction factor can be reduced to ξi = 0.04. Bent elbows with 90° angle. The friction factor for elbows depends on the tube diameter and for tube sizes up to 0.05 m it can be estimated from the following equation

where di is the tube inside diameter and d50 is the upper limit of this diameter of 0.05 m. Welded elbows. In the following formula the angle δ stands for the angle which gives the deviation from the straight line. In this equation it will be used in degrees:

Valves. Because a wide variety of several valve types exists on the market, figure D11.1.6 is presented to help to estimate the friction factor of a valve. The pressure drop is always calculated with the inlet conditions of the fluid. (f) Pressure drop in CICCs CICCs are described in detail in chapter B6. Below, only the general correlation for the friction factor in CICCs is given. This formula (equation (D11.1.19)), which was derived from standard formulae used for pebble-bed applications, was published by Katheder (1994).

where Re is the Reynolds number and v is the void fraction. Usually the void fraction in CICCs is about 40%, therefore this equation can be written in a simplified way

Later pressure drop measurements on the CICCs ( Maekawa et al 1995) which were developed for the Next European Tokomak (NET) confirmed equation (D11.1.19).

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Figure D11.1.6. Friction factors for valves of different types at a nominal size of 100 mm.

(g) Pressure drop in forced-ff low superfluid helium For forced-flow superfluid helium only a few references are available (Dorra et al 1985, Hofman 1995). As confirmed by the authors, the so-called superfluidity under turbulent flow conditions no longer exists. Therefore, by applying the appropriate physical properties of superfluid helium and by means of the empirical formulae for the friction factor, the pressure drop of superfluid helium can be estimated by equation (D11.1.1). D11.1.2.2 Heat transfer in forced-flow systems Since the discovery of high-temperature superconductors in 1986, a large effort to fabricate superconductor wires of this material has been made; nevertheless for the time being the only coolant for industrially produced superconducting magnets is helium in the form of liquid or supercritical helium. For a reliable operation of these systems, a knowledge of the coolant behaviour is extremely important. Because most of the formulae are empirical and have been developed for steady-state conditions, the layout of a system should avoid the critical condition of the coolant, i.e. the coolant outlet pressure should remain above the critical pressure or the coolant condition should not cross the saturation line. In the case of helium the pressure should be higher than 2.5 bar (2.5 × 105 Pa); in other cases the system becomes more and more unstable. In general heat transfer is a convective process. There is a driving temperature difference between the tube wall and the bulk of the coolant causing the removal of heat from the wall or vice versa. Heat transfer by convective processes within flowing fluids occurs in an enormous variety of practical applications. However, it is the rate of the convective heat transfer within a fluid per unit area of the boundary surface which is of primary importance in the design of equipment, since it directly affects the surface area of tubing required to transfer all the necessary heat and thus determines the capital cost of equipment. In cryogenic equipment, however, low-temperature differences as well as low pressure drops are required. The heat flux, the rate of heat transfer by a convective process to or from a fluid per unit surface area of boundary, is dependent on the driving temperature difference ∆T between the bulk temperature of the fluid and the temperature of the wall, and it can be expressed by the following simple equation

Here, Q• is the heat transferred per unit time and A is the surface area involved. The equation may be thought of as describing the heat flow under a temperature potential with a constant of proportionality h which is called the heat transfer coefficient. The magnitude of this coefficient reflects the ability of a

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particular convective process in a body of fluid to transfer heat between the bulk and the boundary. The use of the heat transfer coefficient is an important concept, and a great deal of experimental effort has been expended in understanding the factors which control it and in obtaining its magnitude for specific conditions of heat transfer. In fluid dynamics and heat transfer estimations most of the formulae are empirical. Very often measurements are carried out on downscaled components. By means of similarity laws (Beitz and Küttner 1987) the studied pressure drop or heat transfer coefficient can be calculated. In similarity mechanics, nondimensional parameters are used to describe thermo- and fluid-dynamic processes. Therefore, for helium and for other cryogens in single-phase conditions it is also convenient to describe the heat transfer by means of dimensionless parameters. In the case of forced-flow heat transfer processes, the principal scaling parameters are the Prandtl number, Pr, the Nusselt number, Nu, and the already used Reynolds number, Re. The Reynolds similarity law states that two viscous fluid flows, which are under the impact of inertial and friction forces, are mechanically similar if their Reynolds numbers correspond to each other. The Reynolds number is described by the fluid mass flow m• , by the dynamic viscosity η and by the wetted and heated perimeter U of the channel and can be written as

The Prandtl number is a nondimensional factor of three physical properties of the fluid and is characterized by the dynamic viscosity η, by the specific heat at constant pressure Cp and by the heat conductivity coefficient k of the fluid, and can be given by

The Nusselt number is a nondimensional heat transfer coefficient and it is specified by the real heat transfer coefficient h in a thin layer undergoing a steady-state heat transfer process, further, by a characteristic length, like the hydraulic diameter of the channel dh , which can be calculated by equation (D11.1.9), and finally by the heat conductivity k of the fluid:

For heat transfer processes, in general, there are three sets of equations which must be solved to determine the behaviour of a viscous fluid such as gaseous or liquid helium. A complete development of these equations is available in numerous studies on transport phenomena (see e.g. Bird et al 1966). The first set is the continuity equation, the second one is derived from conservation of momentum and the third one is the energy conservation equation. By suitable normalization it can be shown that the average Nusselt number is a general function of Reynolds number and Prandtl number, that is

(a) Forced flow steady-ss tate heat transfer In numerous experiments scientists have investigated correlations between the above-mentioned dimensionless scaling parameters. The basic equation for all these correlations may be written in the form

In cryogenic applications the simple Dittus—Boelter equation is frequently used

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For temperature differences between wall and bulk higher than 10% of the absolute temperature, Giarratano et al (1971) developed an equation which slightly differs from (D11.1.27)

In the above equations Re and Pr are calculated for the average bulk temperature as already mentioned before. This simplification is valid because in superconducting magnet cooling systems the differences in temperature and pressure between inlet and outlet are usually small. Since all these equations have been developed empirically, their predicted accuracy is about 15%. This should be considered in the layout of a cryogenic system. Since the Prandtl number for cryogens, apart from in the critical region, is nearly constant over a wide range, the Nusselt number is only a function of the Reynolds number. However, the heat transfer coefficient cannot be considerably increased by higher mass flow because the pressure drop limits the mass flow increase drastically. The practical heat transfer coefficient in superconducting magnets is usually less than 1500 W m−2 K−1. (b) Heat transfer in curved pipes Analogous to the consideration in section D11.1.2.1(d), the superposed circumferential flow also has an impact on the heat transfer coefficient. Earlier measurements made by Jeschke (1925) delivered a simple correlation. Later on this equation was supplemented by several authors. Hausen and Linde (1985) compared measured values and found that they can be described by the following equation

Here again d is the tube diameter and D the bending diameter. hs t is the heat transfer coefficient in a straight tube of the same dimensions and at the same Reynolds number. (c) Impact of the surface roughness In tubes with a surface roughness, but otherwise adequate flow conditions, the heat transfer coefficient is essentially higher than in hydraulically smooth tubes. Hausen (1976) proposes the following formula to estimate the heat transfer coefficient in tubes with a rough inner surface

Here the nondimensional heat transfer coefficient in a smooth tube, Nus m o o t h , is calculated from equation (D11.1.26) or (D11.1.27), K is the average height of the unevenness and dh is the tube inside diameter or the hydraulic diameter of the flow cross-section in the case of noncylindrical cross-sections. D11.1.3 Forced two-phase flow Another method of cooling superconducting magnets is to circulate two-phase helium inside cooling tubes. Because of gas plugs in vertically curved pipes the direction of the tube should be either horizontal or vertical. Transfer lines for cryogenic fluids and long dipoles for high-energy physics accelerators like DESY (Horlitz 1984) are examples of two-phase flow in horizontally oriented tubes. The thermo-siphon cooling of large solenoid magnets like Aleph (Lottin and Duthil 1988) at CERN and the TPC magnet at LBL (Green et al 1980) are examples of vertical flow. At the latter the density difference between downstream and upstream flow is used for the fluid circulation. The big advantage of two-phase cooling is the constant surface temperature produced by using the evaporation enthalpy.

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Whether it is a result of unavoidable heat input to the cooling tubes or the effect of cool-down, two-phase flow is a common occurrence in the transfer of cryogen liquids. The existence of vapour in a tube may critically reduce the carrying capacity of the circuit. Therefore, the presence of vapour must be minimized or the tube diameter must be increased to an adequate size. The reduction of the circuit capacity occurs because the mixture of vapour and liquid, having a lower density than that of the liquid alone, must have a greater velocity in order to achieve the same liquid flow rate. D11.1.3.1 Pressure drop in two-phase flow A thorough investigation has been undertaken by Martinelli and Nelson (1948) and by Lockhart and Martinelli (1949) for frictional pressure drop resulting from two-phase flow in pipes. Unfortunately, however, the problem of analytically predicting two-phase flow behaviour is complicated by the mass transition, usually from the liquid to the vapour, that results from the heat input into the pipe and from pressure changes. Moreover, the liquid and vapour are quite frequently not in equilibrium, and this introduces additional variables into the prediction. In attempting to describe two-phase flow in a cryogenic system, one must first realize that the flow in the vapour phase may be laminar or turbulent, and the flow in the liquid phase may be similar to or different from the gas. Furthermore, this flow pattern may be altered by the inclination of the pipe and by the heat input or pressure drop. Of the several different types of flow pattern that may exist, seven distinct varieties are apparent as shown in figure D11.1.7. Some useful empirical correlations at ambient conditions have been developed by Martinelli and Nelson (1948) and by Lockhart and Martinelli (1949) for frictional pressure drop resulting from two-phase flow in pipes, which are summarized by Timmerhaus and Flynn (1989) to describe two-phase flow. The two assumptions made in developing the correlations are:

Figure D11.1.7. Flow patterns in two-phase flow.

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(i) the static pressure is constant in a cross-section of the pipe; (ii) the sum of the liquid and vapour volumes always equals the total volume of the pipe. Thus, the correlations are not valid for slug or stratified flow. From the first assumption, it was proposed that the two-phase pressure drop could be determined from the pressure gradient of either the liquid or vapour phase flowing independently. For the case of adiabatic flow (no heat input to the tube) this results in the following relation

where (∆p/∆L)T P is the total pressure drop per unit length of the tube for two-phase flow, (∆p/∆L)L is the frictional pressure drop per unit length of the tube for one phase (liquid in this case) flowing independently, and φL is an empirical parameter. To obtain a value for this parameter, one must first evaluate another parameter X from the following empirical relation

In this relation, (∆p/∆L)L and (∆p/∆L)G are the pressure drops per unit length of the tube that would exist if the tube were filled completely with liquid or vapour. The factor x is the local value of the vapour content and it is dependent on the length of the channel L. In this case the flow rate of the iquid is given by the equation: m• = ρ wA(1–x). The Reynolds numbers for the liquid ReL and for the gas ReG are also calculated, assuming that the flow in the tube is either all liquid or all vapour, from

where m• is the mass flow rate for either the liquid or the gas, U is the wetted perimeter of the flow crosssectional area and η the viscosity. The constants CL , CG , m and n in equation (D11.1.32) are given in table D11.1.2 and vary with respect to the type of flow exhibited by each phase. There are four possible combinations, namely: laminar liquid, laminar vapour; laminar liquid, turbulent vapour; turbulent liquid, laminar vapour; and turbulent liquid, turbulent vapour. The two Lockhart—Martinelli parameters are related by the following relation

where the constant C also depends on the type of flow exhibited by the liquid and vapour phase. For laminar liquid and laminar vapour, C = 5; for turbulent liquid and laminar vapour flow, C = 10; for laminar liquid and turbulent vapour flow, C = 12; and for turbulent liquid and turbulent vapour flow, C = 20, as given by Timmerhaus and Flynn (1989).

TableD11.1.2. Costants in the Lockhart-Martinellli correlation.

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The frictional pressure drop per unit length of the tube for an independently flowing phase as referred to in equation (D11.1.2) may be calculated from

The friction factor ξ has to be determined in accordance with section D11.1.2.1. If the frictional pressure drop per unit length of the tube for the gas phase is desired, it will be necessary to utilize a mean density of the vapour. The mean density can be approximated as the average of the inlet and outlet densities in the tube. For the case of polytropic processes (involving heat input to the tube surface), Martinelli and Nelson showed that a modification of equation (D11.1.31) can be used to relate the total pressure drop in the tube to that experienced by a flow that is only liquid with a total flow rate equal to the sum of m•L and m• G . The modification has the form

In this equation, x is the local value of the vapour content and it is dependent on the length of the channel L, as described above. The constant n is given in table D11.1.2 and, as shown, it varies with the flow velocity. The parameter ΦL can be determined from equation (D11.1.34) and (∆p /∆L )L is calculated by using equation (D11.1.35) and assuming only liquid flow with a total flow rate equal to m• L plus m• G . It is necessary to numerically integrate equation (D11.1.35) in order to determine the total frictional pressure drop, ∆pf , because the fluid quality varies with the pipe length due to the heat input along its length. Thus

Assuming the heat input normal to the axis of the pipe to be constant such that

we can solve for dL and substitute this result into equation (D11.1.36) to obtain

The change in fluid quality provides an increase in bulk fluid velocity, resulting in an additional pressure drop due to a change in the momentum of the fluid. The momentum pressure drop may be determined from

where the momentum pressure parameter Φm is defined as

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such that RL is the volume fraction of the liquid phase and subscripts in the case of RL1 , x1 and of RL 2, x2 refer to the inlet (1) and to the outlet (2) conditions in the pipe respectively. The volume fraction of liquid phase RL is again a function of the Lockhart—Martinelli parameter X through the relation

where the constant C again depends on the type of flow exhibited by the two phases and is the same value as given earlier for the adiabatic case. The total pressure drop for two-phase polytropic flow is then obtained from

The Lockhart—Martinelli correlation, as noted earlier, was developed for estimating the pressure drops associated with two-phase flow at ambient conditions. However, because of the increased vaporization tendency of cryogenic fluids, this correlation has the tendency to underestimate the pressure drop by as much as 10 to 30% with low-temperature flows. D11.1.3.2 Heat transfer in two-phase flow The empirical curve relating heat flux from superheated wall to liquid is commonly used to characterize boiling conditions on an unrestricted surface immersed in a pool. Such curves are available for many cryogenic fluids in combination with specific metallic surfaces. In figure D11.1.8, as an example, a typical pool-boiling curve for liquid helium is given, which is taken from the article by Giarratano and Frederick (1980).

Figure D11.1.8. Typical empirical pool boiling curve for liquid helium. From Giarratano and Frederick (1980.)

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In cryogenic installations, such as transfer lines or long dipole superconducting magnets, the helium cross section is usually given by the demanded low pressure drop. On the other hand, the temperature difference between wall and bulk is of the order of 1–2 K. Under such conditions the heat transfer can be estimated in a similar way as described in section D11.1.2.2(a).

Figure D11.1.9. The heat transfer coefficient in a two-phase fluid.

In a schematic graph (figure D11.1.9) (Schlünder 1988) heat transfer coefficients for nucleate boiling (hB ) and for convective heat transfer (hk ), together with flow patterns and saturation temperature (TS ) distribution, are shown over a horizontally oriented evaporator tube. The heat input (q•) and the fluid mass flow (m• ) are constant over the tube length (z) and over time. Furthermore the vapour content of the flowing fluid with (x• Gl ) and without (x• ) thermodynamic equilibrium is given. hGO is the heat transfer coefficient at the saturated vapour condition without liquid particles in the vapour. Although the heat transfer coefficient is higher in two-phase flow (hB ) than in single-phase flow (hk ), for a system layout it is usually sufficient to calculate the steady-state heat transfer into a single-phase fluid as given in equation (D11.1.27) or (D11.1.28). By the Lockhart—Martinelli prediction method as shown in section D11.1.3.1, the vapour fraction can be estimated. According to Collier (1972) the presence of the vapour can be regarded as increasing the local velocity of the liquid phase. Hence, a simple augmentation of the single-phase heat transfer coefficient can be obtained

D11.1.4 Transient conditions D11.1.4.1 Transient heat transfer Events which happen in a short time are described as transient processes. In superconducting magnets flux jumping and Lorentz-force-induced mechanical movements can initialize peak disturbances. Because of the extremely low specific heat of the metals used in superconducting magnet design, such a disturbance can lead to a quench of the magnet.

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However, the helium in internally cooled superconducting cables can absorb small disturbances produced by heat deposition, because the specific heat per unit volume of supercritical helium at 4 K is about 600 times higher than the specific heat of a metal under similar conditions. Furthermore, in connection with studies of stability and quench behaviour in CICCs (Lue et al 1978, Miller et al 1979) it was shown that normal zones of some metres are not stable, because the local heat deposition causes, because of the compressibility of helium, strong transient flow increase, which greatly enhances the heat transfer coefficient and extinguishes the normal zone. Krafft (Hands 1986) gives an overview about work done on transient heat transfer, especially for pool-boiling helium. There is little work available on transient heat transfer phenomena in forced-flow cooling. Bloem (1986) carried out basic measurements and compared his results with computed heat transfer coefficients. In the case of forced-flow cooling, transient processes are two to three orders of magnitude faster than forced flow. For estimating transient heat transfer the flow velocity can be neglected. Because the local temperature increase of the wall, i.e. the superconductor temperature, is expected to be in the range 2–5 K, due to the safety margin of the superconductor critical temperature, further simplifications of the estimation are permitted: (i) physical properties are constant; (ii) the transient conduction heat transfer only takes place in the laminar layer of a turbulent flow, i.e. within about 15% of the tube inside radius, as shown in figure D11.1.10; (iii) furthermore, in this short time of 10–20 ms the temperature step between wall and fluid remains constant. Following the analytical calculation for transient heat transfer of Schlünder (1988) and considering the above-mentioned simplifications at the very end one obtains the following formula for the time-dependent momentary transient heat flux

Here the transient heat flux is proportional to the square root of the physical properties (k is the heat conductivity, ρ is the density and Cp is the specific heat at constant pressure) of the fluid and to the temperature step (ϑs 0 − ϑb ) at t = 0, where ϑs 0 is the surface temperature at t = 0 and ϑb the bulk temperature. Furthermore the momentary transient heat flux is a function of the reciprocal square root of the time. Under the assumption that the temperature step takes place in the outermost layer of the helium, from equation (D11.1.45) a heat transfer coefficient can be estimated. Using the physical properties at 4 bar

Figure D11.1.10. Laminar and turbulent flow profiles.

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(4 × 105 Pa) and 4 K, one can see that after about 3 ms the transient heat transfer reaches a steady-state condition. To estimate the heat transfer it is more convenient to use the heat transfer coefficient averaged over the duration of the transient

From Bloem’s measurements it can be concluded that the transient heat transfer coefficient ht in the usually used regime of supercritical cooling at a transient time duration of 4–5 ms is about 2000 W m−2 K−1. D11.1.4.2 Heat-induced transients As a result of magnetic or mechanical disturbances or local heat input, the conductor temperature exceeds the superconductor critical temperature and the superconducting magnet becomes normal. Because the stabilizer inside the conductor is able to take over the full current for a limited time only, a fast discharge of the magnet must be induced. Depending on the stored energy, the quench detection system, the quality of the high-voltage insulation and the external dump resistors, about 70% of the stored energy can be deposited in the external dump resistors. The remaining energy in the coil results in a temperature increase of the coil which induces a pressure increase in the cooling system. One of the most important questions in designing large superconducting magnets with forced-flow cooling is the estimation of the maximum pressure and temperature after an emergency discharge of the magnet system. The institutes involved in the IAE Large Coil Task (Conn et al 1988) carried out fundamental work on this question and developed several computer programs to analyse the quench behaviour of forced-flow-cooled superconducting magnets (Benkowitsch and Krafft 1980, Marinucci 1983, Miller et al 1980). These experimental and theoretical investigations demonstrate that it is possible to estimate all heat-induced transients in a cooling system with stagnant or forced-flow helium. Miller et al (1980) proposed a simple formula to calculate the peak pressure arising from quenchinduced thermal heat load in a forced-flow-cooled superconducting magnet system

where ξ the nondimensional friction factor, Q• (W m−3 ) the heat input to the helium per unit volume and dh (m) the hydraulic diameter of the cooling channel. For a cooling channel closed at one end Pm a x is the peak pressure at the closed end and l (m) is the overall channel length. For a channel open at both ends Pm a x is the peak pressure at the middle of the cooling channel and then l is the half length. D11.1.4.3 Examples for forced flow problems Remarks: physical properties of helium for the following estimations are taken from McCarty (1980) and McCarty and Arp (1980). Example 1 The helium cryostat return line of a bath-cooled superconducting solenoid is directly connected to the low-pressure inlet of a refrigerator by a 150 m long straight transfer line. The inlet pressure into the refrigerator is 1.18 bar (1.18 × 105 Pa) and the helium mass flow is 2.5 × 10−3 kg s−1. If the bath temperature must be kept constant at 4.5 K and the transfer line has no heat losses, determine the inside diameter of the cylindrical return line, which is made of stainless steel. At the very beginning one has to determine the saturation pressure corresponding to a bath temperature of 4.5 K. This is 131 kPa. Now the differential pressure along the transfer line, which can be considered,

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is 13 kPa. For further calculation with the averaged temperature and pressure the physical properties of helium have to be determined. With an assumed inside tube diameter of 0.02 m and by means of equation (D11.1.3) the Reynolds number is calculated as 1.17 × 106. For extruded stainless steel the relative surface roughness in accordance with table D11.1.1 is about 4 × 10–5 m, i.e. in accordance with figure D11.1.4 the pressure drop inside the tube has to be calculated using equation (D11.1.9). Finally, after iterations the minimum inside diameter of the tube has been calculated as 0.023 m. Example 2 An 8 m long superconducting solenoid, built out of 16 coils, with a central induction of 5 T is cooled by forced-flow supercritical helium. The helium circulates in 5 × 10−3 m inside diameter copper tubes wound around the winding and impregnated together with the superconducting winding. The cooling circuit is subdivided into four sections, consisting of four coils each. After each section the helium is cooled back to 4.5 K in a helical heat exchanger. In steady-state operation all coils and heat exchangers are series connected. The bending radius of the cooling tubes is 0.095 m and the tube length per coil is 40 m. Determine the pressure drop per section, if the helium mass flow is 2.5 × 10−3 kg s−1, the inlet pressure is 6.0 bar (6.0 x 105 Pa) and the heat input into the coils warms up the helium at the section outlet to 5.0 K. With an assumed pressure drop over four coils of 0.5 bar (0.5 × 105 Pa) the physical properties of helium have to be determined. The equation (D11.1.2) delivers a Reynolds number of about 1.6 × 105, which is higher than the limit for laminar flow (equation (D11.1.15)). The calculation of the friction factor is carried out by means of (D11.1.4), because the dimensionless roughness for copper pipes of this size is about 0.0003 and the above Reynolds number is low enough to consider the flow as ‘smooth’. The impact of the curved pipe on the friction factor is calculated with equation (D11.1.14). After several iterations the pressure drop in one section is determined to be 40 kPa. Example 3 A 0.3 m diameter helical heat exchanger is immersed in a liquid-helium bath. The bath temperature, which is dependent on the helium compressor suction pressure and on the pressure drop in the refrigerator, is taken as 4.3 K. The heat exchanger is made of a copper tube of 0.01 m inside diameter. Determine the length of the copper tube if the heat exchanger is designed to cool back 3.0 × 10−3 kg s−1 from 5.0 K to 4.5 K. The operation pressure is 500 kPa and the tube inside temperature is the same as the helium bath temperature. The total heat load which has to be taken over by the liquid-helium bath is 7.2 W and the average temperature between pressurized helium and bath is 0.45 K. The heat transfer coefficient is calculated from equation (D11.1.28) to be 486 W m−2 K−1. From equation (D11.1.29) the impact of the curved pipe is 14%. The helical tube length immersed in liquid helium is determined to be 0.92 m. Example 4 The HERA (Horlitz 1984) superconducting magnet storage ring is installed in a tunnel of 6336 m circumference. The cooling system is subdivided into eight octants. The temperature of each superconducting component of the system is kept at about 4.3 K by a two-phase helium flow. The mass flow per octant is 4.5 × 10−2 kg s−1. If the heat losses per octant are about 800 W and the helium inlet conditions before the Joule—Thomson valve are pressure 4.0 bar (4.0 × 105 Pa) and temperature 4.4 K, determine the pressure drop in the two-phase fluid and the helium quality at the inlet and outlet of the octant in the 0.06 m inside diameter cooling tube. We start with the determination of the physical properties of liquid helium at 120 kPa and at 4.4 K and 4.3 K, for inlet and outlet conditions. The estimated Reynolds number for the liquid, ReL , in accordance with equation (D11.1.32) is 3.06 × 105, i.e. the flow is in the transition region towards flow in rough pipes and the pressure drop has to be estimated by equation (D11.1.9). The calculated pressure drop under the

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mentioned conditions is 519 Pa. In the same way the pressure drop for the vapour has to be calculated. Doing so, the equations give ReG = 7.456 × 105 and a pressure drop of 3.6372 kPa. By means of equation (D11.1.34) the first Lockhart—Martinelli parameter X 2 can be calculated as 0.1427[(1–x)/x]2. To determine the second parameter ΦL equation (D11.1.34) has to be used with the constant C = 20 for turbulent—turbulent flow. The calculation provides for ΦL2 = 1 + 52.9456[x/(1–x)] + [x/(1–x)]2/0.1427. With the helium conditions (p = 400 kPa and T = 4.4 K) in front of the Joule—Thomson valve the vapour inlet content in the line is determined as xl = 3.37%. With a heat input of 800 W to the 1.6 km long line the vapour content at the outlet is x2 = 94.07%. By means of the above formula equation (D11.1.38) has to be integrated between x2 and x1. With a liquid pressure drop of 520 Pa the frictional pressure drop in the line is determined as 7.57 kPa. The momentum pressure drop is calculated from equation (D11.1.40). Equations (D11.1.41) and (D11.1.42) provide Fm = 2.245 and the momentum pressure drop is determined as 4.7 Pa. Finally the total pressure drop in the two-phase flow as found from equation (D11.1.43) is 7.575 kPa. References Arp V 1972 Forced flow, single-phase helium cooling systems Adv. Cryogen. Eng. 17 342 Arp V 1975 Thermodynamics of single-phase one-dimensional fluid flow Cryogenics 15 285 Baze J M et al 1988 Design, construction and test of the large superconducting solenoid ALEPH IEEE Trans. Magn. MAG-24 1260 Beitz W and Küttner K-H 1987 Dubbel Taschenbuch für den Maschinenbau 16th edn (Berlin: Springer) p B62 Benkowitsch J and Krafft G 1980 Numerical analysis of heat-induced transients in forced flow helium cooling systems Cryogenics 20 209–15 Bird R B, Steward W E and Lightfoot E N 1966 Transport Phenomena (New York: Wiley) Bloem W B 1986 Transient heat transfer to a forced flow of supercritical helium at 4.2 K Cryogenics 26 300 Collier J G 1972 Convective Boiling and Condensation (New York: McGraw-Hill) Conn R W et al (ed) 1988 The IEA Large Coil Task Fusion Eng. Design 7 Nos 1 & 2 Dorra L, Horvath I, Kwasnitza K and Deotto T 1985 Forced flow superfluid helium tests at SIN Proc. 9th MT Conf (Villigen, Switzerland: Paul Scherrer Institute) p 794 Dresner L 1980 Stability of internally cooled superconductors: a review Cryogenics 20 558 Dresner L 1984 Superconductor stability: a review Cryogenics 24 283 Eck B 1978 Technische Stromungslehre (Berlin: Springer) Giarratano P J, Arp V D and Smith R V 1971 Forced convection heat transfer to supercritical helium Cryogenics 11 385–93 Giarratano P J and Frederick N V 1980 Transient pool boiling of liquid helium using a temperature-controlled heater surface Adv. Cryogen. Eng. 25 455–66 Green M A et al 1980 The operation of a forced two phase cooling system on a large superconducting magnet LBL-10976 (May 1980); 8th ICEC (Genoa, 1980) (Guildford: IPC Science and Technology Press) Hands B A (ed) 1986 Cryogenic Engineering (New York: Academic) Hausen H 1976 Wärmeübertragung im Gegenstrom, Gleichstrom and Kreuzstrom 2nd edn (Berlin: Springer) Hausen H and Linde H 1985 Tieftemperaturtechnik (Berlin: Springer) Hoenig M O 1980 Internally cooled cabled superconductors Cryogenics 20 373, 427 Hofmann A 1995 Thermomechanische Pumpen zur Kühlung mit erzwungener Strömung von superfluidem Helium DKV Tagungsbericht vol 1, pp 83–95 Horlitz G 1984 Refrigeration of a 6.4 km circumference, 4.5 tesla superconducting magnet ring system for the electron proton collider HERA at DESY Proc. ICEC-10 (Helsinki, 1984) (Guildford: Butterworth) Horvath I L, Vecsey G and Zellweger J 1981 The PIOTRON at SIN—a large superconducting double torus spectrometer IEEE Trans. Magn. MAG-17 (MT-7) Ivanov D P et al 1979 Some results from T-7 tokomak superconducting magnet test program IEEE Trans. Magn. MAG-15

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Jeschke H 1925 Wärmeubertragung and Druckverlust in Rohrschlangen Technische Mechanik VDI 69 25–8 Katheder H 1994 Optimum thermohydraulic operation regime for cable in conduit superconductors (CICS) Cryogenics 34 595 Kolm H H, Leupold M J and Hay R D 1966 Heat transfer by the circulation of supercritical helium Advances in Cryogenic Engineering vol 2 (New York: Plenum) Lockhart R W and Martinelli R C 1949 Proposed correlation of data for isothermal two-phase, two-component flow in pipes Chem. Eng. Prog. 45 39 Lottin J C and Duthil R 1988 ALEPH solenoid cryogenic system Proc. 12th ICEC Conf (Southampton, 1988) (Guildford: Butterworth) Lue J W, Miller J R and Dresner L 1978 Vapor locking as a limitation to the stability of composite conductors cooled by boiling helium Adv. Cryogen. Eng. 23 226 Maekawa R, Smith M R and Van Sciver S W 1995 Pressure drop measurements of prototype NET and CEA cablein-conduit conductors (CICCs) IEEE Trans. Appl. Supercond. AS-5 Marinucci C 1983 A numerical model for the analysis of stability and quench characteristics of forced-flow cooled superconductors Cryogenics 23 579 Martinelli R C and Nelson D B 1948 Prediction of pressure drop during forced-circulation boiling of water Trans. ASME 70 695 McCarty R D 1980 Thermophysical properties of helium-4 from 2 to 1500 K with pressures to 1000 atmospheres NBS Technical Note 1025 McCarty R D and Arp V 1990 User’s Guide to HEPAK (Niwot, CO: Cryodata) Miller J R, Dresner L, Lue J W, Shen S S and Yeh H T 1980 Pressure rise during the quench of a superconducting magnet using internally cooled conductors Proc. 8th Int. Cryogenic Engineering Conf. (Genoa, 1980) Miller J R, Lue J W, Shen S S and Lottin J C 1979 Measurements of stability of cabled superconductors cooled by flowing supercritical helium IEEE Trans. Magn. MAG-15 351 Mishra P and Gupta S N 1979 Momentum transfer in curved pipes. 1. Newtonian fluids Indust. Eng. Chem. Process Design Dev. 18 130 Morpurgo M 1970a Review of work done at CERN on superconducting coils cooled by forced circulation of supercritical helium Proc. 3rd Int. Conf, on Magnetic Technology (Hamburg, 1970) Morpurgo M 1970b The design of the superconducting magnet for the OMEGA project Particle Accel. 1 Schlunder E-U (ed) 1988 VDI-Wärmeatlas Berechnungsblätter für den Wärmeübergang (Düsseldorf: VDI) Schmidt E F 1967 Wärmeübergang and Druckverlust in Rohrschlangen Chem. Ing. Technik 39 781 Timmerhaus K D and Flynn Th M 1989 Cryogenic Process Engineering (New York: Plenum) Tsuji H et al 1989 Evolution of the Demo Poloidal Coil Program Proc. 11th Int. Magnet Technology Conf. (London: Elsevier Applied Science) vol 2, p 806 Van Sciver S W 1986 Helium Cryogenics (New York: Plenum) Vecsey G, Horvath I L and Zellweger J 1975 The superconducting muon channels Proc. 5th Int. Conf on Magnet Technology (MT-5) (Rome, 1975)

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D11.2 Some aspects of cryogenic pumps A Hofmann

D11.2.1 Cooling by forced flow of superfluid helium (He II) Helium in the superfluid state (He II) is an extraordinarily good fluid for cooling superconducting magnets. This is because of its extremely high thermal conductivity combined with high specific heat, and also because of its unique hydraulic properties. The drawback of very low operational temperature may be compensated for by the increase of the critical current for superconductors with low critical temperatures. Typically, the field of an Nb—Ti magnet can be increased from 8 to 11 T when the temperature is decreased from 4.2 K to 1.8 K. In most cases, such He II-cooled magnets are of pool-cooled design such that a helium transparent winding structure is immersed into a pool of subcooled He II. This is a well proven technique for small and medium-sized magnets (Claudet et al 1986, Turowski and Schneider 1990, Van Sciver et al 1994). For the very large magnets used in fusion technology, the use of internally cooled conductors embedded in a tight electrical insulation structure is considered indispensable. Such designs need forced flow of the coolant within channels of several hundred metres in length. Solenoid and D-shaped coils of such types are most advantageously built up from double-disc (doublepancake) winding units in the middle of which are hydraulic inlets. By this design, the helium is fed in in the region of maximum magnetic field and it escapes at relatively low field. Hence, the fluid temperature is allowed to increase appreciably during its passage through such channels. Temperature increase may be caused by absorption of heat or by frictional heating. The electrical insulation provides sufficient thermal insulation between the layers. Such magnets can either be operated with He I or with He II. No major modification is required for He II operation. Mixed-state operation with He II only in the high-field layers and with temperature increase up to a level tolerable in the low-field region might be most economic. This has the additional advantage that parasitic heat load such as heat flow through mechanical supports and through current feed-throughs will be removed at the more economical He I temperature level. Moreover, the warm return flow can be used to drive the helium convection by making use of the thermomechanical effect (fountain effect). Some more general remarks on pumps for forced-flow-cooled magnets will be made prior to a detailed discussion on thermomechanical pumps. D11.2.2 Helium circulation by warm compressors Refrigeration systems for He II-cooled magnet operation are in most cases provided with two liquid-helium reservoirs, a He I pool operated at atmospheric pressure, p0 , and a saturated He II pool with vapour pressure of about 15 mbar (1.5 kPa) when 1.8 K is to be achieved. The pressure difference between both pools may be used to drive the flow through the magnet. The respective refrigeration scheme is shown in

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figure D11.2.1. The helium to be supplied to the magnet is taken from the lower end of the He I pool. When passing the counterflow heat exchanger HX3 and the He II bath heat exchanger HX4, the fluid temperature is decreased to To. By this, the magnet is fed with a subcooled (pressurized) fluid which is called He IIp. The pressurized helium leaving the magnet is cooled-back by expansion in EX2. During steadystate operation, the re-liquefaction rate should be in equilibrium with the evaporation rate in the saturated He II pool, but for outlet temperatures below 4.2 K, the liquefaction rate will always be much higher than the evaporation rate. Therefore, this concept where the flow is driven by the main compressor is reasonable only when appreciable additional heat load is being applied to the He IIS pool. This can also be realized when several objects are operated in series as shown by figure D11.2.2. The number of objects will be limited by the overall pressure drop. This operational mode is preferred in accelerator technology where groups of bending magnets may be operated in series (Lebrun 1994).

Figure D11.2.1. A schematic diagram of an He II cooling loop with forced convection driven by the main compressor.

Figure D11.2.2. Series connection of N objects with a thermal load.

D11.2.3 Coolant loop with cold circulator Much greater flexibility is obtained by installation of a pump for circulating He II in a separate loop as shown schematically in figure D11.2.3. Experience (Berndt et al 1990, Lehmann et al 1984, Morpurgo 1977, Weisend and Van Sciver 1987, Zahn et al 1992) has shown that most pumps which work in He I can also be used for He II circulation. However, the demand on efficiency is more critical when the losses are to be removed from the He II temperature level. In this context, one might ask whether it would be more economic to operate such pumps at a higher temperature level as indicated by position P’ or by immersing the circulator into the He I pool. A simple analysis will show that there would not be enough gas flow in the counterflow heat exchanger HX to provide efficient backcooling in the circulator loop. Hence, the thermal load at the He II level will not be reduced effectively by such simple means; more complicated and more expensive refrigerator loops would be necessary. However, when operating with He II, the temperature difference established between the inlet and outlet of the coil can also be used to drive the convection.

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Figure D11.2.3. An He II cooling system with a mechanical pump in the secondary loop.

Even today, there is relatively limited experience of long-term operation of liquid-helium pumps and there is still some margin for improvement. A modern design of a centrifugal pump is described by Forsha et al (1994). Surveys on earlier developments are given in reports from the National Bureau of Standards (NBS) (McConnel 1973). One of the most powerful pumps designed for a flow rate of about 500 g s−1 is being used in the Japanese magnet test facility (Kato et al 1992).

D11.2.4 A coolant loop with a thermomechanical pump Figure D11.2.4 shows schematically how a continuous flow of subcooled He II can be driven by a thermomechanical pump (TMP). The main element of the pump is a so-called superfilter (SF), a porous plug permeable only for the superfluid component of He II, in the ideal case. Such filters can be made of

Figure D11.2.4. A cooling system with a ‘self-driven’ thermomechanical pump for circulation of He II.

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materials with about 0.1 µm pore size. The pressure difference

is established when the end temperatures are kept at T5 and T6 respectively. This pressure head can amount to about 0.5 bar (0.5 × 105 Pa) when the low temperature is 1.8 K. The filter is enclosed by two heat exchangers. At the cold end, CHX provides thermal anchoring to the temperature of the boiling pool, and at the warm end, the major fraction (about 70%) of the heat to be removed from the magnet is transferred by WHX to the pump. The outflows from WHX (paths 7–1 and 3–4) are at intermediate temperature, and they are cooled back by additional heat exchangers in the saturated He II pool. The absolute pressure in the secondary loop can be at any level. It appears most attractive to keep it at the pressure of the He I pool, but also supercritical pressure is advantageous when two-phase flow in the secondary loop must be avoided. Much higher pressures are disadvantageous because of deterioration of superfluid He properties associated with the decrease of the lambda-transition temperature. A reasonable limit might be around 10 bar (10 × 105 Pa). As in natural convective (buoyancy driven) systems, the flow depends strongly on the heat load, but the pressure head caused by the fountain effect is much higher. Nevertheless, it must be checked whether the pressure head is sufficient for operating large coils with internally cooled conductors. It is very convenient to use the charts given by figure D11.2.5 for such analyses (Hofmann 1992). The abscissa is the product of heat load and a factor β that characterizes the flow resistance of the loop. For channels with rough walls, turbulent flow with

can be assumed, where ξ is a constant friction factor and A is the flow cross-sectional area of the channel with hydraulic diameter D and length L. Then the scaling factor is given by

The usage of such charts is explained by the example of the Euratom/Large Coil Task (LCT) coil, a D-shaped Nb—Ti coil with about 40 t weight (Bird et al 1988). During its first operation with He I, a pressure drop of ∆p = 0.1 bar (0.1 × 104 Pa) at m• = 14 g s−1 was found. This yields

The operational state assumed for Q• = 30 W of heat load is β Q• = 2.6 × 106 W m−2. For this value, figure D11.2.5(a) yields the coil outlet temperature T2 ~ 2.1 K when the inlet temperature is T0 = 1.8 K. Figure D11.2.5(b) yields the resultant temperature established at the warm end of the porous plug, namely T6 ~ 1.9 K, and finally figure D11.2.5(c) yields the pressure head resulting from this temperature. The mass flow rate is expected to adjust to m• = 23 g s−1 (according to equation (D11.2.4)). It is interesting to notice that such ‘self-driven’ thermomechanical pumps can be operated with coil outlet temperatures appreciably above Tλ . The flow rate will increase up to outlet temperatures of about 3.5 K†. The porous plug, however, must be designed such that there is frictionless flow within its channels even when the temperature at its warm end approaches Tλ .

† Experience has shown that the pump also works when helium at even higher temperature is returned from the magnet. This may be explained by the fact that there will always be a thin layer of superfluid helium at the outlet of the filter.

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Figure D11.2.5. Diagrams of a TMP coolant loop operated at different base temperatures T0 . The abscissa is the heat load scaled with the flow impedance resulting for fully turbulent flow (constant friction factor): (a) coil outlet temperature T2 ; (b) temperature T6 established at the warm end of the porous plug; (c) pressure difference caused by the thermomechanical effect (fountain effect).

As for electric current in superconductors, there is a critical mass flow in porous plugs. It depends on the size of the intrinsic microchannels and on the warm-end temperature T6. This dependence is plotted in figure D11.2.6. Rather high critical mass flow rates ranging up to about 1.5 g s−1 cm−2 can be obtained with plugs made from 10 mm long stacks of organic material filter membranes, called Sartorius(R) microfilters, with 10 nm and 50 nm filtration grade respectively. For technical applications the more solid sintered Al2O3 ceramic ( i.e. KPM† P80) is preferred. Typically, a pump meeting the LCT requirements has been fabricated with a filter composed of 27 elements each with a length of 25 mm and a diameter of 15 mm (Hofmann 1992). The term superfluidity might be misleading in the context of frictional pressure drop as mentioned above, so a few comments might be helpful. Frictionless flow is given only in very narrow channels such as found in the porous plug. In all other tubes and in flow channels within the magnet, the critical velocity is exceeded. In those cases, the classical pressure drop correlations can also be applied to He II flow (Rousset et al 1994). The respective viscosity term governing those correlations for He I and He II flow is given in figure D11.2.7. This plot results from Arp and McCarty (1989) for He I data and from some

† Königlich Preussische Porzellanmanufaktur, Berlin.

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Figure D11.2.6. Critical He II mass fluxes in different porous plugs as a function of its warm-end temperature T6 .

Figure D11.2.7. Shear viscosity of He I and He II (Arp and McCarty 1989, Brewer and Edwards 1963).

transformations of He II data given by Brewer and Edwards (1963). The thermal conductivity, however, is not affected by the turbulence (Kraemer 1988, Rousset et al 1994). Hence, all properties which are responsible for cryogenic stabilization of the superconductor are also maintained for forced-flow-cooled systems. The proper design of other components such as the various heat exchangers and the connection tubes is also important. Details are given by Hofmann et al (1992).

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References Arp V D and McCarty R D 1989 Thermophysical properties of helium-4 from 0.8 to 1500 K with pressures to 2000 MPa NIST Technical Note 1334 Berndt H, Doll R and Wiedemann W 1990 Two years experience in liquid helium transfer with a maintenance free centrifugal pump Adv. Cryogen. Eng. B 35 1039–43 Bird D S, Klose W, Shimamoto W and Vecsey G (ed) 1988 The IEA Large Coil Task Fusion Eng. Design 7 1–232 Brewer D F and Edwards D O 1963 The effect of pressure on the normal component viscosity and critical velocities of liquid He II Proc. 8th Int. Conf on Low Temperature Physics (LT8) (1962) (Guildford: Butterworths) pp 96–9 Claudet G, Bon Mardion G, Jager B and Gistau G 1986 Design of the cryogenic system for the TORE SUPRA Tokamak Cryogenics 26 443–9 Forsha D H, Nichols K E and Beale C A 1994 High efficiency, variable geometry centrifugal cryogenic pump Adv. Cryogen. Eng. 39 925 Hofmann A 1992 Design of a fountain effect pump for operating the EURATO-LCT coil with forced flow of helium II Adv. Cryogen. Eng. A 37 139–46 Kato T, Ishida H, Tada E, Hiyama T, Kawano K, Sugimoto M, Kawagoe E, Yoshida J, Kamiyauchi Y, Tsui H, Saji N, Asakura H and Kubota M 1992 Development of a large centrifugal cryogenic pump Adv. Cryogen. Eng. B 37 845 Kraemer H P 1988 Heat transfer to forced flow helium II Proc. ICEC12 (Guildford: Butterworths) p 299 Lebrun P 1994 Superfluid helium cryogenics for the large hadron collider project at CERN Cryogenics 34 (ICEC Suppl.) 1–6 Lehmann W and Minges J 1984 Operating experience with a high capacity helium pump Adv. Cryogen. Eng. 29 813–20 McConnel P M 1973 Liquid helium pumps NBS Report 73-316 Morpurgo M 1977 Design and construction of a pump for liquid helium Cryogenics 21 224 Rousset B, Claudet G, Gautier A, Seyfert P, Martinez A, Lebrun P, Marquet M and Van Weelderen R 1994 Pressure drop and transient heat transport on forced flow single phase helium II at high Reynolds numbers Cryogenics 34 (ICEC Suppl.) 317–20 Turowski P and Schneider Th 1990 Design and operation of a 20 T superconducting magnet system and future aspects Physica B 164 3–7 Van Sciver S W, Miller J R, Welton S, Schneider Muntau H J and McIntosh G E 1994 Cryogenic system for the 45 tesla hybrid magnet Adv. Cryogen. Eng. 39 375–80 Weisend J G and Van Sciver S W 1987 Characterization of a centrifugal pump in He II Adv. Cryogen. Eng. 33 507 Zahn G, Hofmann A, Bayer H, Berndt H, Doll R, Herz W, Süsser M, Tumwald E, Vogeley B and Wiedemann W 1992 Test of three different pumps for circulating He I and He II Cryogenics 32 (ICEC Suppl.) 100–3

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D12 Safety with cryogens

R N Richardson

D12.0.1 Introduction The safe use of cryogens requires a knowledge of their properties and an understanding of the effect they may have on materials with which they come into contact. Whilst there is a general appreciation of the extremely low temperatures associated with cryogens, the full extent of the potential hazards involved may not be immediately apparent. However, provided appropriate precautions are taken, correct procedures followed and common sense exercised, cryogenic substances and systems may be handled easily and safely. Cryogens are the liquid (or under appropriate conditions solid) form of substances more usually encountered as a gas. Cryogens are therefore distinguished by their extremely low boiling temperatures at ambient pressures (figure D12.0.1). Strictly, the term cryogen only applies to the liquid or solid form of the so-called permanent gases; that is those that cannot be liquefied by the application of pressure alone at ambient temperatures. This includes helium, hydrogen, neon, nitrogen, argon, oxygen, methane and krypton which all have critical temperatures well below ambient and also xenon, ethylene and ethane which although having critical temperatures a little above 0 °C (273 K) still cannot be liquefied by the application of pressure at most ambient temperatures. Whilst not conforming to this definition, solid carbon dioxide is sometimes classed as a cryogen because of its low sublimation temperature at atmospheric pressure. A summary of the thermophysical properties of these cryogens is given in table D1.1.1. The types of cryogen encountered in practice will depend on the particular application. A wide range of cryogens may be used in research although the quantities involved will probably be small. In an industrial environment it is more likely that a large quantity of a single cryogen will be encountered. Although the difficulties associated with handling bulk quantities of cryogen are not insignificant, safety procedures are well established and in general rigorously enforced. It is, therefore, perhaps important to emphasize that the majority of safety incidents involve only relatively small quantities of cryogen. The main factors to be considered in assessing the difficulties likely to be encountered with cryogens are • • •

properties of the cryogen—thermophysical and chemical physiological hazards material compatibility.

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Figure D12.0.1. Saturation temperatures (boiling points) are for a pressure of 101.3 kPa (1 bar). Absolute zero corresponds to 0 K or –273.15°.

Two hazards common to all cryogens are the effect of the extremely low temperatures, which may cause embrittlement of materials and cold burns to personnel, and the increase in specific volume associated with phase change from liquid to vapour which can result in a significant pressure hazard. All other hazards are dependent on the properties of the particular cryogen. The hazards may be summarized as follows • • • • •

extreme low temperatures high specific volume ratio (over-pressurization) flammability oxygen enrichment oxygen depletion (asphyxiant).

The problem of flammability is of concern when handling hydrogen, methane (LNG), ethane and ethylene but this hazard is usually recognized since the intrinsic flammability of these gases is well known. A potentially greater hazard exists due to oxygen enrichment since although oxygen itself is not flammable it will support vigorous combustion in many otherwise inert or slow-burning materials and may even react explosively under certain conditions. An oxygen enrichment hazard may exist even in the absence of the bulk liquid due to the condensation of air which will occur on the uninsulated surfaces of vessels containing nitrogen, neon, hydrogen or helium. An oxygen deficient atmosphere whilst not, in general, posing any problems of material compatibility does create a danger of asphyxiation. All cryogens, with the obvious exception of oxygen, present an oxygen depletion hazard which results primarily from the large volumes of vapour which may be evolved from a relatively small quantity of liquid. A more detailed explanation of the potential hazards involved in using cryogens together with procedures for safe design and handling are given in the following sections.

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D12.0.2 Safety legislation and codes of practice It is very important that users of cryogens are aware of any national legislation or local rules governing their activities. The extent and applicability of such legislation varies from county to country and there is likely to be an even greater range of regulations imposed at a local level by the particular organization or institution in which cryogens are used. Unfortunately, legislation is often not very specific and where codes of practice exist these usually relate to the design and construction of particular equipment rather than the use of cryogens. For example, in the UK there is virtually no specific legislation covering the use of cryogens and only a few dedicated standards or codes of practice. This does not, however, mean that there are no accepted standards or applicable legislation. The implications of the UK Health and Safety at Work Act (HSWA) are far reaching and certain sections are relevant to the use of cryogens even if they are not explicitly mentioned. It is also possible that the Control of Substances Hazardous to Health (COSHH) regulations may apply to the use of certain cryogens although it is interesting to note that despite their properties, cryogens as a group are not listed under the COSHH regulations. Special regulations apply to liquefied natural gas (LNG—major constituent methane) and liquefied petroleum gas (LPG—not a cryogen but sometimes grouped with cryogens) because of their flammability but it is not anticipated that these substances will be used to cool superconductors so they need not be considered further here. The UK HSWA places a duty on employers, employees and third parties to ensure safe working ‘as far as reasonably practicable’. There can be no abdication of responsibility: each individual must take all reasonable precautions to ensure both his or her own safety and that of others. Whilst this may seem obvious when handling cryogens or operating low-temperature plant it also applies to the design and construction of equipment. The test of what is reasonably practicable under the HSWA is primarily by reference to codes of practice and other approved sources. Unfortunately, the Health and Safety Commission, the statutory body in the UK, has not issued a code of practice for cryogens. In the absence of such a code, established good practice must be followed and guidance sought from appropriate British Standard/International Standards Organisation (BS/ISO) standards and publications issued by authoritative bodies. For the purposes of law such information will be deemed to be the appropriate standards. The UK Health and Safety Executive (HSE) publish a general guide (HS(R)6) to the regulations contained in the HSWA which, whilst not mentioning cryogens specifically, is obviously relevant. HSE8, which deals with fire and explosion hazards due to misuse of oxygen, may also be appropriate where this cryogen is present. The most relevant official document is BS5429 ‘Code of practice for safe operation of small-scale storage facilities for cryogenic liquids’. In this context small scale is defined as storage capacities from 0.5 t to 135 t, the size of typical commercial bulk storage vessels, which for many users will actually represent a significant capacity. The British Cryogenics Council publishes ‘The Cryogenics Safety Manual—A Guide to Good Practice’ which is generally accepted in the industry as the reference to established practice. In addition, suppliers of cryogens, and most large users, produce their own safety literature. The design of equipment for use with cryogens presents special safety problems because of the extremely low temperatures involved and the very large specific volume increase as a cryogen vaporizes which can lead to significant pressure hazards. BS5429 does mention these problems but for detailed guidance appropriate reference books and design guides must be consulted. In the case of dewars and other vessels used to contain cryogens the codes relating to pressure vessels and compressed gas storage and transport may be relevant (e.g. BS4741, BS7777, BS.5355). In particular, the ‘Pressure systems and transportable gas containers regulations’ (Statutory Instrument 1989 No 2169) should be consulted. A thorough assessment of potential hazards is vital to the safe use of cryogens. If there is any doubt or uncertainty about a system or procedure further guidance must be sought before a cryogen is introduced. Further, it must be appreciated that simply because no specific legislation or code covers a particular application involving cryogens this does not mean that certain parts of other codes are not applicable. The

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bibliography at the end of this chapter includes various sources of reference but should not be considered exclusive. D12.0.3 Physiological hazards and personal safety D12.0.3.1 Cold burns Direct contact with cryogens or a surface at cryogenic temperatures can produce effects on skin tissue similar to bums (hence the term cold burns). The severity of injury will depend on temperature, the rate of cooling and time of exposure. Injury results from the freezing, and consequent expansion, of water in the body tissue which causes irreversible damage to the cell structure. Whilst the tissue remains frozen no pain is experienced (the technique is employed in cryosurgery). The rate of heat loss is a critical factor. Momentary contact between dry skin and cold vapour or even a small quantity of liquid may not result in damage since thermal contact will be poor and the heat capacity of the skin sufficient to prevent the tissue temperature falling below 0° C. However, if the skin is wet, improving thermal contact, or the exposure prolonged then tissue damage is inevitable. Even brief exposure to cold vapour may cause damage to delicate tissue such as the eyes or lungs. Inhalation of cold vapour will result in some discomfort in breathing but prolonged exposure can cause respiratory failure. Continued exposure to chilled atmospheres not necessarily cold enough to cause immediate damage may result in frostbite or cause hypothermia. Naked or inadequately protected tissue, including that covered by wet clothing, which comes into contact with a cold surface may stick fast due to freezing and the flesh may be tom in removal. Little discomfort will be felt whilst the damaged area remains frozen but on thawing severe pain may be experienced and the wound will be very susceptible to infection. It should also be remembered that objects which are pliable at room temperature, including body tissue, become hard and brittle at low temperatures and can easily be shattered or broken. D12.0.3.2 Oxygen deficiency Asphyxiation due to lack of oxygen is responsible for virtually all the fatalities resulting from accidents involving cryogens. Atmospheric air normally contains about 21% oxygen and 79% nitrogen by volume. Although healthy individuals can tolerate oxygen concentrations in the range 13%–60% asphyxia of increasing severity will be experienced as the concentration in an atmosphere falls below about 18% (high levels of oxygen are not usually a serious physiological hazard although prolonged exposure to concentrations in excess of 60% can cause lung damage; a much greater danger to life is posed by enhanced flammability of clothing and even the body itself, particularly hair, in the presence of oxygen). The asphyxiation hazard results primarily from the large amount of vapour that is produced by an evaporating cryogenic liquid. For example, 1 1 of liquid nitrogen at atmospheric pressure will yield 678 1 of vapour. The potential danger is compounded by the fact that in most instances the release of vapour and consequent oxygen depletion takes place slowly often without any indication of the danger until it is too late. The symptoms of oxygen deficiency become progressively more acute as the percentage of oxygen reduces as shown in the following • • • • •

12%–14% breathing more difficult, increased pulse rate, poor coordination 10%–12% judgement/speech impaired, giddiness, lips blue 8%–10% nausea, vomiting, unconsciousness 6%–8% 8 min exposure 100% fatality, 6 min 50% fatality <4% single breath probably fatal.

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The insidious nature of asphyxiation is such that the victim may initially be completely unaware that anything is wrong. Subsequently the victim will collapse. This is often the first and only warning by which time it is already too late for self-help. Instances of asphyxiation most commonly involve nitrogen and argon which have been used to purge vessels or have accumulated in service ducts or excavations, although the possibility of a build-up in inadequately ventilated rooms or vehicles must not be discounted. In such cases asphyxiation may be almost instantaneous upon entering the oxygen-deficient atmosphere. This point should be borne in mind by potential rescuers. Attempting to recover an asphyxiation victim without adequate breathing equipment will in most cases simply increase the number of fatalities. When entering a potentially deficient atmosphere a safety line should be used (see HSE GS5 ‘Entry into confined spaces’). There is a very real danger that because inert gases such as argon and nitrogen are nontoxic and nonflammable they will be considered intrinsically ‘safe’ and the asphyxiation hazard overlooked. It is generally considered that two breaths of nitrogen vapour will almost certainly prove fatal. The asphyxiation hazard may be avoided simply and with almost complete reliability by employing a personal oxygen monitor and a strict adherence to good practice. D12.0.3.3 First aid In the event of cryogenic liquids (or solid CO2) coming into contact with the skin or eyes the affected area should be flooded with copious quantities of cold or tepid water. Hot water or dry heat should never be used. Skin which is stuck fast to a cold surface may need to be eased away with the help of a blunt instrument such as a spatula. Damaged tissue should be loosely covered by a dry sterile dressing. Patients should be kept warm. In all cases specialist medical advice should be sought and the nature of the incident explained since those treating the casualty may not be familiar with cryogenic injuries. If a person feels dizzy or loses consciousness whilst working with cryogens he or she should be removed immediately to a well ventilated area. In more acute cases of asphyxia it may be necessary to apply artificial respiration or administer oxygen. Following any loss of consciousness, however brief, the patient should be medically examined. With the exception of the hydrocarbons (and CO2) all the cryogens referred to in this publication have very low toxicity. Inhalation of the hydrocarbons can cause headaches, dizziness, drowsiness and even loss of consciousness but only in high concentrations. D12.0.3.4 Storage and handling of cryogens Great care should be taken when handling cryogens and low-temperature equipment. It should be remembered that the toughness of many materials may be considerably reduced at low temperature and that in order to minimize thermal conduction paths much of the equipment designed for use at low temperatures, for example dewars, may include fragile glass and thin metal sections. The advice of both the equipment and cryogen suppliers should always be sought. In the case of a large storage facility (e.g. a bulk liquid nitrogen tank) the system will usually have been designed and installed by the supplier of the cryogen in which case full operating instructions should be provided. Similarly, in the case of dewars or liquid cylinders the manufacturer would be expected to provide information about their use and maintenance. Cryogens should only be handled in well ventilated areas and in the presence of materials which will not be unduly affected by a minor spillage. If a liquid spillage occurs onto a warm surface the cryogen will disperse into small drops and spread rapidly over the surface. The drops do not evaporate immediately. Each drop is surrounded by a layer of vapour which insulates the liquid of the drop from the surroundings and increases mobility. A relatively small quantity of liquid may cool a large area.

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Under appropriate atmospheric conditions spillage of a cryogen may result in the formation of a dense cloud. This cloud is not the cryogen itself but rather the condensation produced when water vapour in the atmosphere comes into contact with the cold cryogen. Whilst the cloud may in itself create a hazard due to reduced visibility it is important to realize that a potential hazard may exist well beyond the cloud. The boundary of the visible cloud does not necessarily indicate the extent of the cryogen vapour. Warm vapour may be present beyond the limit of the cloud and depending on the cryogen this may pose an asphyxiation or fire hazard. Considerable splashing may occur due to vigorous boiling when transferring cryogenic liquids to a warm container or when inserting warm objects into the liquid. All such operations should be performed slowly to minimize these effects. Remote handling equipment should be used where appropriate. When inserting open-ended pipes into a liquid a stream of liquid and cold gas will spurt from the warm end which must either be directed to eliminate any hazard or blocked until the cold end reaches temperature equilibrium with the bulk liquid. Suitable precautions must be taken when transporting cryogens to ensure adequate ventilation particularly in the case of vehicles. Cryogens should never be transported in the passenger compartment of a vehicle. A build-up of oxygen would present a significant fire hazard whilst excessive concentrations of the inert gases could result in asphyxiation. It must be ensured that the dewars being used are indeed suitable and that they are securely restrained. Larger storage vessels mounted on wheels should be manoeuvred with extreme care. Where gradients are to be encountered it is vital that the dewar can be braked. Similarly, when stationary the dewar must be secured against movement. Cryogens being transported by lift should not be accompanied and it should be ensured that no other persons can enter the lift at intermediate levels. If cryogens must be stored in rooms not equipped with forced ventilation (not to be recommended) it is essential that warning notices are posted. This applies equally to confined spaces which have been purged using any of the cryogenic gases. Disposal of cryogens must take place under carefully controlled conditions paying due attention to the low temperatures involved, large amounts of vapour that will be generated and the properties of the cryogens. D12.0.3.5 Protective clothing No one should be allowed to handle cryogens without first ensuring that they are suitably protected. Clothing or overalls should be loose fitting and without pockets, cuffs or turn-ups in which liquid might accumulate. Wet clothing must not be worn since this provides no insulation and may become frozen to the skin. Eyes should be protected by safety glasses, goggles or a visor. Thick leather or synthetic gloves should be used when handling cold objects or liquids. However well insulated, gloves will only provide a temporary barrier against the cold and should not be used for extended periods. It is particularly important that gloves should be a loose fit in order that they can be removed easily in the event of a liquid spillage. In circumstances where large quantities of inert vapour are likely to be evolved, for example purging a tank with nitrogen or pipe freezing below ground level, breathing apparatus should be available. In all situations where there is any danger of oxygen deficiency a personal oxygen monitor must be carried. D12.0.4 Pressure hazard The increase in specific volume of a cryogen as it changes phase from liquid to vapour is very large. For example, the expansion ratio of nitrogen is 678 whilst that of neon is even greater at 1417. If due allowance is not made for this, very high pressures may be generated and in the worst case catastrophic failure may result. Any situation in which a quantity of liquid may become trapped in a small volume represents a potential hazard. Irrespective of the level of insulation, heat in-leak will eventually cause the liquid to vaporize with consequent increase in specific volume. A particular danger exists where a

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Fire and oxygen hazards

881

cryogen may be isolated between valves. If no means of venting is provided a dangerous overpressure will be created in the isolated section. Overpressure protection in the form of pressure relief valves or bursting discs should be included in all systems. These safety devices must be checked regularly. It has been known for the operation of a safety valve to be impeded by a build-up of ice rendering it useless. Special precautions apply in the case of dewars and other equipment employing vacuum insulation. These are discussed in more detail in section D12.0.6. D12.0.5 Fire and oxygen hazards D12.0.5.1 Fire Hydrogen, methane, ethylene and ethane are intrinsically flammable within given concentration limits but this is, of course, a property of the gas not the cryogen. An additional hazard associated with storing these gases as a cryogenic liquid is the large quantity of potentially flammable vapour that would be released from the spillage of a small amount of liquid. In the case of a fire involving a large quantity of liquid gas it may prove very difficult to extinguish the flames. In such circumstances the only effective action is to isolate the liquid source and let the fire burn itself out whilst attempting to confine the fire by cooling the surroundings. Particular care must be taken with hydrogen since this has such a low ignition energy that ignition may occur without any apparent cause. Typical sources of ignition are minute static discharges, slight friction or the presence of a contaminant in otherwise insignificant quantities. Once ignited a hydrogen flame will be virtually impossible to extinguish. The situation is not made any easier by the fact that a hydrogen flame is almost invisible and can often only be detected by the heat haze it produces. Consideration should be given to the selection and operation of electrical equipment in the vicinity of potentially flammable cryogens. All such equipment should be intrinsically safe and reference should be made to HSE publication HS(G)22(1984) ‘Electrical apparatus for use in potentially explosive atmospheres’. Fire fighting in the presence of a cryogen is made more difficult by the fact that water cannot be used with safety although water-based foams may be used and are often the most effective agent. Water sprayed onto apparatus containing a cryogen will create more flammable vapour (by supplying the heat required to vaporize the liquid) and may freeze. Indiscriminate use of water may lead to a significant build-up of ice which can block vents and safety valves, creating an explosion hazard due to build up of internal pressure, and may even cause collapse of a structure due to the sheer weight of ice. Water in the form of a fine spray can, however, be useful in maintaining material adjacent to a cryogen-fed fire below the ignition point particularly in an oxygen enriched atmosphere. Foams can be used to blanket a burning flammable liquid. In the case of relatively small fires CO2 or powder extinguishers will prove the most effective. The use of nitrogen or argon to create an inert atmosphere is an effective fire prevention measure but care must then be taken to avoid asphyxiation. D12.0.5.2 Oxygen hazard The presence of oxygen in excess of the normal atmospheric concentration of 21% creates a particularly acute fire hazard. Since gaseous oxygen is colourless, odourless and tasteless an enriched atmosphere cannot be readily detected without monitoring equipment. Whilst not itself flammable, oxygen will support combustion in many materials and may react explosively with organic materials. In the presence of excess oxygen many materials will ignite well below their normal ignition temperatures. In a 25% oxygen atmosphere the rate of burning of readily combustible materials will increase significantly. At a concentration of 30% many materials which are slow burning or considered fire retardant in a normal atmosphere may burn fiercely. In addition, materials which have apparently been extinguished may burst

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into flames again even after a considerable period of time has elapsed. Combustion becomes more intense as oxygen concentration, pressure and temperature increase. In many situations it may prove impossible to extinguish an oxygen-fed fire. Liquid oxygen in combination with a wide range of materials will form a violent explosive. In particular, oxygen must never be allowed to come into contact with hydrocarbonbased oils and greases. Under appropriate conditions even metals may burn in the presence of oxygen, including the stainless steels and aluminium alloys which are otherwise eminently suitable for use in cryogenic systems. Personnel working with oxygen should be aware that danger may persist for some time after they leave an oxygen atmosphere due to saturation of their clothing with the gas. Clothing should be ventilated in air for at least 15 min before any source of ignition is approached.

D12.0.5.3 Air liquefaction hazard Any situation involving the use of helium, hydrogen, neon or nitrogen presents the possibility of oxygen enrichment due to air liquefaction. Atmospheric air (dew point at 1 bar approximately 81 K) will condense on an exposed surface cooled by any of these cryogens. Such a situation may arise in the case of inadequate or damaged insulation. Although the volumetric composition of dry air is 21% oxygen and 79% nitrogen the higher-boiling-point oxygen component will condense first. The equilibrium composition of liquid air at the dew point is such that the drops of liquid formed are approximately 50% oxygen (see figure D12.0.2). The oxygen content of any accumulation of liquid air will increase further as the lower-boiling-point nitrogen component vaporizes. In only a short period the liquid air may be transformed into almost pure oxygen with all its associated hazards. The real danger lies in the fact that an oxygen hazard was not anticipated.

Figure D12.0.2. Temperature—composition diagram for nitrogen/oxygen mixtures. (Note that at the dew point the vapour ratio is 79% N2/21% O2 but the corresponding liquid composition is 50% O2 .)

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Materials and equipment

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D12.0.6 Materials and equipment D12.0.6.1 Materials and design The physical properties of many materials are significantly affected by low temperatures. Of particular concern are thermal contraction and the reduction in toughness that may occur as temperature is reduced (figures D12.0.3–D12.0.5). In addition, thermal cycling may lead to fatigue failure. In general, materials tend to become less ductile as the temperature is reduced although the ultimate yield strength of many structural materials actually increases. The change in properties may be abrupt with the transition from ductile to brittle behaviour occurring over a very narrow temperature range characterized by a well defined transition temperature. Below the transition temperature toughness is considerably reduced and even low levels of stress or shock may cause brittle fracture. Internal stresses resulting from contraction may initiate failure if allowance is not made for thermal movement.

Figure D12.03. Coefficient of expansion at low temperature for various materials. (Values may vary by as much as ±25% depending on composition and treatment of material.)

Figure D12.0.4. Coefficient of expansion at low temperature for various materials. (Values may be as much as ±25% depending on composition and treatment of material.)

The design of cryogenic systems is fundamental to safety; a poorly designed system employing inappropriate or overstressed materials would present a significant hazard. Design considerations will vary with each particular application but the same basic materials criteria will apply. (i)

Strength/toughness. A material must be sufficiently strong to withstand anticipated stresses (including an allowance for overload) and must be tough enough to absorb impact loads at low temperature without suffering brittle failure. (ii) Stability. Allowance must be made for thermal expansion and contraction and the possibility of differential movement where a number of different materials are employed. Thermally induced stresses may easily exceed the yield stress of a material and even if failure does not occur there is the danger of leakage due to joint movement.

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Figure D12.05. Toughness at low temperature for various materials (transition temperatures). (These are typical data taken from a number of different sources and should be used for comparative purposes only.)

(iii) Compatibility. Materials must be chemically compatible with all the fluids with which they may come into contact. This is particularly important where oxygen service is involved. Consideration should also be given to the electrochemical compatibility of dissimilar metals if corrosion is to be avoided. Other properties to be considered include thermal capacity (which will influence cool-down behaviour), thermal conductivity (heat leak and thermal gradients, hence thermal stresses) and electromagnetic behaviour. Although these may not be primary safety considerations it must be remembered that a system is only safe if each element in the system is safe; no aspect of a design should be considered too insignificant in the context of safety.

D12.0.6.2 Metals The metals most suitable for use in cryogenic systems are those with face-centred cubic structures since the mechanical properties of these exhibit only limited temperature dependence. They include the pure and alloyed forms of aluminium, copper and nickel and the austenitic stainless steels. Metals with a bodycentred cubic structure including plain carbon steels and low and medium alloy steels (e.g. nickel steels, ferritic stainless steels) are not in general suitable for use at cryogenic temperatures because of their relatively high transition temperatures (figure D12.0.5). However, at moderately low temperatures certain of the nickel-alloy steels are used for structural purposes with 9% nickel steel being accepted as suitable for use down to 73 K in sections up to 50 mm thick. Metals with a hexagonal close-packed structure show a wide variation in properties at low temperature: for example zinc is completely brittle just below room temperature, but titanium and its alloys offer particularly good strength to weight ratios at temperatures down to 20 K.

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D12.0.6.3 Nonmetals With the exception of PTFE virtually all polymers and elastomers have transition temperatures above 150 K and are unsuitable for use at cryogenic temperatures. This severely limits the choice of materials which may be used for seals and gaskets at low temperature (in many applications metal seals made of lead, indium or even gold are employed). Although PTFE retains some flexibility even at temperatures approaching absolute zero its tendency to ‘cold flow’ means that careful design is required if a PTFE sealed joint is not to leak. D12.0.6.4 Oxygen compatibility The possibility of materials burning or reacting explosively in an oxygen-enriched atmosphere has already been highlighted in the context of fire hazards. Oxygen compatibility is a measure of the resistance of a material to ignition in an oxygen-enriched atmosphere. Only those materials that have been tested and approved under appropriate conditions of pressure, temperature and concentration must be used for oxygen service. Fully oxidized materials such as glass, quartz and refractories are safe under virtually all conditions. Similarly, most metals have relatively high ignition temperatures (typically > 1000 °C) and in solid form will only ignite under extreme conditions. In contrast, organic materials have very low ignition temperatures (<400 °C) and may burn fiercely or react explosively (hydrocarbon oils/greases) in the presence of oxygen. Metals which are highly resistant to ignition include gold, silver, copper, nickel and many of their alloys (brass, bronze, Monel and nonferrous nickel alloys). Metals having lower ignition temperatures but which are still resistant to ignition unless finely divided (high surface area/volume ratio) include carbon and alloy (stainless) steels, cast iron, aluminium and zinc. Steels may be used for oxygen service provided flow velocities are limited and the system is clean. Similarly, although aluminium will burn in oxygen under appropriate conditions ignition is only likely to occur under extreme conditions or in the presence of contaminants. Only a very few nonmetals can be used safely in oxygen service. PTFE and certain other polymers with high ignition temperatures may be used but only at moderate temperatures and pressures. The ignition temperature of polymers and elastomers can vary significantly with composition. Although any oxygen-enriched atmosphere is potentially hazardous, combustion will not take place without a source of ignition. The energy required to initiate combustion decreases as the oxygen concentration increases. In addition to a flame or incandescent source, ignition may be caused by mechanical friction or impact, adiabatic compression, static electricity and rapid oxidation as in the case of hydrocarbon oils. Whilst the possibility of ignition and subsequent combustion can never be completely eliminated, the hazard may be minimized by appropriate design and operating procedures. Table D12.0.1 ranks materials in order of increasing oxygen compatibility. It should be noted that compatibility is not determined solely by ignition temperature: factors such as heat of combustion, thermal properties and surface characteristics must also be considered. Useful references to oxygen compatibility include the publications of the American Society for Testing and Materials Committee G4, Industrial Gas Committee documents and BS.3N100 (see bibliography at the end of this chapter). D12.0.6.5 Equipment and storage vessels Only equipment specifically designed for use at low temperatures should be employed in cryogenic systems. If experimental systems are to be constructed it is vital that the design be supervised or approved by an individual having the necessary experience. The design of cryogenic systems is a subject area in its own right.

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Safety with cryogens Table D12.0.0.1 Oxygen compatability of various materials (in order of increasing compatability).

It should be appreciated that equipment specified for nitrogen service may not necessarily by suitable (or safe) for use with helium or neon, for example, because of the lower temperatures involved. Conversely, because of the very low density of liquid helium (125 kg m−3 ) helium dewars may be of particularly light internal construction, in order to minimize conduction paths and thermal mass, and may not therefore be suitable for denser cryogens such as nitrogen (804 kg m−3 ) or neon (1200 kg m−3 ). All but the very smallest vessels, which may be of moulded polystyrene, are of multi-wall design involving sophisticated insulation techniques employing vacuums and solid insulants either singly or in combination. Although reasonably robust, care must be taken to ensure that dewars are not subject to undue mechanical load or shock. In order to minimize thermal shock and also reduce splashing it is important that a dewar should be filled slowly. Wide necked or shallow containers should be partly covered during a transfer process to minimize splashing and loss of liquid. With the exception of the very smallest vessels decanting of liquid from a dewar should be accomplished by tilting, assuming the vessel is designed for this, or the use of a transfer tube. The majority of dewars are not designed to operate at pressures significantly above ambient (the

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References

887

Figure D12.0.6. Typical liquid storage dewar vessel and transfer tube.

exception being liquid cylinders) or vacuum. With the exception of wide-necked open dewars all vessels containing cryogens should incorporate a safety vent and pressure relief valve, bursting disc or similar device. Figure D12.0.6 shows a typical arrangement. Blockage of the neck or vent or a sudden loss of vacuum will lead to an increase in pressure. If the pressure cannot be released in a controlled manner the vessel is likely to explode. Plugging of a neck or vent is most likely to occur at high rates of boil-off. Most plugs will be of ice but in the case of dewars containing helium, hydrogen or neon it is possible for air to freeze solid. Fortunately the rate of pressure increase within a plugged dewar is usually quite slow provided the insulation has not been damaged. In many instances the blockage may be cleared by melting the plug using a hot tube or rod. Partial loss of vacuum will lead to an increased rate of boil-off but may also result in air being condensed in the vacuum space of a multi-wall dewar. As the dewar warms up the trapped liquid will evaporate producing vapour which cannot escape rapidly enough to prevent an increase in pressure sufficient to burst the vacuum shell. The vacuum space of vacuum-insulated dewars and vessels should therefore be protected by a bursting disc or similar failsafe pressure relief device. References ASTM1 Committee G4 Design guide for oxygen ASTM Committee G4 Guide for evaluating metals for oxygen service ASTM Committee G4 Guide for evaluating non-metallic materials for oxygen service BCGA2 Code of practice (1984) A method for estimating the offsite risks for bulk storage of liquefied oxygen BCGA Code of practice CP19 (1990) Bulk liquid oxygen storage at users premises BCGA Code of practice CP20 (1990) Bulk liquid oxygen storage at production sites Barron R F 1985 Cryogenic Systems (Oxford: Oxford University Press) British Cryogenics Council3 Cryogenics Safety Manual —A Guide to Good Practice British Cryogenics Council Symposium 1988 Safe storage and handling of cryogenic liquids Cryogenics 28 December BS4 5355 Specification for filling ratios for liquefied and permanent gases BS 5429 Code of practice for safe operation of small-scale storage facilities for cryogenic liquids BS 6364 Specification for valves for cryogenic service BS 7777 (four parts) Flat-bottomed, vertical, cylindrical, storage tanks for low temperature service

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BS 3N 100 General design requirements for aircraft oxygen systems and equipment Health and Safety Executives5 HS(R)6 (1989) A guide to the health and safety at work act Health and Safety Executive HSE8 Fires and explosions due to misuse of oxygen Health and Safety Executive HSE GS5 (1995) Entry into confined spaces The Factories Act 1961, Amendment SI 1989/2169 Pressure systems and transportable gas containers regulations Health and Safety Executive HS(R)30 A guide to the pressure systems and transportable gas containers regulations (1989) EIGA6 Technical Note 23/79 Periodic inspection and testing of cryogenic pressure vessels EIGA Document 16/85 Liquid oxygen storage installations at user premises EIGA Document 33/87 Cleaning of equipment for oxygen service guidelines Wigley D A 1978 Materials for Low Temperature Use, Materials Engineering Design Guide (Oxford: Oxford University Press)

Addresses

American Society for Testing & Materials (ASTM), 1916 Race Street, Philadelphia, PA 19103-1187, USA British Compressed Gas Association (BCGA), 14 East Links, Tollgate, Eastleigh, S053 3TG, UK 3 British Cryogenics Council (BCC), c/o Institute of Cryogenics, University of Southampton, Southampton, SO17 1BJ, UK 4 British Standards Institution (BSI), Linford Wood, Milton Keynes, MKI4 6LE, UK 5 Health and Safety Executive (HSE), Baynards House, 1 Chepstow Place, Westboume Grove, London, W2 4TF, UK 6 European Industrial Gases Association (EIGA), Avenue des Arts, 345 BTE16, Brussels, B1040, Belgium 1 2

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El Cryogenic fluid-level indicators

Juan Casas

E1.0.1 Introduction Cryogenic fluid-level indicators are necessary for monitoring and controlling the remaining quantity of the cryogen contained inside a vessel. In superconducting applications the cooling power is usually provided by either helium or nitrogen and only level sensors that apply to these fluids will be covered in this chapter; often the same measuring techniques can be used for other cryogens like argon, hydrogen, oxygen or neon. The most appropriate measurement technique depends on a number of factors such as the environmental characteristics, the maximum tolerated heat leakage per sensor and the cryogen physical properties. For instance, what would be adequate for measuring the liquid level in a laboratory cryostat might not be appropriate for space applications where gravity is zero and no cryogen refilling is possible. Also when working with devices that are very sensitive to magnetic fields (e.g. SQUIDs) the use of electrical probes should be considered with care. Most of the cryogenic fluid-level indicators rely on the measurement of the liquid—vapour interface to calculate the respective volume of each component. This interface should be deduced by measuring an appropriate physical parameter whose value depends strongly on whether the cryogenic fluid is in the liquid or vapour phase: typical parameters could be, for instance, the dielectric constant, heat transfer characteristics, acoustic or optical impedance, etc. At present many commercial devices exist for measuring the level of various cryogenic fluids, and for many applications it is difficult to justify the inhouse fabrication of level meters. E1.0.2 Float sensors This method is based on Archimedes’ principle and, apart from a direct observation of the liquid level when using glass cryostats with an observation slit, it is probably the simplest and cheapest measuring technique. The probe is made of a float attached to the end of a straight and rigid wire or tube (figure E1.0.1(a)), by using a suitable mechanism this method has been applied to monitor continuously the liquid level in large reservoirs (Townsley and Cockett 1971) although this type of apparatus should be considered as an indicator. For liquid helium this method is impractical because of its low liquid density: for comparison the density at saturation and at atmospheric pressure is ≈ 124 kg m− 3 for liquid helium versus ≈ 800 kg m-3 for liquid nitrogen.

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Figure E1.0.1. Liquid level sensors: (a) float; (b) vibrating membrane; (c) weight; (d) differential pressure; (e) capacitance; (f) superconducting.

E1.0.3 Vibrating membranes

This device is also known as a flutter or thumper tube. The measuring method is based on the thermoacoustic oscillations that can occur spontaneously in any tube that is closed at the hot end and open in the cold end (figure E1.0.1(b)); these oscillations are accompanied by a large transport of heat and if not taken into account when designing a cryostat they can cause an embarrassingly large liquid-helium boil-off. When used as a level meter a diaphragm made of thin rubber or plastic closes a cone attached to the hot end of a long, thin tube. When the tube is moved vertically inside the cryostat the oscillation frequency, which can be heard without the need for any conditioning device, changes abruptly when crossing the liquid—vapour interface (see figure E1.0.2, Conte 1970); the claimed accuracy for liquid helium is to within a few millimetres (Conte 1970).

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Figure E1.0.2. The oscillation frequency of a vibrating membrane probe versus depth.

E1.0.4 Thermal point level sensors These sensors are based on the difference with which heat is transferred from the sensor to the cooling medium depending on whether the latter is in the liquid or vapour state. For instance if a resistive thermometer is used as a point level sensor the bias should be chosen in such a way as to maximize the sensor temperature excursion observed when crossing the liquid—vapour interface: the sensor temperature is always higher than that of the environment because of the Joule self-heating. Recently there has been some interest concerning the use of arrays of point level sensors for the measurement of the cryogen content in space applications (Shirron et al 1994, Siegwarth et al 1992) where a micro-gravity environment is present and as a consequence it is impossible to predict the topology of the liquid—vapour interface. The sensors that have been studied include most types of cryogenic thermometer (carbon resistor, diode, platinum, etc) and their operation point has to be deduced experimentally through a calibration procedure. ‘Continuous’ level gauges can be made by using several sensors along a vertical line and many authors report correct operation of this technique in conventional Dewars. A major drawback of this type of level gauge is the need to calibrate and maintain an appropriate bias point that can have a strong dependence on external parameters such as the fluid pressure or temperature. For instance, the author has found it very difficult to obtain an appropriate bias point when using carbon resistors as wetting indicators inside a tube in which the gas and the liquid are at the same temperature and there is a varying vapour flow-rate. This is a very different operation regime to that found in conventional Dewars where the gas temperature increases with the distance from the liquid—vapour interface. Nevertheless it is important to note that silicon sensors have operated as designed in space applications (Shirron et al 1994) where no temperature gradients are expected. To obtain such a successful result a great amount of work was necessary in order to have a good understanding of the physical phenomena occurring at the sensor-liquid interface. This type of sensor could be considered in applications involving liquid helium and strong magnetic fields that prevent the use of superconducting probes. For helium it is also possible to make point sensors by using a short superconducting wire that remains in the normal state unless completely immersed in the liquid bath: commercial devices based on this principle exist and they are intended for measuring the level of liquid helium in storage Dewars that are accessible from the top. E1.0.5 Weight If the exact weight of the empty vessel is known it is in principle very easy to calculate the amount of remaining cryogen (figure E1.0.1(c)). Actually this method is difficult to implement because often the tare weight is high when compared with the net weight of the vessel contents; the measurement is

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furthermore perturbed by connecting pipework that may restrain the vessel movement and by wind forces when installed outdoors. E1.0.6 Hydrostatic gauges This is a relatively simple method that relies on the measurement of the differential pressure between the top and bottom parts of a container. It is widely used for large reservoirs, and it is a robust and reliable method because the sensing element is at room temperature and can be exchanged if necessary. The hydrostatic head ∆p inside a vessel can be calculated as follows

where ρ is the density that depends on the vertical position h, T is the temperature, p is the absolute pressure, g is the acceleration due to gravity and the subscripts refer either to vapour or liquid. Equation (E1.0.1) can be approximated as shown on the right when the cryogen is well separated into a liquid and a vapour fraction. For calculating the total mass of the remaining cryogen it is necessary to know exactly the internal dimensions of the reservoir. Equation (E1.0.1) does not include corrective factors that might be necessary to account for hydrostatic heads along the pressure-sensing capillaries. Care is necessary when designing these capillaries (ideally the temperature gradient should occur mainly along a horizontal line close to the bottom of the reservoir) and if not properly thermalized they can spoil the performance of a vessel. Depending on the operation conditions, a detailed calculation is necessary. If this is not done properly errors as high as 11% can occur when using a cryogen in the supercritical state (Zhang and Varghese 1992). Differential pressure measurements have also been applied to study the liquid fraction inside pipes transporting two-phase flow. For these applications cold pressure transducers are used and individual calibrations are probably necessary. E1.0.7 Continuous electrical liquid level gauges A continuous level gauge is a device whose output signal is proportional to the liquid level. As previously mentioned, a linear array of point sensors can approximate a continuous gauge at the expense of a certain discretization of the level measurement. E1.0.8 Capacitance This technique is based in the different dielectric constant of the fluid depending on whether it is in the liquid or vapour phase. The total capacitance C is given by

where ε (8.859 × 10−12 A s V−1 m−1 ) and εr are respectively the vacuum and relative dielectric constants, h is the height, r1 and r2 are respectively the outer radius of the central cylinder and the inner radius of the external cylinder and the subscripts l and υ refer respectively to the liquid and vapour. From equation (E1.0.2) and by using the appropriate relative dielectric constants (see table E1.0.1) it is possible to deduce the liquid fraction contained inside a vessel. This type of sensor is not adequate for measuring with helium because of the small difference between the relative dielectric constants observed in vapour and liquid. Commercial devices based on this principle are available from different manufacturers.

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Table E1.0.1. Vapour and liquid relative dielectric contant for various cryogenic fluids.

E1.0.9 Superconducting probes for liquid helium This sensor is made of a superconducting wire (used as a linear sensing element) enclosed inside a protective tube with perforations distributed longitudinally to allow for the wetting of the active element. The superconducting wire is fed with an electrical current whose value is chosen so that it is sufficiently low to recover the superconducting state when immersed in liquid helium. A heater is used to destroy the superconducting state at the top of the probe and the resistive state propagates along the superconducting wire until it reachs the liquid-vapour interface (see figure E1.0.1(f)). Commercial devices typically dissipate about 1–2 W m−1 in the gas and only a fraction of this power goes through thermal conduction into the liquid. Some electronic read-out units can also be configured to excite the wire at predetermined time intervals if the power dissipated in a continuous-operation mode is too large for the application. These gauges give accurate measurements in both normal and superfluid helium. However, measurement errors can occur for a saturated normal helium bath whose temperature is between the lambda point (2.17 K) and about 2.5 K. In this interval the level is often strongly underestimated but the exact explanation for this phenomenon will not be given here. Possible reasons for the propagation of the resistive region inside the liquid could be: (i) that the maximum nucleate boiling heat flux is reached (Van Sciver 1986) creating a gaseous envelope around the wire or (ii) a higher wire temperature at the liquid— vapour boundary is present thus enhancing the propagation of the normal zone into the bath. The higher temperature is the result of a poorer thermal heat exchange with the gaseous helium because of its lower density. Once the superfluid helium phase is reached the cooling power of the bath is enhanced and the superconducting state is fully recovered until the separation interface is again reached. For temperatures above 4.6 K level under-readings have also been reported and of course it is not possible to employ this technique above the helium triple point. A superconducting level gauge should not be used when operating in magnetic fields comparable with the wire critical field. The advent of hightemperature superconductors opens up the possibility to extend this measurement technique to other cryogens with higher saturation temperatures. E1.0.10 Conclusion Different measurement techniques are presented to determine the cryogen contained inside a vessel or cryostat. Most of them rely on the detection of the liquid—vapour interface. When a single—phase state is present a net weight measurement or a differential pressure technique can be employed to determine the mass of the cryogen contained inside a vessel. At present commercial fluid-level gauges are applicable to a variety of cryogenic fluids, and they have sufficient accuracy for most applications. For some types there is long operational experience, no particular calibration procedure with the working fluid is necessary and

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furthermore the cost is relatively low when compared with the effort necessary for developing an ‘in-house’ solution. References Conte R R 1970 Éléments de Cryogénié (Paris: Masson) Shirron P J, DiPirro M J and Tuttle J G 1994 Performance of discrete liquid helium/vapour and He-I/He-II discriminators Adv. Cryogen. Eng. 39 1105–12 Siegwarth J D, Voth R 0 and Snyder S M 1992 Resistive liquid-vapour surface sensors for liquid nitrogen and hydrogen J. Res. Nail Inst. Stand. Technol. 97 563–77 Townsley H and Cockett A H 1971 Instrumentation Cryogenic Fundametals ed G G Haselden (New York: Academic) Van Sciver S W 1986 Helium Cryogenics (New York: Plenum) Zhang B X and Varghese A R 1992 Calibration scenario of level gauge for pressurized cryogenic vessels Adv. Cryogen. Eng. 37 1471–7

Further reading Apart from the references given above the following books cover to some extent the subject of this chapter. Barron R F 1985 Cryogenic Systems (Oxford: Oxford University Press) Hands B A (ed) 1986 Cryogenic Engineering (New York: Academic) Timmerhaus K D and Flynn T M 1989 Cryogenic Process Engineering (New York: Plenum)

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E2 Gas flow-rate meters

Juan Casas and Luigi Serio

Flow-rate metering is an essential tool for many industries like public utilities, chemical or petroleum that use the information obtained from these devices for billing, control, indication or alarm purposes. Because of its enormous economic impact there is an important market for flow metering devices and a great effort has been invested to improve their accuracy. Many of the commercially available instruments can be used in cryogenic facilities as long as the operating temperature is close to ambient conditions. A recent and complete review of flow measurement has been published elsewhere (Spitzer 1991). As far as cryogenics is concerned, gas flow metering devices are used for: (i) assessment of cryogenic heat loads; (ii) process monitoring and control; (iii) detection of faults and wear on pumping machinery. Apart from oil contamination in the ambient temperature circuits, most of the gaseous fluids used in cryogenics are pure, clean and are well described by the ideal gas law. However, when a fluid rate is to be measured at cryogenic temperatures care is necessary in order to avoid the presence of two-phase flow or cavitation that makes the interpretation of data very difficult and can also compromise the reliability of the flow meter (Rivetti et al 1996). E2.0.1 Fundamentals of flow measurement When using flow metering devices it is important to become familiar with the various fluid† properties and with the equations related to the measurement of matter in motion. The main fluid properties for gas flow-rate meters are: temperature, pressure, density and viscosity, and are usually well described by the ideal gas law. It is important to note that the following equations are valid as long as the velocity gradients are not very important, the velocity is much lower than the velocity of sound in the medium and thermal exchange is ignored (Landau and Lifchitz 1971). The transport of fluid through a pipe is described by the equation of continuity, Bernoulli’s theorem and the Reynolds number Re. The equation of continuity states the conservation of mass

† A fluid is a substance, such as a liquid or gas, that can flow, has no fixed shape and offers little resistance to an external stress.

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where ρ is the fluid’s density, t the time and υ the velocity. When considering incompressible fluids flowing along a pipe the previous equation can be simplified and represents the conservation of volumetric flow, Q

where S is the pipe inner cross-sectional area, ~υ is the average velocity in the flow cross-section and the subscripts refer to different positions along the pipe, see figure E2.0.1.

Figure E2.0.1. Conservation of volumetric flow and Bernoulli’s theorem.

The equation of Bernoulli is the result of the conservation of energy along a line of flow and in the absence of viscosity is

where p is the pressure, g is the acceleration due to gravity and z is the vertical distance from a reference horizontal plane. At this point it is possible to introduce the concept of head or energy that is often used by engineers; in equation (E2.0.2) the first term is the velocity head (kinetic energy), the second is the pressure head (static energy) and the last one is the elevation head (potential energy), see figure E2.0.1. All flowmeters based on the measurement of differential pressure apply equation (E2.0.2) to infer the mass flow rate. When a real fluid flows through a pipe, energy is dissipated by irreversible processes, one of the main causes being the fluid’s viscosity that produces both longitudinal and transverse velocity gradients in the piping. At low velocities the flow is laminar with a parabolic velocity profile as shown in figure E2.0.2. When reaching a critical fluid velocity, turbulence appears and the velocity profile is to some extent squared up (figure E2.0.2). A dimensionless quantity called the Reynolds number Re defines whether the flow

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Differential pressure flowmeters

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Figure E2.0.2. Longitudinal velocity profiles for laminar and turbulent flow.

inside a pipe is in the laminar, transition or turbulent regime

where D is the internal pipe diameter, µ is the absolute viscosity and v = µ/p is the kinematic viscosity. For Reynolds numbers higher than 4000 the flow is turbulent, if it is lower than 2000 the flow is laminar and if the value is in between the flow may be turbulent or laminar and may change at random from one flow condition to the other. E2.0.2 Differential pressure flowmeters Differential pressure flow-rate meters are the most common measuring devices employed because of the large number of different types industrially available (orifice, Venturi, V-Cone, Pitot tube, etc), the reasonable cost and good accuracy and the intrinsic robustness (simple construction, no moving parts, external instrumentation and low maintenance). Their operating principle is based on the measurement of a differential pressure across a restriction. From Bernoulli’s equation (equation (E2.0.2)) we can derive that if the velocity (kinetic energy) decreases between two flow cross-sections, the pressure (static energy) increases accordingly. Since the equation of continuity for incompressible fluids states that the volumetric flow rate is constant (equation (E2.0.1b)), a change in section would give a change in velocity, hence a change in pressure. The volumetric flow rate as a function of pressure is given by

The volumetric flow rate Q is then proportional to the square root of the differential head or differential pressure. Equation (E2.0.4) overestimates the actual flow rate because irreversible processes are ignored; dimensionless correction factors are then necessary to improve the quality of the measurements and they are: (i) the discharge coefficient factor Cd and (ii) the gas expansion factor Y that is equal to unity for incompressible fluids. As a general rule the differential pressure read-out depends on the location of the pressure taps and as a consequence individual calibrations are necessary for optimum performance. Pressure taps that are perpendicular and flush with the pipe inner surface are used to measure the static head. Otherwise, if both

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the kinetic and static heads need to be measured, the pressure tap is parallel to the pipe with an opening opposite to the flow direction. E2. 0.2.1 Orifice flowmeters Orifice flowmeters (figure E2.0.3(a)) are probably the most commonly used flowmeters in cryogenic industrial applications because of their robustness, low cost and relatively good accuracy. They consist of a plate with an opening that can be concentric or eccentric depending on the cleanness of the fluid (i.e. a concentric orifice has impaired performance once there is some build-up of dirt at the plate, but is more accurate than an eccentric one).

Figure E2.0.3. Differential-pressure-type flow-rate meters: (a) orifice; (b) Venturi; (c) V-cone; (d) Pitot.

A fluid passing through an orifice sees its velocity increasing first and decreasing again after the restriction (equation (E2.0.1b)) and from equation (E2.0.2) the increase in kinetic energy (velocity) corresponds to a decrease in static energy (pressure) and vice versa. Because real gases are not perfect fluids some static energy is lost due to friction (permanent pressure losses) and this is a function of the ratio of the orifice bore d to the pipe diameter D otherwise called the beta ratio

the permanent pressure losses can be expressed as

where ∆p is the pressure drop across the restriction. For a real fluid equation (E2.0.4) thus becomes

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Variable-area flowmeters

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The discharge coefficient is given by the constructor and is deduced from laboratory tests. These flowmeters have an accuracy† of 2–5% of the full scale and the rangeability is limited by the accuracy of the differential pressure measuring device employed, as for other differential-pressure-type flowmeters. Flow rangeability of 10:1 can be achieved by using ‘smart’ differential pressure transmitters, i.e. whose output is proportional to the square root of the measured differential pressure and the span automatically adjusted according to the measured pressure. E2.0.2.2 Venturi flowmeters Venturi flowmeters are based on the same principle but are made of a relatively long passage with smooth entry, giving the advantage of reduced permanent pressure losses and better working performances in dirty fluids (figure E2.0.3(b)). Of course this makes the flowmeter more expensive, but accuracy† down to 1% of the full scale can be achieved. E2.0.2.3 V-Cone flowmeters In this flowmeter the restriction in the fluid stream is achieved by the insertion of a V-shaped cone element at the centre of the pipe (figure E2.0.3(c)). The beta ratio in this case is

and the differential pressure generated is lower than an orifice flowmeter comparable in size, beta ratio and flow rate. Another important advantage is that the V-shaped cone acts also as a flow conditioner, increasing the range of Reynolds number exploitable and reducing the length of upstream and downstream straight piping. E2.0.2.4 Pitot tube A Pitot tube is a device that measures the local fluid velocity by pressure difference between the static pressure and the total pressure measured at an impact port (figure E2.0.3(d)). The total pressure is the sum of the kinetic energy transformed into static energy and the static pressure. This type of flow meter is very sensitive to nonuniform velocity profiles; however, it allows local exploration of flow velocities in the vein. E2.0.3 Variable-area flowmeters This is one of the simplest and cheapest types of meter, it is made of a float and a tube whose internal diameter increases downstream of the flow (figure E2.0.4). The measurement principle is similar to that of a differential pressure flowmeter but in this case the differential pressure is constant and the flow rate is deduced from the position of the float along the measuring tube. Floats tend to be unstable on low-pressure gas service and when ordering it is then imperative to indicate the lowest operating pressure. The accuracy (as defined earlier) is usually stated as a function of the full scale and it varies between ±5% and ±20% depending on size and on whether a calibration is being used. Although electrical read-outs of the float position can be added as accessories, this type of apparatus should rather be considered as a flow indicator.

† Accuracy is intended as real performance in normal working conditions.

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Figure E2.0.4. Variable-area flow-rate meter.

Gas flow-rate meters

Figure E2.0.5. Positive Displacement flow-rate meters: (a) rotating piston, (b) sliding vane.

E2.0.4 Positive displacement flowmeters A positive displacement flowmeter is a device that divides the incoming flow into a number of known volumetric segments (figure E2.0.5). The main components are a housing, a moving mechanism (different types are piston, sliding vane, disc, etc) and a counter. The principal parameters are the fluid viscosity that seals the clearance between the moving parts and the housing, as well as the temperature and pressure that affect the precision via dimensional variation of the housing and rotating mechanism. This type of meter is useful for totalling the gas consumption in small installations. The size of the instrument should be chosen according to the expected flow rates and the information can be used to cross-check data from other types of measurement device like thermal or differential pressure meters. Typical accuracy (as defined earlier) can be as low as 1% of the full scale and the measuring range is typically 5:1, but can be increased up to 10:1 with a lower accuracy because of nonlinearity at high and low flow rates. E2.0.5 Thermal mass flowmeters Thermal mass flowmeters have been widely used in numerous applications ranging from space applications to pollution monitoring systems and nowadays the semiconductor manufacturing industry to measure and control accurately small flows of clean gases. In cryogenics such flowmeters have direct application in liquid mass flow-rate measurements, although their use should be considered with care because of their potential impact on the total heat load budget. They are also widely employed to make boil-off measurements of cryogenic fluids in their gaseous form at ambient temperature. There are two such types of flowmeter, both based on the variation of the thermal characteristics of the measured fluids: those that measure the heat loss from a heated element (resistance wire, thermocouple, etc) to the flow stream and those that measure the temperature rise of a fluid passing over a heated element (figure E2.0.6). The first type of flowmeter is based on the King equation for a hot wire

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Thermal mass flowmeters

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Figure E2.0.6. Thermal mass flow-rate meter.

where q is the rate of heat loss per unit of time, ∆T and d are respectively the element’s mean temperature elevation and diameter, κ is the conductivity of the fluid stream and Cυ the specific heat of the fluid at constant volume. The advantages of this type of sensor are compactness, few power requirements and fast response times, but on the other hand they are nonlinear and depend on the gas thermal conductivity, temperature and velocity profile. The second type deduces the mass flow rate m• from the fluid’s enthalpy variation ∆H after absorption of a heating power W

where Cp is the specific heat of the fluid at constant pressure and ∆T the temperature increase across the heating element. Usually the heating and sensing elements are placed outside the thin wall of a capillary tube carrying all or only a well known fraction of the fluid (by using a suitable bypass element) thus allowing a wide range of sizes (10 sccm to 15 000 slpm, which are the units employed in most of the manufacturers’ data sheets, or in SI units about 0.16 × 10−6 to 0.25 m3 s−1 ). The small diameter and thin wall capillary tube is necessary to heat the fluid homogeneously and to reduce the response time, the main drawback being the risk of obstruction while using unclean fluids. The use of heating and sensing elements inserted in the fluid stream is not recommended because of concerns of leak tightness and risk of explosion with hazardous gases. The sensing element is often a thermocouple, while the heating is obtained by either wrapping some insulated resistive heater or by inductively heating the capillary. A more sophisticated and compact version is achieved by etching the capillary on an integrated chip whose semiconductor materials act as a temperature sensor. The flowmeter body acts as a heat sink keeping both ends of the capillary at ambient temperature. Without flow a symmetrical temperature gradient is established across the heater and as the flow increases the temperature profile is deformed and a measurement of ∆T gives the mass flow. Thermal mass flowmeters in contrast with most of the other devices have the advantage of measuring directly the mass flow rate which determines energy content. Many of the commercial devices operate at

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constant ∆T by adjusting the heating power and as a result the time response is improved. In general, thermal flowmeters are comparatively expensive, they give accurate and repeatable results if they are calibrated with the working fluid and are operated in a clean fluid stream. Care is necessary because the read-out can be affected by oil contamination coming from pumping machinery. Also many of these flowmeters are sold as a single package in conjunction with a flow control valve and a feedback circuit to deliver a programmable steady flow rate.

E2.0.6 Turbine flowmeters A turbine flowmeter is probably the most accurate flow-rate metering device and it is sometimes available as a certified instrument, i.e. an instrument traceable to a national institute of standards. It is made of a rotor mounted on a bearing mechanism, a housing and a device for measuring the rotational frequency that is proportional to the flow rate (figure E2.0.7). The turbine blades are designed to operate in a ‘freespin’ mode and to have a minimal sensitivity to changes in Reynolds’s number, resulting in an inefficient blade design with respect to power producing turbines that operate only in a much narrower rotational velocity range. Because a turbine flowmeter has a rotational component it is also very sensitive to swirling, therefore care is necessary to condition the fluid flow. Typically at least ten pipe diameters are required upstream of the meter, keeping in mind that swirl has been measured in a pipe after the fluid has travelled for more than 100 pipe diameters. For gas service this type of device is capable, after calibration, of having an accuracy (as defined earlier) down to 0.3% of reading and a repeatability of 0.1%. These instruments have a typical measuring range of 10:1 that on request can be increased to 25:1, or even higher, but the data on the extended range are valid only for the exact viscosity and temperature of the fluid that was used for the calibration, and the increased rangeability generally affects the output linearity.

Figure E2.0.7. Exploded view of a turbine flowmeter. Courtesy of Ketema Inc. (previously Ametek), Schutte & Koerting Division.

The performance of a turbine flowmeter is often represented by the K factor, which is defined as the turbine rotational frequency f divided by the flow rate, versus the Reynolds number. For achieving the ultimate performance it is necessary to take into account the effects of temperature on the meter body; this

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is done by including in the calibration data the Strouhal number

where D0 is the meter diameter at the reference temperature T0 , α is the linear expansion coefficient of the meter body (typically 0.003% K−1 for austenitic steel). Then the turbine meter calibration data can be presented as a function of St versus Re St , where the former quantity can also be expressed as f D 2/v. E2.0.7 Vortex flowmeters A vortex flowmeter is based on the vortex-shedding mechanism that occurs when a flowing fluid hits an obstacle, the results being the creation of vortex swirls that separate from the object on alternating sides (figure E2.0.8); an illustration of this effect is the waving of a flag due to the wind. A vortex- shedding flowmeter is made of a bluff or impact body and a vortex detector. The detector should be capable of measuring the vortex creation frequency; this is the most difficult part to build and most of the designs are proprietary and, depending on the manufacturer, they employ a piezoelectric crystal, a sonic beam or the torque measurement of the bluff body, etc. The ratio of the frequency of the shedding f to the velocity of the fluid is given by the Strouhal number (equation (E2.0.11)) where D in this case is the width of the bluff body. As for the turbine flowmeter, it is common to receive from the manufacturer a K factor, the units of which are in this case the shedding frequency over flow rate.

Figure E2.0.8. Vortex-shedding flowmeter: operation principle.

Vortex flowmeters should be used in the range of Reynolds number specified by the manufacturer (the minimum boundary being typically above 10000) and after calibration a 0.75% accuracy (as defined earlier) on the reading can be obtained with a typical measuring range of 10:1. Among the advantages of the vortex meter one can mention that there is very little maintenance because there are no moving parts and the pressure loss is relatively low. However, during installation care is required in order to guarantee a smooth inner surface in the hydraulic connections, otherwise vortex shedding can be created at any discontinuity affecting the read-out of the vortex detector. E2.0.8 Target flowmeters A target flowmeter is based on the measurement of drag created when a moving fluid crosses an obstacle: the obstacle or target is typically a circular disc mounted concentrically in a pipe (figure E2.0.9). The upstream face of the target sees a higher pressure than the opposite face since there the flow is abruptly stopped. The result is a force Ft , applied on the target; the force can be measured by different means such as by a force balance mechanism, strain gauges, etc. The mass flow rate m• is given by

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Figure E2.0.9. Target flowmeter.

where K is a constant (no relationship with the K-factor introduced previously) which is typically determined experimentally. A target flowmeter produces a relatively high pressure loss and its accuracy (as defined earlier) is lower when compared with other devices, varying from 2% to 5% of the full-scale reading with a typical rangeability of 5:1. Maintenance requirements are very low because there are no moving parts and it is only necessary to control the target geometry from time to time. E2.0.9 Flow conditioning and installation Even the best type of flowmeter, accurately calibrated by a certified laboratory, can give poor results if it is not installed and operated properly. Most of the flowmeters measure a flow velocity that can vary greatly if the flowmeter is not immersed in a uniform velocity profile that depends not only on the Reynolds number but also on the roughness of the inside pipe wall. Bends, changes in diameter, rough surfaces, pipe fittings and welding can produce a distortion and/or swirl of the velocity profile upstream and downstream. Upstream and downstream straight pipes of equal diameter, whose length is expressed as a multiple of the inside pipe diameter, are suggested by the manufacturer of the flowmeter together with the use of piping with smooth inner surfaces. Care is necessary during welding operations in order to limit welding burrs that in principle should be reamed off. Some flowmeters require less attention because their construction is capable of reshaping a nonuniform velocity profile (i.e. V-cone), others can give completely wrong results in a nonuniform velocity profile (i.e. thermal mass or turbine flowmeters). Flow conditioners should be considered in case the required pipe conditions cannot be achieved, taking into account that the pressure losses would increase and that to obtain optimum performance clean fluids are a must. In cryogenic installations the mass flow rate can depend on the gas temperature: it is therefore advisable to perform absolute temperature and pressure measurements of the metered fluid to take into account variations of the fluid characteristics that might yield errors as high as 20% in the measured mass flow rate. E2.0.10 Summary In this chapter we have tried to list what the authors think are the most practical gas flow-rate meters for cryogenic installations. Table E2.0.1 summarizes qualitatively the main features of the flowmeters presented and it can be used as a guide to optimize the selection procedure. If ultimate performance is desired a turbine-type flowmeter is the flowmeter of choice with the drawback of higher maintenance requirements and cost. Otherwise, for lower cost and maintenance, differential flowmeters can be used with lower accuracy and rangeability. For large cryogenic facilities with a large helium inventory, fluid management is a critical issue and of course fluid-rate metering devices are widely used. Most of the

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References

907

Table E2.0.1. A qualitative comparison of various gas flow-rate meters. The ultimate accuracy can be obtained only if the instrument is installed and maintained according to the manufacturers’ specifications and is individually calibrated.

instruments are of the differential pressure type because of the requirements of industrial robustness and low maintenance. When nonsteady flow conditions are present (i.e. pulsating flow) mass flow measurement data are extremely difficult to interpret because apart from equation (E2.0.1a) the theory introduced previously is not valid as it ignores the fluid compressibility. There also exist other types of metering device that have not been included because they are less frequently used, for instance the Coriolis flowmeter which can be applied to gas mass flow metering where the process pressure is above 5 bar (5 × 105 Pa) and also some more exotic techniques such as laser Doppler velocimetry. References Landau L and Lifchitz E 1971 Mécanique de Fluides (Moscow: Mir) Rivetti A, Martini G and Birello G 1996 LHe flowmeters: state of the art and future developments Adv. Cryogen. Eng. B 41 1789–96 Spitzer D W 1991 Flow Measurement (Research Triangle Park, NC: Instrument Society of America)

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E3 Pressure measurements of cryogenic gases and liquids J A Zichy

The pressure of cryogenic gases and fluids is, besides the temperature, the most important variable of state to be determined when a piece of cryogenic apparatus is operated. Furthermore, vapour pressure thermometry is used to determine the temperature of cryogenic liquids (N2 , 3He and 4He) in containers. Since pressure is normalized force (force/unit area), it can be measured only by the displacement caused by the applied force. Most pressure transducers can be used either as an absolute or as a differential gauge. In the latter case the relative displacement caused by the pressure change is metered. Several survey articles have been published on pressure measurement, in which one may find detailed descriptions of the different transducers used mainly for laboratory purposes (Adams 1993, Arvidson and Brennan 1976, Pavese and Molinar 1992). E3.0.1 Choice of measuring method The displacement caused by pressure may be measured either at room temperature with conventional pressure transducers or at cryogenic temperatures with gauges developed in the last decade. The decision concerning which method should be applied depends on the required precision and stability of the transducer as well as on the requirements for frequency response and on the geometry of the measuring apparatus. E3.0.1.1 A transducer at ambient temperature The simple conventional method is to run a capillary pipe—i.e. a long, thin-walled tube of a few millimetres inner diameter—from the point where the pressure measurement is desired, to a convenient location at ambient temperature, where a suitable pressure-measuring device is attached to the capillary pipe. The measuring device may have an electrical output like a conventional pressure transducer or, for instance, the familiar Bourdon gauge may be used. Such a system works well in most applications; however, this approach may introduce, via the capillary pipe, additional problems into the cryogenic system. Measuring the pressure at room temperature has the following advantages: (i) access to the transducer during measurement (ii) the calibration is repeatable at any time

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(iii) the transducer can be shielded from irradiation (iv) it can be located outside magnetic fields. As mentioned, the capillary pipe connecting the cold volume with the transducer at room temperature introduces the limitations of this measuring method: (i) (ii) (iii) (iv)

heat leakage thermoacoustic oscillations leakage or blockage limited frequency response.

To reduce heat influx a thin-walled pipe should be used. To avoid thermoacoustic oscillations special attention should be paid to the installation of the capillary pipe. It is beneficial to anchor the pipe at an intermediately cold surface before it is routed along the shortest path to ambient temperature. Leakage of helium to the surrounding volume is avoided by using hard-soldered stainless steel pipes. A blockage can be avoided by blowing out the pipe with clean helium, and purging the helium system several times. A transducer at room temperature is useful to observe the steady-state pressure; however, rapidly varying (v > 10 Hz) pressure introduces large distortions and delays in both the amplitude and shape of the signal. This effect is qualitatively explained as follows. The signal at ambient temperature is the convolution of the pressure wave and of the temperature-dependent velocity of sound in the capillary pipe. Unfortunately the original shape and amplitude of the pressure signal cannot be calculated reliably, because the temperature gradient along the capillary pipe is not well enough known, and therefore the variation of the velocity of sound with temperature cannot be deduced (Haug and McInturff 1990). Some simplified equations to predict the response of the transducer to step inputs and to calculate the effect of temperature gradients along the tube axis have been developed for pneumatic systems (Hord 1967, Howard 1959). Installing an appropriate temperature sensor at the cold end of the capillary pipe gives the opportunity to measure two variables of state simultaneously (integrated sensors). The lead wires of the temperature sensor can be routed through the capillary pipe to the signal conditioning at room temperature, and a conventional vacuum-tight feed-through can be used. For the wiring it is recommended that wires which reduce changes in lead resistance with temperature are used. However, the use of manganin sensor leads limits the accurate reading of low-level signals. E3.0.1.2

A transducer at cryogenic temperatures

The shortcomings of warm transducers may be avoided with the so-called in situ pressure transducers, which are either directly attached to the cold component or submerged into the cryogenic fluid. Since the widely used cryogenic pressure transducers were developed for room temperature applications they represent a low-cost approach to pressure measurement. A measurement is performed by sensing a displacement caused by a force or pressure. Therefore cryogenic pressure and displacement gauges are very similar, using common subassemblies. Pressure transducers have the following main components: (i)

first a sensing element (diaphragm, strain gauge, crystal, etc) to convert the pressure into a displacement (ii) then a mechanical or electrical link to transmit the displacement and possibly to amplify it (iii) and finally a converting device to produce an electrical signal. Cryogenic pressure transducers eliminate the previously listed shortcomings of the measuring system at ambient temperature; however, they introduce new sources of error: (i) the calibration of the transducer at cryogenic temperatures is indispensable (ii) temperature effects influence the calibration curve (iii) radiation may alter the calibration of the transducer

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(iv) the resistance of the transducer may introduce some Joule heating (v) the accuracy is limited because of magnetic induction (vi) thermal cycling of the transducer. In order to measure the pressure it is necessary to determine its relationship to the parameter being measured, i.e. the transducer must be calibrated. Up to now each laboratory calibrated its own cryogenic pressure transducers at low temperature as a function of different parameters. Recently, however, well calibrated cryogenic pressure transducers have become commercially available. Several test facilities have been proposed and built (Adams 1993, Arvidson and Brennan 1976, Boyd et al 1990, Clark 1991, Cerutti 1983, Kashani et al 1990, Pavese 1984, Pavese and Molinar 1992, Pavese et al 1980). Capacitive (Adams 1993, Cerutti 1983) and piezo-resistive (Boyd et al 1990) transducers were extensively measured; they are reliable in the measured temperature range and have a well behaved, reproducible output signal. In a carefully designed calibrating procedure the following parameters should be measured: at several temperatures the calibration factor, the shift of the zero-signal and of the thermal sensitivity; the repeatability, the nonlinearity, the hysteresis and the long-term stability of the signal; the magnetic field effects at the operating temperature and aging by thermal cycling of the transducer between ambient and operating temperature. The electric signal of the transducers depends not only on the pressure, but also on the effect of the temperature on its various components. The so-called steady-state temperature effect is seen on the change of sensitivity and on the zero shift of the calibration curve. The sensitivity change is caused by the different thermal expansivity of the various materials used to make the transducer and by the change of the elastic modulus of its components with temperature. The zero shift is partially due to the differences in thermal expansivity of the various materials; however, the dimensional change of the components with temperature is a larger effect. The sensitivity change can be compensated electrically, the zero shift by proper design of the transducer. A further cause of error could be a thermal shock, creating a temperature gradient in the transducer and changing thereby simultaneously the sensitivity and the zero shift of the output signal. Therefore it is essential to avoid thermal shocks in order to perform valid measurements. Another well known effect, which must be taken in account, is the change of the signal reproducibility after the transducer has been thermally cycled. This thermal effect, together with the influence of irradiation and of magnetic induction on the output signal, must be evaluated individually to estimate the accuracy of a measurement. E3.0.2 Measuring devices E3.0.2.1 At room temperature The most advanced systems measure the capacitance change created by a flexible membrane, placed between two pressurized chambers. For an absolute measurement one chamber is sealed and the second is connected by a pipe to the cold volume where the pressure must be metered. Additional electronics converts the capacitance change to a standard output signal. The transducer must be calibrated. However, it can be located in an accessible position, far away from any disturbing influences. E3.0.2.2 At cryogenic temperature Capacitive transducers Many of the in situ cryogenic pressure gauges are based on a diaphragm and a capacitive measurement system to detect its deflection (Jacobs 1986). They consist of a metal body, copper-beryllium or metal-coated sapphire electrodes, insulating material and adhesives. The Cu-Be transducers are considered by most experts as too delicate for industrial applications; however, they are successfully applied in the laboratory for gas and helium-3 melting curve thermometry. To detect the capacitance changes the capacitor

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is either used as the frequency-determining element in an oscillator (Thornton 1978) or it is compared in a capacitance bridge to a reference capacitor (Adams 1993). The capacitance changes at cryogenic temperatures are very small, therefore the signal must be amplified near to the transducer, preferably using cold electronics. Some future work must focus on the development of temperature-compensated signal conditioning for capacitive transducers to minimize the temperature dependence of the sensitivity and the influence of thermal cycling. A sensitivity of 1.51 pF MPa−1 was achieved with a multilayer capacitive transducer (Lawless et al 1985). Below 10 K the output of this transducer is independent of temperature and magnetic field. The measuring error is, over a range of some hundreds of megapascals, for a one-point pressure calibration, less than ±0.1% and the resolution is of the order of some hundreds of pascals. A small absolute pressure transducer, 0.61 mm thick and 8 mm diameter, has been developed, with 5 pF MPa−1 sensitivity and 1 Pa resolution at 3.5 MPa (Griffioen and Frossati 1985). It is built from two sapphire discs soldered together with glass in such a way that between them an evacuated space is left. The platinum electrodes are sputtered onto the outer surface of the discs, as shown in figure E3.0.1. The dielectric constant of the measured medium does not affect the result, since the transducer is pressurized from outside. No thermal shift was detected during cooling it down to 1 K; thermal cycling caused a pressure change of 4 Pa at 3.5 MPa, see the capacitance versus pressure curve in figure E3.0.2. A similar capacitive transducer, with Al electrodes and capable of measuring differential pressure too, has a sensitivity of better than 0.1 pF Pa−1 (Polunin et al 1986). A novel capacitive pressure transducer with a resolution of ∆p/p = 8 × 10−8 was recently developed, using silicon micromachining techniques, see figure E3.0.3. This technique enables one to choose the gap between the electrodes and fabricate it precisely, allowing the tailoring of the shorting pressure and of the dynamic range of the transducers (Echternach 1996).

Figure E3.0.1. Sketch of a capacitive pressure transducer (Griffioen and Frossati 1985): (a) platinum electrode; (b) the sapphire discs; (c) electric contacts; (d) ducts for the electrodes; (e) glass soldering of the discs.

Inductive (Validyne type) transducers The transducer is usually of an all-welded construction with a stainless steel diaphragm. The diaphragm is clamped between two metal blocks, each containing an inductance coil. The balanced diaphragm provides equal reluctance for the magnetic flux path of both coils. When a pressure difference deflects the diaphragm the magnetic reluctance changes thereby varying the inductance of the coils. The coils are connected in a bridge circuit, excited by an a.c. current, and the output voltage of the circuit is proportional to the applied pressure difference. The calibration curve is nearly linear, but its sensitivity and zero shift change with the temperature. The characteristics of the transducer remain linear and repeatable to better than ±1% after temperature cycling, over-pressurizing and mechanically vibrating the transducer (Kashani et al 1988). Detailed calibration tests qualified this type of transducer for application in space (Kashani et al 1990).

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Figure E3.0.2. Characteristic behaviour of the capacitance versus pressure in a capacitive transducer (Griffioen and Frossati 1985).

Figure E3.0.3. Sketch of the silicon micromachined capacitance pressure transducer (Echternach 1996).

Piezoelectric transducers When pressure oscillations with frequencies up to 100 kHz must be measured piezoelectric transducers are the right choice. They must be calibrated at the operating temperature but their characteristics are nearly

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linear. They are not suited for static measurements, because of the slow electric discharge of the crystal. Piezoresistive transducers The transducer consists of four strain gauges, mounted on a diaphragm and connected as a full resistive bridge (Bukovich et al 1977, Kraft et al 1980, Walstrom and Maddocks 1987). A sketch of a typical transducer is shown in figure E3.0.4. The transducer converts a small pressure change into a large change of resistance, which in turn is easy to measure by applying a constant current source of about 1.0 mA. Since the transducer is small, the stresses caused by differential thermal contraction are also small. If the transducer is encased in an iron can, it can only be used in a magnetic field if an adequate correction is applied. Up to now this transducer was mainly used for laboratory purposes, because the electrical contacts of the silicon strain gauges are sensitive to heat and very delicate to handle.

Figure E3.0.4. Sketch of a typical piezoresistive transducer: (a) the implanted Si resistance, (b) Si substrate; (c) metal can; (d) electric lead. The reference pressure is p0 and the pressure to be measured is p.

Some years ago rugged and calibrated cryogenic pressure transducers became available (produced by Pressure Systems Inc. in Virginia, USA), offering a solution to the problems concerning reliability and reproducibility of pressure measurements. The transducers are based on a silicon piezoresistance sensing element. The transducers are also small and robust with low hysteresis, nearly linear characteristics and good repeatability. The transducers, covering the pressure range from 7 to 3500 kPa, are calibrated in the temperature range 1.5–373 K. The measurements are repeatable to within ±0.1% and ±0.5% of full scale depending on the pressure range. Since the hermetically sealed pressure transducer dissipates less than 6 mW power it can even be used by submerging it within the cryogenic fluid. The output voltage versus pressure, and the normalized sensitivity of these industrial-type piezoresistive transducers are shown in figures E3.0.5 and E3.0.6 (Boyd et al 1990). It would be expected that the effect of a magnetic field on four closely matched resistors should be about the same as long as no iron casing is used. Tests in magnetic fields perpendicular to the transducer confirmed this assumption. In a magnetic field of <1.0 T only a small effect of 0.2% was detected on the output signal of the bridge. The effects of irradiation have also been studied on this type of transducer, and for radiation doses up to 108 rads, the variation of the pressure signal remained within the calibration accuracy (Juanarena and Rao 1992). By thermally coupling the pressure transducer to the cryogenic vessel the accuracy of the measurement is increased in a wide temperature range to better than 0.5% of the maximum pressure. Additionally the heat leak, associated with the sensing lines extending to ambient temperatures, is also significantly reduced (Clark 1992). Recently, integrated pressure—temperature transducers have been introduced on the market for use in the temperature range 1.5–375 K (Juanarena and Rao 1992). These transducers measure simultaneously two thermodynamic variables of state and they are sufficient to calculate any other thermodynamic quantity at a given moment. As a basic temperature-sensing element, an Si diode is combined with the previously discussed calibrated silicon piezoresistance into a dual transducer. However Si diodes are known to be

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Figure E3.0.5. Output voltage versus pressure for a 1000 kPa absolute, piezoresistive transducer at 4.2 and 300 K (Juanarena and Rao 1992).

Figure E3.0.6. Normalized sensitivity—S (T)/S (300)—versus temperature for a 1000 kPa absolute, piezoresistive transducer (Juanarena and Rao 1992).

highly effected by strong (>1.0 T) magnetic fields, so the dual transducer is not recommended for use in such an environment. Fibre-oo ptic transducers A very promising new technique, developed for high-pressure measurements, is the fibre-optic transducer, recently successfully tested at 77 and 4.2 K (van Oort and Kate 1993). The sensing element is a Fabry—Pérot interferometer, formed by two cleaved fibre surfaces close to each other. The light reflected from these two surfaces interferes in the fibre lead. The readout system utilizes either monochromatic or white laser light. The latter readout provides an absolute strain and pressure measurement. The transducer is not sensitive to magnetic induction up to about 14 T, and its sensitivity to temperature changes can be compensated. The pressure sensitivity between 0 and 150 MPa is about 25 nm MPa−1, the maximum resolution 20–30 nm and when cycling the pressure the reproducibility is of the order 2–3 MPa. The drawback of commercially available transducers is that they must be calibrated at low temperature by the user and they must be installed together with an appropriate temperature transducer in order to perform accurate measurements. The irradiation effects of fibre-optic transducers must be studied further, in order to gain a wider field of application.

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E3.0.3 Thermometry with pressure transducers Thermometry with in situ pressure transducers has recently been extended from the kelvin into the millikelvin range (Pobel 1992). Constant-volume gas thermometry is based on the relation for an ideal gas; however, a correction must be applied, because real gases deviate from the ideal. Vapour pressure thermometry relies on the behaviour of the gas phase above a cryogenic liquid and on the Clausius— Clapeyron equation. The advantages of this type of thermometry are that no thermomolecular corrections are needed and higher resolution and stability can be achieved with a carefully calibrated transducer. Melting pressure thermometry has been developed in the last decade for 3He and recently also for 4He (Adams 1993). The 3He thermometry is used now as an interim standard to compare data of different laboratories. The melting pressure, described by the Clausius—Clapeyron equation, exhibits a pronounced temperature dependence for 3He, in the range 1 mK < T < 320 mK. In addition its melting curve has three easily verifiable, well defined fixed points below 2.5 mK, which may also be used for pressure calibration (Greywall 1986). The 4He melting curve can be used in a similar fashion for precise temperature measurements at around 1 K (Adams et al 1987).

References Adams E D 1993 High-resolution capacitive pressure gauges Rev. Sci. Instrum. 64 601-11 Adams E D, Tang Y H, Uhlig K and Haas G E 1987 Thermodynamics of freezing and melting of 4He in Vycor J. Low Temp. Phys. 66 85–98 Arvidson J M and Brennan J A 1976 Pressure measurement at low temperature Advances in Instrumentation vol 31(II) (Pittsburgh, PA: ISA) pp 607/1–9 Boyd C, Juanarena D and Rao M G 1990 Cryogenic pressure sensor calibration facility Advances in Cryogenic Engineering vol 35, ed R W Fast (New York: Plenum) pp 1573–81 Bukovich R A, Smith J L Jr and Tepper K A 1977 A bounded-strain-gauge pressure transducer for high-speed liquid helium temperature rotors Advances in Cryogenic Engineering vol 23, ed K D Timmerhaus (New York: Plenum) pp 140–5 Cerutti G, Maghenzani R and Molinar G F 1983 Testing of strain-gauge pressure transducer up to 3.5 MPa at cryogenic temperatures and in magnetic fields up to 6 T Cryogenics 23 539–45 Clark D L 1991 Thermally-coupled cryogenic pressure sensing Application of Cryogenic Technology vol 10, ed J P Kelley (New York: Plenum) pp 109–16 Clark D L 1992 Temperature compensation for piezo resistive pressure transducers at cryogenic temperatures Advances in Cryogenic Engineering vol 37, ed R W Fast (New York: Plenum) pp 1447–52 Echternach P M, Hahn I and Israelsson U E 1996 A novel silicon micromachined cryogenic capacitive pressure transducer Advances in Cryogenic Engineering vol 41/13, ed P Kittel (New York: Plenum) Greywall D S 1986 3He specific heat and thermometry at millikelvin temperatures Phys. Rev. B 33 7520–38 Griffioen W and Frossati G 1985 Small sensitive pressure transducers for use at low temperatures Rev. Sci. Instrum. 56 1236–8 Haug F and McInturff A 1990 Measurements of pressure transmission Advances in Cryogenic Engineering vol 35, ed R W Fast (New York: Plenum) pp 1583–91 Hord J 1967 Response of pneumatic pressure-measurement systems to a step input in the free molecule, transition, and continuum flow regimes ISA Trans. 6 252–60 Howard W M 1959 The effect of temperature on pressure measured in hypersonic wind tunnel J. Aero/Space Sci. 26 764 Jacobs R 1986 Cryogenic applications of capacitance-type pressure transducers Advances in Cryogenic Engineering vol 31, ed R W Fast (New York: Plenum) pp 1277–84 Juanarena D B and Rao M G 1992 Cryogenic pressure-temperature and level-temperature transducers Cryogenics 32 39–43

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Kashani A, Spivak A L, Wilcox R A and Woodhouse C E 1988 Performance of Validyne pressure transducers in liquid helium Proc. 12th Cryogenics Engineering Conf. ed R G Scurlock and C A Bailey (Guildford: Butterworths) pp 379–83 Kashani A, Wilcox R A, Spivak A L, Daney D E and Woodhouse C E 1990 SHOOT flowmeters and pressure transducers Cryogenics 30 286–91 Kraft G, Spiegel H J and Zahn G 1980 Suitability of commercial strain gauge pressure transducers for low temperature application Cryogenics 20 625–8 Lawless W N, Clark C F and Samara G A 1985 Quantum-ferroelectric pressure sensor for use at low temperatures Rev. Sci. Instrum. 56 1913–6 Pavese F 1984 Investigation of transducers for large-scale cryogenic systems in Italy Advances in Cryogenic Engineering vol 29, ed R W Fast (New York: Plenum) pp 869–77 Pavese F, Cerutti G, Ferrero C and Rivetti A 1980 Automatic test facility for cryogenic transducers in the range 2 K to 300 K and up to 7 T Proc. 8th ICEC vol 8, ed C Rizzuto (Guildford: IPC Business) pp 812–6 Pavese F and Molinar G 1992 Modern Gas-Based Temperature and Pressure Measurements (New York: Plenum) Pobel F 1992 Matter and Methods at Low Temperature (Berlin: Springer) Polunin S P, Astrov D N, Belyanskii L B, Dedikov Yu A and Zakharov A A 1986 Precision differential capacitance manometer Instrum. Exp. Technol. 29 501–3 Thornton G K 1978 A high resolution pressure transducer operating at liquid helium temperatures Proc. 7th ICEC vol 7 (Guildford: IPC Business) pp 608–14 van Oort J M and Kate H H J 1993 A fiber optics sensor for strain and stress measurements in accelerator magnets IEEE Trans. Magn. MAG-30 3600–3 Walstrom P L and Maddocks J R 1987 Use of Siemens KPY pressure transducers at liquid helium temperature Cryogenics 27 439–41

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E4 Thermometry

F Pavese

The range of temperatures considered in applied superconductivity appears to be quite small, a few hundred kelvins, when compared with the millions of degrees encountered in, for example, the plasma region. However, much fundamental work in superconductivity was done at temperatures well below 1 K. Therefore a logarithmic scale of temperature values is a more realistic way of portraying the temperature scale (figure E4.0.1) than a linear scale. Since absolute zero is approached logarithmically, it becomes similar to the upper boundary of the temperature scale. In the very low-temperature range, conditions far away from human experience occur. Temperatures far below the minimum existing in nature (background cosmic radiation, 2.75 K) are commonly attained in laboratories, where equilibrium temperature values may be different for different components of the solid substance and substantial temperature gradients may occur during nonequilibrium conditions. These conditions directly affect the measurement of temperature. E4.0.1 Temperature scales E4.0.1.1 Thermodynamic temperature In principle, any thermodynamic law of physics or chemistry can be used to obtain a thermodynamic temperature. One such law governs the behaviour of the ideal gas

Kelvin advocated the assignment of 273 as the temperature of the freezing point of water in order to define a value for the gas constant. E4.0.1.2 Empirical temperature The determination of thermodynamic temperature from first principles is, in general, a difficult experiment. In addition, every measurement is only an approximation, because of imperfections in the model used for describing the basic thermodynamic law, insufficient control of the secondary experimental parameters which are included in the model (the so-called ‘corrections’) or because of experimental random errors. In nature, however, a temperature-independent phenomenon is rather the exception than the rule. Any temperature-dependent physical quantity can, in principle, be used to measure temperature: thermometers were invented long before the beginning of the science of heat. Whether a physical quantity is suitable or

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Figure E4.0.1. The logarithmic natural scale of temperature.

not as a thermometer is a matter of convenience; a choice may depend as well on the required precision of the resulting temperature scale. An important element to consider for the evaluation of a candidate empirical scale, apart from the trivial necessity for the physical quantity to behave monotonically with temperature (e.g. water density is not), is whether or not it is a linear transformation of absolute temperature—which must always be taken into account. If it is not, the relationship between the physical quantity and the thermodynamic temperature must be carefully specified; this specification is often difficult to find, especially over long periods of time. Most empirical temperature scales are defined in terms of the temperature dependence of a different quantity—electrical resistance, thermal expansion, vapour pressure, etc. The definition of the empirical scale will include the value of the measured parameter at various specified reference temperatures. Two defining points at least are used in the empirical scale definition. They are called ‘fixed points’ of that scale. None needs to be the defining point of the kelvin scale. The scale definition assigns them conventional temperature values. Though these values typically have been assumed to be exact thermodynamic temperature values, they are actually the best approximations achieved so far. The empirical temperature θ defined by these scales has been considered to be satisfactory as an approximation of the thermodynamic temperature T. This point will be made clearer by the example of the platinum resistance thermometer. In the first place, the experimental data pertain exclusively to a specific substance, platinum, as they cannot be extended, by means of a theory, to be valid for other metals. Other restrictions must be considered as well, for instance the bulk material must be in well defined chemical (oxidized, etc) and physical (strain-free, etc) states, and the results are valid only for a given range of chemical purity. When some of the characteristics of the interpolating instrument can be specifically quantified, the range of the accepted values for these parameters must be defined as well in the scale, to enable one following this prescription to reproduce the results, namely the scale. If this is not possible, with, for example, diode thermometers or thermistors, results and that scale will be valid only for the specific

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production-lot of devices from which results were obtained. The thermometer used in scales of these types is known as an ‘interpolating instrument’, because it is not itself required to reproduce a thermodynamic property, but only precisely the smooth selected function of the empirical temperature θ, y = f (θ), which is used to interpolate temperature values between the values assigned to the fixed points. To summarize, there are three constitutive elements associated with an empirical scale: (i) an interpolating instrument (ii) a mathematical definition (iii) a set of fixed points. The interpolating instrument has already been discussed. The mathematical definition must be the simplest mathematical function (or set of functions) representing, within the stated accuracy, the relationship between the measured quantity and the thermodynamic temperature; it has a number of free parameters (this is characteristic of an empirical scale), whose numerical values must be obtained from a ‘calibration’ of the interpolation instrument at a number of reference points). The number of these defining fixed points must be equal to the number of free parameters and their position must be such as to permit the best compromise between mathematical requirements and what is available in nature. The temperature value of the reference point is defined, i.e. fixed by the scale definition. A scale of this type, which is the most common, is affected by a specific shortcoming, which must be kept in mind, called nonuniqueness, a term expressing the concept that in each particular unit implementing the interpolating instrument (i.e. the thermometer) the relationship between the measured quantity and temperature is slightly different. Therefore, the mathematical interpolating procedure defined by the scale cannot adequately represent the measurements performed with each individual unit of the thermometer, but can only approximate the physical behaviour of these units. In other words, if the scale definition is applied to two specific units of the thermometer placed in an isothermal enclosure, there will be a measurable difference between the temperature values supplied by their calibration tables and the temperatures actually measured. This is true at every temperature except at the fixed points, at which the units have originally been calibrated. Calibration at the fixed points means, by definition, associating the measured numerical value of the property (e.g. of electrical resistance) with each temperature value of the calibration table (expressed in the scale, e.g. T90 ). The nonuniqueness effect cannot be reduced, since it is a systematic error intrinsic to thermometer fabrication. Consequently, it must not be mixed up with errors in the scale realization, nor must it be ascribed to the limited ability of the mathematical definition to match the physical behaviour of the thermometer; it is due, instead, to technical limits of the interpolating instrument itself. E4.0.1.3 International temperature scales Since the measurement of temperature is a necessity in a laboratory, ‘laboratory scales’, more or less based on thermodynamic determinations, have always proliferated. In this connection, after it has been demonstrated how delicate a matter the temperature definition is, it seems now appropriate to draw attention to the great care that accurate temperature measurements demand. As a consequence of the accuracy requirements, an international temperature scale was agreed upon quite early. At its first meeting after the ‘Convention du Métre’, the Conférence Générale des Poids et Mesures in 1887 endorsed the ‘normal hydrogen scale’. A historical review of the process of periodic revision of the scale can be found in many texts (e.g. BIPM 1990). The most recent version of the temperature scale, the ITS-90, was adopted to begin on 1 January 1990. The meaning of an ‘official’ scale is not bureaucratic, but scientific and technical. As in all types of measurement, to improve accuracy not only must random errors (type A in the terminology internationally adopted) be reduced, but also the largest possible number of causes of

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systematic deviations (type B errors) have to be eliminated. This requires an international body, where recognized specialists in the temperature measurement field meet and discuss their experiences and those of their colleagues in order to reduce the error limits. Another equally important role of this body is to assess the traceability of lower-level approximations of the thermodynamic scale to the ITS, and to fill a gap of information between the ITS definition and the international codes developed for industry (e.g. by Organization Internationale de Metrologie Légale (OIML), International Electrotechnical Commission (IEC )) or prepared by other international bodies (e.g. IUPAC, IUPAP). An outline of the basic definition below 273.16 K of the scale-in-charge, the ITS-90, is given in table E4.0.1. All the temperature values in this chapter will be given on this scale (T90 ), unless otherwise specified. It may prove desirable to amplify briefly the discussion of table E4.0.1, since the measurement methods given there may not be familiar to the reader. E4.0.2 Reference points for thermometry First of all, let the term ‘reference point’ be given a meaning more restrictive than is usually associated with it. A thermometric reference point is an equilibrium state of a specified substance, the realization of which only depends on the measurement of its temperature, purity and composition. Accordingly, neither ‘boiling points’ nor the vapour-pressure functions mentioned in table E4.0.1 should be termed reference points, as their temperature is defined only through a temperature-pressure relationship, so that pressure must be measured too. From Gibbs’ rule, F = c − φ + 2 (where F are the degrees of freedom, c the number of chemical substances and φ the number of phases involved ), it follows that, for a pure substance, this definition applies only to ‘triple points’, a thermodynamic state in which three phases are co-existing. This implies that both temperature and pressure values at a triple point are unique. Reference points are of great importance in thermometry, not only because they allow empirical scales (where they become ‘fixed points’: see section E4.0.1.2) to be defined, but because each of them provides

Table E4.0.1. Summary of the ITS-90 definition below 273.16 K.

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a reference temperature, at which the calibration of a thermometer of any type can be checked in any laboratory without resorting to the help of another. The three coexisting phases can be combined in different ways, not all being equally suitable as fixed points for thermometry. Only the solid—liquid—vapour type is the triple point of a gas. Solid to solid and liquid to liquid transitions are also used as reference points, though of lesser quality. E4.0.2.1 Ideal substances versus standard reference materials A reference temperature can be obtained by simple reference to the ideal substance or, instead, by specifying a sufficient number of relevant physical properties of it to define a single temperature-dependent state of the substance. The difference between the two alternative definitions is equivalent to that existing between a thermodynamic and an empirical temperature scale. In the first case, the substance is defined only by means of its basic physical parameters so that anyone can reproduce the thermodynamic state in an independent way. Such parameters need to be listed and defined, specifying also the uncertainty of the temperature value. These parameters can be the maximum level of impurities (or of a single impurity), the physical state (annealed, etc), the composition (including the isotopic one, when relevant) and its tolerance in the case of mixtures. Each experiment aimed at reproducing the real thermodynamic state and complying with the above specifications is expected to reproduce the same temperature value within the given uncertainty. Each such experiment is independent of the others and no specific procedures need be defined. In this respect, the exercises usually performed by major laboratories, especially metrological, such as round-robin tests or intercomparisons of results, are normally to be considered as checks of their ability to perform the experiment correctly, not as checks of the quality of the substance employed. They are not aimed either at ‘calibrating’ the fixed points. In the second case, in contrast, a certain amount of a substance is prepared to have uniform characteristics and is stored and made available by an organization. The physico-chemical characteristics of the substance, and its temperature value—besides its reproducibility between samples—are determined by measuring samples taken from the batch and are assigned to the whole batch of material considered.

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Subsequently, samples of that batch are made available to other laboratories together with a certificate of compliance with the characteristics of the batch. The materials used for this purpose are called ‘reference materials’ (RM), and the word ‘standard’ (SRM) is added when they are certified by an official body. In principle, they need not comply with any of the specifications that an ‘absolute’ reference point has to. The batch of material needs only to be uniform enough to allow the desired reproducibility of certain property characteristics to be obtained between samples, in order to achieve the required accuracy of the temperature value. The users of these materials must rely on the certificate issued by the supplier, and so, consequently, must the temperature scale based on these reference materials. Sometimes, when it is not even possible to establish a certified value of the property for a whole batch of a material, each sample is certified individually. An example is provided by the realization of the fixed points based on superconducting transitions (section E4.0.2.2(b)). The transition temperature value cannot be guaranteed to represent the value of a physical state; it is therefore usually certified as the transition value realized by the individual device and is said to be a ‘device temperature’ value. Such a type of SRM relies only on the stability in time of the certified value of the property. E4.0.2.2 Types of cryogenic reference point In the region of interest to studies on applied superconductivity, the reference points mainly belong to these groups, one based on the triple points, a second on superconducting transitions and a third including the superfluid transition of 4He and solid to solid transitions. In the first case, several reference temperatures are available above 13.8 K. Superconducting transitions are useful for reference point realizations below 9 K, as, so far, no attempt has been made to use the new high-Tc , materials for this purpose. The 4He superfluid transition occurs near 2.2 K and solid to solid transitions are available above ≈20 K. (a) Realization of the triple point of gases as thermometric fixed points A comprehensive discussion on a correct realization of fixed points based on the triple point of gases is given by Pavese and Molinar (1992). During a traditional realization of a triple point, after the desired amount of gas has been condensed in a container fitted into the cryostat, the sample of gas is sealed into the apparatus by closing the valve connecting the cryostat to the gas filling system. This sample can be further manipulated in different ways by evaporating it back to the filling system, but, at the end of the experiment, the cryostat is warmed up to room temperature and the whole sample is usually discarded. If at a subsequent time, from a few days to many years, another measurement has to be carried out, another sample of gas will be used, generally taken from another storage bottle. In the meantime, the results of the previous measurements remain ‘stored’ and available only by means of thermometers used during the measurements with the reference point and they are supposed to remain stable in time. This old procedure has many disadvantages. Firstly, the fixed point is not readily available at all times for a new realization, except during the short period when the measurements are carried out. The storage of results in thermometers has proved to be not reliable, as the thermometers often are not stable enough in time. In addition, the old procedures are not routine measurements, but each time they are actually new experiments, as a new sample of gas is generally used and, consequently, the degree of its purity has to be ascertained anew, a far from trivial task. As already pointed out, because of the necessary new manipulation of the gas, it is again possible that the sample will become more contaminated than certified by the supplier. Consequently, it is not surprising that in the past the measurements at the fixed points were not made routinely, and that only calibrated thermometers were used instead, until the technique of permanently sealing the sample in a cell was perfected. This new method for gases has only developed since the beginning of the 1970s, though there exist a few examples of an earlier method that was used previously with liquids and solids in calorimetry. With gaseous substances, purity preservation, as made possible by permanent sealing and the consequent

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elimination of further gas manipulation, has to be proved. The container must not contaminate the sample, which is relatively simple to achieve with most solid or liquid samples, but is not, generally speaking, easy to achieve with gases for the reasons already mentioned. Almost 20 years of experience has shown, however, that the method can be used with most gases, and the values of the triple-point temperature (Tt p ) obtained with sealed cells have proved to remain stable within a few tenths of a millikelvin over a period of decades (Pavese et al 1992). The design of a permanently sealed cell, in comparison with a traditional experiment, differs in a substantial feature: the realization of an accurate and stable fixed point is permanent. A sealed cell can be made small and fully self-contained, so that it can be mounted in the working chamber of most types of commercial cryostat (figure E4.0.2). The cryostat itself is drastically simplified and, since the cells are independent of the cryostat, they can be interchanged in a single cryostat.

Figure E4.0.2. Typical IMGC-type (Istituto di Metrologia ‘G Colonnetti’) sealed cell.

The three constitutive elements of a sealed cell are (i)

Container. Any material can be used for its fabrication, provided it withstands the high internal roomtemperature pressure. It accounts for most of the cell mass. The use of materials having high thermal conductivity promotes uniform temperature throughout the entire sample, but increases the thermal coupling between the heater and the thermometers. (ii) Inner body. This element may contain the thermometers, as in figure E4.0.2, or it may be used only to transfer the interface temperature to an external thermometer block. It is usually made of OFHC copper or stainless steel and needs fins or baffles to ensure a good thermal contact with the solid phase for as large a melted fraction as possible; however, the larger the inner cell surface is, the more critical the problem of sample contamination becomes. (iii) Sealing device. The seal must remain leak-proof for a very long time, under thermal cycling from room temperature to that of the cryogenic environment during each cool down. Pinch-weld seals have successfully been used.

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The equilibrium technique is used for performing the fixed point and requires the use of an adiabatic calorimeter. It consists of intermittent exchanges of energy with the sample. After each energy exchange cycle, when the thermal flow produces temperature gradients in the sample which produce an increase of the melted fraction F, the sample is left undisturbed under adiabatic conditions for a time sufficient to attain thermal re-equilibration. The temperature versus 1/F plot must be used to define the reference temperature, as follows. The temperature value Tt p of a triple-point realization, when used as a reference fixed point, is defined by the equilibrium value obtained when melting a sample using the adiabatic calorimetric technique at the liquidus point, defined as 1/F = 1 and obtained by extrapolation from the melting plateau, within the melting range, by means of the T versus 1/F plot. Sealed cells realizing the triple point (and the solid to solid transitions) of pure substances such as hydrogen, neon, nitrogen, oxygen, argon and carbon dioxide are commercially available. (b) Reference points based on superconducting transitions Superconductive transitions are solid-state phase changes of second order (no enthalpy change), in contrast to most fixed points used in thermometry. Essential peculiarities of the superconductive transitions are the larger influence on them of impurities, crystal defects and physical strain, which affect both the transition widths and temperatures (Ts c ). For application as thermometric fixed points, therefore, special sample preparation techniques and special methods for the detection of the superconductive transitions must be used. So far superconductive temperatures, as realized, for example, by the NIST (the American National Institute of Standards and Technology, Boulder, CO) devices SRM 767 and SRM 768 (Schooley et al 1980, Schooley and Soulen 1982) were only used in the sense of a reference material (see section E4.0.2.1). None of these reference points is included in the definition of the ITS-90. Modem sample preparation and handling techniques (Fellmuth et al 1987), in conjuction with convenient material characterization methods, are sufficient to guarantee an accuracy and stability of Ts c values of these reference points within about ±1 mK if definite specifications concerning sample preparation are fulfilled. Further improvement is possible only by carefully annealing the samples and by selecting them on the basis of maximum allowed transition width, since the main difficulty is the influence of impurities and crystal defects. The influence of impurities is typically 1 mK ppma−1. The residual electrical resistance is an excellent indicator of this influence, except in the few cases where localized moments exist; in this case the influence can be much larger so the concentration of the magnetic impurities must be specifically determined. The effect of crystal defects can be reduced to a few tenths of a millikelvin by using suitable preparation techniques, which may differ for different elements. For checking the magnitude of this effect it is important that the transition width be always smaller than the change in Ts c , caused by the defects. Hence, for the realization of superconductive reference points, detailed information concerning both the physical properties of the materials and the preparation and characterization methods is necessary. A simple and effective method for avoiding the summing up with time of strain in the samples is that of sealing each of them in thin and small glass ampoules together with some helium exchange gas (Lipinsky et al 1990 ). In table E4.0.2(a) the Ts c , best estimates (as of 1987) of superconductive transitions with their total estimate of the uncertainty are shown. Also the dependence of the Ts c , value on the method used for its detection has been taken into account. The mutual inductance technique is most effective because of its relatively easy application, the minimum stress transmitted to the sample, and the negligible sample contamination. Thermal anchoring of the samples and of the electrical leads is critical for the observation of nonhysteretic and reproducible transitions. Because of the absence of enthalpy change in transition, a high-quality adiabatic cryostat is necessary for accurate Ts c measurements. To check that the thermal conditions are adequate, the difference between the values obtained by increasing and decreasing the temperature of the sample through the transition range should always be measured. Then the temperature should be

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stabilized at the midpoint of the transition before thermometer calibrations are carried out. Furthermore, external magnetic fields earth’s field and measuring field) can have an influence of a few millikelvin on Ts c ; the effect can be corrected for, or, better, compensated for earth’s magnetic field by means of magnetic shields or Helmholtz coils—see chapter E5. Table E4.0.2(b) reports the temperature values assigned to NIST devices SRM 767 and 768. The values are assigned, since they are reference materials only; for the same reasons, the values can be slightly different from those of table E4.0.2(a). Other SRM devices are available from Physikalische-Technische Bundesanstalt (G) (PTB) (niobium, Fellmuth et al 1985) and Institut Niskich Temperatur i Badan Strukturalnych, Polish Academy of Sciences, Poland (INTiBS) (indium in glass ampoules, Lipinsky et al 1990).

Table E4.0.2. Superconductive transition temperatures.

(c) The 4 He liquid to superfluid liquid transition and solid to solid transitions Among liquid to liquid transitions, it is only worthwhile considering the 4He λ-transition from normal to superfluid liquid at 2.1768 K. It is easily found to very high accuracy, in a temperature region where no other references of comparable quality are available. Its temperature value falls half-way between that of the superconducting transitions of aluminium (1.1810 K) and of indium (3.4145 K). The reader is directed to texts on helium properties, e.g. Van Sciver (1986), for further reading about its physical properties. The λ-transition in the p —T plane is actually a line, as the two liquid phases exist in the pressure range below the solid-phase boundary. The state being discussed is the lower end point of this line,

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where it joins the saturated vapour-pressure line. It is always crossed when lowering pressure on a 4He liquid bath. The transition is a second-order one, as entropy shows only a slope discontinuity and no enthalpy of transition is involved. The value of specific heat shows a logarithmic infinity at Tλ. Thermal conductivity varies from moderate values in normal liquid (0.02 W K−1 m−1 in 4HeI, but 0.5 W K−1m−1 just below Tλ ), to extremely high values in the superfluid phase (>1000 W K−1 m−1), much higher than those exhibited by the best conducting metals. In bulk liquid, the transition takes some time to involve all the liquid, as normal liquid helium (socalled liquid 4HeII) shows thermal layering, and the superfluid liquid (so-called liquid 4HeII) has its maximum density near Tλ . Therefore liquid II tends to leave the surface, where refrigeration takes place, and be replaced by liquid I. Besides, since no thermal gradients can exist in liquid II, all liquid II will remain at exactly Tλ until some liquid I is present. For this reason a flat temperature (and pressure) plateau is observed, lasting for some time and looking like that of a first-order transition. A thermometer immersed in liquid II will accurately measure Tλ (account being taken of the Kapitza effect). Care must also be taken to limit the hydrostatic temperature gradient in the liquid II (about 100 mK MPa−1, equivalent to about 1.5 mK per metre of liquid depth) if the bath is deep. The effect of impurities is not critical, equivalent only to –1.4 µK ppm−1 (where ppm is parts per million) with 3He, which is the only likely impurity (at a level of 0.5–2 ppm in natural 4He). This reference point can also be realized using a sealed device (Lin Peng et al 1990; Song et al 1991). For more detail see also Pavese and Molinar (1992). In solid to solid transitions only the solid and vapour phases are present; only those occurring along the saturated solid-vapour line are suitable as reference points. First-order transitions, such as the β –γ in oxygen, which exhibits an enthalpy of transition, behave like, and in principle are realized in the same way as, the triple points. However, the quality required for the calorimeter is much higher, and the equilibration time much longer. Second-order transitions have no enthalpy of transition but only show a heat capacity peak. Therefore the transition, in the T versus time representation, does not show a flat plateau but only an S-shaped behaviour, with a derivative value at the inflection point, (dT/dt )I ≠ 0 (Cp,I ≠ ∞). A transition width can be defined from the transition duration at Tt r , obtained from the extrapolated lines for constant heat capacity before and after the transition, or between the first derivative maxima. Though not of the same quality of triple points, these transitions can be used—and are used—as reference points. For gases exhibiting such transitions (e.g. O2, N2, CH4), they are an additional ‘bonus’ in the use of sealed cells. E4.0.2.3 The ITS-90 between 13.80 K and 273.16 K, and scale approximations using only sealed fixed points The ITS-90 has been conceived in order to reproduce as closely as possible the thermodynamic temperature and to interpolate between fixed points with a nonuniqueness limited to ±0.5 mK above 25 K and ±0.1 mK below 25 K (table E4.0.1). These requirements set the number and the position of the fixed points above 13.8 K and required the use of the interpolating constant-volume gas thermometer (ICVGT) between 3 K and 25 K (see section E4.0.3). In the range from 13.80 K to 273.16 K considered in this section, the approach of using only triple points of gases as fixed points has been followed wherever possible and was prompted, especially, by the experience gained in an international intercomparison of fixed points of this type (Pavese et al 1984 ). One to four triple points of gases, which can be realized in sealed cells, are necessary to produce the ITS-90, depending on the subrange. For the lowest subrange (below 25 K, down to 13.8 K), two points of e-H2 vapour pressure are also used. Additionally, the triple point of water and the triple point of mercury must be used. Today, it seems advisable to consider, unless a refrigerator is used, two different cryostats

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for the realization of gas triple points and for the realization of the triple points of the substances which are liquid at room temperature, and two different types of device for their production. The sealed cells already described are by far the best means to realize the gas triple points. The tendency to make all low-temperature fixed points available as sealed devices is likely to change completely the attitude of users and their attitude to the ITS-90. It becomes a simple measurement of a self-contained device always ready for use, according to a simple procedure which can be carried out by a computer. This change should encourage a wider dissemination of the temperature scale since it can be realized in a direct way from its definition, i.e., independently. The high accuracy and reproducibility that the ITS-90 provides are not always needed, and, in such a case, an approximation of the ITS-90 within a specified degree of traceability is a better solution. In other words, if the required uncertainty of the approximation is, say, ±0.01 K, the approximation should never deviate from ITS-90 by more than ±0.01 K. The subject is treated in a monograph published by the Working Group 2 of the Comite Consultatif de Thermometrie of the Bureau International des Poids et Mesures (BIPM) (Bedford et al 1990): it does not endorse ‘secondary definitions’ of the scale, but states that the traceability to ITS-90 of these scale approximations is known within a given uncertainty. In the range from 13.80 K to 273.16 K, there are several possible alternative definitions for temperature scales approximating the ITS-90, which still use standard platinum resistance thermometers (SPRTs) of the quality specified by the ITS-90 and provide an accuracy not much lower than that of ITS-90. Table E4.0.3 summarizes these possibilities. On the other hand, there are no possibilities extending a scale below 84 K with the use of only one gas and water. At present, no calculations exist for ITS-90 approximations confined to lower temperatures, e.g. only below 84 K or only below 54 K. For an accuracy level of ±10–20 mK, as provided by the scales of the third group in table E4.0.3, some industrial-grade platinum resistance thermometers (IPRT) seem suitable as interpolating devices. However, a sufficient stability was observed only in selected units and it would be, therefore, unwise to rely on a generic IPRT for realizing a scale, unless its quality is certified. E4.0.3 Gas thermometry below 273.16 K Traditionally, gas thermometry is considered an issue reserved for specialists. Its foundations are briefly reported here not only because it is part of the ITS-90 definition, but also because some simplified realizations are now available. Until recently all research work on gas thermometry below 273.16 K was based only on 4He. However, with the use of 3He, gas thermometry can be carried out at temperatures lower than 2.5 K, so fully covering the range of 4He vapour-pressure thermometry and overlapping also most of that of 3He. The use of 3He can be afforded today, as the lighter isotope is available at a reasonable price and with sufficient purity and because its virial (nonideality) corrections are now known to a high accuracy down to 1.5 K. Both isotopes are considered in the ITS-90 definition for use in gas thermometry between 3 K and 24.6 K. The basic equation of an ideal-gas thermometer is given in equation (E4.0.1). A full discussion on gas thermometry can be found in the book by Pavese and Molinar (1992). The uncertainty of the value of R presently limits the accuracy of a temperature measurement to ±3 mK at 300 K and to ±0.3 mK at 30 K but this uncertainty is suppressed by measuring temperature ratios with respect to a reference condition. When, most commonly, the volume V and the molar amount of gas n are assumed to remain rigorously constant, the thermometer is called a ‘constant-volume gas thermometer’ (CVGT): T = Tr e f (p/p0 ). The constant-volume assumption concerns the requirements of the gas container, which are relevant only from the standpoint of the technical realization. In contrast, the requirement for the amount n of gas to remain constant (as implied in equation (E4.0.1)), involves physical properties of the thermometric substance itself, i.e. the number of its ‘active’—i.e. freely moving in the volume V—molecules N. The grouping

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Table E4.0.3. Some approximations of the ITS-90 in the range from 13.8 K to 273.16 K using different sets of fixed points.

into ‘physical’ and ‘technical’ involves all the ‘corrections’, i.e. the parameters which contribute to the overall uncertainty. Figure E4.0.3 gives a picture of the number and size of these influence parameters, the so-called ‘corrections’, for both the physical type (a) and the technical type (b, c). Only the uncertainties in these parameters, not their values, are obviously relevant to the overall uncertainty.

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Figure E4.03. Size of the main corrections in a CVGT. (a) Physical: virial or non-ideality. Technical: (b) for Tr e f = 27.1 K; (c) for Tr e f = 273.16 K.

E4.0.3.1

Gas thermometers with a built-in cryogenic pressure-measuring device

The general layout of a gas thermometer using a cryogenic pressure transducer is shown in figure E4.0.4, which includes all possible options to be discussed in the following. A design of this type is not the traditional one and has been implemented, so far, for 4He gas thermometers by Astrov et al (1969) in the 4–20 K range, by Astrov et al (1989) in the 12–300 K range and by Van Degrift et al (1978) for measurements below 10 K. All major corrections in the traditional design (the one with the manometer at room temperature), except the virial and adsorption correction, are avoided with this design, because they are a result of the pipe connecting the bulb to the manometer, the so-called ‘capillary’. On the other hand, the required pressure transducer must be of a special type. For a full discussion see the book by Pavese and Molinar (1992). The use of a differential pressure transducer for use at cryogenic temperatures is the least demanding

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design for a CVGT of this type, but still requires a pressure measuring line from the differential transducer to room temperature (figure E4.0.4). Only the zero stability of the transducer is relevant and must be of the same order of magnitude as that required for the room-temperature types. Design options depend on the actual characteristics of the transducer employed. If the transducer has a small and reproducible temperature coefficient of the diaphragm zero, it can be mounted directly as part of the gas bulb, thus eliminating the dead volume completely. If this is not the case, the transducer can be attached to the bulb by means of a short tube (which constitutes a small dead volume), and maintained at a constant temperature. Thermal cycling to 300 K is unavoidable. If the diaphragm zero is stable enough on thermal cycling, no checks are necessary during operation, and the ‘bulb + transducer’ system can, in principle, be sealed off if it can withstand a pressure rise up to 0.4–0.5 MPa at 300 K. If the transducer zero is not stable enough on thermal cycling, an in situ check is necessary, which is a difficult technical problem to solve. Being a differential transducer, it must be connected to the standard pressure-measuring apparatus at room temperature by means of a tube, which can be filled only with helium gas. Therefore, if the manometer uses a working gas different from helium, a second diaphragm transducer of suitable precision— though of the conventional type—is necessary in the room-temperature manometric circuit. The pressure tube, however, is a much less a crucial element than in the case of a traditional CVGT, since its design is the same as used for vapour-pressure measurements (section E4.0.4), where large bores can be used to minimize the thermomolecular pressure effect.

Figure E4.0.4. General layout of a gas thermometer with a cryogenic diaphragm pressure transducer.

The use of an absolute cryogenic pressure transducer is much more demanding and it has never been attempted. The measurement uncertainty of this transducer must range from ±0.005% to ±0.02% over a pressure interval from about 0.1 MPa to less than 1 kPa respectively. Its sensitivity must be better by about one order of magnitude. A transducer having a good zero reproducibility in thermal cycling is likely to be a good one also when pressure is cycled. Linearity is not strictly a requirement, as nonlinearity complicates the transducer calibration only slightly. In any case, in situ calibration against an absolute pressure standard is required.

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With the use of an absolute pressure transducer a tube connecting the gas thermometer to room temperature is no longer necessary, but the device becomes fully self-contained, only if it can be permanently sealed off. This is allowed, however, only by a device withstanding a high room-temperature pressure. The use of a pressure transducer built into the gas thermometer bulb allows a complete change in the bulb design. The most outstanding modification is the bulb size, which can be very small, since the dead-volume effect is zero or greatly reduced. The volume of the gas bulb might reduce to that of the pressure transducer itself. However, a question arises as to whether there is a minimum usable volume, a question which is essentially equivalent to asking whether there is a minimum amount of gas sufficient to keep the uncertainty due to unwanted changes in the active amount of gas n in a CVGT within a stated limit: see the book by Pavese and Molinar (1992) for a discussion of this point. Obviously, even with a self-contained small gas thermometer, one cannot avoid corrections for the virials, gas purity and the deformation of the bulb volume. In addition, the volume variation due to the diaphragm deflection of the pressure transducer, which usually can be neglected, being only a few cubic millimetres, should be carefully corrected when the bulb volume reduces, and the deflection itself must be minimized. E4.0.3.2 Interpolating constant-volume gas thermometers In section E4.0.1 the meaning of an ‘interpolating instrument’ was introduced as one of the three elements necessary to define an empirical temperature scale. The decision to use a CVGT for defining an empirical scale instead of directly measuring the thermodynamic temperature is prompted by different considerations. One is, for instance, the interest in improving measurement reproducibility beyond the thermodynamic accuracy limit. Another is the simplification when the gas thermometer is used as an interpolating instrument. Now CVGT is part of the ITS-90 definition between 3 K and 24.6 K. In general terms, an ICVGT differs from a CVGT in that it is calibrated at more than one fixed point (i.e. pressure is measured at temperatures whose values are determined by independent means, for example by a CVGT). Then a functional relationship is used for interpolating between these fixed points, or for limited extrapolation. There is a substantial difference and simplification with respect to using a gas thermometer for thermodynamic measurements, since the corrections for the technical parameters or for the measured pressure values are no longer calculated or measured. In fact, all corrections are taken into account in the ‘calibration’ of the ICVGT at the fixed points (except, obviously, the corrections related to the whole manometric system, including the separation diaphragm pressure transducer, which must still be applied). At the same time, one has to assume that the physical conditions of the experiments and the thermal conditions of the apparatus are reproduced in subsequent measurements and maintained exactly the same as they were at the moment of the calibration. This basic assumption involves, for example, the temperature distribution in the capillary , which is not measured with an ICVGT. The reproducibility of an empirical, ICVGT-defined scale, depends both on the selection and reproducibility of the technical parameters of the ICVGT noted previously, and on the reproducibility of the fixed points, as well as on the capability of the stipulated interpolating functional relationship to approximate the thermodynamic (T, p) relationship in different experiments (the so-called nonuniqueness: see section E4.0.1.2). The accuracy of the scale depends on the accuracy of the thermodynamic temperature values assigned to the fixed points. For a full treatment of the ICVGT see the book by Pavese and Molinar (1992). E4.0.3.3 Gas thermometer realizations As described in E4.0.3.1, a CVGT is fabricated from essentially three parts (figure E4.0.4):

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(i) the measuring bulb, whose temperature defines the thermodynamic temperature (ii) the pressure capillary, connecting the bulb to the pressure-measuring system, when it is placed at room temperature (iii) the pressure-measuring system. When the manometer is at room temperature, a 1 L OFHC copper volume is typical for top-accuracy apparatus, but volumes smaller by a factor of two or three have been used, yielding the same accuracy. A larger bulb volume makes the effect of the dead volume less critical, but the temperature uniformity is more difficult to achieve, the surface which adsorbs the thermometric gas becomes larger and so does the size of the entire experimental apparatus. A capillary tube is unnecessary only for the case of the CVGT design equipped with a cryogenic pressure transducer. In all other cases, including the ICVGTs, its design involves many of the most difficult compromises between conflicting requirements. Especially in the use of the ICVGT where no capillary parameters are measured, the capillary must be designed to operate in stable and controlled thermal conditions, in order to have a stable and reproducible temperature distribution. This goal may be achieved by building an isothermal calorimetric environment around this tube. This is not the usual practice, but several thermometers (generally thermocouples) are generally placed on the capillary in order to map its temperature distribution, a method far from satisfactory. Another rule is to keep the capillary as short as practical, since there is no advantage in a long capillary section (Pavese and Molinar 1992). The pressure-measuring apparatus is made of a differential pressure gauge (typically at room temperature), of very small internal volume, which is necessary to isolate the thermometric gas from the manometric gas, often of a different type and, in any case, less pure. The operation of this gauge needs a check of the zero, which must be carried out by directly applying the same pressure to both sides by opening a bypass valve. This operation risks contamination of thermometric gas with some of the manometric gas filling in the bypass section, or loss of some of the thermometric gas. The valve sealing the bulb during these operations and the bypass valve must have a reproducible volume, when open and when closed, since the dead volume must be reproducible. Obviously, the gas trapped in the circuit becomes a mixture of the thermometric and the manometric gases, and therefore must be pumped away after each bypass operation; in the case of the costly 3He, in contrast, a cryogenic adsorption pump (integral with the cryostat) should be used to recover the gas for subsequent purification. Before the bulb is re-opened, the tubes must be refilled with fresh pure gas at the working pressure. The reader is directed to the relevant literature for details on the modern high-accuracy realizations (see Pavese and Molinar 1992). See section E4.0.3.1 for the advantages of having a cryogenic pressuremeasuring system. It is also useful to become acquainted with some realizations of CVGTs intended for use when the required accuracy is much lower (Bedin et al 1980, Winteler 1981). Any CVGT instrument can also, in principle, be used as an ICVGT if an interpolating function is selected and the instrument is calibrated at the corresponding fixed points, simplifying its use. However, no such specific use has yet been attempted, except for a few preliminary studies (Nara et al 1989, 1990, Van Degrift et al 1978). Some types of gas thermometry use an intrinsic property instead of an extrinsic property like pressure, namely the dielectric constant, refractive index or sound velocity. It cannot be affirmed, at present, whether an instrument based on these properties has superseded the CVGT. The reader is directed to consult the relevant literature (see Pavese and Molinar 1992). E4.0.4 Vapour-pressure thermometry Vapour pressures have long been used for temperature measurements or for calibrating thermometers against a physical property, and are still at present very popular in the temperature region of interest of

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applied superconductivity, since the saturated vapour pressure of a pure substance above its liquid phase depends only on temperature and the measurements are relatively easy. Vapour pressures are also very commonly used for the realization of the fixed points called ‘boiling points’. These points are simply specific points on the vapour pressure line, generally those at 101 325 Pa (normal boiling point), and do not deserve special attention. Present knowledge of the physics is adequate to provide an accurate analytical description of the relationship existing between pressure p and thermodynamic temperature T. The following equation was established in the 1950s and can be found in many textbooks (e.g. Keller 1969)

where ∆υ a p Hm , 0 K is the molar enthalpy of vaporization at 0 K; i0 defined by ln[ gσ(2πma )3/2k 5/2h −3] is the chemical constant, with ma being the mass of a single atom and go the degeneracy due to nuclear spin; ε(T) defined by In(pVυ /nRT ) –2B(T)(n/Vu ) – C(T)(n/Vυ )2 is the vapour virial correction, with B(T) and C(T) being the virial coefficients; SL(T) is the molar entropy of the liquid along the saturation line and Vυ (p) and V L(p) are the volumes of the vapour and the liquid along the saturation line. However, not all the parameters involved in equation (E4.0.2) can be calculated from first principles with the accuracy desired for high-precision thermometry. Below the λ-point with 4He and below ≈1 K with 3He calculations of a vapour-pressure scale allow an accuracy to within ±0.5 mK, with only one unknown parameter, ∆υ a p Hm, 0 K. Only the first three terms are significant contributors, but, for higher temperatures, the other three terms must be taken into account as well, especially the contribution of entropy. With 4He, a 1% uncertainty in these three terms contributes to the uncertainty in the vapour-pressure scale by ≈3 mK at 3 K and by ≈9 at 4 K. For most substances, the uncertainty is much higher than with helium. Therefore, vapour-pressure scales cannot be used, at present, as high-accuracy thermodynamic thermometers. On the other hand, in the implementation of such a thermometer, the choice of the technical parameters is not as critical as for gas thermometry. Therefore, the exact relationship of equation (E4.0.2) is replaced by a semi-empirical, or even empirical, equation. The following is often used when only low accuracy is required

This relation is based on the Clapeyron equation and assumes that the heat of vaporization is a linear function of temperature and that V L << Vυ and the vapour is an ideal gas. The parameters α , β and γ are found experimentally for each substance. Vapour-pressure equations are expected to be valid for a broad variety of experimental conditions and, in general, no calibration is necessary at any fixed point, as the values of all coefficients are specified (and tabulated as defining values). In fact, the p-T relationship of a specific pure substance, after being determined experimentally and being compared with a thermodynamic thermometer (such as a gas thermometer), is assumed to be valid, within the stated accuracy, irrespective of any specific implementation. At present, the definitions of 3He and 4He vapour-pressure scales are part of the ITS-90. A scale of this type differs from an empirical scale in that the realization of the latter requires a specific interpolating instrument, as discussed in section E4.0.1. With the vapour-pressure scale definition, no fixed points are needed as all the values of the parameters of the defining equation are defined. The basic criteria of the vapour-pressure scale also differ from those of gas thermometry, in which the technical design may seriously affect the accuracy obtainable with an individual piece of apparatus. For this reason, when a gas thermometer is used as an interpolating instrument, the value of a certain number of parameters of the defining equation must be obtained from calibration. No such requirement exists with vapour-pressure thermometry, since only the physical properties of a substance are significant.

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Figure E4.0.5. (a) Range for vapour pressure thermometry of various gases. (b ) Sensitivity d p/dT of vapourpressure thermometry for selected gases. The shaded parts indicate regions where it is less common or less accurate hatching—not available; 0—lower accuracy; • —critical point; ο —triple point; —λ-point.

On the other hand, each substance allows a scale of this type to be realized only over a very narrow temperature interval, as shown in figure E4.0.5(a); unfortunately, over certain temperature intervals vapourpressure scale realization is not even possible or, at least, not to a high accuracy. Both liquid—vapour and solid—vapour equilibria have to be considered, though with solid—vapour equilibria greater thermal problems are involved, which may limit the accuracy of the realization.

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E4.0.4.1 The influence of physical parameters With an ideally pure gas, when some liquid has been condensed in the bottom of the bulb, after crossing the dew-point line, the pressure remains at a fixed value if the temperature is maintained at a fixed value— by removing the heat of condensation—during further condensation of gas in the bulb from an external cylinder. The temperature may become unstable only if so much liquid is condensed that it overflows the bulb, simply because there will be no more vapour in equilibrium with the liquid at the temperature of the bulb. When this condition is avoided, the vapour pressure in the bulb realizes the same and unique vapour-pressure scale of a given pure real gas. Virial corrections are always the same on the saturated line, and saturation always occurs, at equilibrium, when some liquid is present. However, no substance is perfectly pure, but it forms a mixture with other substances which have, at each temperature, different vapour pressures. Impurities are a major source of uncertainty in vapour-pressure thermometry (Pavese and Molinar 1992). The effect of most nonvolatile impurities on p = f (T), as observed in specific experiments (e.g. Ancsin 1978), thus involves a change in all equation coefficients. The effect of impurities provides a possible explanation for discrepancies, as high as hundredths of a kelvin, that have often been observed between the temperature scales based on vapour pressures realized in different laboratories. They can also be due to different isotopic and isomeric composition. The pressure dependence on the vapour—liquid ratio, which is zero for an ideally pure substance, can be used as a check to determine whether the effect of impurities is small enough for the required accuracy. Vapour-pressure thermometry (like gas thermometry) is well suited and advisable for use in high magnetic fields—considering the persistent difficulties of employing electrical thermometers for this purpose—since it is not greatly affected by magnetic fields, except when a few substances such as O2 and 3 He are used.

E4.0.4.2 The influence of technical parameters A vapour-pressure thermometer consists of a bulb, where the temperature is measured, and a manometer, connected to the bulb with a tube. In most cases, when the manometric gas is different from the thermometric substance, and when the high purity of the latter must be preserved, a differential diaphragm pressure transducer is used to separate the two gas circuits. As purity requirements are as stringent as those for the realization of triple points, from the standpoint of reliability and simplicity, for a vapour-pressure thermometer one should also consider the advantage of using a sealed device, but solutions must be different (Pavese and Molinar 1992). The design of a vapour-pressure thermometer appear to be similar to that of a gas thermometer (figure E4.0.4), but the design parameters are substantially different, as they play a very different role in the accuracy of the thermometer. The thermometer volume is irrelevant within a wide range of values with respect to both the bulb volume and the total volume, i.e. the bulb plus all the addenda. The amount of liquid that avoids bulb overfilling must be carefully computed: for details see the book by Pavese and Molinar (1992). It is necessary that all of the liquid phase remains in the thermometer bulb where the temperature T is intended to be measured, otherwise, pressure values will be altered. The most critical effect is the so-called ‘cold spot’, a zone of the thermometer wall outside the bulb where the temperature is lower than that of the bulb. This cold surface exerts an extremely effective pumping action on the liquid in the bulb, since pressure always tends to equilibrate at a value corresponding to the lowest (wall) temperature, and since the large change in volume due to condensation at the cold spot brings about a steady vapour flow. The mass flow, i.e. the mass of liquid transported, is very large when compared with that obtained by a pumping action from room temperature, since the low-temperature density is much

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higher and the vapour flow takes place inside a much shorter tube. Even when mass transfer is very small, pressure alteration because of ‘cold spots’ has very serious consequences in the measurement accuracy. In vapour-pressure thermometry the bulb, which must be massive to ensure temperature uniformity, does not require a large volume, as is necessary in gas thermometry where the effect of the dead volume must be minimized. The typical size for the bulb is a few cubic centimetres. Accordingly, the volume of the connecting tube is insignificant to a large extent, and the tube need not be a ‘capillary’, as is required for a gas thermometer. The diameter can then be several millimetres and, if necessary, be widened in steps up to several centimetres, according to the density decrease, in order to limit the effect of thermomolecular pressure difference. An upper value for this volume is mostly a matter of convenience, or can be determined by the need to limit thermal exchange by convection or by the reponse time. Pressure-measurement accuracy requirements increase with decreasing temperature, because the thermometer sensitivity quickly decreases. Figure E4.0.5(b) shows the dependence of sensitivity dp/dT on temperature. The upper limit of the thermometer range is set by the maximum pressure that the manometer can measure, or by the value of the critical pressure (always less than 10 MPa). The lower limit, in contrast, is set by the absolute measurement uncertainty of the manometer, as sensitivity decreases with pressure (though relative sensitivity may increase). Of course a lower-range manometer could be used for this purpose, but a lower limit for absolute uncertainty is generally set at about 0.1 Pa, at best, by other unavoidable factors such as the value of the static vacuum limit. Compared with the sensitivity required for a gas thermometer that of the vapour-pressure thermometer at any temperature is higher by a factor of at least two. The temperature value is obtained from a p(T) relationship, where p is the pressure value at the liquid—vapour interface. The pressure value measured with the room-temperature manometer is different from p, because of the aerostatic and thermomolecular effects, which must be corrected for. With respect to gas thermometry, the connecting-tube design can be optimized more satisfactorily and more easily, since there are no trade-off problems. Thermal problems are much more critical with the solid phase, because of the very poor thermal characteristics of the solid phase. It is therefore comparatively much more difficult to obtain true thermal equilibrium and, consequently, to obtain accurate values for vapour pressures of the solid—vapour interface. However, this is not impossible and, with some substances, the extension of vapour—pressure measurements down to solid—vapour equilibrium appreciably extends the range, and enhances the usefulness, of a vapour-pressure thermometer. These extensions are possible, in particular, when the triple-point pressure is high and also they are the only way of allowing temperature scales based on this type of principle to cover certain temperature intervals. For details see the book by Pavese and Molinar (1992). E4.0.4.3 The realization of vapour-pressure temperature scales Since vapour-pressure thermometry is semi-empirical, an interpolating instrument is not defined, and only an equation is specified. Should some of its coefficients not be defined, their value has to be determined at an equal number of fixed points, in this way ‘calibrating’ the specific apparatus used. If all of the coefficients are defined, fixed points are no longer necessary. The latter solution is generally preferred by international bodies, when they officially endorse a vapour-pressure scale. When no such endorsement exists for a given substance (actually for any substance except helium isotopes), the first concern for a user must be to select an equation suitable to represent the p = f (T) relationship, whose traceability to the thermodynamic temperature is reliably established to sufficient accuracy. The next concern for a laboratory (or an industry) that wants to use a vapour-pressure thermometer is how to design an apparatus convenient to use and ensuring the required accuracy level. The aim is not to realize an empirical scale, but to reproduce thermodynamic equilibrium states; therefore, nonequilibrium

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conditions or experimental artefacts must not be allowed to affect the measurements within the stated uncertainty. The criteria and precautions deriving from the discussion in earlier sections are the basis for such a design, irrespective of the accuracy level. Design can be less careful (e.g. lower substance purity) when lower accuracy is acceptable. For liquid-vapour equilibrium, the semi-empirical equation (E4.0.3) has been modified in several ways by adding or suppressing terms. However, these changes cause the model to switch from a semiempirical to a purely empirical model, a field where a wide set of mathematical tools is available for experimental data fitting. In fact, the literature abounds with far more papers on data fitting than on new experimental data. Very few of the experimental papers specify the temperature scale. The lack of such an indication does not mean, in general, that the temperature values are obtained directly on the thermodynamic scale. Most often, practical thermometers calibrated on an empirical scale are used for measurements. Very seldom is there an explicit reference to any version of the International Scale, so that traceability to thermodynamic values is questionable. The equations derived from such papers are themselves affected by the same lack of traceability. Therefore, when vapour-pressure measurements are used in thermometry, it is advisable to use those equations that have been subjected to international peer review, even in the absence of a formal endorsement. This type of review for several gases has been performed specifically by IUPAC and the Comite Consultatif de Thermometrie (CCT) (Bedford et al 1996). The resulting equations are not endorsed by these international bodies, but they are, indeed, ‘recommended’. They concern liquid—vapour pressures of e-H2 , Ne, N2 , Ar, O2 and solid—vapour pressures of Ne, N2 and Ar. At present, these equations and other selected equations from Pavese and Molinar (1992) are also available recalculated in the ITS-90 (Bedford et al 1996, Pavese 1993). A special case is helium vapour-pressure thermometry, because it has been officially endorsed by international bodies since the beginning of the 1950s (T58 and T62 scales: for details see Pavese and Molinar (1992)). In the ITS-90, 3He and 4He vapour pressures constitute the only definition below 3 K and must also be used for the determination of the lower fixed point of the ICVGT. An accurate measurement of helium vapour pressure involves specific precautions, especially when superfluid 4He is measured below 2.1768 K and when 3He is measured below ≈1 K. The reader is directed to the ‘Supplementary Information for the ITS-90’ (BIPM 1990); the ‘Techniques for Approximating the ITS-90’ (Bedford et al 1990) and the relevant literature in the book by Pavese and Molinar (1992).

E4.0.5 Electrical thermometers and their use Nearly all types of thermometry in the field of applied superconductivity are accomplished using electrical thermometers. The types of thermometer used at low temperatures show extreme variety (table E4.0.4) (Pavese 1990). This variety reflects the difficulty of finding a single type with a broad range of applications. Figure E4.0.6 shows the change in sensitivity with temperature for the different types. The characteristics of the thermometers listed in table E4.0.4 will be reviewed in the next section with the aim of providing a guide to their selection in view of each particular application and for each accuracy level (tables E4.0.5 and E4.0.6). This will be done by taking account of the parameters generally used for the characterization of sensors and giving particular consideration to the typical problems in cryogenic applications. A discussion of the calibration methods and of the accuracy which can be obtained with each type, with respect to the International Temperature Scale, can be found in the book by Bedford et al (1990).

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Table E4.0.4. Types of cryogenic thermometers.

E4.0.5.1 The choice of cryogenic thermometers The temperature-dependent electrical parameter that is measured is different for different types of thermometer: resistance, capacitance, voltage, electromotive force (emf). It is related to temperature by a functional relationship, which may itself affect the overall precision when using the sensor. In the worst case, nonmonotonic behaviour, as found in some capacitance thermometers (figure E4.0.6(f)), produces a gap in the useful temperature range, which must be spliced into two separate subranges. A similar

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Figure E4.0.6. Typical characteristics of cryogenic thermometers. The behaviour with temperature of the sensitivity and of the relative sensitivity is indicated. (a) Thermocouples: for the types refer to table E4.0.4; KP = Chromel P. (b) Platinum resistor. (c) Rhodium—iron resistor. (d ) Platinum—cobalt resistor. (e) Germanium (G) and carbon-glass (CG) resistors: carbon resistors show a behaviour similar to CG. (f) Capacitance thermometer; the sensitivity change relative to the value at 4.2 K is shown: notice the change in sign. (g) Gallium arsenide diode. (h) Silicon diode.

case occurs when a monotonic behaviour shows a sudden change in slope, because of a change of the underlying physical law, as with the silicon diode thermometers (figure E4.0.6(h)) again the characteristics must be spliced into two separate subranges, and in the narrow region where the slope shows the sharp change the accuracy will be adversely affected. The change in the underlying physical law is a very frequent reason for thermometers to be used in both the liquid nitrogen and the liquid helium temperature ranges. Therefore, it is quite common to experience difficulty in fitting the whole working range with

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Table E4.0.5. Accuracy required in applied superconductivity.

Table E4.0.6. Accuracy requirements for the measuring instrument.

a single equation. It is well known that high-order polynomials are needed to represent accurately the characteristics of germanium thermometers (figure E4.0.6(e)) (Bedford et al 1990). With the gallium arsenide diode thermometers (figure E4.0.6(g)), the accuracy of fitting with a single equation from 4 to 300 K requires polynomials of an even higher order.

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The usual temperature range of use of each thermometer, taking into account these difficulties—and the achievable accuracy, which will be discussed later—is shown in table E4.0.4. The functional relationship relating the thermometric parameter to temperature is valid only for each well defined thermometric substance. Therefore, the purity in the case of pure elements (e.g platinum), or the composition in the case of alloys (e.g. rhodium—iron, gold—cobalt), must be precisely specified for the interchangeability of sensors nominally of the same type to be preserved or else each unit must be calibrated. This is an important parameter in industrial applications, and is one weak point for most of the cryogenic thermometers, including the platinum type, which is one of the best in this respect above room temperature. For the solid-state thermometers, interchangeability is even more difficult to achieve, as it depends on the uniformity and reproducibility of the doping. An insufficient degree of interchangeability can indirectly limit the overall accuracy of the measurements due to an insufficient flexibility of the measuring instrument in accounting for the spread of the sensor characteristics. Today a sensor cannot be simply evaluated for its use in manual measurements. It is important also to consider, when selecting a type, the requirements set by it for the associated electronic instrumentation. In this respect, measuring a resistance or a voltage is simpler than measuring a capacitance, and electronic or resonant thermometers (noise, quartz, quadrupole) require specialized and costly equipment: for this reason they are not considered here. In table E4.0.5 the requirements for relative and absolute accuracy of the measuring instrument are reported for each type of sensor: some of the best accuracy levels cannot actually be obtained for other reasons, as indicated in table E4.0.4. The relationship between the thermometric parameter and temperature may also depend on other physical parameters, which are to be considered as spurious. The thermometers must be as insensitive to them as possible. However, some of these effects are intrinsic to the thermometric material and cannot be avoided or minimized, unless another material is used. This is the case of the effect of magnetic fields, which is a situation very specific to cryogenics, because most of the high-field magnets are of the superconducting type (see table E4.0.7). In addition, the thermometers cannot generally be considered as isotropic in this respect. Therefore, the direction of the field lines with respect to the axis of the element must be known for the effect of the field to be specified. In table E4.0.6 (Pavese 1990), the change relative to the calibration in zero field is reported as an equivalent temperature change, but this is not entirely correct when the effect of the magnetic field on the thermometer is very large, because in this case it is also nonlinear as the thermometer sensitivity is changed by both temperature and magnetic field. This has been observed, for example, with the platinum—cobalt thermometers: the magnetic field affects the sensitivity so much that below 4 K and above 5 T it goes to zero. The change in the thermometer sensitivity, due to the effect of the magnetic field, is more or less present in all cases, but is generally not reported in the literature. Some cryogenic thermometers have been specifically developed for minimizing the calibration change in high magnetic fields, for example the capacitance, the carbon-glass and the compensated germanium resistance thermometers. Sometimes the resulting shape of the characteristics is quite complicated, reducing the possibility of achieving a high-accuracy fit and, therefore, reducing the overall accuracy obtainable from the thermometers. The thermometric material must be mounted in a device to form a sensor. This fabrication process usually determines most of the overall quality of the sensor, especially its reproducibility on thermal cycling. In cryogenic sensors, the main source of instability comes from the mechanical effects induced by the different thermal expansion coefficients of different parts, which produce a change each time the sensor is cooled down. Strain is introduced in the wire-wound or film types of resistance thermometer and in thermocouples; in the compound types (carbon or cermet thermometers) the grains of conductive material tend to move relative to each other; in the solid-state types, problems with the connecting leads are considered the main single source of instability. The reproducibility levels for the different commercial types are given in table E4.0.4: they are realistic figures, even if some better levels can often be suggested by the manufacturers.

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E4.0.5.2 Thermometer mounting A good thermometer must minimize the effect of spurious parameters which have an influence on the thermometric parameter. Some of these parameters, such as the sensitivity to magnetic fields and to strain, have already been discussed. Among others, it is worth mentioning, for cryogenic sensors, the spurious thermal emfs which are easily induced in thermocouples (see note b in table E4.0.4). However, if care is not taken, these emfs also develop as a result of the Seebeck effect at the connections between the sensing element and the leads, especially in semiconducting thermometers, and this may be a limiting factor for the achievement of the highest accuracy. Another limiting factor for resistance thermometers, when using an a.c. bridge for its measurement, is the size of the imaginary part of the actual thermometer impedance, which depends on the thermometer itself and on its connecting wires to the measuring apparatus.

Figure E4.0.7. A thermometer for measuring the temperature T0 of the body B. It requires, in general, a fitting F, which is connected to the body B through a mounting element M. The thermometer generally shows a thermal resistance to its fitting, so that T3 can be different from the temperature T0 to be measured. If the sensing element itself dissipates energy, its temperature T will be higher than T3.

In cryogenic sensors these errors can be very high, unless great care is taken, because of the increasing difficulty in keeping the thermal resistance small between the sensing element and the body whose temperature is being measured and because the sensor is often working in vacuo. In figure E4.0.7, a thermometer is shown mounted in a body B. The sensing element, with characteristics f (T), is connected via two or four leads, which can be thermally anchored at temperature TA to room temperature: the measured T is an estimate of T0. In general, the thermometer is inserted in the body B through a fitting F at temperature T2 via a mounting M. All precautions must be taken so that T2 = T0. It is not always possible to screw the thermometer itself to its mounting M. In laboratory use, in particular, it is most often inserted in a well drilled in the mounting and some conductive media, generally vacuum grease, is used. With dissipative thermometers, T3 tends to be higher than T2 because of this thermal resistance. For the same reason T > T3: this is the overheating error, which can be much higher than the thermal resistance error, because the sensing element is thermally connected to the sensor case mainly through some gas

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Thermometry Table E4.0.7. Magnetic-field-dependent temperature errors(Pavese 1990).

contained in the sensor, more than through the connecting leads. For operation in vacuo, the sensor must therefore be sealed, which is a requirement generally unnecessary and uncommon for use above room temperature. The size of the overheating error and of the heat introduced in the experiment by the sensor limits the use of some types of thermometer to temperatures above 1 K. When small, these errors can be detected and corrected for, only with resistance thermometers, by measurement at two different current levels.

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947 TableE4.0.7. Continued.

E4.0.6 Future trends in thermometry The gas-based types of thermometry will greatly benefit from the development of sealed devices, which can be of practical use, but this development depends upon the availability of cryogenic pressure transducers of higher quality, which are still under development. A more widespread use of fixed points for checking thermometer calibration or for their (re-)calibration even outside metrological institutions can be foreseen, since the development of sealed fixed points for

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Thermometry Table E4.0.7. Continued.

use in the cryogenic temperature range is already a reality. The goals for the improvement of electrical thermometers can be summarized as follows: (i) availability of a single thermometer of good reproducibility in the whole cryogenic temperature range (ii) availability of a reproducible thermometer which is also insensitive to magnetic fields at least up to 12 T (iii) improvement of the interchangeability (iv) lowering the cost of each unit (v) development of simple devices for in situ calibration or check of the calibration stability. The research efforts are mainly concentrated on three different approaches. One is to develop solidstate devices matching a better interchangeability with higher sensitivity and good reproducibility, such as silicon diodes or germanium resistors which additionally are less affected by high magnetic fields. Another is to develop film thermometers made of pure metals, like platinum, or of stable metallic alloys.

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References

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like rhodium—iron, or of conductive oxides, like RuO2 for high magnetic fields. The third is to improve the technology of wire-wound thermometers or to use it with new alloys. References Ancsin J 1978 Vapour pressures and triple point of neon and the influence of impurities on these properties Metrologia 14 1–7 Astrov D N, Orlova M P and Kytin G A 1969 PRMI temperature scale in the range from 4.2 K to 20 K Metrologia 5 111–8 Astrov D N, Belyanski L B, Dedikov Yu A, Polunin S P and Zakharov A A 1989 Precision gas thermometry between 2.5 K and 308 K Metrologia 26 151–66 Bedford R E, Bonnier G, Maas H and Pavese F 1990 Techniques for Approximating the International Temperature Scale (Monograph No 1) (Sèvres: Bureau International des Poids et Mesures) Bedford R E, Bonnier G, Maas H and Pavese F 1996 Recommended values of temperature on the International Temperature Scale of 1990 for a selected set of secondary reference points Metrologia 33 133–54 Bedin E, De Combarieu A and Doulat J 1980 Un thermomètre à gaz à volume constant à basse température (4–300 K) Bulletin BNM 5–13 BIPM 1990 Supplementary Information for the ITS-90 (Sèvres: Bureau International des Poids et Mesures) Cetas T C and Swenson C A 1972 A paramagnetic salt temperature scale 0.9 K to 18 K Metrologia 8 46–64 Fellmuth B, Maas H and Elefant D 1985 Investigation of the superconducting transition point of niobium as a reference temperature Metrologia 21 169–80 Gershanik A P, Glikman M S and Astrov D N 1978 On the sorption correction for the precise gas thermometer Doc. CCT/78-45, Annexe T11, Comptes Rendus CCT (Sèvres: Bureau International des Poids et Mesures) pp 93–7 Hudson R P 1972 Principles and Application of Magnetic Cooling (Amsterdam: North-Holland) Keller W E 1969 Helium-3 and Helium-4 (The International Cryogenics Monograph Series) (New York: Plenum) Lin Peng, Mao Yuzhu, Hong Chaosheng, Yue Yi and Zhang Qinggeng 1990 Study of the realization of 4-He lambdatransition point temperature by means of a small sealed cell Proc. ICEC 13 Cryogenics 30 special issue Mangum B W and Bowers W J 1978 Two practical magnetic thermometers for use below 30 K J. Physique Coll. C6 1175 Lipinsky L, Manuszkiewicz H and Szmyrka-Grzebyk A 1990 Sealed cells containing superconducting samples for the realization of thermometric fixed points IV Symp. on Temperature and Thermal Measurements in Industry and Science, TEMPMEKO 90 (Helsinki, 1990) pp 126–30 Nara K, Rusby R L and Head D I 1989 The interpolation characteristics of a sealed gas thermometer Doc. CCT/89–21 (Sèvres: Bureau International des Poids et Mesures) Nara K, Rusby R L and Head D I 1990 Study of the interpolation characteristics of a sealed gas thermometer Cryogenics 30 952–8 Pavese F 1990 Fields of application: cryogenics Sensors—a Comprehensive Book Series in Eight Volumes (Thermal Sensors 4) ed T Ricolfi and H Scholz (Weinheim: VCH) Pavese F 1993 Recalculation on ITS-90 of accurate vapour-pressure equations for e-H2, Ne, N2, O2, Ar, CH4 and CO2 J. Chem. Thermodyn. 25 1351–61 Pavese F, Ancsin J, Astrov D N, Bonhoure J, Bonnier G, Furukawa G T, Kemp R C, Maas H, Rusby R L, Sakurai H and Ling Shankang 1984 An international intercomparison of fixed points by means of sealed cells in the range 13.81–90.686 K Metrologia 20 127–44 Pavese F, Ferri D, Giraudi D and Steur P P M 1992 Long-term stability of permanent realizations of the triplepoint of gases in metal sealed cell Temperature, its Measurements and Control in Science and Industry vol 6, ed R E Bedford (New York: AIP) pp 251–6 Pavese F and Molinar G F 1992 Modern Gas-Based Temperature and Pressure Measurements (The International Cryogenic Monograph Series) (New York: Plenum) Rubin L G and Brandt B L 1986 Adv. Cryogen. Eng. 31 1221–30 Rusby R L 1972 Temperature, its Measurement and Control in Science and Industry vol 4, ed H H Plumb (Pittsburg, PS: Instrument Society of America) pp 865–9

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Schooley J F, Evans G A Jr and Soulen R J Jr 1980 Preparation and calibration of the NBS SRM 767: a superconductive temperature fixed point device Cryogenics 20 193–9 Schooley J F and Soulen R J Jr 1982 Superconductive thermometric fixed points Temperature, its Measurements and Control in Science and Industry vol 5, ed J F Schooley (New York: AIP) pp 251–60 Song Naihao, Hong C S, Mao Yuzhu, Ling Peng, Zhang Qinggeng and Zhang Liang 1991 Realizazion of the lambda transition temperature of liquid helium-4 Cryogenics 31 87–93 Van Degrift C T, Bowers W J Jr, Wilders D G and Pipes P B 1978 A small gas thermometer for use at low temperatures ISA Annu. Conf (New York: ISA) pp 33–8 Van Rijn C and Durieux M 1972 A magnetic temperature scale between 1.5 K and 30 K Temperature, its Measurement and Control in Science and Industry vol 4, ed H H Plumb (Pittsburgh, PA: Instrument Society of America) pp 73–8 Van Sciver S W 1986 Helium Cryogenics (The International Cryogenic Monograph Series) (New York: Plenum) Winteler H R 1981 High-pressure gas-filled thermometers Proc. I Symp. IMEKO TC12, CSVTS, Dum Techniky (Prague, 1981) pp 162–8

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E5 Methods and instrumentation for magnetic field measurement K N Henrichsen, C Reymond and M Tkatchenko

E5.0.1 Introduction Hans Christian Oersted discovered electromagnetism in June 1820. He observed that an electric current in a platinum wire, placed near and parallel to a magnetic compass needle affected the compass reading. By September of the same year Andre-Marie Ampère had already formulated the quantitative law of electromagnetism. Shortly after, in 1831, Michael Faraday discovered electromagnetic induction. The first laboratory electromagnets were soon in use in many places. So, these events mark the beginning of magnet measurement technology. Before computers became common tools, electromagnets were designed using analytical calculations or by measuring representative voltage maps in electrolytical tanks and resistive sheets. Magnetic measurements on the final magnets and even on intermediate magnet models were imperative at that time. Nowadays it has become possible to calculate strength and quality of magnetic fields with an impressive accuracy. However, the best and most direct way to verify that the expected field quality has been reached is to perform magnetic measurements on the finished magnet. It is also the most efficient way of verifying the quality of series-produced electromagnets in order to monitor wear of tooling during production. It is curious to note that while the measurement methods have remained virtually unchanged for a very long period, the equipment has been subject to continual development. In the following only the more commonly used methods will be discussed. It is noticeable that these methods are complementary and that a wide variety of equipment is readily available from industry. For the many other existing measurement methods, a more complete discussion can be found in the two classic bibliographical reviews (Germain 1963, Symonds 1955). An interesting description of early measurement methods can be found in the article by McKeehan (1929). Much of the following material was presented at the CERN Accelerator School on Magnet Measurement and Alignment (Turner 1992). Those proceedings contain a recent compendium of articles in this field and form a complement to the classic reviews.

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E5.0.2 Measurement methods E5.0.2.1 Choice of measurement method The choice of measurement method depends on several factors. The field strength, homogeneity and variation in time, as well as the required accuracy all need to be considered. Also the number of magnets to be measured can determine the method and equipment to be deployed. As a guide, figure E5.0.1 shows the accuracy which can be obtained in an absolute measurement as a function of the field level for a few commonly used methods.

Figure E5.0.1. Measurement methods: accuracies and ranges.

E5.0.2.2 The fluxmeter method This method, which is based on the induction law, is the oldest of the currently used methods for magnetic measurements, but it can be very precise (Coupland et al 1981). It was used by Wilhelm Weber in the middle of the last century (Weber 1853) when he studied the variations in strength and direction of the earth’s magnetic field. Nowadays it has become the most important method for particle accelerator magnets. It is also the most precise method for measuring the direction of the magnetic flux lines; this being of particular importance in accelerator magnets. Measurements are performed either by using fixed coils in a dynamic magnetic field or by moving the coils in a static field. The coil movement may be a rotation through a given angle, a continuous rotation or simply a movement from one position to another. The coil geometry is often chosen to suit a particular measurement. One striking example is the flux ball (Brown and Sweer 1945) whose complex construction made it possible to perform point measurements in inhomogeneous fields. Very high accuracy may be reached in differential fluxmeter measurements using a pair of search coils connected in opposition, with one coil moving and the other fixed, thus compensating fluctuations in the magnet excitation current and providing a much higher sensitivity when examining field quality. The same principle is applied in harmonic coil measurements, but with both coils moving. A wide variety of coil configurations are used, ranging from the simple flip-coil to the complex harmonic coil systems used in fields of cylindrical symmetry.

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Induction coils The coil method is particularly suited for measurements with long coils in particle accelerator magnets (Brown 1986, Finlay et al 1950), where the precise measurement of the field integral along the particle trajectory is the main concern. Long rectangular coils were usually employed and are still used in magnets with a wide horizontal aperture and limited gap height. In this case, the geometry of the coil is chosen so as to link with selected field components (de Raad 1958). The search coil is usually wound on a core made from a mechanically stable material, in order to ensure a constant coil area, and the wire is carefully glued to the core. Glass or ceramics with their low thermal dilation are often used as core materials. During coil winding the wire must be stretched so that its residual elasticity assures a well-defined geometry and mechanical stability of the coil. Continuously rotating coils with commutating polarity were already employed in 1880 (McKeehan 1929). The harmonic coil method has now become very popular for use in magnets with circular cylindrical geometry and has been developed since 1954 (Dayton et al 1954, Elmore and Garnett 1954). The coil support is usually a rotating cylinder. The induced signal from the rotating coil is often transmitted through slip rings to a frequency selective amplifier (frequency analyser), thus providing analogue harmonic analysis. The principle of the harmonic coil measurement is illustrated in figure E5.0.2. The radial coil extends through the length of the magnet and is rotated around the axis of the magnet. As the coil rotates, it will cut the radial flux lines. A number of flux measurements are made between predefined angles and will permit the precise determination of the strength, quality and geometry of the magnetic field. A Fourier analysis of the measured flux distribution will result in a precise description of the field parameters in terms of the harmonic coefficients

where B0 is the amplitude of the main harmonic and r0 is a reference radius. bn and an are the harmonic coefficients. In this notation b1 will describe the normal dipole coefficient, b2 the normal quadrupole

Figure E5.0.2. The principle of a harmonic coil measurement.

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coefficient, etc. The corresponding skew field components are described by the coefficients a1, a2 etc. It should be mentioned that this is the European notation. Those definitions are slightly different in North America where the normal dipole coefficient would be described by a coefficient b0, the normal quadrupole coefficient by a coefficient b1, etc. With the advent of modern digital integrators and precise angular encoders, the harmonic coil measurement method has improved considerably and is now considered as the best choice for most types of particle accelerator magnet, in particular those designed with cylindrical symmetry. In practice the coil is rotated one full turn in each angular direction while the electronic integrator is triggered at the defined equidistant angles by an angular encoder connected to the axis of the coil. In order to speed up the calculation of the Fourier series, it is an advantage to choose a binary number (e.g. 512) of measurement points. This method provides the advantage of simultaneous measurements of field strength, quality and geometry. The reproducibility and resolution of the flux measurement can be of the order of 100 ppm of the main field component. A compensating coil, connected in series and rotated with the main coil, may be used to suppress the main component and thus increase the sensitivity of the system for precise measurements of field quality. More than an order of magnitude may be gained on the measurement of the higher-order harmonics in this way. The field geometry is determined with an angular precision of 0.1 mrad and the magnetic centre in multipole magnets may be detected with an impressive accuracy of 0.01 mm (Pagano et al 1984). This accuracy is unfortunately often reduced to about 0.1 mm as the mechanical reference point is transferred from the measurement coil to the outside of the magnet itself. Dynamic fields are measured with a static coil linking to selected harmonics (Morgan 1972). The harmonic coil measurement principle and its related equipment were described in detail by Walckiers (1992). A thorough description of the general theory including detailed error analysis can be found in the article by Davies (1992). The practical use of the harmonic coil method for large-scale measurements in superconducting magnets was described by Brück et al (1991) and Schmüser (1991) and more recent developments by Billan et al (1994), Green et al (1993) and Thomas et al (1993). Finally it should be mentioned that complete systems for harmonic coil measurements are now available from industry. Another induction measurement consists of moving a stretched wire in the magnetic field, thus integrating the flux cut by the wire. It is also possible to measure the flux change while varying the field and keeping the wire in a fixed position. Tungsten is often selected if the wire cannot be placed in a vertical position. The accuracy is determined by the mechanical positioning of the wire. Sensitivity is limited, but can be improved by using a multiwire array. This method is well suited to geometry measurements, to the absolute calibration of quadrupole fields and in particular to measurements in strong magnets with very small aperture. The choice of geometry and method depends on the useful aperture of the magnet. The sensitivity of the fluxmeter method depends on the coil surface and the quality of the integrator. The coil—integrator assembly can be calibrated to an accuracy of a few tens of parts per million in a homogeneous magnetic field by reference to a nuclear magnetic resonance (NMR) probe, but care must be taken not to introduce thermal voltages. Not only the equivalent surface of the search coil must be measured, but also its median plane which often differs from its geometric plane due to winding imperfections. In the case of long measurement coils, it is important to ensure very tight tolerances on the width of the coil. If the field varies strongly over the length of the coil, it may be necessary to examine the variation of the effective width. The main advantage of search coil techniques is the possibility of a very flexible design of coil. The high stability of the effective coil surface is another asset. The linearity and the wide dynamic range also play important roles. The technique can be easily adapted to measurements at cryogenic temperatures. After calibration of the coils at liquid-nitrogen temperature, only a minor correction has to be applied for use at lower temperatures. On the other hand, the need for relatively large induction coils and their related

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mechanical apparatus, which is often complex, may be a disadvantage. Furthermore, the measurements with moving coils are relatively slow. The flux measurement Induction coils were originally used with ballistic galvanometers and later with more elaborate fluxmeters (Grassot 1904). The coil method was improved considerably with the development of photoelectric fluxmeters (Edgar 1937) which were used for a long period of time. The measurement accuracy was further improved with the introduction of the classic electronic integrator, the Miller integrator. It remained necessary, however, to employ difference techniques for measurements of high precision (Green et al 1953). Later, the advent of digital voltmeters made fast absolute measurements possible and the Miller integrator has become the most popular fluxmeter. With the development of solid-state d.c. amplifiers, this integrator has become inexpensive and is often used in multicoil systems. Figure E5.0.3 shows an example of such an integrator. It is based on a d.c. amplifier with a very low input voltage offset and a very high open-loop gain. The thermal variation of the integrating capacitor is the most critical problem. The integrating components are therefore mounted in a temperature-controlled oven. Another problem is the decay of the output signal through the capacitor and the resetting relay. Careful protection and shielding of these components is therefore essential in order to reduce the voltages across the critical surface resistances.

Figure E5.0.3. An analogue integrator.

The dielectric absorption of the integrating capacitor sets a limit to the integrator precision. A suitable integrating resistor is much easier to find. Most metal-film resistors have stabilities and temperature characteristics matching those of the capacitor. The sensitivity of the integrator is limited by the d.c. offset and low-frequency input noise of the amplifier. A typical value is 0.5 µV which must be multiplied by the measurement time in order to express the sensitivity in terms of flux. Thermally induced voltages may cause a problem, so care must be taken in the choice of cables and connectors. In tests at CERN the overall stability of the integrator time constant proved to be better than 50 ppm over a period of three months. A few electronic integrators have been developed by industry and are commercially available. In more recent years, a new type of digital integrator has been developed, which is based on a highquality d.c. amplifier connected to a voltage-to-frequency converter (VFC) and a counter. The version shown in figure E5.0.4 was developed at CERN (Galbraith 1986) and is now commercially available. The input of the VFC is given an offset of 5 V in order to provide a true bipolar measurement. This offset is balanced by a 250 kHz signal which is subtracted from the output of the VFC. Two counters are used in order to measure with continuously moving coils and to provide instant readings of the integrator. One

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Figure E5.0.4. A digital integrator.

of the counters can then be read and reset while the other is active. In this way no cumulative errors will build up. This integrator has a linearity of 20 ppm. Its sensitivity is limited by the input amplifier, as in the case of the analogue amplifier. This system is well adapted to digital control but imposes limits on the rate of change of the flux since the input signal must never exceed the voltage level of the VFC. The accuracy is currently better than 100 ppm, but can be improved. A good example is given by the CERN SPS dipole magnetic field cycle measurement (Di Cesare et al 1989). A static coil placed in the SPS reference magnet constantly monitors the field and two NMR measurements performed at two points in each cycle allow us to continuously correct the offset and gain of the digital integrator. In such an arrangement, the resolution in time is 1 ms and the accuracies reach 10 ppm for the field and 1 µs for the time. E5.0.2.3 The Hall generator method Hall discovered in 1879 that a metal strip immersed in a transverse magnetic field and carrying a current developed a voltage mutually at right angles to the current and field that opposed the Lorentz force on the electrons (Hall 1879). In 1910 the first magnetic measurements were performed using this effect (Peukert 1910). It was, however, only around 1950 that suitable semiconductor materials were developed (Pearson 1948, Welker 1952, 1955) and since then the method has been used extensively. It is a simple and fast measurement method, providing relatively good accuracy, and therefore the most commonly used in large-scale field mapping (Acerbi et al 1981, Bazin et al 1981, Swoboda 1981). The accuracy can be improved at the expense of measurement speed. Theory of the Hall effect The effect can be explained with the help of figure E5.0.5. Supposing a semiconducting plate with a current Ix along the x axis is exposed to a transverse magnetic field Bz. Then a voltage Vy (also called Hall voltage VH) can be detected on the y axis at both ends of the semiconductor plate. Under the assumption that the semiconductor is infinitely long one has to distinguish two phenomena: the longitudinal magnetoresistivity ρ = ρ0(1 + M B 2) where is ρ0 the resistivity at zero magnetic field and M is the coefficent of the magnetoresistivity; due to ρ and the current density J an electrical field Eρ = ρJ is generated; the transverse magnetoresistivity (Hall resistivity) yielding, in the presence of a current density J, a Hall electrical field of EH = RHB × J where RH is the Hall constant. The physical reason for the

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Figure E5.0.5. The basic arrangement for the measurement of the Hall effect.

transverse magnetoresistivity is the fact that there is a force F acting on the conduction electrons F = —e υ × B resulting in a redistribution of electrical charges (e is the electric charge of an electron and υ is its velocity). The sum of both electrical fields is

According to figure E5.0.6 the measured voltage in the y direction (the Hall voltage) is

where RH is the Hall constant, Bz the z component of the magnetic field, Ix the x component of the current (also called the Hall current) and d is the thickness of the semiconductor plate defined by Jx = Ix/bd.

Figure E5.0.6. Electrical field E and current density J in a semiconducting plate of width b. The applied magnetic field is directed upwards (out of the paper plane).

Hall probe measurement The Hall generator provides an instant measurement, uses very simple electronic measurement equipment and offers a compact probe, suitable for point measurements. A large selection of this type of gaussmeter is now commercially available. The probes can be mounted on relatively light positioning gear (Swoboda 1981). Considerable measurement time may be gained by mounting Hall generators in modular multiprobe arrays and applying multiplexed voltage measurement. The wide dynamic range and the possibility of static operation are other attractive features.

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However, several factors set limits on the obtainable accuracy. The most serious is the temperature coefficient of the Hall voltage. Temperature stabilization is usually employed in order to overcome this problem (Brand and Brun 1979), but increases the size of the probe assembly. The temperature coefficient may also be taken into account in the probe calibration by monitoring the temperature during measurements (Poole and Walker 1981). It depends, however, also on the level of the magnetic field (Poole and Walker 1981), so relatively complex calibration tables are needed. Another complication can be that of the planar Hall effect (Goldberg and Davis 1954), which makes the measurement of a weak field component normal to the plane of the Hall generator problematic if a strong field component is present parallel to this plane. This effect limits the use in fields of unknown geometry and in particular its use for determination of field geometry. Last but not least is the problem of the nonlinearity of the calibration curve, since the Hall coefficient is a function of the field level. A Hall generator of the cruciform type (Hauseler and Lippmann 1968) shows a better linearity and has a smaller active surface than the usual rectangular generator. Its magnetic centre is, therefore, better defined, so it is particularly well suited for measurements in strongly inhomogeneous fields. Special types, which have a smaller temperature dependence, are available, but these show a lower sensitivity. The measurement of the Hall voltage sets a limit of about 20 µT on the sensitivity and resolution of the measurement if conventional d.c. excitation is applied to the probe. This is mainly caused by thermally induced voltages in cables and connectors. The sensitivity can be improved considerably by application of a.c. excitation (Cox 1964, Donoghue and Eatherly 1951). A good accuracy at low fields can then be achieved by employing synchronous detection techniques for the measurement of the Hall voltage (Dickson and Galbraith 1985). Special Hall generators for use at cryogenic temperatures are also commercially available. Although they show a very low temperature coefficient, they unfortunately reveal an additional problem at low temperatures. The so-called ‘Shubnikov—de Haas effect’ (Babiskin 1957, Frederikse and Hosier 1958) shows up as a field-dependent oscillatory effect of the Hall coefficient which may amount to about one per cent at high fields, depending on the type of semiconductor used for the Hall generator. This adds a serious complication to the calibration. The problem may be solved by locating the Hall generator in a heated anticryostat (Polak 1953). The complications related to the planar Hall effect are less important at cryogenic temperatures and are discussed in detail by Polak and Hlasnik (1970). Altogether, the Hall generator has proved very useful for measurements at low temperature (Kvitovic and Polak 1993). Calibration Hall generators are usually calibrated in a magnet in which the field is measured simultaneously using the NMR technique. The calibration curve is most commonly represented in the form of a polynomial of relatively high order (7 or 9) fitted to a sufficiently large number of calibration points. This representation has the advantage of a simple computation of the magnetic induction from a relatively small table of coefficients. A physically better representation is the use of a piecewise cubic interpolation through a sufficient number of calibration points which were measured with high precision. This can be done in the form of a simple Lagrange interpolation or even better with a cubic spline function. The advantage of the spline function comes from its minimum curvature and its ‘best approximation’ properties (Walsh et al 1972). The function adjusts itself easily to nonanalytic functions and is very well suited to interpolation from tables of experimental data. The function is defined as a piecewise polynomial of third degree passing through the calibration points such that the derivative of the function is continuous at these points. Very efficient algorithms can be found in the literature (Ralston and Wilf 1967). The calculation of the polynomial coefficients may be somewhat time-consuming but need only be done once at calibration time. The coefficients (typically about 60 for the bipolar calibration of a cruciform Hall generator) can

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be easily stored in a microprocessor device (Brand and Brun 1979) and the subsequent field calculations are very fast. The quality of the calibration function can be verified from field values measured between the calibration points. A well-designed Hall-probe assembly can be calibrated to a long-term accuracy of 100 ppm. The stability may be considerably improved by powering the Hall generator permanently and by keeping its temperature constant. E5.0.2.4 Magnetic resonance techniques The magnetic resonance technique is frequently used, not only for calibration purposes, but also for high-precision field mapping. This measurement method was first used in 1938 (Rabi et al 1938, 1939) for measurements of the nuclear magnetic moment in molecular beams. A few years later the phenomenon was observed in solids by two independent research teams (Block et al 1946a, b, Purcell et al 1946). Since then, the method has become the most important way of measuring magnetic fields with very high precision. It is now considered as a primary standard for calibration. Based on an easy and precise frequency measurement it is independent of temperature variations. The theory and applications of NMR are widely covered in the literature. Let us just recall that in the presence of a static magnetic field B0, a nucleus with magnetic moment µ can take (2I + 1) distinct energy states, I being the spin quantum number. The separation of these states is ∆E = B0/I. Transitions between levels can be induced by applying an alternating magnetic field perpendicular to the static field if its frequency equals the resonance frequency f = ∆E/h = γ B0, with γ = µ/hI. For magnetic fields of the order of 1 T, the NMR frequencies lie in the radiofrequency (RF) region. For protons, γ is known very precisely. In practice, a sample of water is placed inside an excitation coil, powered from an RF oscillator (figure E5.0.7). The precession frequency of the nuclei in the sample is measured either as nuclear induction (coupling into a detecting coil) or as resonance absorption (Bloembergen et al 1948). The measured frequency is directly proportional to the strength of the magnetic field with coefficients of 42.576396(13) MHz T–1 for protons (Cohen and Taylor 1987, Taylor and Cohen 1990) and 6.53569 MHz T–1 for deuterons. The magnetic field or the RF is modulated with a low-frequency signal in order to determine the resonance frequency (Borer and Fremont 1977).

Figure E5.0.7. The schematic arrangement for the measurement of a magnetic field by NMR.

Commercially available instruments measure fields in the range from 0.04 T up to 14 T with an absolute accuracy better than 5 ppm and a reproducibility of 0.1 ppm. The advantages of the method are its very high accuracy, its linearity and the static operation of the system. The most important disadvantage is the need for an almost homogeneous field in order to obtain a sufficiently coherent signal (see table E5.0.1). Note also that this technique does not allow us to determine the direction of a magnetic field.

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Table E5.0.1. Required homogeneity in ppm cm-1 of the field the most common commercially available NMR magnetometer.

A small compensation coil, formed on a flexible printed circuit board and providing a field gradient, is often placed around the probe when used in a slightly inhomogeneous field. A correction of the order of 0.35 T m–1 may be obtained (Borer and Fremont 1977). The limited sensitivity and dynamic range also set limits to the suitability of this method. It is, however, possible to use several probes with multiplexing equipment if a range of more than half a decade is needed. NMR magnetometers are used for magnetic field measurement and stabilization of mass spectrometers, particle accelerators dipoles and gaussmeter calibration bench magnets. Magnetic resonance imaging has been proposed for accelerator magnet measurement (Gross 1986). It is a very promising technique, which has proven its quality in other applications. However, the related signal processing requires powerful computing facilities, which were not so readily available in the past. Low-field probes (0.5 to 3 mT) have been realized using the electronic paramagnetic resonance (EPR) of special narrow linewidth sample material, to extend the range of current NMR magnetometers below 40 mT. They are used in calibration bench measurements and to perform warm measurements on superconducting coils to detect any faults in order to avoid the costs of installation inside a cryostat and of the liquid helium required for cooling down. In magnetic resonance imaging (MRI) systems used in medicine, the magnets are mapped and shimmed to a few parts per million in a sphere of 40 to 50 cm diameter, with a commercially available dedicated multiprobe NMR magnetometer. This instrument is based on the continuous wave (CW) principle (figure E5.0.8). One or several probes are fed with a modulated radio frequency. Each time this frequency matches the sample nucleus

Figure E5.0.8. A multiprobe NMR magnetometer based on the CW principle.

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Figure E5.0.9. The measurement sequence of a multiprobe NMR magnetometer based on the CW principle.

resonance frequency, an absorption signal is detected and amplified (figure E5.0.9). When the signal crosses a given threshold, a pulse triggers an individual probe counter. The content of this counter is proportional to the field offset of the probe. All probes are simultaneously read. One measurement run lasts about 50 modulation periods. Each signal occurrence is monitored by a microprocessor to eliminate perturbing signals, taking the average of all good values and performing a root mean square (rms) calculation in order to estimate the credibility of the results. To measure the field on the surface of a spherical volume (e.g. diameter 500 mm), up to 24 probes are mounted on a semicircular plate installed on a rotating holder positioned inside the magnet. With such an arrangement, an MRI magnet can be measured at several hundred points, in a couple of minutes, with a field accuracy of 0.1 ppm and a probe position uncertainty smaller than 0.5 mm. The set of data is processed in a computer in order to express the field in the form of a spherical harmonics deconvolution. The resulting set of coefficients representing the inhomogeneity of the field is used in the shimming process. E5.0.2.5 Flux-gate magnetometer The flux-gate magnetometer (Kelly 1951) is based on a thin linear ferromagnetic core on which detection and excitation coils are wound. The measurement principle is illustrated in figure E5.0.10. In its basic version, it consists of three coils wound around a ferromagnetic core: an a.c. excitation winding A, a detection winding B that indicates the zero-field condition and a d.c. bias coil C that creates and maintains the zero field. In practice the coils are wound in subsequent layers. The core is made up from a fine

Figure E5.0.10. The measurement principle of a flux-gate magnetometer. A—a.c. excitation coil; B—detection coil; C—d.c.-bias coil.

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wire of Mumetal or a similar material that has an almost rectangular hysteresis curve. The method was introduced in the 1930s and was also named ‘peaking strip’. It is restricted to use with low fields, but has the advantage of offering a linear measurement and is well suited for static operation. As a directional device with very high sensitivity, it is suitable for studies of weak stray fields around magnets. Much more complex coil configurations are applied for precision measurements and in cases where the measured field should not be distorted by the probe. The most interesting application is now in space research and important developments of this technique have taken place over the last decade (Gordon and Brown 1972, Nielsen et al 1992, Primdahl 1979). The use of modern materials for magnetic cores has increased the sensitivity to about 20 pT and can assure a wide dynamic range. The upper limit of the measurement range is usually of the order of a few tens of millitesla, but can be extended by applying water cooling to the bias coil. Flux-gate magnetometers with a typical range of 1 mT and a resolution of 1 nT are commercially available from several sources. They have many other practical applications, for example for navigation purposes. E5.0.2.6 Magnetoresistivity effect The magnetoresistivity of bismuth was exploited quite early and a commercial instrument already existed at the end of last century. Technical problems were, however, important (Kapitza 1928). Dependence on temperature and mechanical stress, combined with difficulties of manufacture and problems with electrical connections, caused a general lack of reliability in this measurement method. As with the Hall generator, it was only when semiconductor materials became available that the method turned into a success. Then inexpensive magnetoresistors came on the market and were used also for magnetic measurements (Welch and Mace 1970). A more recent application for field monitoring was implemented in one of the large electron-positron spectrometers (Brouwer et al 1992). E5.0.2.7 Visual field mapping The best known visual field mapper is made by spreading iron powder on a horizontal surface placed near a magnetic source, thus providing a simple picture of the distribution of flux lines. Another very classical way of observing flux-line patterns is to place a free-moving compass needle at different points in the volume to be examined and note the direction of the needle. This compass method was applied, long before the discovery of electromagnetism, for studies of the variations in the direction of the earth’s magnetic field. Another visual effect may be obtained by observing the light transmission through a colloidal suspension of diamagnetic particles, subject to the field. Faraday effect The magneto-optical rotation of the plane of polarization of polarized light (Faraday effect) is a classical method for the visualization of magnetic fields. A transparent container filled with a polarizing liquid and placed inside the magnet gap may visualize for example the field pattern in a quadrupole by observation through polarization filters placed at each end of the magnet. The rotation of the plane is proportional to the field strength and the length of the polarizing medium. This can give a certain indication of the field geometry. This measurement principle has proved useful for measurements of transient magnetic fields (Malecki et al 1957). It is less convincing when applied to the precise determination of magnet geometry, even though modern image-processing techniques might improve the method substantially. Floating wire method Floating wire measurements were quite popular in the past (Ratner and Lari 1965). If a current-carrying conductor is stretched in a magnetic field, it will curve subject to the electromagnetic force and describe the path of a charged particle with a momentum corresponding to the current and the mechanical tension

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in the wire. A flexible annealed aluminium wire was used in order to reduce effects of stiffness and gravity. This method has now been entirely replaced by precise field mapping and simulation of partcle trajectories by computer programs. E5.0.3 Conclusion Proven measurement methods and powerful equipment is readily available for most of the measurement tasks related to beam-guidance magnets as well as for spectrometer magnets. It is therefore prudent to examine existing possibilities carefully before launching the development of a more exotic measurement method. The measurement methods described above are complementary and the use of a combination of two or more of these will certainly meet most requirements. Even at an early stage of the system design, particular attention must be drawn to definitions of geometry and the future alignment considerations. In the field of new technologies, there are two methods which merit consideration. Magnet resonance imaging is a promising technique which could find a lasting application. Also the use of superconducting quantum interference devices (SQUIDS) might in the long run become an interesting alternative as an absolute standard and for measurements of weak fields (Drung 1993, Romani 1985). The complexity of these methods is still at a level which prevents current laboratory use.

References Acerbi E, Faure J, Laune B, Penicaud J P and Tkatchenko M 1981 Design and magnetic results on a 3 tesla, 10 weber spectrometer magnet at Saclay IEEE Trans. Magn. MAG-17 1610–3 Babiskin J 1957 Oscillatory galvanomagnetic properties of bismuth single crystals in longitudinal magnetic fields Phys. Rev. 107 981–92 Bazin C, Costa S, Dabin Y, Le Meur G and Renard M 1981 The DM2 solenoidal detector on DCI at Orsay IEEE Trans. Magn. MAG-17 1840–2 Billan J, Buckley J, Saban R, Sievers P and Walckiers L 1994 Design and test of the benches for the magnetic measurement of the LHC dipoles IEEE Trans. Magn. MAG-30 2658–61 Bloch F, Hansen W W and Packard M 1946a Nuclear induction Phys. Rev. 69 127 —1946b The nuclear induction experiment Phys. Rev. 70 474–85 Bloembergen N, Purcell E M and Pound R V 1948 Relaxation effects in nuclear magnetic resonance absorption Phys. Rev. 73 679–712 Borer K and Fremont G 1977 The nuclear magnetic resonance magnetometer type 9298 CERN 77-19 Brand K and Brun G 1979 A digital teslameter CERN 79-02 Brouwer G, Crijns F J G H, König A C, Lubbers J M, Pols C L A, Schotanus D J, Freudenreich K, Ovnlee J, Luckey D and Wittgenstein F 1992 Large scale application of magnetoresistors in the magnetic field measuring system of the L3 detector Nucl. Instrum. Methods A 313 50–62 Brown B C 1986 Fundamentals sof magnetic measurements with illustrations from Fermilab experience Proc. ICFA Workshop on Superconducting Magnets and Cryogenics, Brookhaven National Lab. (Upton, 1986) pp 297–301 Brown W F and Sweer J H 1945 The Fluxball Rev. Sci. Instrum. 16 276–9 Brück H, Meinke R and Schmüser P 1991 Methods for magnetic measurements of the superconducting HERA magnets Kemtechnik 56 248–56 Cohen E R and Taylor B N 1987 The CODATA recommended values of the fundamental physical constants J. Res. NBS 92 85 Coupland J H, Randle T C and Watson M J 1981 A magnetic spectrometer with gradient field IEEE Trans. Magn. MAG-17 1851–4 Cox C D 1964 An a.c. Hall effect gaussmeter J. Sci. Instrum. 41 695 Dayton I E, Shoemaker F C and Mozley R F 1954 Measurement of two-dimensional fields, part II: study of a quadrupole magnet Rev. Sci. Instrum. 25 485–9 Davies W G 1992 The theory of the measurement of magnetic multipole fields with rotating coil magnetometers Nucl. Instrum. Methods A 311 399–436

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de Raad B 1958 Dynamic and static measurements of strongly inhomogeneous magnetic fields Thesis Technische Hogeschool Delft pp 55–67 Dickson R and Galbraith P 1985 A digital micro-teslameter CERN 85-13 Di Cesare P, Reymond C, Rottstock H and Sommer P 1989 SPS magnetic field cycle measurement system CERN SPS/PCO/Note 89-9 Donoghue J J and Eatherly W P 1951 A new method for precision measurement of the Hall and magneto-resistive coefficients Rev. Sci. Instrum. 22 513–6 Drung D 1993 Recent LTS SQUID developments Eur. Conf. on Applied Superconductivity (EUCAS) (Gottingen, 1993) pp 1287–94 Edgar R F 1937 A new photoelectric hysteresigraph Trans. Am. Inst. Elect. Eng. 56 805–9 Elmore W C and Garrett M W 1954 Measurement of two-dimensional fields, part I: theory Rev. Sci. Instrum. 25 480–5 Finlay E A, Fowler J F and Smee J F 1950 Field measurements on model betatron and synchrotron magnets J. Sci. Instrum. 27 264–70 Frederikse P R and Hosier W R 1958 Oscillatory galvanomagnetic effects in n-type indium arsenide Phys. Rev. 110 880–3 Galbraith P 1986 private communication Germain C 1963 Bibliographical review of the methods of measuring magnetic fields Nucl. Instrum. Methods 21 17–46 Goldberg C and Davis R E 1954 New galvanometric effect Phys. Rev. 94 1121–5 Gordon I and Brown R E 1972 Recent advances in fluxgate magnetometry IEEE Trans. Magn. MAG-8 76–82 Grassot M E 1904 Fluxmetre J. Physique 4 696–700 Green G K, Kassner R R, Moore W H and Smith L W 1953 Magnetic measurements Rev. Sci. Instrum. 24 743–54 Green M I, Sponsel R and Sylvester C 1993 Industrial harmonic analysis system for magnetic measurements of SSC collider arc and high energy booster-corrector magnets Supercollider 5: Proc. 5th Int. Industrial Symp. on the Supercollider (San Francisco, 1993) pp 711–4 Gross D A 1986 Magnetic field measurement with NMR imaging Proc. ICFA Workshop on Superconducting Magnets and Cryogenics, Brookhaven National Lab. (Upton, 1986) pp 309–11 Hall E H 1879 On a new action of the magnet on electric currents Am. J. Math. 2 287–92 Hauesler J and Lippmann H J 1968 Hallgeneratoren mit kleinem Lineariserungsfehler Solid State Electron. 11 173–82 Kapitza P 1928 The study of the specific resistance of bismuth crystals and its change in strong magnetic fields and some allied problems Proc. R. Soc. A 119 358 Kelly M 1951 Magnetic field measurements with peaking strips Rev. Sci. Instrum. 22 256–8 Kvitkovic J and Polak M 1993 Cryogenic microsize Hall sensor Eur. Conf. on Applied Superconductivity (EUCAS) (Gottingen, 1993) pp 1629–32 Malecki J, Surma M and Gibalewicz J 1957 Measurements of the intensity of transient magnetic fields by the Faraday effect Acta Phys. Polon. 16 151–6 McKeehan L W 1929 The measurement of magnetic quantities J. Opt. Soc. Am. 19 213–42 Morgan G H 1972 Stationary coil for measuring the harmonics in pulsed transport magnets Proc. 4th Int. Conf. On Magnet Technology, Brookhaven National Lab. (Upton, 1972) pp 787–90 Nielsen V, Johansson T, Knudsen J M and Primdahl F 1992 Possible magnetic experiments on the surface of Mars J. Geophys. Res. 97 1037–44 Pagano O, Rohmig P, Walckiers L and Wyss C 1984 A highly automated measuring system for the LEP magnetic lenses J. Physique Coll. Cl (MT-8) 949–52 Pearson G L 1948 A magnetic field strength meter employing the Hall effect in germanium Rev. Sci. Instrum. 19 263–5 Peukert W 1910 Neues Verfahren zur Messung magnetischer Felder Elektrotechn. Z. 25 636–7 Polak M 1973 Low temperature InSb Hall plate with suppressed de Haas—Shubnikov effect Rev. Sci. Instrum. 44 1794–5 Polak M and Hlasnik I 1970 Planar Hall effect in heavy doped n-InSb and its influence on the measurement of magnetic field components with Hall generators at 4.2 K Solid State Electron. 13 219–27 Poole M W and Walker R P 1981 Hall effect probes and their use in a fully automated magnetic measuring system IEEE Trans. Magn. MAG-17 2129–32 Primdahl F 1979 The flux-gate magnetometer J. Phys. E: Sci. Instrum. 12 241–53

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Purcell M, Torrey H C and Pound R V 1946 Resonance absorption by nuclear magnetic moments in a solid Phys. Rev. 69 37–8 Rabi J, Millman S, Kusch P and Zacharias J R 1939 The molecular beam resonance method for measuring nuclear magnetic moments Phys. Rev. 55 526–35 Rabi J, Zacharias J R, Millman S and Kusch P 1938 A new method of measuring nuclear magnetic moment Phys. Rev. 53 318 Ralston A and Wilf H (ed) 1967 Mathematical Methods for Digital Computers vol 2 (New York: Wiley) pp 156-68 Ratner G and Lari R J 1965 A precision system for measuring wire trajectories in magnetic fields Proc. Int. Symp. on Magnet Technology (Stanford, 1965) pp 497–504 Romani L 1985 SQUID instrumentation in the measurement of biomagnetic fieldsProc. 9th Int. Conf. on Magnet Technology (MT-9) (Zurich, 1985) pp 236–42 Schmüser P 1992 Magnetic measurements of superconducting magnets and analysis of systematic errors CERN Accelerator School CERN 92-05 (Montreux, 1992) pp 240–73 Swoboda D 1981 The polar co-ordinate magnetic measurement system for the axial field spectrometer magnet at the ISR-CERN IEEE Trans. Magn. MAG-17 2125–8 Symonds J L 1955 Methods of measuring strong magnetic fields Rep. Prog. Phys. 18 83–126 Taylor B N and Cohen E R 1990 Recommended values of the fundamental physical constants: a status report J. Res. Natl Inst. Stand. Technol. 95 497 Thomas R, Ganetis G, Herrera J, Hogue R, Jain A, Louie W, Marone A and Wanderer P 1993 Performance of field measuring probes for SSC magnets Supercollider 5: Proc. 5th Int. Industrial Symp. on the Supercollider (San Francisco, 1993) pp 715–8 Turner S (ed) 1992 Proc. CERN Accelerator School, Magnetic Measurement and Alignment CERN 92-05 (Geneva, 1992) Walckiers L 1992 The harmonic-coil method CERN Accelerator School CERN 92-05 (Montreux, 1992) pp 138–66 Walsh L, Ahlberg J H and Nilson E N 1962 Best approximation properties of the spline fit J. Math. Mech. 11 25–234 Weber W 1853 Ueber die Anwendung der magnetischen Induction auf Messung der Inclination mil dem Magnetometer Ann. Phys., Lpz. 2 209–47 Welch E and Mace P R 1970 Temperature stabilized magneto-resistor for 0.1% magnetic field measurement Proc. 3rd Int. Conf. on Magnet Technology (Hamburg, 1970) pp 1377–91 Welker H 1952 Ueber neue halbleitende Verbindungen Z Naturf. a 7 744–9 —1955 Neue Werkstoffe mit grossem Hall-Effekt und grosser Wiederstandsanderung im Magnetfelt Elektrotechn. Z. 76 513–7

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F1.1 Basic aspects of tensile properties B Obst

F1.1.1 Introduction The most challenging needs for cryogenic materials relate to structures and devices that operate at temperatures of 4 K or below, commonly in large-scale superconducting magnets for controlled thermonuclear fusion and high-energy physics, in magnets for medical diagnosis (nuclear magnetic resonance (NMR)), and in superconducting machinery. These cryogenic applications (see volume 2) comprise essentially three different classes of materials: (i) the superconductors themselves; (ii) normal metals and high-strength steels as part of superconducting cables and support structures respectively; (iii) insulating nonmetallics and structural composites. This work will be entirely limited to experiments on normal metals and alloys, in particular on facecentred cubic (f.c.c.) metals for reasons to become clear in section F1.2. The adequate safety and long service lifetime demanded of the superconducting applications mentioned above and the increasing sophistication of modern technology have created the need for new and more accurate data on mechanical properties of existing construction materials and the development of new cryogenic alloys. Because mechanical properties change dramatically with temperature, test procedures have to be specified for low temperatures to measure the suitability of a material for service under particular combinations of applied stress, temperature, deformation rate and other relevant parameters, and to study the basic mechanisms controlling temperature-dependent mechanical behaviour. We present current techniques for mechanical testing along with recent results and we intend to provide a guide to the significance of the information obtained from these tests. This will be followed by a condensed treatment of salient theoretical features pertaining to deformation at low temperatures. F1.1.2 Deformation mechanism maps The mechanical properties of materials describe the relations between forces acting on a test piece and its resistance to changes in form or dimensions and fracture. Of all the test methods used to characterize the macroscopic mechanical behaviour, the uniaxial tensile test, in which a specimen is pulled to failure in a relatively short period of time, is still the most widely employed technique. Here, unless otherwise specified, the cross-head of the tensile testing apparatus is driven at constant speed and the load developed in the specimen is measured as the dependent variable. The change in length of the specimen can, if only low accuracy is required, be derived from the cross-head extension after correcting it for the extension due to the machine itself—if greater precision is required, some form of sensitive extensometer is needed. (The equipment and techniques developed in recent years and successfully used for measurements at

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Figure F1.1.1. Engineering stress—strain curves for a typical ductile material: (a) elastic region and initial plastic region showing 0.2% offset yield strength; (b) complete diagram.

cryogenic temperatures will be discussed in section F1.1.5.) Consequently, from any complete test record, we are now able to get information about the material’s elastic properties, the stress that produces elastic failure (elastic limit, yield point or yield strength), the character and extent of plastic deformation and the toughness, i.e. the energy absorbed before fracture. Figure F1.1.1 gives a brief description of the glossary commonly used to present the results of a tensile test. As far as the elastic properties are concerned, we will focus our attention only on Young’s modulus, E = σε–1. As the constant of proportionality between normal elastic strain ε and uniaxial tensile stress σ, E is a most sensitive ‘diagnostic parameter’, capable of testing the accuracy of the total experimental set-up. To get more physical insight into the subject of elastic constants and their intimate relationship to various fundamental solid-state phenomena (including a variety of measurement methods), we must refer the reader to the appropriate literature (see, in particular, Ledbetter 1983). With regard to the plastic (i.e. irreversible) properties, it is well known that the microscopic mechanisms controlling the initia tion of plastic flow are also involved in sustaining plastic flow as the specimen is further extended. For this reason, a study of the elastic limit (R e ) and the influence of temperature and other variables on it might be expected to yield important information on the rate-controlling mechanisms occurring during plastic deformation and fracture; we will stress this point later. Engineering materials are invariably polycrystalline. In principle, there are six or more distinguishable and independent ways (giving additive strain rates) in which a polycrystal can be deformed (Ashby 1972). Fortunately, at temperatures below which diffusion-controlled plastic flow is no longer dominant or, in fact, no longer able to permit steady-state deformation, the major mechanisms by which a polycrystalline solid may plastically deform are the glide motion and mutual interaction of lattice imperfections, called dislocations, and twinning. Ashby (1972) was the first to introduce ‘deformation-mechanism maps’, displaying the fields of stress and temperature in which a particular mechanism supplies a greater strain rate than any other (of the six) mechanism. These plots show that, in practice, at temperatures below ambient temperature two main types of behaviour are observed, the first occurring for f.c.c. metals and the second being shown by body-centred cubic (b.c.c.) metals. Figure F1.1.2 makes these facts clear. It illustrates in a simplified form the quintessence of the tensile properties of pure metals, together with the failure mode at ordinary strain rates (Read 1983).

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Figure F1.1.2. Simplified deformation behaviour maps: (a) f.c.c. material; (b) b.c.c. material. Reproduced from Read (1983) by permission of ASM International.

The most striking point in figure F1.1.2 now is that metals of b.c.c. crystal lattice type (but not the alkali metals) show a tendency towards brittleness, i.e. tensile failure with limited crack tip plasticity, as the temperature is lowered. In contrast, f.c.c. metals remain ductile down to liquid-helium temperature, accommodating a large amount of plastic deformation before fracture (except when they are subjected to severe environmental conditions). This makes them inherently reliable for use at low temperatures. That is why, in the following, emphasis is on f.c.c. metals and alloys, the discussion being confined to basic aspects that govern their strength in tension and, apparently, their fatigue strength and hardness. Al, Ni, Cu, Au, Pb are examples of pure metals having f.c.c. crystal structure. The best known technical alloys are Al—Cu, Mg, Pb solid solutions, CuZn40 (α -brass), CuSn6 (bronze) and the series of austenitic stainless steels (17–25% Cr, 8–20% Ni with or without nitrogen alloying), all of which are conventional low-temperature materials. In recent times, a Ni-based superalloy (Incoloy 908: 49 Ni, 4 Cr, 1 Al, 1.5 Ti, 3 Nb, 41.5 Fe) designed for fusion-reactor components has become a material of interest to researchers in the field. F1.1.3 Some fundamental properties of dislocations We assume that the reader is familiar with elementary dislocation theory, the crystallography of slip in f.c.c. lattices included, and with the fundamentals of strain and solution hardening (see the reference list for relevant literature). In this chapter, we will revise only those facts and theoretical concepts that relate to a reasonable understanding of the temperature dependence of tensile properties. Plastic deformation at low temperatures (T ≤ 0.4Tm with Tm = melting point in K) results from the passage of dislocations through a regular crystal (and, as circumstances dictate, also to some extent from twinning). A glide dislocation is a line defect that marks the boundary between the slipped and unslipped regions within the crystal (figure F1.1.3). The unit vector b of slip is called the Burgers vector of the dislocation; it is the same at all points along the line. Since this is the case, the nature (‘character’) of the dislocation is fully defined by the direction of b relative to the unit tangent vector t of the dislocation line. If b is parallel to t , we have a pure screw (¬) dislocation; if b is perpendicular to t , it is a pure edge (⊥) dislocation. For any other orientation of b relative to t the dislocation possesses both screw and edge components and is said to be of the mixed type. Depending on the character which a dislocation or a segment of it has, its slip behaviour is seriously affected. The pure screw dislocation, since its slip plane is not uniquely defined, can move on all slip

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Figure F1.1.3. A schematic representation of a dislocation loop in a slip plane, showing regions of pure screw (b || t ) and pure edge (b ⊥ t ) dislocation. Slip has occurred only across the hatched area. The Burgers vector b defines the magnitude and direction of the slip.

Figure F1.1.4. Formation of a partial dislocation connected by a stacking fault in an f.c.c. crystal. A complete dislocation dissociates into two partial dislocations whose Burgers vectors b 1 and b 2 add up to the unit of slip, b. The partials put a local perturbation in the f.c.c. stacking into brackets, involving the formation of a region of h.c.p. structure; the free-energy difference is related to the stacking-fault energy, γ.

systems containing the Burgers vector. Hence, at high stresses it may leave the glide plane (e.g. to circumvent a barrier), making further deformation easier. The movement of the edge or mixed dislocation is, by contrast, under normal conditions constrained to the slip plane defined by b and t. The atomic configuration around a dislocation line is closely related to the crystal structure of the solid. In f.c.c. crystals, a unit (‘complete’) dislocation is, for geometrical and energetic reasons, normally dissociated into two parallel partial dislocations, connected by a stacking fault, a local change in stacking from f.c.c. to h.c.p. (figure F1.1.4). The strip of hexagonal-close-packed (h.c.p.) material that is intro duced will raise the energy of the crystal (otherwise the metal would be normally h.c.p.), keeping the repulsive partial dislocations together. The whole configuration is called an extended dislocation. In edge dislocations the equilibrium distance of the partials is about twice as large as in screw dislocations.

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The role of the stacking fault energy (γ), associated with this local change in the stacking from f.c.c. to h.c.p., is considered important with regard to strengthening mechanisms and shape changes resulting from deformation twinning, making this material parameter the most significant one, aside from the shear modulus (see equation (F1.1.2) in section F1.1.4). As mentioned earlier, twin modes may be activated in some materials under cryogenic temperature test conditions, where slip becomes more and more constrained or even blocked. Mechanical twinning is a homogeneous shear deformation parallel to certain crystallographic planes in certain directions which converts, on an atomic scale, a portion of the crystal into a mirror image of the parent lattice. It occurs in a discontinuous manner, being occasionally indicated by load drops on the stress—strain curve (see section F1.1.8). The twinning displacement is small. Hence, the importance of twinning in plasticity is primarily not because of the deformation it produces, rather twinning provides the mechanism necessary to reorient parts of the deformed lattice more favourably for slip to continue. These facts point to the need for large stress concentrations to nucleate twin bands; the stress required to propagate twinning is appreciably less. F1.1.4 The temperature sensitivity of the flow stress In order to deform a crystalline material plastically we have to make dislocations move across their slip plane. Since a dislocation, by its nature, is a line imperfection associated with large localized stresses, there will always be ‘obstacles’ to its motion. Hence, from the very beginning of deformation, the yield process in a real metal is affected by several factors, including (i) the intrinsic lattice friction (called the Peierls— Nabarro stress), (ii) stress fields of parallel dislocations, dislocation tangles or cell walls opposing the motion of the glide dislocation and (iii) ‘forest’ dislocations (‘trees’) threading through the slip plane— both (ii) and (iii) are sources of structural hardening which results from the deformation process itself— and finally (iv) solute atoms impeding the dislocation motion by their stress fields due to a size misfit (or electric field, in some materials); frequently the solute additions also decrease the stacking fault energy which makes the dislocations more extended (see figure F1.1.4). The obstacles (i)–(iv) acting in combination may be thought of as being a measure of the overall resist ance of the lattice to dislocation motion, characteristic of the alloy system under consideration. To com plete the picture, we mention that in polycrystals the presence of grain boundaries introduces additional constraints to the movement of glide dislocations in the initial stages of plastic deformation. However, in the cold-worked state of most engineering materials this effect diminishes compared with the effects of multiple slip in the grains themselves right from the onset of plastic flow. When moving glide dislocations meet short-range energy barriers in the slip plane, as shown in figure F1.1.5, the reaction rate theory of plas tic deformation takes into account that, over small volumes of the crystal and at ‘high’ temperatures, thermal fluctuations will assist dislocations in overcoming the barriers. Hence, the applied stress will be less than what it would be without this ‘extra’ (thermal) energy. The more extended the barrier forces are, however, the more highly resistant to the effects of such fluctuations the barriers will be. Accordingly, the yield or flow stress of f.c.c. metals is formally separated into two components at any point along the stress—strain curve (Seeger 1955, 1956)

Above a critical, material-specific temperature Tc the flow stress is supposed to be almost constant, i.e. σ * = 0 for T ≥ Tc . σ * is called the effective stress . This thermally sensitive component is associated with short-range obstacles or stress fields such as the Peierls—Nabarro stress (i), forest dislocations (iii), dislocation core dissociation (see figure F1.1.4) and resistance to cross-slip of screw dislocations. σG is called internal stress . This athermal component is presumed to relate to long-range stress fields surrounding a moving dislocation, but its details are still under discussion (see section F1.1.6.1).

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Figure F1.1.5. A schematic diagram showing the variation of resistance F (force per unit length of dislocation line) to dislocation glide with area A swept out along the slip plane. The dislocation (⊥) can overcome the energy barrier by applied stress τ and thermal energy ∆G, if ∆G can be supplied by thermal fluctuations of the crystal. Barriers with the highest ∆G values determine the temperature sensitivity of the strain rate.

It is found that

where G and b are the usual symbols for shear modulus and magnitude of the Burgers vector, respectively, and ρ is the total dislocation density. Strain-hardening, then, since it is related in some unique way to an increase in ρ, produces an increase in σG. The proportionality between σG and G emphasizes the importance of the shear modulus in crystal plasticity, as mentioned earlier. As is obvious from equation (F1.1.2), σG would be independent of temperature, except for the fact that G depends on T. Hence, the temperature dependence of the flow stress (correlated with it is the strain-rate dependence) is nearly totally involved in the temperature dependence of σ * (see equation (F1.1.1). A number of dislocation interaction mechanisms have been advanced to evaluate the effect of tem perature and strain rate on the flow stress in terms of an Arrhenius-type rate equation; for a discussion in more detail we refer to the extensive literature (e.g. Nabarro 1980). Setting aside the question of the correctness of one or the other idea, we will describe here only the fundamentals and predictions of the theory of Seeger (1955, 1956) which still provide the closest approximation to recent experimental results (see section F1.1.6.1). In Seeger’s approach (figure F1.1.6), the main processes which contribute to the σ * of f.c.c. crystals are the intersection of dislocation lines (thereby forming jogs, i.e. geometric steps in the dislocation line) and the production of point defects (by jogs in screw dislocations; see ‘Further reading’ list at the end of this chapter). The intersection process thus requires an addition to the work of plastic distortion, depending on the dislocation character (edge or screw, see figure F1.1.3) and the stacking fault energy (see figure F1.1.4). All in all, the theory of dislocations cutting through the forest is quite complex (for a lucid discussion of the problems, see Dorn 1968). In the simplest version—neglecting statistics and assuming the quantities that enter the equation for σ * to be appropriately smeared average values—Seeger finds for the yield strength, depending on the stacking-fault energy, a dependence on temperature as schematically shown in figure F1.1.7. Two important features are to be seen in the diagrams in figure F1.1.7. First, a greater and greater external stress must be applied to deform a crystal as the temperature is reduced, indicating that less and less thermal energy is available to enable dislocations to overcome short-range obstacles which give rise to σ * (see figure F1.1.5). Second—and herein lies the significance of Seeger’s prediction—as it becomes increasingly difficult to drag along jogs in screw dislocations as thermal activation becomes less, below some critical temperature Tc (T0 , T1 ) dislocations with predominant edge character (see figure F1.1.3) move at lower stresses than screw dislocations. In other words, at Tc a transition to a low-temperature mechanism of plastic deformation occurs which is accompanied by a change in the lattice defects created during deformation. We will show later (in section F1.1.6.2) that this might enable us to understand serrated

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Figure F1.1.6. The superposition principle for the yield strength: σ = σ * (thermally activated) + σG (not thermally activated), σ * is associated with the cutting of repulsive forest dislocations and σ G is the stress required to extrude the glide dislocations through long-range internal stress fields of neighbouring dislocations, the main contribution coming from dislocations which are roughly parallel to the glide dislocations and having the same Burgers vector.

Figure F1.1.7. The yield strength versus temperature for f.c.c. metals with high (HSFE) and low (LSFE) stacking fault energy γ (Seeger 1955, 1956, schematic representation).

yielding, a typical low-temperature strain effect, and that the change in dislocation character (•→ ⊥ at Tc ) also affects the material’s toughness (energy absorbed before fracture) and fracture appearance. Tc is, according to theory, correlated with the stacking fault energy (γ) in such a way that

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This relation and figure F1.1.7 make clear the importance of the stacking-fault energy in controlling the characteristic types of temperature sensitivity of the yield strength (σ y ). Since σy by its nature is structure sensitive and governed by the dislocation configurations (the ansatz equation (F1.1.1)), it may be considered to be a fundamental mechanical property of the material. A study of the influence of temperature on σy (i.e. on σ *) might thus be expected to yield information on the dislocation mechanisms occurring during deformation at low temperatures. F1.1.5 Tensile tests at low temperatures As already mentioned in section F1.1.2, the uniaxial tensile test, investigating the effect of temperature and strain rate on the tensile response, is still the technique most widely used both to learn about the basic mechanisms and to get engineering design data. Here, only a brief outline is given of the main types of apparatus that have been employed most successfully in obtaining the results discussed in section F1.1.6. F1.1.5.1 A tensile cryostat for temperatures between 293 and 5 K The main item of a cryogenic tensile-testing apparatus is its tensile cryostat. Figure F1.1.8 shows a recent model of a continuous flow cryostat in the static mode (CFCS), capable of obtaining fundamental data at any temperature from 293 K down to 5 K. The salient point of the CFCS is that the specimen is surrounded by an isolated volume of helium exchange gas at reduced pressure (∼8 kPa) making use of its high thermal diffusivity, α , to preserve thermal equilibrium between the heat sink (the inner wall of the specimen chamber) and the material to be tested ( α is equal to κc-1v ρ -1, where κ is the thermal conductivity for heat transfer, cυ the specific heat and ρ the density). Accordingly, under nonsteady-state conditions, α provides a measure of the ratio of the amount of heat transport out of a volume to the amount of heat stored in it, in other words α controls the time in which thermal equilibrium is established. As figure F1.1.9 shows

i.e. gas cooling is superior to cooling with liquid helium 4. To set out the significance of this result in more detail, we mention that during plastic deformation about 90% of the expended energy is dissipated as heat. Since, as is known, the lattice specific heat at low temperatures of most solids used in construction is by a factor of ∼104 less than the room-temperature value, very small amounts of heat will considerably raise the temperature. Hence, in tensile tests great importance must be attached to a coolant with a very large α to cope adequately with the heat generated during straining of a specimen, especially in the case of serrated yielding (see section F1.1.6.2). The thermal diffusivity of gaseous He coolant is of the same order of magnitude as that of the test ed metallic materials. Therefore, in principle, heat removal from the specimen may occur in two different ways, first by heat losses taking place down the specimen into the pull rods and second by heat that is lost radially to the surroundings. Since, however, with the usual geometry of tensile specimens (ratio of the reduced length l 0 to diameter d0 ≥ 8 ), the surface area exceeds the area of cross-section by a factor of 30 to 50, so cooling by He gas would be the greatest sort of heat loss. Details of the construction of thermal anchoring and shielding may be inferred from figure F1.1.8. We will just describe some further key features of the tensile cryostat and the techniques used in testing mechanical properties which we think are of general interest. (i)

The specimen is linked with the grips of the pull rods by a spherically seated connection, shown in figure F1.1.10, largely avoiding bending moments that might result if the assembly is nonuniaxial. (ii) The whole tensile test apparatus is mounted on four air pads (eigenfrequency ∼ 2 Hz) to keep away mechanical disturbances from the surrounding laboratory.

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Figure F1.1.8. A continuous flow cryostat in the static (CFCS) mode for material testing in the range 5–293 K.

(iii) During the cool-down procedure, the upper pull rod is not tightly connected with the fixed upper cross-head of the screw-driven load frame to allow for thermal shrinkage without exerting a force on the specimen. Only when the test temperature is reached is connection made. (iv) The temperature setting is performed by adjusting a constant helium gas flow rate through the cooling system (see the helium inlet in figure F1.1.8) with a needle valve (∆T ≤ ±0.1 K after 1 h). The liquid-helium consumption for a 6 K test is about 15 1. The uniaxial tensile test is conducted by downward motion of the lower cross-head at constant speed and the load developed in the specimen is measured. For fundamental research, the only consistent way to do this is to take the data in place, as indicated in figures F1.1.8 and F1.1.10. This arrangement obviates the need for an error correction regarding the influence of factors such as the flexible bellows and feedthroughs. The load cell used here is a 20 kN quartz force transducer of great mechanical stiffness (∼1.3 kN µm-1) providing high resonance frequency (∼50 kHz) the resonance frequency of a commercial straingauge-based load cell typically is <1 kHz. We emphasize that this great stiffness and the measuring arrangement are prerequisites in order to observe the kinetics of high-velocity processes (see section F1.1.6.2(c))—it

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Figure F1.1.9. Thermal diffusivity µ for helium 4: ¢—liquid helium and O —gaseous helium at normal pressure; •—gaseous helium at reduced pressure ( p = 0.1 × 105 Pa). Data plotted according to Arp and McCarty (1989).

Figure F1.1.10. Lower gripping fixture mounted on the quartz force transducer.

is not sufficient just to have a machine that is ‘as stiff as possible’. To clarify this problem we present a simple example. Figure F1.1.11 displays two time-resolved load signals in response to brittle fracture of a ferritic steel one (denoted with Fe x t ) is from a commercial 20 kN transducer positioned in the external load frame as usual, the other (denoted with Fi n t ) is from the internal quartz directly contacting the gripping fixture. While Fint shows a load drop to nil within ≤40 µs (corresponding to a slope of ∼2 × 10 7 MPa s −1 ), the Fe x t signal does not even show the beginnings of a decay during this time interval. Obviously the inert mass of the massive pull rod works as a kind of ‘low-pass filter module’, choking high-frequency events.

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Figure F1.1.11. Mechanical testing for brittle fracture in a ferritic steel: response times of a commercial strain-gaugebased transducer positioned in the external load frame ( Fe x t ) in comparison with a very rigid internally mounted quartz force transducer (Fi n t ). (Note: the abrupt load drop to zero at fracture of the specimen starts the tensile machine vibrating, hence the oscillations in the Fi n t signal.)

From this it must be concluded that it makes no sense to derive much information about dynamic effects (e.g. the sharp yield drop or serrated strain phenomena) from measurements using a load cell at a remote position, even if it is sufficiently stiff. Figure F1.1.11 also demonstrates that in studies of dynamic effects it may be quite natural to display the output signal of the quartz-force transducer against time (much like the output of a seismograph), corresponding to the conditions of the tensile test (straining the specimen with constant speed). F1.1.5.2 Fundamentals of data acquisition The example of a fast load drop shown in figure F1.1.11 necessitates a recording system that is commensurate with the dynamic range of the quartz transducer. The most common way to do this is by a PC-based data acquisition system; figure F1.1.12 shows a typical block diagram. Each of the many different components involved in a workstation acts as a source of errors. Here, we place special emphasis on two critical aspects of data acquisition hardware: resolution and speed. The resolution, usually specified in the rated number of ‘bits’ of the analogue-to-digital converter (ADC) (see figure F1.1.12), indicates the smallest measurable change in the input signal. The conversion from bits of resolution to actual resolution is derived by dividing 2(n u m b e r o f b i t s ) into 1. For example, a board featuring 16-bit resolution and a full-scale input range of 10 V ideally could detect a change of one part in 216, or 10 V full-scale/65.536 ~ 0.15 mV. In fact, this calculation presupposes that the peak value of noise (caused by external sources or by components within the actual system) does not exceed one least significant bit (LSB). Other factors reducing the system’s performance are gain-error, nonlinearity and drift. The sampling rate (‘speed’) is also an important specification of data acquisition and is expressed in frequency (in Hz). General sample theory (the Nyquist theorem) states that an input signal must be sampled at greater than twice its highest frequency component in order to retain all necessary information. Typically, a factor of three or four, or even higher is used in real applications (the sampling rate used in figure F1.1.11 is 100 kHz). Finally, to take advantage of the many capabilities of the workstation.

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Figure F1.1.12. A block diagram of a typical high-performance PC-based data acquisition system. Reproduced by permission of Keithley Instruments.

you have to choose high-speed disk access software, allowing not only real-time data acquisition but also conditional recording (pre- and post-trigger), signal processing and presentation (see figure F1.1.12). F1.1.5.3 Measurement of strain In a tensile test, the extension of the specimen produced by the downward motion of the lower cross-head at constant speed (see figure F1.1.8) must be converted into a strain of the specimen. When only the total permanent deformation of a ductile material is to be measured, it may be sufficient to obtain this property from the movement of the cross-head, as described in section F1.1.2, or by fitting together the fracture halves and measuring the elongation of the specimen relative to the initial gauge length. If, however, Young’s modulus or the load producing elastic failure (Re ) has to be determined to study the fundamental mechanisms involved in plastic deformation, extensometry is needed. As for the load measurement, very accurate strain measurements require an in-place testing technique. Of the many instruments that have been made for use at room temperature—mechanical, optical and electrical ones (for a compilation of the different sensing techniques the reader is referred to Roark and Young (1975))—most are usually impracticable for low-temperature applications. The extensometers successfully employed for cryogenic operation use the principle of placing a beam in bending, the surface of the beam having resistive strain gauges attached to it. The essential assumption then is that the strain developed by the specimen is accurately transmitted to the resistance element. Therefore, care has to be taken to keep the influence of adhesive and substrate on the elastic properties of the bent beam as small as possible, e.g. by choosing a very small gauge geometry (of the order of millimetres). Today’s extensometers, shown schematically in figure F1.1.8, are of the double-cantilever beam type, a principle which was developed by NASA engineers in the early 1960s. To meet the most exacting requirements for low-noise strain measurements, extensometers with a single beam as sensing element (inherently less susceptible to transient thermal conditions) also proved a success. Resistance changes of the strain gauges caused by the mechanical deformation are converted into an electrical signal by a Wheatstone bridge circuit. To ensure accuracy at low temperatures, the extensometer is calibrated in the existing test facility (figure F1.1.8), attaching it across two halves of a dummy specimen (overlapping cylinders) and correlating the output signal of the bridge at a gain of up to 10 000 with that of a linear variable differential transformer (LVDT) or an optical instrument. Employing low-noise amplification and

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read-out apparatus, changes in gauge length of ∼0.2 µm or strains in the range of 10−5 can be measured (the maximum measurable extension being several millimetres). With the use of two extensometers, elongation on both sides of a sample is recorded, thus allowing any bending of the sample, e.g. due to nonaxial loading, to be detected. Averaging both signals to obtain a mean strain eliminates the influence of this effect. The initial slope of the stress-strain curve is then calculated by linear regression of the corresponding data, the ratio of tensile stress to strain representing the modulus of elasticity (Young’s modulus). F1.1.6 Tensile properties of f.c.c. metals and alloys Plastic deformation is a strongly irreversible process, as discussed in section F1.1.3, proceeding far from thermodynamic equilibrium. For this reason, a mechanical equation of state does not exist. The salient point is that the stress or strain characterizing the dislocated state of a crystal will depend strongly on the temperature and strain rate of the prior deformation. The simple stress—strain relation σ = σ (ε ), extended to include temperature and time

is generally referred to as the constitutive equation of the material (see equation (F1.1.1)). These few remarks should make clear that only those specimens having the same thermal and mechanical history (heat treatment and cold work) will yield the same functional relations when the deformation conditions to which the material is subjected are the same (in other words: the plastic strain εp is not a variable of state). Hence, to study the physical mechanisms of deformation from effects of tem perature or strain rate on the flow stress, at the beginning of a series of measurements all specimens must be in the ‘virgin’ state. This is best approximated by preparing as many specimens as needed in a test from the same batch of material (‘multispecimen’ technique). It is evident from equation (F1.1.5) or (F1.1.1) that the commonly used technique of subjecting a single specimen to multiple loadings at different temperatures inevitably leads to unreliable results. The investigated materials all have f.c.c. crystal structure, as set out in section F1.1.2, ranging from pure metals to austenitic stainless steels with high and low stacking fault energies respectively. The specimens were prepared from commercially supplied plates or rods in ‘half-hard’ condition. The latter guarantees a sufficiently high density of forest dislocations (see figure F1.1.6) to ‘probe’ the basic mechanisms of dislocation motion; furthermore, a pre-deformed sample can be handled with less care than a well-annealed one. F1.1.6.1 The effect of temperature on the yield and flow First evidence of the plastic deformation of a material depends on the sensitivity of strain measurements. In the following, deviations from proportionality of stress and strain specified at an offset of 0.001% (see section F1.1.5.3) are taken as criteria for the initiation of yielding. The corresponding stress level is called the macro-elastic limit , Rp0

Rp 0 is by definition larger than the ideal point of elastic failure (Re ). In figure F1.1.13, values of the R p 0-value and the flow stress at various offset strains (x = 0.005–2%) of cold-worked nickel are shown for temperatures ranging from 6 to 300 K. Ni has an intrinsically high stacking-fault energy, γ300K ∼ 120 mJ m−2 (the stacking-fault energies of a number of f.c.c. metals and alloys are listed, for example, in Remy et al 1978, Schramm and Reed 1975 and Nabarro 1985; see also

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textbooks). It turns out that the type of curve shown here is representative of other high-γ materials, too, irrespective of whether it concerns pure metals or alloys. Therefore, the principal features of the yield and flow of high-γ specimens may be discussed from figure F1.1.13. Let us first focus on the Rp 0(T) curve. It shows more clearly than any of the other curves that three regions exist—I, II and III—which greatly differ in their temperature coefficient Θ0 ≡ dRp 0 /dT. Making a comparison with the upper part of figure F1.1.7, a close agreement between the experimental findings and the theoretical prediction may be noticed if we disregard region II. Relating to this, it must be remem bered that Seeger’s approach is based on the assumption of a single thermally activated rate-controlling process (that of jog formation; see section F1.1.4) for the motion of a dislocation over the entire low-tem perature range. However, it appears that two processes have to be taken into account in order to interpret the experimental results. The considerable increase in Θ0 at T0 (≈20 K for Ni ) and the sharpness of the transition are indicative of a change in the deformation mechanism when T falls below T0 — in excellent agreement with the theory. We will show later (see section F1.1.6.2) that it is this range T < T0 in which a distinctive deformation behaviour (serrated stress-strain curves) is observed. The ‘transition’ temperature T0 depends on the

Figure F1.1.13. Tensile yield strength Rp x versus temperature at various specified plastic strains (per cent) of community pure, cold-worked nickel (F46 denotes the ultimate tensile strength Rm at room temperature to be ≥460 Mpa).

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Figure F1.1.14. Transition temperature T0 (see upper part of figure F1.1.7) of aluminum, nickel and an austenitic stainless steel (material number 1.4845, equivalent to AISI 310S) in decreasing order of γ . The Rp 0.2 /1.4845 curve is to demonstrate that the temperature dependence of the flow stress at an offset strain of less than 0.2% is necessary to identify T0 .

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stacking-fault energy (γ) in the same way as predicted by equation (F1.1.3), i.e. T0 increases when γ decreases. Figure F1.1.14 verifies this result with the example of aluminium, nickel and an austenitic stainless steel (AISI-type 310S) in decreasing order of γ. In figure F1.1.14, the ‘proof’ stress (specified as the ‘offset yield’ stress at a permanent set point of 0.2% plastic strain) of the austenitic steel is also plotted in order to demonstrate that in the R p 0.2(T) curve the transition temperature T0 is hard to make out, underlining once again the need for accurate and very sensitive extensometry in fundamental studies and for taking note of the multispecimen technique. Just as the R p x (T) curves of nickel (figure F1.1.14) stand for the whole class of high-γ materials, it is found that within the class of low-γ materials, too, the tensile properties between 6 and 300 K show close similarities. We will again discuss some characteristics using just one example. Figure F1.1.15 shows the low-temperature mechanical behaviour of a nitrogen-strengthened fully austenitic stainless steel (CSUS-JN1), recently developed in Japan for cryogenic structural applications. It is well known that nitrogen, an interstitial solute, improves the strength of the austenite phase (f.c.c.) of Fe-Cr-Ni alloys, particularly at low temperatures, as is obvious from figure F1.1.15. A reasonable solubility of N in austenite is achieved by adding manganese without leading to precipitations (e.g. of Fe4N) of the nitrogenated steel. Presumably, due in large part to the manganese and nitrogen additions, the stacking-fault energy of CSUS-JN1 is very low (the estimated value is γ < 10 mJ m− 2 ). Again, the R p 0(T) curve closely approaches the theoretical curve given in the lower part of figure Fl.1.7. At T = T1 , the transition to the predicted low-temperature deformation mechanism is very

Figure F1.1.15. The effect of temperature on the tensile properties of a nitrogen-strengthened low-γ stainless steel(CSUS-JN1: Fe-25 Cr-15 Ni-0.35 N-4 Mn; C, Si, P, S).

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Figure F1.1.16. Transition temperatures T1 of some LSFE materials: T1 increases as γ decreases.

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distinctive (CSUS-JN1: T1 ≈ 35 K). The transition temperatures of all the investigated low-γ materials meet equation (F1.1.3), too, with the effect that T1 increases as γ decreases (figure F1.1.16). We note that, despite the successful description of the macro-elastic limit (R p 0 ) as a function of temperature by Seeger’s model, the higher strained states (R p x ) are not in full agreement with it. As is obvious from the temperature range T ≥ 60 K, a thermally activated stress-assisted mechanism is superimposed on the athermal stress level, making unclear the physical meaning of σG in equation (F1.1.1). This effect is more pronounced the higher the degree of alloying is (see figures F1.1.14 and F1.1.16). Regardless of this shortcoming, it is well established that a critical temperature Tc (T0 , T1 ) exists below which edge dislocations move more easily than screw dislocations. Clearly, such a change in the character of a glide dislocation must have an effect on the plastic flow curve. This is discussed below. F1.1.6.2 Serrated stress-strain curves Except for effects produced by impurities and solute interactions with lattice dislocations in metals and alloys, respectively (yield point phenomena, Portevin-LeChatelier effect), the typical tensile stress-strain curve (figure F1.1.1) is smooth and unbroken all the way to failure at ambient temperature. At very low temperatures, however, the onset of unusual modes of deformation is observed which act very rapidly, causing a distinctly different type of ‘serrated’ stress-strain curve, often associated with audible clicks emitted from within the sample. (Recall that in the tensile test the cross-head of the tensile machine is driven at constant velocity, so a drop in load actually results from a sudden strain increment (‘strain burst’) producing a momentary elongation rate of the specimen which exceeds the imposed cross-head speed.) Figure F1.1.17 illustrates this behaviour for CSUS-JN1 stainless steel at various test temperatures. At 6 K, serrated yielding starts in the early stage of plastic flow. As the specimen temperature increases the critical strain εi that must have been produced to initiate load drops successively increases, too

until, finally, at some characteristic temperature Ts there is just one load drop left, resulting in fracture (CSUS-JN1: Ts ≈ 35 K ). Above Ts deformation proceeds in what appears to be a smooth type of plastic flow. The εi = εi (T) dependence is found to be typical for all metals and alloys, irrespective of the value of the stacking-fault energy. Figure F1.1.17 makes clear some further details. (i)

Once the system is in the mode of catastrophic flow, the magnitude of the load drops (∆σ ) is virtually uninfluenced by the temperature (apart from marginal effects) which is considered to be evidence in favour of an essentially athermal initiation event. (ii) ∆σ grows with increasing strain-hardening, ∆σ ↑ = ∆σ (ε ↑). Conversely, this correlation makes it quite plausible that reducing the strength of the specimen, e.g. by some annealing process, will result in correspondingly low values of ∆σ (ε ). As already mentioned in section F1. 1.5.1, during plastic deformation most of the work done by the external forces on the specimen (~90%) is dissipated as heat within the crystal. Therefore, plastic instabilities—since they are known to be intimately connected with strong localization of plastic shear— must be traceable through localized temperature increases (‘thermal spikes’). Figure F1.1.18 shows a plot of stress and surface temperature versus time obtained in nickel during deformation at 5.5 K. Clearly, on reaching εi , the temperature profile reflects a deformation process that proceeds for the most part in discrete strain increments. The T profile further shows that, once the ultimate tensile strength (R m) is exceeded (at a maximum in the upper envelope of the serrations), catastrophic flow becomes localized in the necked region of the specimen. In this region of areal contraction no strain-hardening takes place, hence the small and constant temperature peaks in figure F1.1.18. The maximum rise in temperature observed at

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Figure F1.1.17. Engineering stress-strain curves for CSUS-JN1 stainless steel at temperatures between 6 and 35 K; see figure F1.1.18.

the surface of a nickel specimen was ∆T ~ 17 K, that of a high-strength stainless steel ~55 K. These values should be regarded only as a lower limit of a ∆ T estimation. A closer estimation shows that the temperature at the centre of the specimen (in the slip band) must rise to a much greater value than that measured at the surface with a thermocouple. The salient point in serrated yielding now is (shown in the enlarged detail in the lower part of figure F1.1.18) that the increase in temperature occurs only after the load has started to fall. This fact has trig gered much speculation as to the physical origin of serrations. It is beyond the scope of this article to review all the numerous works. We, therefore, outline here just two models, each one of which describes an aspect of a load drop quite correctly, but only in combination—to come to the most important point first—do they explain consistently the appearance of catastrophic flow. (a) Thermal instability of plastic flow In the ‘adiabatic heating’ model (Basinski 1957, 1960), the best developed model to date, a nucleating deformation will first have to occur (not visible in the stress-strain curve), perhaps at a stress concentration within the specimen and aided somewhat by thermal fluctuations. Combined with relatively low specific heats and an appreciable slope of the flow stress against temperature (see figures F1.1.13 to F1.1.16), local hot spots might be able to develop, lowering the required flow stress and permitting considerable localized plastic deformation to continue.

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Figure F1.1.18. Serrations and surface temperature profile (thermocouple output) for nickel at 5.5 K; sampling rate f = 40 kHz/channel.

The weak point of this concept of a ‘thermal’ instability is certainly the amount of fast deformation that must have occurred before the temperature will rise, and the details of the nucleation process have never been conclusively explained on a thermal or mechanical basis. (b) Dislocation (athermal) instability of plastic flow Unlike Basinski’s model in which heat is of major importance to initiate jump-like deformation, the variants on the ‘mechanical’ (athermal) models associate serrated yielding with an instability on the scale of dislocation groups—local heating, clearly, is only a consequence of energy dissipation during this localized strain burst that could aid in the subsequent deformation process. The dislocation dynamical concept (Seeger 1956) is based on equation (F1.1.1) and the assumption that the increase in flow stress occurring with plastic deformation to a large extent results from the extensive formation of dislocation pile-ups against unyielding obstacles (see figure F1.1.6). Dislocation-interaction effects give rise to the heterogeneity of intercrystalline stress fields and a dispersion of the levels of energy barriers governing dislocation motion. By the high stresses reached in low-temperature deformation a breakaway through the obstacles may be triggered at the head of only a few piled-up dislocations, freely generating new dislocations which then result in the ‘collective’ process of plastic flow. The problem with Seeger’s model in its original form is that it assumes glide dislocations to have predominant screw character. However, as was shown in section F1.1.3, screw dislocations can circumvent

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internal stress maxima by cross-slipping out of their slip plane, without giving rise to some sort of catastrophic process, e.g. the collapse of the obstacle. Hence, under these conditions unstable flow as such is difficult to imagine, to say nothing about the distinct feature that serrations exist only below some critical temperature Ts . However, in practice things are quite different, recalling that for T < Tc (T0 , T1 ) the process of plastic deformation occurs basically by the motion of dislocations with predominant edge (⊥) character, as discussed in section F1.1.4. Since edge (or mixed) dislocations, in contrast to screw dislocations, are forced to move on their slip plane they might not leave a pile-up straight away. So, by taking into account explicitly the dislocation character, the mechanical model is able to explain a burst-type formation of dislocations accompanying a drop in load in the way described above. Schwarz and Mitchell (1974) were the first to prove the ‘near-edge’ orientation of dislocations involved in a fast localized relaxation process, testing single crystals of Cu—Al alloys at 4.2 K. More recently, further experimental evidence on an edge dislocation mechanism of serrated yielding was derived from tensile tests with a series of f.c.c. metals and alloys (Obst and Bauriedl 1988, Obst and Nyilas 1991). Here, we give only a few examples which clearly demonstrate the fundamentals of plastic deformation at low temperatures. As already described in section F1.1.6.2, figure F1.1.17 shows that for each material a characteristic temperature Ts exists (depending only on the strain rate of the tensile test) above which plastic deformation is a continuous process. Comparing Ts obtained in this way with the ‘transition’ temperature Tc derived from the temperature dependence of the macro-elastic limit (figure F1.1.15), one finds as a general result (equally valid for low- and high-γ materials)

From this and what has been said in section F1.1.4 it follows that serrated yielding in f.c.c. crystals at low temperatures is controlled by the edge component of the glide dislocations. (c) The low-temperature plastic instability—a coupled two-stage process To study in more detail the kinetics of this high-velocity dislocation phenomenon, we have further enlarged the time axis of figure F1.1.18, a typical result being plotted in figure F1.1.19. From the load—time curve at the position of the quartz force transducer (see figures F1.1.8 and F1.1.10) a ‘load drop’ emerges as a coupled two-stage process: (I) a linear macroscopic drop at very high strain rate bearing all the features of a viscous rather than a thermally activated dislocation motion (∆•σI ∼ 1.3 × 105 MPa s–1 ) and (II) in the wake of process (I) a thermal softening going off much more slowly (∆σ• II ~ 4 × 10 3 MPa s –1). The temperature—time curve shows that no significant rise (‘hot spot’) occurs until the load has started to fall. To sum up all the experimental findings and theoretical results, one may think of assigning stage I of the discontinuous yield process to the pile-up instability of edge dislocations described above. Under the thermal conditions prevailing at that moment (low thermal capacity and diffusivity), the catastrophically developing local slip will cause a large temperature rise, which is indeed observed, heating up adjacent regions of the crysta (figure F1.1.20 is an attempt to portray this situation). In stage II, the high tensile stresses are still at work on the specimen due to the elastic compliance of the sample-machine system, and they are going to produce—combined with the strong temperature dependence of the flow stress (see figure F1.1.13)—additional excessive plastic deformation in the heated zone at stresses much lower than the level of stress at the onset of the load drop. To complete the picture we point out that the abrupt local release of dislocations in stage I, overcoming a region of high internal stress, will cause compressive stress waves (an audible ‘click’) to propagate outside the specimen along the pull rods. The resulting periodic oscillations superimpose on the applied load, leading to stress modulations observed in ∆σI I (see figure F1.1.19). The oscillation period is determined by the dynamical properties of the testing system.

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Figure F1.1.19. Load (stress, MPa)—time and temperature—time records observed during deformation of nickel at 5.5 K (enlarged detail of figure F1.1.18).

In summary, serrated yielding in f.c.c. crystals obviously is of mechanical (athermal) origin (∆σI ) due to the nature of the low-temperature plasticity (predominant edge dislocation glide) and the burst-type motion of dislocation pile-ups resulting from that plasticity operates as a ‘nucleating deformation’ (see Basinski’s model) to trigger a thermal instability (∆σI I ). The latter is compliance controlled. This clear hierarchy in a real load drop, involving both an athermal and a thermal component, has led to some misinterpretations in the literature. F1.1.7 Dynamical dislocation pile-ups—an electronic response As set out in section F1.1.3, a dislocation in a crystalline solid is a line imperfection. Using its core, which is a most seriously disorganized region of the lattice (see figure F1.1.4), or, to a much greater extent, using its outer strain field (differences depending on whether the dislocation is of edge, screw or mixed character) a dislocation is able to scatter phonons (the quantized equivalents of lattice vibrational modes) and electrons. Conversely, it is this interaction that leads to an energy-dissipative process for moving dislocations. Since at low temperatures the probability for collective thermal excitations decreases sharply (the density of phonons is ∼ T 3 ), in metals the scattering of electrons by the elastic fields of dislocations tends to dom inate all other mechanisms. The dislocations, in overcoming the viscous drag, will lose some of their momentum and cause the electron system to drift into the direction of the dislocation velocity. As a result, a potential difference ∆ϕ arises between two opposite side edges of the sample to provide overall local compensation for the electronic and lattice charges. Kravchenko (1966, 1967) has estimated that, for detecting the electrical effect, a sufficiently high flux density of dislocations moving in a uniform direction is required. Strain rates of ε• 10 2 s–1 are expected to meet this condition. Such large values of ε• are, of course, hard to achieve in steady-state deformations. However, during ‘catastrophic’ slip as described above, shear rates of γ• ≤ 104 s–1 are reached temporarily in the slip bands.

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Figure F1.1.20. Model of the coupled two-stage process of a low-temperature load drop (schematic): I—a pile-up instability at very high strain rates occurs first, generating localized heating; II—smearing of the ‘thermal spike’ with time (t0 < t1 < t2 …) along the sample length will then lead to an increase in the rate of thermally activated slip in the heated zone.

Hence, apart from the load—time characteristic of a load drop (figure F1.1.19) giving rise to the assumption of two different mechanisms of plastic flow, the burst-type deformation should also generate electronic signals, yielding important information about the dynamic dislocation phenomena. Figure F1.1.21 shows the result of such investigations with an aluminium single crystal deformed by compression at 4.2 K (Lebyodkin and Bobrov 1994). During a large load drop a series of short pulses (of the order of microseconds) is observed on the background of a longer pulse (of the order of milliseconds). Both groups of pulses are not significantly influenced by the test temperature, but depend strongly on the magnitude of the load jumps. It has been established that the microsecond pulses are of nonthermal origin, each of them resulting from the motion of separate dislocation pile-ups (at velocities close to that of sound). The occurrence of a series of such pulses having different amplitudes, polarities and shapes reflects the collective character of the unstable glide process. In contrast to these microsecond pulses, the millisecond pulse (background) is associated with the thermo-electromotive force at thermal excitations inside the specimen, caused by the dynamic motion of the individual dislocation pile-ups. Because of that, the occurrence of the thermo-electromotive force is always preceded by an initial microsecond pulse (see figure F1.1.21). Summarizing the result of the electrical measurements, discontinuous deformation must be viewed as

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Figure F1.1.21. Examples of short (microsecond) electrical pulses and the thermo-electromotive force background signal (millisecond) at the moment of a large load jump of an A1 single crystal, deformed by compression at 4.2 K. Reproduced from Lebyodkin and Bobrov (1994) by permission of Scitec Publications Ltd.

a combination of a dynamic (athermal) and a thermal effect, the latter emerging from the former. This scenario is in full agreement with the interpretation of the mechanical data (see section F1.1.6.2( c)). Fl.1.8 Modes of plastic deformation other than slip Although slip is the most common mechanism of plastic flow in crystalline solids, twin modes may be activated in some metals to contribute to the deformation when slip is constrained, especially at low temperatures (see section F1.1.3). Mechanical twinning, a homogeneous shear process which is always preceded by some slip, occurs within microseconds, originating with a sharp (but small) release of load on the stress-strain curve. Figure F1.1.22 shows a typical deformation curve of a copper polycrystal strained at 5.5 K. In the highresolution plot of a part (inset) two types of jerky flow at high stresses can be observed. Superimposed on the ‘normal’ serrations are very small load drops which we attribute to mechanical twinning. To all appearances, when the sample has completely twinned at local stress concentrations, (discontinuous) slip once more occurs, as expected. This is strong evidence that lattice rotations by twinning, as stated in section F1.1.3, may help to provide the mechanism to make shear band formation easier. F1.1.9 Effect of plastic instabilities on failure Figure F1.1.1 schematically illustrated that the ability of a ductile material to undergo substantial amounts of plastic deformation is terminated by an inhomogeneous form of deformation termed fracture. According to the deformation maps (see figure F1.1.2), in the class of f.c.c. materials plastic flow occurs readily at all temperatures, that is by the shear mechanisms of slip or twinning. As figures F1.1.17 and F1.1.18 show, at test temperatures T < Tc (T0 , T1 ) the current deformation mode is progressively replaced by nonuniform deformation modes until, finally, one of the discontinuities serves as a direct precursor to failure, thus affecting the material properties. (It has never been observed that a serrated stress-strain curve becomes smooth again just before fracture.) The commonly observed stages in ductile fracture are: (i) nucleation of microvoids or pores at inclusions or second-phase particles (in engineering alloys) or by complex dislocation reactions (in very high purity metals); (ii) void growth by means of plastic strain (taking place during necking of the tensile specimen); (iii) crack formation (by local necking instabilities between the voids and coalescence along planes of maximum shear stress) and propagation. If ductile failure occurs as a result of a plastic instability as in figures F1.1.17 and F1.1.18 that produces a localized band of intense shear (shear rate •γ ∼ 104 s–1 ; see section F1.1.6.2(c)), then the voids are distorted by the strong shear components present and shear

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Figure F1.1.22. The load—time record observed during deformation of polycrystalline Cu at 5.5 K (sampling rate = 80 kHz). In the enlarged detail (inset) two types of jerky extension at high flow stresses can be made out. After each large ‘normal’ load drop, with the departure from elastic reloading, small-amplitude discontinuities occur which are attributed to mechanical twinning.

rupture is possible. In this case, failure occurs by necking on a microscopic scale, but the final mechanism of fracture is irrelevant and the polycrystalline sample separates macroscopically at an angle of 45° to the tensile axis (on a plane of maximum resolved shear stress) by a sliding-off mode. Figure F1.1.23(a) shows such a shear rupture at T = 6 K (< T1 ) in 316LN stainless steel, a widely used structural material for cryogenic applications, which exhibits relatively little necking before fracture. In figure F1.1.23(b) we illustrate the well known cup and cone fracture surface, obtained in a uniaxial tensile test at 50 K (> T1 ), the stress— strain curve at that temperature being smooth throughout its length. (For more details of the complicated system of tensile and shear stresses set up in this mode of fracture, we must refer the reader to the literature in the ‘Further reading’ list.) F1.1.10 Fracture-control design planning As mentioned above (in section F1.1.1), there is a great need for data on the mechanical properties of construction materials. Uniaxial tensile tests are, as shown, the most widely used approach to structural design and material selection, providing values of the yield strength or ultimate tensile strength. Moreover, in com bination with temperature and strain rate these parameters may give deep insight into the fundamental mechanisms controlling time-dependent mechanical behaviour. Regarding fracture behaviour in particular, most failures of engineering components or structures can only be vaguely understood solely in terms of the ‘strength of materials’ approach. Leonardo da Vinci 500 years ago demonstrated what we call nowadays the size effect. He found the strength of long iron wires to be inferior to that of smaller sections. These results implied that evidently an additional structural variable hitherto not taken into account into the theories of plastic flow controlled the strength: these additional variables were preexisting defects.

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Figure F1.1.23. Ductile fracture in 1.4429 (316LN) austenitic stainless steel: (a) shear rupture at 6 K; (b) cup and cone appearance at 50 K.

It is the merit of modern fracture mechanics to approach structural design planning from this point of view, quantifying the critical combinations of applied stress, flaw size and fracture toughness (replacing fracture strength as the relevant material property). It is, however, beyond the scope of this chapter to give an outline of this discipline that has matured since World War II.

References Arp D and McCarty R D 1989 Thermophysical properties of helium4 from 0.8 to 1500 K with pressures to 2000 MPa US Department of Commerce/National Institute of Standards and Technology (NIST) Technical Note 1334 Asby M F 1972 A first report on deformation mechanism maps Acta Metall. 20 887–97 Basinski Z S 1957 The instability of plastic flow of metals at very low temperatures I Proc. R. Soc. A 240 229–42 Basinski Z S 1960 The instability of plastic flow of metals at very low temperatures II Aust. J. Phys. 13 354–8 Dorn J E 1968 Low temperature dislocation mechanisms Dislocation Dynamics ed A R Rosenfield et al (New York: McGraw-Hill) pp 27–55 Keithley MetraByte 1992 Data Acquisition and Reference Guide vol 25, p 15 Kravchenko V Ya 1966 Influence of electrons on the deceleration of dislocations in metals Sov. Phys.—Solid State 8 740–5 Kravchenko V Ya 1967 Using electrical effects for observing the motion of dislocations in conducting crystals Sov. Phys.—Solid State 9 823–7

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Lebyodkin M A and Bobrov V S 1994 Role of dynamical processes at discontinuous deformation of aluminium Solid-State Phenom. 35/36 411–6 Ledbetter H M 1983 Elastic properties Materials at Low Temperatures ed R P Reed and A F Clark (Metals Park, OH: American Society for Metals) pp 1–45 Nabarro F R N (ed) 1980 Moving Dislocations (Dislocations in Solids 3) (Amsterdam: North-Holland) Nabarro F R N 1985 Work hardening of face-centered cubic single crystals Strength of Metals and Alloys (ICSMA 7) vol 3, ed H J McQueen et al (Oxford: Pergamon) pp 1667–700 Obst B and Bauriedl W 1988 The instability of plastic flow at low temperatures—an explanation from a new point of view Advances in Cryogenic Engineering Materials vol 31, ed A F Clark and R P Reed (New York: Plenum) pp 275–82 Obst B and Nyilas A 1991 Experimental evidence on the dislocation mechanism of serrated yielding in f.c.c. metals and alloys at low temperatures Mater. Sci. Eng. A 137 141–50 Read D T 1983 Mechanical properties Materials at Low Temperatures ed R P Reed and A F Clark (Metals Park, OH: American Society for Metals) pp 237–67 Rémy L, Pineau A and Thomas B 1978 Temperature dependence of stacking fault energies in close-packed metals and alloys Mater. Sci. Eng. 36 47–63 Roark R J and Young W C 1975 Formulas for Stress and Strain 5th edn (Tokyo: McGraw-Hill Kogakushua) Schwarz R B and Mitchell J W 1974 Dynamic dislocation phenomena in single crystals of Cu–10.5 at.%-Al alloys at 4.2 K Phys. Rev. B 9 3292–9 Schramm R E and Reed R P 1975 Stacking fault energies of seven commercial austenitic stainless steels Metall. Trans. A 6A 1345–51 Seeger A 1955 The generation of lattice defects by Moring dislocations and its application to the temperature dependence of the flow-stress of f.c.c. crystals Phil. Mag. 7 1194–217 Seeger A 1956 The mechanism of glide and work hardening in face-centered cubic and hexagonal close-packed metals Dislocations and Mechanical Properties of Crystals (New York: Wiley)

Further reading Newby J R (coordinator) 1995 ASM Handbook Vol 8: Mechanical Testing (Metals Park, OH: ASM International) Hayden W, Moffatt W G and Wulff J 1965 Mechanical Behaviour (New York: Wiley) Hertzberg R W 1983 Deformation and Fracture Mechanics of Engineering Materials 2nd edn (New York: Wiley) Hirth J P and Weertman J (eds) 1968 Work Hardening (New York: Gordon and Breach) McClintock F A and Argon A S 1966 Mechanical Behaviour of Metals (Reading, MA: Addison-Wesley) Reed R P and Horiuchi T (eds) 1983 Austenitic Steels at Low Temperatures (New York: Plenum) Wigley D A 1971 Mechanical Properties of Material at Low Temperatures (New York: Plenum)

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F1.2 Structural stainless steel materials A Nyilas

F1.2.1 Preface The large-scale application of superconducting coils necessary for high magnetic fields in various technological areas demands a sound mechanical data base for the materials used. The design engineer, however, often does not possess a profound understanding of the test methods and the limits of applicability of the resulting values. Handbooks of material data commonly do not provide all the information needed for safe manufacturing of structures. The problem of stress relief during the manufacturing process of complex components, to mention but one example, requires detailed knowledge of the thermal and mechanical history of the used materials. This point is even more relevant for welded structures owing to their eventual sensitization behaviour, i.e. their susceptibility to intergranular corrosion by carbide precipitation in austenite grain boundaries. The significant impact of this behaviour on fracture toughness and a knowledge of the relevant data at temperatures below 77 K are important. Therefore, it is a large task for a material scientist to give a comprehensive solution and touch on all the various aspects of this problem. Such an attempt, being partly the aim of this section, can only give a rough orientation in the rapidly evolving technical world, and it should stimulate the scientist in the area of cryogenic materials testing to investigate the future demands of our technical society. After a brief outline of the increasingly successful methods of mechanical testing at cryogenic temperatures, a survey of some important structural materials will be given, dealing with tensile and fracture properties. The materials treated here have been investigated in detail and the results have already been used in several engineering components working at low temperatures. F1.2.2 Stress—strain measurements at cryogenic temperatures and related problems The stress—strain behaviour of materials is a key engineering feature in the design of machines working in ambient as well as in cryogenic environments. The uniaxial tensile test delivers important engineering material parameters: the 0.2% offset yield strength, for example, which directly enters the design rules, determines the allowable stress limitation in a structure under operation. To obtain the correct values of the yield strength it is necessary to measure the initial straight line proportionality between stress and strain. The measured deviation from proportionality gives the macroelastic limit and depends on the resolution, and the ratio of stress to strain represents Young’s modulus. The accuracy of the offset slope thus determines all the quantities derived from the stress—strain curve. In a tensile test, both the properties of the load cell and the displacement transducer as well as their positioning greatly determine the accuracy of the measurement. In the case of slow rate testing the load cell is usually placed outside the cryostat. In contrast, accurate strain measurements always require the displacement transducer (extensometer) to be directly clamped to the reduced section of the specimen

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under test to avoid additional readings owing to the rest of the sample, the load train and the testing machine. Since bending strains, due, for example, to misalignment of grips, nonaxiality of cryogenic rigs, specimen nonuniformities and process tolerances, can never be excluded the use of two extensometers on opposite sides of a sample is recommended. Using the averaged signals to obtain a mean strain value thus ensures precise measurement. For calibration of the extensometer system, micrometer-type devices or linear variable differential transformers (LVDTs) are widely used (for design and application of LVDTs, see ASM Handbook 1995). Finally, in obtaining meaningful load-elongation data the choice of proper grips is critical. Pin-loaded mechanical grips, threaded joints and serrated jaws are the most commonly used fixtures in tension testing for flat or round specimens. They must be stiff enough and work reliably during loading to prevent any slipping. Slipping may cause instant shocks, influencing the attached extensometer (knife-edge motion, etc). For the same reason vibrations coming from the environment (cryogenic liquid convection or bubbling, step-motor vibrations and others) should be avoided or properly damped down. F1.2.2.1 Clamp-on extensometers For tension testing of metallic materials at low temperatures, an extensometer must work in the same cold environmental chamber as the specimen. Moreover, because of limited space in most cryostats, the specimens often can no longer satisfy precisely the ASTM E 8-81 standard, their smaller dimensions imposing additional requirements on the extensometer as far as size and resolution are concerned. One common sensing device successfully employed in cryogenic applications is the extensometer applied with strain gauges, which makes use of the change in resistance of a material in the presence of mechanical deformation. Four gauges, generally of the metal-foil type with an integral polymeric backing, are bonded on a metallic cantilever beam and connected to a bridge circuit. Deflection of the beam when the specimen is strained produces an electrical signal. The electrical read-out can be amplified to as high as 10 000 to 1, allowing a variety of sensitivity ranges. The extensometers used at present are manufactured by electro-discharge machining (EDM) to minimize additional residual stresses in the frame upon machining. The frame itself is the sensing region and is not bolted or welded. This type of extensometer works, therefore, completely within the elastic regime and shows a high degree of linearity and reproducibility within the given travel length (±0.2 µm deviation from linearity, see figure F1.2.1). By special tailoring, such extensometers are capable of measuring strains up to 70%. The frame material is made of a special Ti alloy (Ti-6A1-4V), having a hardness in the range of 1200 HV, a yield strength of >1000 MPa and an elastic Young’s modulus of about 105 GPa. Owing to the high hardness value of this alloy, the end of each beam is given a knife-edge shape. The dual extensometer system is attached by a spring-wire technique, the knife edges being set to the exact gauge length (GL) with a special mounting block. For wiring between the terminals and connector, Teflon insulated 0.125 mm diameter wires are recommended. Strain-gauge extensometers of this kind are small, lightweight (mass < 3 g with GL = 15 mm), and easy to handle. Application of nickel-chrome foil strain gauges (GL = 1 mm) on the Tialloy frames results in marginal changes of the calibration factor between ambient temperature and 4 K, thus aiding the calibration procedure. The dual symmetrical sensing system is—apart from the already mentioned purpose of averaging the signals—beneficial in that it balances the specimen. This point becomes more important when measuring small diameter samples (e.g. wires). F1.2.2.2 Signal conditioning, data acquisition and evaluation The electrical signals from the load cell (typically of the strain-gauge type) and the extensometers must be conditioned into output signals that are proportional to the applied force and extension of the specimen

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Figure F1.2.1. The calibration curve of an extensometer at 7 K using a gain factor of 1000. The computed standard error of estimate is <0.1 µm with a reproducibility of ∼99.96%.

respectively. This requires a two-channel signal conditioning operating in low-noise mode at linear gains up to 10000. With today’s chip technology it is possible to reduce electronic noise below 1.5 mV at 10 V output, working at a bandwidth of about 10 kH.z. These details may serve as a guideline to obtain highquality measurements. The load—displacement data are either presented on a recorder chart or can be transmitted to a computer. With current computer technology, high-performance data acquisition boards are commercially

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available with many different functionalities providing maximum flexibility and matching the needs of most applications. Low-noise boards with an effective resolution of nearly 16 bits (i.e. the peak-to-peak noise added to the signal is <0.15 mV at 10 V full-scale input range) at a maximum sampling rate of 200 kHz are the best recommended models today; boards with 18- or even 21-bit resolution at 60 Hz sam pling rate are available, too. Clearly, this development aims to spare analogue signal-conditioning peripherals and make the computer itself with the software into a general-purpose instrument. Hence, software is required that easily integrates any or all of the hardware components into a single system and is capable of adding data evaluation and the calculation of test results. Programming, of course, offers the most flexibility to exactly match the particular application needs, but one may also purchase complete software packages supporting all aspects of data acquisition, analysis and presentation. Probably, most applications are best served by a combination of both. F1.2.3 Cryogenic structural materials Ferritic steels, like other body-centred-cubic (b.c.c.) metals and alloys, show a tendency towards brittleness as the temperature is reduced. In contrast, the face-centred-cubic (f.c.c.) materials remain ductile down to liquid-helium temperatures, accounting for their overwhelming importance as cryogenic engineering materials. The austenitic stainless steels are Fe–(16–26%) Cr alloys, plus sufficient (8–24%) Ni, and Mn or N to make them austenitic (f.c.c.) at room temperature. Although not specifically developed for cryogenic use, they are the conventional low-temperature materials. Utilizing the known relations between composition and phase stability, thermomechanical processing and microstructure, their mechanical properties can be tailored for 77 K and 4 K applications. These steels are easily welded if attention is paid to weld sensi tization. In addition, these Cr—Ni stainless steels are nonmagnetic provided that the f.c.c. structure is preserved. In the following, low-temperature tensile properties of the most commonly used engineering stainless steels are given. F1.2.3.1 Material AISI 316LN (Werkstoff No 1.4429) This material, a low (L) carbon and nitrogen (N) alloyed Cr—Ni stainless steel, attracted those industries manufacturing highly loaded structures at cryogenic temperatures. The history of this alloy began about eight decades ago where it was recognized for the first time that nitriding of Fe—Cr—Ni alloys increases the yield strength at room temperature significantly. Apart from the marked solution strengthening of N (the effect being twice the effect of C) interstitial nitrogen causes an increase of the austenite stability, there by permitting the replacement of part of the nickel (0.1% N addition is equivalent to ~4% Ni). With the beginning of the 1960s the cryogenic community paid attention to these nitrogenated steels because of the beneficial increase of the mechanical yield strength particularly at low temperatures. Later, the search for a candidate material suitable for structural applications in magnetic fusion energy focused attention on a nitrogen-strengthened stainless steel designated ‘316LN’. The material chemical composi tion is given in table F1.2.1 and shows the high nitrogen content of over 0.17% with a low carbon content. The low carbon content is necessary to prevent sensitization of the material during the welding or long heat-treatment process. As given in table F1.2.1, the chemical analysis shows a spatial variation of composition of the manufactured ingot. Note the nitrogen variation between 0.175% and 0.19%. Since nitrogen additions significantly increase the strength of austenitic stainless steels (but thereby decrease the toughness, see below), these variations in N, though marginal, will strongly influence the tensile properties of the material. Therefore, it is necessary to quantify the material properties by machining tensile specimens from different

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Mechanical properties of engineering metals and alloys Table F1.2.1. Results of vendor’s chemical heat analysis in wt% for a typical 316LN ingot.

regions of the heat. In addition, the last fabrication stage may change the mechanical properties drastically compared with the solution-heat-treated material. Figure F1.2.2 shows the tensile response of the material 316LN given in table F1.2.2, revealing sig nificant differences in the annealed and after-manufacturing (extrusion) conditions. Solution heat treatment of the machined specimen from the extruded section at 1050°C for 0.5 h yields material properties which are considerably different from the last fabrication stage.

Table F1.2.2. Tensile properties of 316LN mateiral at different conditions measured 7 K.

Changing the metallurgical steel-making process also affects the plastic flow behaviour of this steel as shown in figure F1.2.3 and table F1.2.2. The differences in the yield strengths between cast 316LN stainless steel and 316LN plate material are marginal. However, the elongation at fracture, the ultimate tensile strength and the stress—strain behaviour depend significantly on the steel-making process. In addition, the effect upon 4 K yield strength (Hall—Petch relation) is marginal although the grain sizes of both differ approximately by an order of magnitude. However, the flow properties after the general yielding reveal the high influence of the material microstructure characterized by the grain size. We finally mention that 316LN is a ‘metastable’ austenitic stainless steel, transforming martensitically to the ∈ phase (hexagonalclose-packed (h.c.p.)) or to α’ (b.c.c.) during straining at low temperatures (T < 160 K). The parameter controlling the ease of transformation, the type of martensite and its substructure, work-hardening and deformation textures is the stacking-fault energy (see section F1.1).

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Figure F1.2.2. The stress—strain response of 316LN tested at 7 K: run 1—solution-heat-treated material; run 2—the same material after extrusion; run 3—material extruded and subsequently heat treated at 650°C for 200 h.

F1.2.3.2 Material SUS 316 (~AISI 316) In several cryogenic applications the material 316 has been selected as a first-choice material if the structural requirements are not appropriate for high-strength (>1000 MPa) materials. Several tonnes of this material are used widely throughout the world. Table F1.2.3 gives the material heat composition. The main difference in composition of this material compared to 316LN is the high carbon and very low nitrogen content, the latter considerably decreasing the yield strength (see figure F1.2.4). Again, the stress—strain behaviour can be changed drastically by the fabrication process. A fabricated round bar

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Figure F1.2.3. The stress—strain response of 316LN at 7 K manufactured by different steel making processes: run 1—316LN manufactured by vacuum induction melting; run 2—316LN manufactured by casting.

Figure F1.2.4. The stress—strain response of SUS 316 at 7 K with different fabrication conditions: run 1—SUS 316 solution-heat-treated plate material; run 2—SUS 316 round bar material fabricated by the drawing process.

Table F1.2.3. Chemical composition of SUS 316 (AISI 316).

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Table F1.2.4. Tensile properties at 7 K of SUS 316 material at different conditions.

Table F1.2.5. Chemical composition of AISI 321.

product of material SUS 316 with similar chemical composition as given in table F1.2.4 shows a reduced elongation at fracture and a significant difference in the strain hardening behaviour compared with the SUS 316 plate material. Although the yield strengths of these two materials do not differ markedly (see table F1.2.4) the plastic deformation influences the fracture properties of both materials. F1.2.3.3 Material ~AISI 321 (Werkstoff No 1.4541) The material AISI 321 is widely used in European countries for several applications, especially at 77 K. The composition of the material is given in table F1.2.5. This material is suitable for cryogenic applications in which sound welding is an important requirement. To prevent weld sensitization caused by high carbon content, this material is additionally alloyed with titanium. The low-temperature tensile properties of this low-strength material show a reduced elongation at fracture behaviour compared with AISI 316 or AISI 316LN materials (see figure F1.2.5). Table F1.2.6 gives the tensile results at 4.2 K (liquid helium) obtained with tests of this material in the as-received and aged condition. Heat treatment at 800°C for 0.5 h greatly changes the stress—strain behaviour. The material yield strength increases markedly after aging (precipitation of σ phase) and the onset of the serrations starts earlier than in the as-received condition.

Table F1.2.6. Tensile properties at 4.2 K of AISI 321 material at two different conditions.

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Figure F1.2.5. The stress—strain response of AISI 321 (Werkstoff 1.4541) at 7 K under different conditions: run 1—material solution-heat-treated; run 2—same material heat-treated for 0.5 h at 800°C.

F1.2.3.4 Material SUS JN1 This material is a new Japanese cryogenic steel developed with the purpose of meeting the design requirements of high strength and high toughness in the field of fusion energy applications. SUS JN1 is fabricated by the Nippon steel company in the form of a 100 mm thick plate. The chemical composition of this alloy is given in table F1.2.7. The striking difference of this steel in comparison with 316LN-type nitrogen-strengthened stainless steels is the high nitrogen content and the remarkably high content of manganese. Both these elements increase the austenite stability, therefore the alloy preserves its fully austenitic (f.c.c.) structure during straining at low temperatures. Table F1.2.8 gives the results of two tests performed with SUS JN1, one being tested at 4.2 K in a liquid-helium environment and the other at 8 K in a gaseous-helium environment.

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Table F1.2.7. Chemical composition of SUS JN1.

Table F1.2.8. Tensile properties of SUS JN1 at 4.2 K and at 8 K.

Figure F1.2.6. The stress—strain response of SUS JN1 tested in two different environments: run 1—SUS JN1 tested at 8 K in gaseous helium; run 2—same material tested at 4.2 K in liquid helium.

As with all other materials the stress—strain response shows a clear dependence of the tensile flow properties on the coolant (see figure F1.2.6). The material tested in gaseous helium (~50 mbar (5 kPa)) has a lower temperature owing to the higher thermal diffusivity of gaseous helium than that of liquid helium (see section F1.1). As a consequence, the onset of the serrations shifts to smaller strains if the

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material is tested in a gaseous-helium environment at 8 K than if it were tested in liquid helium; also the total number of serrations is greatly increased. These tests confirm the high cooling performance of the gas at low pressure (∼5–10 kPa) during low-temperature tests. F1.2.4 The strength—toughness relationship High strength is an essential requirement of practical materials for structural use. In view of long service lifetime, fracture toughness (i.e. resistance to the extension of a crack usually described in terms of the plane strain critical stress intensity factor, KI C ) is a key material property. With design stress, preexisting flaws, fissures, cracks or other stress concentrators, it controls the conditions for failure in a component. (For the procedures for fracture toughness measurements, we refer the reader to the literature given in the ‘Further reading’ section at the end of this section.) To be suitable for cryogenic applications, structural alloys should fail in a ductile mode, characterized by expenditure of considerable energy. When appreciable plastic deformation accompanies the progressive fracture process, toughness depends on the same basic mechanism of strength and resistance to dislocation motion as the tensile properties described in section F1.1.3, but the relationship is more complex owing to the multiaxial stress field at a crack tip. According to this interplay between strength and toughness, a trend is present for the toughness to decrease if the yield strength is made stronger (i.e. if crack extension can occur with a lower expenditure of energy). Since for a particular alloy and thermomechanical treatment the yield strength increases automatically as the temperature is lowered, the fracture toughness in general decreases monotonically. Hence, to increase the fracture toughness of an alloy at 4 K, the strength—toughness combination as a whole must be raised. Figure F1.2.7 refers to the measured fracture toughness data

Figure F1.2.7. The strength—toughness relation of several engineering materials measured at 4.2 K. The variation of yield strength directly affects the materials’ toughness behaviour. This diagram also shows the dominating regimes of the ductile and brittle failure of materials.

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versus tensile yield strength for several structural alloys at 4 K. In this diagram the target value (JAERIBox, Japan Atomic Energy Research Institute) is also shown for high-strength, high-toughness structural material applications in the field of fusion energy. The goal is to design materials for 4 K applications in which both parameters are increased.

References Newby J R (coordinator) 1995 ASM Handbook Vol. 8, Mechanical Testing (Metals Park, OH: ASM International)

Further reading Anderson T L 1995 Fracture Mechanics 2nd edn (Boca Raton, FL:Chemical Rubber Company) Collings E W and Gegel H L (eds) 1975 Physics of Solid Solution Strengthening (New York: Plenum) Corten H T 1969 Toughness of materials Ocean Eng. I 261–85 Hertzberg R W 1983 Deformation and Fracture Mechanics of Engineering Materials 2nd edn (New York: Wiley) Mann D (ed) 1977 A User’s Manual of Property Data in Graphic Format (LNG Materials and Fluids) (Boulder, CO: NIST) Papirno R and Weiss H C (eds) 1989 Factors that Affect the Precision of Mechanical Tests, Technical Report No STP 1025 (Philadelphia, PA: ASTM) Read D T 1983 Mechanical properties Materials at Low Temperatures ed R P Reed and A F Clark (Metals Park, OH: American Society for Metals) pp 237–67

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F2 Properties of fibre composites G Hartwig

F2.0.1 Introduction The outstanding features of fibre composites favour their application in low-temperature technology. The high specific strength and stiffness, the high fatigue life, the low thermal conductivity, the electrical insulation capability and special features of thermal expansion behaviour are some of the advantages in their cryogenic use. At present not all features mentioned above are achieved by a single class of composites. Each type has its own merits. Specific cryogenic problems of all fibre composites arise both from thermal stresses between the fibres and the matrix and the brittleness of the polymeric matrix. A great variety of properties is provided by different types of fibres and polymeric matrices as well as by the fibre arrangement. Different fibre types can be combined in a hybride system. The proper choice of the polymer matrix is important, if certain resistances are required. Resistance against radioactive irradiation is important for polymers used in fusion reactors or accelerators. The highest dose is 108 Gy for the most resistant polymers, e.g. polyimide (PI). Resistance against water absorption is another requirement for several applications. There are several moisture-resistant polymers available, such as polycyanates (PUR), polycarbonate (PC), etc. The matrix type generally does not noticeably influence the fibre-dominated mechanical properties of composites. An exception is cryogenic fatigue behaviour. It is astonishing that a composite with a brittle epoxy matrix shows a higher fatigue life than one with a more ductile thermoplastic matrix. Emphasis is put on the influence of the matrix on the composite properties (Hartwig and Knaak 1984). The material components considered are compiled in table F2.0.1. It is the great advantage of composites that they can be tailored to suit the requirements. The fibre arrangement allows a great variability of properties to be achieved. For several applications forces act in one direction only and with unidirectional fibre composites the strength can be concentrated in this direction. Anisotropic arrangements open the possibility of making better use of the material. Carbon or Kevlar fibres arranged at ±30° form a composite with a negative coefficient of thermal expansion (about half that of steel, but negative). This unusual property can be used for compensating thermal mismatch. The scope of this chapter on fibre composites covers the following properties: • • • • •

moduli, strength, fracture strains, interlaminar shear and fatigue behaviour thermal expansion, thermal conductivity and specific heat break-through voltage and dielectric properties gas permeability resistance against radioactive irradiation.

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The errors in the data obtained for fibre composites are either statistical or arise from the measuring method. Further deviations occur as a result of different processing techniques and the quality of the fibre composites. Inhomogeneities or fibre misalignment influence properties, especially thermal expansion and fracture strength. The interfacial bond is an important parameter, which is difficult to quantify but it influences the magnitude of several properties. The data are therefore subject to unknown uncertainties.

Table F2.0.1. Material components considered.

F2.0.2 Applications of fibre composites Important areas of application of fibre composites at low temperatures are: • superconducting magnets (fusion technology, accelerators, energy storage, magnetic levitation,

cryoelectronics)

• support elements for cryogenic devices (in tension or compression) • compensating support elements (support elements with a strong negative thermal expansion coefficient

for compensating unwanted expansion)

• nonmetallic cryostats

(for superconducting quantum interference device (SQUID) detectors or liquefied gases, hydrogen technology) (Callaghan 1991)

• cryogenic vessels (storage and transportation of liquefied gases, hydrogen technology) • space technology (cooling of equipment, cryogenic wind tunnels, space antennas without thermal

expansion) (Bansemir and Haider 1991).

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F2.0.2.1 Superconducting magnet Superconducting magnet are used in fusion technology or accelerators for ‘bearing’ either the plasma or the particle beam. The most important applications of fibre composites in superconducting coils are: • •

electrical insulation structural and support elements.

Electrical insulation of cables or coils is usually achieved using glass fibre—epoxy tapes and Kapton foils. Spacers of glass fibre for liquid helium paths are used in most superconducting magnets. If eddy currents induced by pulsed magnets are a problem, the coil body or casing can be made of glass fibre or carbon fibres. The important design parameters are strength, fatigue behaviour, and thermal expansion. Energy storage systems are designed for feeding current at peak power consumptions, in this way lowering the average level of power supply. Storage can be done: • inductively by large superconducting-coil systems • mechanically by flywheels with superconducting bearings.

Flywheels can be made of carbon fibres supporting a heavy metal ring. Magnetic levitation systems are bound to lightweight constructions with glass fibre or carbon. The tensile strength per weight for carbon is 4–12 times higher than that for steel and about half that for aluminium. The large range is due to the fibre arrangement. The largest value applies to unidirectional carbon fibre composites. Cryogenic devices, e.g. SQUID detectors, are very sensitive to even small magnetic fields. Their structural elements are made of nonmagnetic fibre composites (S- or R-glass). F2.0.2.2 Support elements Support elements for superconducting coils or cryogenic devices require a low thermal conductivity and a high tensile or compressive strength (stiffness). For tensile support usually straps are used. Carbon fibres exhibit a very low thermal conductivity only at very low temperatures; at higher temperatures conductivity is quite high; the contrary is true for glass fibre. Therefore, the best thermal insulation is achieved by combining both fibre types in series. An example is shown in figure 2.0.1. Two straps are connected; a carbon fibre strap in the low-temperature range and glass fibre strap at higher temperatures. The strength of the straps is strongly increased by additional reinforcement with crossplied layers in their circular parts.

Figure F2.0.1. Tensile support straps of carbon fibre and glass fibre.

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The decisive parameter of thermally insulating support structures is the ratio of thermal conductivity per strength or stiffness (see section F2.0.10). There is another class of fibre composites, namely, ceramic fibres (Al2O3 , not SiC). Instead of a combined system, ceramic fibres can be used over the total temperature range from 2 K to room temperature, yielding nearly the same insulating capability (see figure F2.0.42 later). F2.0.2.3 Compensating support elements Compensating support elements with a strong negative thermal expansion coefficient can be applied when cooling of cryo-devices leads to a loosening of the support or to an unwanted geometrical shift. An unwanted geometrical shift may occur for superconducting accelerator magnets. A loosening may happen when energy storage superconducting magnets are cooled. The coil shrinks when cooled and the support to the warm casing might be loosened. The compensating elements involve carbon fibres in a special fibre arrangement which yields a large negative coefficient of thermal expansion. The elements will be described in section F2.0.9. F2.0.2.4 Nonmetallic cryostats Nonmetallic cryostats are usually made of glass fibre. If a nonmagnetic behaviour is required, e.g. for SQUID detectors, iron-free glass fibre (R- or S-glass) is used. A general requirement made on all fibre composites used for building cryostats is a very low gas permeability even if subjected to thermal cycling. Investigations on glass fibre have revealed that neither thermal nor mechanical cycling increases permeability remarkably by formation of additional microcracks. Gas permeation of fibre composites for most applications is sufficiently low (see section F2.0.13). F2.0.2.5 Cryogenic vessels Cryovessels for transportation of liquefied gases (e.g. methane or hydrogen) are made of carbon fibres since these fibres exhibit the highest fatigue life and a sufficiently low gas permeability. This is especially important for pressure vessels. Hydrogen propulsion is becoming important and lightweight tanks are required for vehicles and especially for aeroplanes (cryoplanes). The large specific strength and stiffness of carbon is a great advantage. F2.0.2.6 Space technology Space technology makes use of cryogenics. Cryogenic wind tunnels give higher Reynold’s numbers per volume and power consumption. The paddles are made of carbon fibres because of their high specific strength and fatigue life. Antennas of satellites are exposed to large temperature variations which might cause thermal deformations and defocusing. Carbon fibre composites without any thermal expansion can be made by a special arrangement (see figure F2.0.7 later). This is again a unique feature of carbon fibre composites. F2.0.3 Fibres The fibres used at present in composites are made of glass fibre, carbon, Kevlar or ceramic. Reinforcing fibres are processed by spinning glass fibre directly or aramid fibres (Kevlar) from the nematic liquid crystalline phase of stiff aromatic polyamide segments. Ceramic and carbon fibres are processed by pyrolizing a filamentary precursor. The precursor for SiC fibres is a silicon polymer; alumina fibres are spun from an aqueous slurry mix of alumina particles followed by flame firing.

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Fibres

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For processing of carbon fibres two types of precursors are used: pan-filaments (polyacylonitrile) or pitch filaments. Carbonization and graphitation occur by pyrolysis (Donnet and Bansel 1990). Several features can be deduced from their compositions and structures, which are shown in figure F2.0.2. Fibre glass is an isotropic material, but less stiff because of the oxygen bridge between the Si-atoms. Ceramic materials (not shown in figure F2.0.2) are also isotropic but much stiffer (Davidge 1979). SiC has a structure like diamond with Si and C alternating. AI2O3 has a tight hexagonal structure. Both are polycrystalline materials.

Figure F2.0.2. Composition and structure of fibres: (a) glass fibre; (b) carbon; (c) Kevlar.

Carbon fibres consist of small but strong graphite pieces oriented in the fibre direction in a circular (pan-based) or radial arrangement (pitch-based). The pieces are bonded by van der Waals forces. The anisotropic structure is reflected in most of their properties. High-modulus fibres consist of larger graphite pieces than high-tensile fibres. It should be mentioned that the highest-modulus fibres (K13 fibres) available nearly reach the theoretical stiffness of graphite (E = 1090 GPa). Advanced high-strength carbon fibres have a tensile strength up to 8 GPa. Kevlar consists of stretched aramid molecules which are bonded by van der Waals forces and hydrogen bridges. They are the most anisotropic fibres with a very weak transverse strength. A short summary of the advantages and disadvantages of fibres is given in table F2.0.2 as a guideline for their proper application. General fibre properties are compiled in table F2.0.3. The temperature dependences of both the strength and the modulus are very small for most fibres. An exception is the tensile strength of glass fibre. Well below room temperature there is a strong increase up to 40%. The surface of common glass fibre is coated by several atomic layers of water. This leads to stress corrosion in the microcracks due to the diffusion of hydroxyl groups (–OH) when tensile load is applied. At low temperatures the diffusivity is frozen and stress corrosion ceases. Kevlar fibres also exhibit some temperature dependence since they are polymer fibres. Several prefabricated fibre arrangements are available. (i) Tapes (unidirectional arrangement). (ii) Crossplies (0°/90° lay-up or woven fabric). The wavy shape of woven fabrics reduces stiffness and fatigue life compared to separate 0° and 90° stacking. On the other hand, woven fabrics exhibit a

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Properties of fibre composites Table F2.0.2. Features of fibres.

Table F2.0.3. General fibre properties.

higher interlaminar shear strength. A good compromise is satin weaves where the warp tow crosses over several fill tows and under the following one (usual five to seven harness). (iii) Angleplies (unidirectional layers symmetrically sandwiched at angles ±ω1 ; ±ω2 … ; multilayer composites). A great variation of properties can be achieved by the variation of the fibre angles. (iv) Three-dimensional (3D) fabrics (fibres also in thickness direction; see figure F2.0.3). The weak matrix dominated properties can be improved by even a small through thickness fibre volume (5–10 vol.%). The tensile and compressive strengths and stiffness in the thickness direction are increased. This might be an advantage for spacers used in superconducting magnets. The thermal expansion is decreased and better matched to copper. The interlaminar shear strength and flexural strength or stiffness are considerably increased. Even in-plane compressive strength is improved since buckling is stabilized and shear failure restrained. The failure modes of 3D fibre composites are generally more fibre dominated and therefore less sensitive to radiation damage (Iwasaki et al 1991).

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Structural stainless steel materials

1013

Figure F2.0.3. 3D fabric.

F2.0.4 Polymer matrix It is difficult to provide general rules for the selection of the matrix materials so that the desired composite properties will be achieved. Generally, epoxy resins become more brittle than thermoplastic polymers at low temperatures. This might be considered a disadvantage and it is for several matrix- dominated properties, but there are properties which show a different behaviour. For example, the fatigue behaviour, e.g. especially of carbon fibre composites, is much better with a brittle epoxy matrix than with a ductile thermoplastic one. In the case of environmental influences or radioactive irradiation, highest resistance is given to a composite by a polyimide matrix (admissible radioactive dose ≈108 Gy). The strongest fibre-matrix bond can be achieved with a PEEK matrix, because it crystallizes onto the fibre surfaces. In table F2.0.4 several matrix properties are compiled (moduli, tensile strength σU T , strain εU T , and integral thermal expansion ∆L/L). Liquid crystalline polymers (LCP), e.g. Vectra 50, are self-orienting, high-performance materials, which are waiting to be used commercially as matrix materials in composites. The advantages include a very high environmental resistance, a high working temperature (580 K) and an anisotropic alignment along the fibres in case of special processing. The alignment would be of great advantage for cryogenic use, since most of the thermal stress between the fibre and the matrix would be avoided. In the alignment direction, LCP’s thermal expansion is low and comparable to that of fibres.

Table F2.0.4. Polymer properties at 4.2 K and 77 K (Hartwig 1994).

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Properties of fibre composites

The question arises as to whether or not the polymer properties in a fibre composite are similar to those measured in bulk materials. Mixing rules and lamination theories are applied for calculating the composite properties from those of the fibres and matrices. In several cases the polymers behave differently in a fibre composite. Several matrix properties are changed by the constraints of the fibres. The applicability of some polymer bulk properties is compiled in table F2.0.5.

Table F2.0.5. Applicability of polymer bulk properties

F2.0.5 Manufacturing of composites Several processes have been developed for manufacturing fibre composites. They are different for • •

thermoplastic matrices which can be made viscous by heating; duroplastic matrices which are cured by a chemical reaction of the resin with a hardener. After a curing time of several hours, they are rigid and insensitive to thermal treatment. Important classes of duroplastic matrices are polyester resins and epoxy resins.

The production methods used are as follows. (i) Prepreg hot pressing. Laminates or fabrics are preimpregnated with a matrix and hot pressed to sheets or any shape. The applied pressure and temperature controls the fibre content. This method is well suited for thermoplastic matrices. Preimpregnation with precured epoxy resins, however, involves problems since chemical reactions go on except when cooled. Therefore epoxy preimpregnated laminates cannot be stored over a long time period. (ii) Pultrusion. This is a continuous process for manufacturing bars with any cross-sectional shape. Fibres are pulled through an epoxy resin bath and through a conical of desired dimensions where compression and curing by heat treatment occur. (iii) Hybride yarn. This yarn consists of thin filaments of thermoplastic polymers and fibres. By heating, a tight and homogeneous connection of fibre and matrix is achieved. This yarn can be used in a pultrusion-type technique or for filament winding. (iv) Impregnation with a liquid and outgassed epoxy resin. One possibility is a wet lay-on technique of filaments, braids or laminates; the other is injection of resin into the ready structure of fibres. Best results are achieved if the structure is surrounded by a bag, which can be evacuated before injecting the resin (autoclave-technique). (v) Filament winding.

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Fibre arrangement and properties

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F2.0.6 Fibre arrangement and properties Fibre angles and fibre content control most mechanical and thermal properties of composites. Several properties can be calculated from those of the fibre and the matrix by simple mixing rules. Two examples are: (i) stiffness E|| of unidirectional composites: Ec || = Ef || Vf + Em ( 1–Vf ) (ii) specific heat: Cc , = Cf Vf + Cm ( 1–Vf ) The subscripts are c: composite; f: fibre; m: matrix; Vf is the fibre volume fraction. Most other mechanical, thermal or dielectrical properties are calculated by lamination theories. The most important fibre arrangements are: • • • •

unidirectional (UD) crossplies (0°, 90°); warp- and fill direction might be different mono-angleplies: one angle ω in symmetrical arrangement of ±ω layers multi-angleplies: symmetrical arrangement of ±ω1 , ±ω2 , ±ω3 layers.

Multi-angleplies can be arranged in such a way that nearly isotropic properties exist in-plane. All other arrangements are anisotropic in-plane. Their basic assumptions and formulae are briefly reviewed for elasticity and thermal expansion. (a) Step I UD layers are the basic elements of any angleply composite. They are assumed to be smeared out to homogeneous but anisotropic sheets. Their orthotropic properties (parallel and transverse to the fibre direction) are assumed to be known. The constitutive relations are

{S} is the orthotropic compliance tensor. For thin sheets only in-plane stiffness (x , y) and shear (x y) are considered. (b) Step 2 Off-axis properties in +ω and –ω directions are calculated by the angle transformation tensors T+ω , and T–ω respectively (Chawla 1987, Engineered Handbook Composites 1987, Piggott 1980). The coordinate systems are shown in figure F2.0.4.

Figure F2.0.4. Transformation of orthotropic axes 1;2 to x y coordinates by an angle ω.

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Properties of fibre composites

The transformation of orthotropic properties by an angle ω is given by:

Inserting in equation (F2.0.2) gives

Tensile (compressive) load or thermal expansion of anisotropic material in off-axis directions deforms a specimen also in shear (this is different from isotropic or orthotropic materials). A quadratic piece of a UD composite in the off-axis direction deforms to a rhombic shape, symmetrically mirrored for +ω and –ω directions under off-axis load or with temperature variations (see figure F2.0.5). In the latter case the rhombical shape is enhanced when fibres have a very small (or negative) thermal expansion and the matrix a large one.

Figure F2.0.5. Deformation of off-axis unidirectional composites by external load or temperature variation ∆T.

For tensile (compressive) loading the rhombic shape is increased with the ratio of fibre to matrix stiffnesses. This is especially pronounced for the anisotropic carbon and Kevlar fibres whose transverse stiffness is low and comparable to that of the matrix. Glass fibre, in contrast, has too low a stiffness and too high a thermal expansion in the fibre direction. The previous statements can be expressed in a more general way by using the orthotropic properties of the UD laminates: the rhombical shape is increased by large ratios of E|| / E⊥ and α⊥ /α|| . The rhombically deformed pieces, e.g. by temperature variations, can be restored to a rectangular shape by shear forces (or corresponding tensile and compressive forces). It can be shown that for ω at about 30° those forces cause an elongation in the x direction. This special case means that a negative thermal expansion exists for carbon and Kevlar fibres in the range of ω ≈ 30°. (c) Step 3: formation of mono-aa ngleplies The external forces, which restore the rectangular shape are opposite for +ω and –ω pieces. When glueing together both pieces, the external forces can be removed and internal forces between the ±ω laminates are established. The internal shear forces are the reason for the strong negative thermal expansion coefficient and other unusual properties which will be discussed later.

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Fibre arrangement and properties

1017

The formation of angleplies obeys two basic principles which can be used for the analytical treatment (Rosen 1965): • •

equilibrium of forces (internal forces between laminates) compatibility of all strains of layers, which are involved

This also means that shear deformations are concealed (shear angles occur for single layers but not for ±ω composites: γx y . c o m p = 0 for all layers involved ). Usually an arrangement is chosen which is also symmetric to the midplane of the sheets. Otherwise, bending moments occur by thermal or mechanical loading. F2.0.6.1 Typical results from laminate analysis The analysis, sketched in the previous section, can only be performed by computer programs. Some typical results on elastic and thermal expansion properties are shown in figures F2.0.6 and F2.0.7. They are intended to demonstrate the strong influence of fibre orientation on the properties. The properties considered are defined for the midline between +ω and –ω laminates ( x coordinate ) For the Young’s modulus the values in the y direction are added (Engineered Handbook Composites 1987).

Figure F2.0.6. Young’s moduli Ex and Ey shear modulus G x y and Poisson’s ratio V x y of high-tensile carbon fibre mono-angleplies as a function of the fibre orientation ±ω.

The curves of figure F2.0.6 are due to composites with high-tensile carbon fibres, but the angular dependence is similar for other fibres. Moduli with E-glass fibres are much lower; Kevlar and ceramic are intermediate. Young’s modulus Ex decreases strongly with ω (≈cos4 w) and approaches the transverse modulus Ex at an angle well below 90°. The in-plane shear modulus Gx y is always highest at ±ω = 45°. The very high Poisson’s ratio of up to vx y ≈ 1.5 at ±ω ≈ 30° for carbon is surprising. (The ‘rule’ v ≤ 0.5 is only valid for isotropic materials.) The Poisson’s ratio is defined by

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Properties of fibre composites

Stress analysis reveals that at ±30° a strain εx , leads to a very large negative strain εy . Even more astonishing is the fact that at ±30° the Poisson’s ratio in the thickness direction z of carbon or Kevlar fibre composites is negative

This means when straining in the x direction that the thickness increases. The reason is easy to understand: the volume must remain constant when deformed. Since vx y is extremely large at ±30° the in-plane area is extremely reduced. Constancy of volume is only possible by increasing thickness (for 3D-fibre reinforcement the situation is more balanced and the thickness increase is constrained). At ω ≈ 90° Poisson’s ratio is very small since strain in the x direction leads to a very small transverse strain because this is the fibre direction. The unusual properties at ±30° are especially pronounced for anisotropic fibres such as carbon and Kevlar. They are very stiff in the fibre direction, but weak in the transverse direction. Ceramic fibres are stiff but not anisotropic. They are also stiff in a direction transverse to the fibre direction. Glass fibre shows less unusual behaviour since it is relatively weak and isotropic. This behaviour is especially pronounced for the thermal expansion. In figure F2.0.7 the coefficient of thermal expansion is plotted versus the fibre angle.

Figure F2.0.7. Coefficient of thermal expansion at 293 K versus fibre angle for mono-angleplies.

It can be seen that a remarkable negative thermal expansion exists for ±ω ≈ 30° plies with carbon and Kevlar. Glass fibre expansion, in contrast, is always positive; a small shoulder exists in the range of ±30° arising from internal shear forces. They are, however, small because of the small longitudinal fibre stiffness and the too large transverse stiffness. Furthermore, a negative thermal expansion only exists if the fibre expansion is small enough to be dominated by the effect of internal shear forces which give a negative expansion component. This is true for carbon and Kevlar, but not for glass fibre and ceramic. The magnitude of the negative thermal expansion coefficient is especially high for anisotropic fibres when the ratio E |f| / E ⊥f of longitudinal and transverse fibre stiffnesses is large. In the range of ±ω = 45°, zero expansion exists for carbon and Kevlar. As already mentioned in section F2.0.2, applications exist which require materials with zero expansion. For many applications crossplies are used. Their elastic moduli can easily be calculated from the longitudinal and transverse stiffnesses of UD laminates. Also the in-plane shear modulus Gx y can be determined from the UD shear modulus. It can be shown that the shear moduli are nearly equal in 0° and

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Static mechanical properties

1019

90° directions. Assuming the same laminate thickness for both directions, one obtains for crossplies

where G12 is the orthotropic shear modulus measured for UD composites. Poisson’s ratio for the crossplies can be calculated from the moduli in the 0° and 90° directions and from the orthotropic Poisson’s ratio of a UD laminate

More complicated fibre arrangements are described in references Chawla (1987), Hancox (1975) and Rosen (1965).

F2.0.6.2 Limitations of the lamination theory Many assumptions have to be made when applying lamination theory: no debonding of fibres, no delaminations, no formation of microcracks in the matrix, no misalignment or inhomogeneities of the fibre arrangement. Calculation of elasticity properties by lamination theory is restricted to low loads. Also, thermal stresses are assumed to be low when calculating thermal properties (e.g. expansion). This is not true for calculating fracture properties. High loads might lead to debonding, delamination and to the formation of microcracks. Stress concentrations may occur at microcracks, voids or inhomogeneities (the situation is similar to dielectric properties: the permittivity at low voltage can be calculated by lamination theory, but not the breakthrough voltage). Even if the strength of each UD laminate involved is known separately, it is not sufficient for calculating fracture properties of angleply or crossply composites. There might be an unexpected interaction between laminates, e.g. because of crack propagation between laminates. The stacking sequence of laminates influences fracture properties. There is another complication which has to be incorporated in any theory dealing with cryogenic fracture properties. When cooling, thermal mismatch leads to thermal stresses between fibre and matrix and between differently plied laminates. The thermal stresses might be high, especially for the matrix. The fracture strain of most polymers is 2 to 3.5% at 4 K. Nearly 1% is wasted by cooling to 4 K in composites with carbon or Kevlar fibres. The successful application of semi-empirical fracture theories (Tsai-Wu, Halpin-Tsai) (Hancox 1975, Rosen 1965) is bound to the knowledge of all fracture processes involved. The prediction of fatigue behaviour is even worse. Polymer matrices with a good fatigue behaviour may show poor behaviour when used in a composite and vice versa. The fatigue life even of fibre-dominated UD composites might be controlled by the matrix type.

F2.0.7 Static mechanical properties F2.0.7.1 Survey It is very difficult to extract reliable fracture data on composites, especially at cryogenic temperatures. The components and the processing methods are not well defined, and reliable measurements are difficult. Properties of UD laminates are the basis for calculating properties of crossplies or angleplies. A survey of typical data (index 1 or || in the fibre direction; index 2 or ⊥ perpendicular) of different fibre composites

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Properties of fibre composites

at 77 K is given in table F2.0.6. The values are only meant to be used as a guideline. The shear modulus G12 , shear strength τ12 and shear angle γ12 are the in-plane values (see figure F2.0.4 for the definition of coordinates). Table F2.0.6. Typical mechanical properties of UD composites at 77 K. Fibre volume ~60 vol.%, matrix: epoxies or PEEK.

Another survey is given in table F2.0.7 for the most commonly used crossplies. For crossplies various types of fibre arrangement are applied. (i)

(0°, 90°) lay-up laminates. Under tensile load there might be a destructive interaction between the 90° and the load-bearing 0° laminates which reduces strength. Examples are shown in figure F2.0.8. (ii) Woven cloth (fabric). The waviness of the fibre causes additional bending which reduces tensile strength. Several examples on G10 or G11 and on ceramic cloth composites will be given.

Table F2.0.7. Typical mechanical in-plane properties on crosspiles at 77 K. Fibre volume ~60 vol.%, matrix: epoxies.

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Static mechanical properties

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(a) Temperature dependences Matrix-dominated properties, such as transverse or shear properties, depend on the temperature and the values of fibre-dominated compressive properties increase at low temperatures since the matrix becomes stiffer and this inhibits micro-buckling. Some data of glass/epoxy are given in table F2.0.8.

Table F2.0.8. Temperature dependences of matrix-dominated properties of glass/epoxy (Walsh et al 1995).

For glass and also Kevlar fibre-dominated properties depend on the temperature (Reed and Golda 1994). Mechanical strength of glass fibre, and to some extent the stiffness and compressive strength of Kevlar fibres, increases at low temperatures for reasons mentioned in section F2.0.3. For Al2O3–PEEK composites a temperature dependence of fibre-dominated mechanical properties has been observed (Kritz and McColskey 1990).

Table F2.0.9. Temperature dependencies of fibre dominated mechanical properties of UD composites (Kasen 1981).

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Properties of fibre composites

Tensile properties of carbon UD composites increase only by about 10% between room temperatul and 4.2 K ( Hartwig and Ahlborn 1990). The temperature dependences of glass crossplies (cloth) al summarized in table F2.0.10. Table F2.0.10. Temperature dependencies of mechanical properties of glass fibre crosspiles (G-10 CR) (Kasen et al 1980).

F2.0.7.2 Static mechanical properties of carbon at 77 K (a) Unidirectional composites Some data on orthotropic properties (subscript || or 1 and ⊥ or 2) are given in table F2.0.11. For a definition of Poisson’s ratio see equations (F2.0.5) and (F2.0.8). The first subscript symbolize the stress direction. The static properties in the fibre direction are nearly unaffected by the matrix type (Hartwig and Pannkoke 1992, Pannkoke and Wagner 1991). However, the transverse fracture properties, are higher wit a ductile PEEK matrix than with a brittle epoxy matrix (see table F2.0.12). The transverse bending strength and the interlaminar shear strength τI L S are also much higher with PEEK matrix, as seen from table F2.0.13.

Table F2.0.11. UD carbon properties at 77 K.

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Static mechanical properties

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Table F2.0.12. Transverse mechanical properties of UD carbon at 77 K.

Table F2.0.13. Bending and shear properties of UD carbon at 77 K σLB: longitudinal bending strength; σTB: transverse bending strength.

(b) Carbon crossplies (0°° , 90°° lay up) at 77 K The moduli of crossplies can easily be calculated from the orthotropic UD properties. It is assumed that thicknesses d0 and d90 and their fibre volumes are equal

The tensile strength, however, is more complex since interaction between the 90° and the load-bearing 0° layers might occur. Carbon fibres are brittle in the transverse direction. Transverse cracks from the 90° layers might cut load bearing 0° layers. Glass fibre is less brittle and not sensitive to transverse cracks. Under tensile loading in the 0° direction cracks are induced in the weak 90° layers which propagate to the load-bearing 0° layers. Three types of interaction have been observed (see figure F2.0.8). In the first case tensile strength is reduced by up to 50%. In the second case strength is reduced less but delamination is a serious failure. For the third case the PEEK matrix has an excellent bond. It crystallizes onto the fibre surface and protects the fibres. Transverse cracks are bent before reaching the 0° layers. It holds that

where σ1 and σ2 are the orthotropic strengths. Since σ2 ≈ 0.1σ1

Some results are summarized in table F2.0.14.

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Properties of fibre composites

Figure F2.0.8. Different failure processes in carbon crossplies under tensile loading (Hübne and Hatwig 1996)

Table F2.0.14. Tensile properties of crosspiles with different matrices at 77 K.

(c) Shear loading of carbon crossplies Shear loading of crossplies leads to no interaction of layers. Typical shear properties of carbon crossplies at 77 K are

It can be shown that shear properties of (0°, 90°) crossplies and 0° and 90° UD composites are the same. (d) Tensile loading of carbon ± 45 ° plies Tensile stress σx induces both stress and shear components; the latter are the critical ones (see figure F2.0.4 for the definition of the coordinates)

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Survey on field calculations

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The low orthotropic shear strength τ12 of a UD laminate limits the tensile strength of a ±45° crossply. The stress situation is matrix dominated. As seen from table F2.0.15 the values, especially the fracture strain εx , with a ductile PEEK matrix are higher than with epoxy.

Table F2.0.15. Tensile properties of ±45° plies with different matrices at 44K.

Tension of ±45° plies allows the determination of the shear modulus

(b) Shear loading of carbon ± 45 ° plies Shear loading of ±45° plies stresses laminates in both tension or compression. Since strength is very high in the fibre direction a high shear strength is expected. Stress analysis (Fujczak 1978, Puck 1967) reveals that other stress components establish the limitation, and roughly a quarter of the fracture stress σ1 results for τx y . The values, given in table F2.0.16 have been measured on thin tubes under torsion. This property is fibre dominated. The state of stress is shown in figure F2.0.9. Care should be taken about the stacking of laminates. The outer layers should not be loaded in compression (delamination).

Table F2.0.16. Shear properties of ±45° plies (tubes) at 77 K. (γxy is the shear angle in miliradiants.)

Figure F2.0.9. State of stress in a ±45° composite under shear loading.

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Properties of fibre composites

F2.0.7.3 Ceramic composites As already shown in table F2.0.9 the static properties of alumina composites show a temperatur dependence and, as shown in table F2.0.17, some dependence on the matrix type. Actually the fibre-matri: bond is of different quality for the matrices applied. Scanning electron microscopy (SEM) fractrograph reveal an excellent bond for PEEK or EP and a weaker bond for PESU. The data of table F2.0.17 can be used as a guideline. All values are extrapolated to the same fibre volume of 60%. Transverse tensile and compression properties of UD Al2O3-PEEK, and shear properties extractec from tensile loading of ±45° plies are summarized in table F2.0.18. Tensile properties of Al2O3 crossplies (0°/90° cloth) are compiled in table F2.0.7.19. Comparison of tables F2.0.17 and F2.0.19 reveals that their strength (not the moduli) is much lower than expected from the volume of the load-bearing layers. This is probably caused by the waviness of the cloth.

Table F2.0.17. Static tensile properties of alumina UD composites with 60% fibre volume at 77 K.

Table F2.0.18. Transverse UD properties and shear properties of ±45° plies of A12O3-PEEK (43% fibre volume) (Kritz and McCloskey 1990).

Table F2.0.19. Tensile properties of ceramic crosspiles (0° /90° cloth) at 77 K.

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Static mechanical properties

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F2.0.7.4 Interlaminar shear strength The bond between fibre and matrix or between laminates is given by the interlaminar shear strength τI L S The value of τI L S is controlled by the weakest link of the following parameters: • • •

fibre shearing interfacial bond matrix shear strength.

Several methods are available for their measurement (Becker 1990, Evans et al 1990). The best, but most expensive, is torsion of thin 90° tubes. The most common one is the short beam shear test. At a low span length L to thickness t ratio the interlaminar shear dominates at bending (L/t < 5). The disadvantage is the stress peak at the edges. This is also true for all lap-shear tests. For short beam shear tests the value of τI L S can be determined by the sample width w the thickness t and the bending force P to failure

The value of τI L S increases with decreasing temperatures due to increased shear strength of the polymer matrix. For a very high bond strength and no shear failure of the fibres, the upper limit of τI L S is given by the matrix. At 4.2 K values of up to 200 MPa have been found for UD carbon fibre with a PEEK matrix. The interfacial bond with PEEK is especially high since it crystallizes onto the fibre surface. It should be mentioned that an excellent interfacial bond is not necessarily an advantage. For static strength it is, but not for the fatigue life. A medium τI L S , e.g. with epoxies, helps to reduce stress concentrations. A compilation of τI L S for several composites is given in table F2.0.20. The values were obtained from short beam tests. For some results a reasonable agreement with torsion measurements on tubes has been observed (Hartwig and Ahlborn 1990, Pannkoke and Wagner 1991).

Table F2.0.20. Interlaminar shear strength and failure processes.

As already mentioned the value of τI L S can strongly be enhanced by a small number of throughthickness fibres (Okada and Nishijima 1990). Some results on G11ICR and three UD glass composites measured by the notched shear test (Kasen 1990) are shown in figure F2.0.10. The temperature dependences of τI L S are plotted in figure F2.0.11 for several composites.

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Properties of fibre composites

Figure F2.0.10. Interlaminar shear strength of glass composites: A—G11CR, glass cloth/EP; B—UD-glass/EP (52 vol.%), (CTD/100 epoxy); C—UD-glass/EP (52 vol.%), (CTD/112 epoxy); D—UD-glass/PI (52 vol.%).

Figure F2.0.11. Interlaminar shear strength versus temperature for several UD composites (Ahlborn 1991, Hartwig and Ahlborn 1990).

An interesting correlation between interlaminar shear strength τI L S and bending strength σI B has been observed by Ahlborn. At each temperature (room temperature, 77 K and 4.2 K) there is a linear dependence between both properties (see figure F2.0.12) (Hartwig and Ahlborn 1990). At least for UD carbon it holds at each temperature

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Static mechanical properties

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Figure F2.0.12. Correlation between interlaminar shear strength τI L S and bending strength σB e n d of UD carbon.

F2.0.7.5 Multi-axial loading Stress analysis by the Tsai—Hill criterion can be applied for multi-axial states of stresses. Combination of tensile stress σt , compressive stress σc and shear stress τ can be described by the relation

The stresses are normalized to their ultimate values (superscript *). Investigations of multi-axial loading concentrate mainly on glass cloth. In particular, the combination of in-plane shear and compression in the thickness direction is important for many applications in

Figure F2.0.13. Failure envelopes of G10 and 3D composites (Zi-003) at 77 K (liquid nitrogen temperature—LNT) and room temperature (RT) (Okada and Nishijima 1990).

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Properties of fibre composites

superconducting magnet technology (Fabian et al 1995, Okada and Nishijima 1990). Considering only compression and shear, both properties obey an elliptical relation which constitutes failure envelopes. Typical results are plotted in figure F2.0.13. As already mentioned, both in-plane shear and throughthickness compressive strength can be improved by a small amount (≈8%) through-thickness fibres (3D fibre arrangement). Results on 3D crossplies are included in figure F2.0.13. It is evident that 3D crossplies are superior to two-dimensional (2D) crossplies, such as G10CR, especially at room temperature. At 77 K the matrix is stiffer and increases both properties. F2.0.8 Fatigue behaviour The degradation by fatigue cycling depends on the following parameters: • • • •

fibre type and arrangement matrix type interfacial bond mode of loading: —threshold tension (R ≈ 0.1) —tension—compression (R = –1) —shear or torsion —bending.

For characterization of cyclic loading the ratio is defined

The fatigue strength σD is defined as the highest stress amplitude withstanding N → ∞ of load cycles. In practice N ≈ 107 load cycles are sufficient for characterizing this quantity. The fatigue behaviour is usually presented in so-called S—N diagrams, where the upper stress is plotted versus the number N of load cycles to failure. For UD composites more information, especially on the influence of the matrix, can be obtained when plotting the strain life instead of the stress life (see figure F2.0.14(b)). The conversion to strain diagrams is meaningful since both the fibres and the matrix are loaded in parallel for UD composites and their moduli remain nearly unchanged during fatigue cycling. The general results are as follows. (i)

(ii) (iii) (iv) (v) (vi) (vii)

The frequency of fatigue cycling at cryogenic temperatures can be chosen to be much higher than at room temperature without remarkable heating of specimens. (The mechanical loss factor tan δ is lower by a factor of ≈10 at 4 K, and about 5 at 77 K compared with room temperature; in addition the heat diffusion is higher at low temperatures; see figure F2.0.47 later.) Fatigue strength even of fibre-dominated properties is strongly influenced by the matrix type. Alternating loading leads to a shorter fatigue life than for threshold cycling. Young’s modulus of UD composites is nearly unchanged by fatigue cycling. This is not true for crossplies. Multi-axial loading (e.g. tension—torsion) reduces moduli and fatigue life drastically. The fatigue behaviour of the pure matrix is usually not reflected in the fatigue behaviour of a composite. Under similar conditions, tensile fatigue strength of carbon is highest, followed by Kevlar and glass. Ceramic composites exhibit the lowest fatigue strength. For UD carbon composites a ratio σD/σUT up to 80% can be achieved (see table F2.0.21).

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Fatigue behavior

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F2.0.8.1 Carbon fibre composites Most fatigue data are concentrated on tensile threshold cycling (R = 0.1). Data on tension-compression fatigue (R = –1) are rare. In figure F2.0.14 fatigue data with R = –1 and R = 0.1 are compared. As already mentioned, tension-compression fatigue leads to much lower fatigue strength than for R = 0.1. Extrapolations are made by the Weibull parameters (Hartwig and Ahlborn 1990). A tensile loading frequency up to 100 Hz has been applied for specimens of about 1 mm thickness and a liquid N2 environment. The heating in the midplane was less than 10 K (Ahlborn 1988, 1991).

Figure F2.0.14. Comparison of the fatigue behaviour of UD carbon composites with R = –1 and R = 0.1.

(a) Tensile threshold cycling on UD-cc omposites in the fibre direction The influence of the fibre type on the fatigue behaviour can be seen from figure F2.0.15(a) by stress life curves. Composites with three fibre types, but the same epoxy matrix are considered. It can be seen that composites with high-strength fibres (AS4) degrade more strongly than those with low-strength fibres (T300). A similar behaviour has been found for composites with the same fibres, but a different quality of manufacturing. Poor quality composites exhibit lower static values, but the fatigue strength after 107 load cycles is similar to that of a high-quality composite. Obviously only a certain level of damage is accumulated by fatigue cycling. In figure F2.0.15(b) strain life curves are plotted for the same composites (Pannkoke 1994). It can be seen that only the static strain εU T depends on the fibre type but not the fatigue strain limit. It approaches the same limit irrespective of the fibre type. The influence of the matrix material is obvious from figure F2.0.16. The composites are made up of the same fibre type but the matrices are different and so are the values of the fatigue strain limit. For composites with brittle epoxy matrices (especially a modified epoxy ICI 977-2) the strain life is much higher than that with ductile thermoplastic matrices (e.g. PEEK, PC). The strain life is matrix dominated and hardly influenced by the fibre type. A schematic representation in figure F2.0.17 combines both the results of figures F2.0.14(b) and F2.0.16. The static ultimate strain of UD composites is fibre dominated and hardly influenced by the matrix type, and vice versa for the fatigue strain limit. The question arises about the influence of the matrix on the fatigue behaviour. Some explanation can be found by analysing the failure processes in the matrix of a composite. The fatigue behaviour depends on the capability of shear transfer between the fibres by the matrix, especially in the vicinity of broken fibres. SEM analysis reveals an accumulation of many transverse matrix cracks during fatigue cycling in

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Figure F2.0.15. Wöhler curves of UD carbon fibre composites with different fibres but the same (epoxy) matrix at 77 K and R = 0.1: (a) stress life diagrams; (b) strain life diagrams.

Figure F2.0.16. Strain life diagrams of carbon composites with the same fibre but different matrices at 77 K.

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Figure F2.0.17. A schematic presentation of strain life diagrams.

brittle epoxy matrices. Since many microcracks are created, each has a small stress concentration only and is not dangerous. A single transverse crack would have a high stress concentration, which might lead to the destruction of neighbouring fibres. Despite the many microcracks, the shear transfer seems to remain strong enough. In contrast to this, no transverse cracks are induced in ductile thermoplastic matrices, but longitudinal cracks which might reduce shear transfer appreciably are induced. In addition, longitudinal cracks might be generated in the fibre—matrix interface if the bond is weak, (e.g. for PC matrix). For PEEK, however, the fibre—matrix bond is very strong and longitudinal cracks run through the matrix (Hartwig and Pannkoke 1992, Pannkoke 1994).

Table F2.0.21. Ultimate tensile strength σUT|| and fatigue strength σD at 77 K (Hartwig and Pannkoke 1992).

(b) Shear threshold cycling (torsion) on UD carbon fibre (tubes with 90° fibre arrangement) The importance of shear transfer of the matrix between the fibres can be tested more directly by shear loading. Torsion of circumferentially wound tubes is a method of testing the shear transfer between fibres and measuring the interlaminar shear strength (ILSS). Its shear behaviour reflects the part played by the matrix when UD composites are loaded in tension. As expected, the tubes made with a brittle epoxy matrix exhibit a higher fatigue strength than those with a ductile thermoplastic matrix. Thi illustrates directly the different degradations of shear transfer due to different matrix materials. The results are plotted in figure F2.0.18(a) for composites with different matrices but similar fibre types. The static and fatigue shear strength is controlled mainly by the matrix shear strength and the interface. Both are excellent for PEEK.

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The degradation by fatigue of PEEK-based composites, however, is much stronger than for that with an epoxy matrix. The relative amount of degradation is demonstrated in figure F2.0.1 8(b), where the stress values are related to the static strength. The fatigue strength is noticeably lower for composites with a PEEK matrix (the same tendency has been observed for tensile cycling of UD composites), which indicates that shear transfer of the matrix between fibres is an important parameter for the fatigue behaviour. It should be mentioned that pure PEEK shows a higher shear fatigue strength than EP. The composites with those matrices, however, show the reversed behaviour (Hartwig and HUbner 1995).

Figure F2.0.18. (a) Shear fatigue of UD carbon tubes with different matrices and similar fibres at 77 K. (b) Shear fatigue of UD carbon at 77 K normalized to the static shear strength.

A schematic presentation of the shear fatigue behaviour of UD composites is given in figure F2.0.19. The Young’s modulus of UD composites stays nearly unchanged during fatigue cycling. (c) Tensile threshold fatigue on carbon fibre crossplies (0°° , 90°° lay up) The mechanical properties of crossplies differ from those of UD composites. As already mentioned in many cases the tensile strength of crossplies cannot be calculated from the strength of UD composites measured parallel and vertical to the fibre direction. The cause is a destructive interaction between the 0° and 90° layers (see figure F2.0.8). When load is applied in the 0° direction, microcracks are induced in the 90° layers which might destroy the 0° load-bearing layers (Hartwig and Hübner 1995, Hübner and Hartwig 1996). This is true for both static and fatigue loading. For crossplies with a ductile thermoplastic matrix (e.g. PEEK ) this effect is less pronounced and does not cause a noticeable reduction of static strength. It is true that microcracks are also induced in the 90° layers of a PEEK matrix, but they are bent before reaching

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Fatigue behaviour

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Figure F2.0.19. A schematic presentation of the shear fatigue behaviour of 90° carbon composites.

load-bearing 0° layers. Thus, static strength is higher for composites with a PEEK matrix than for those with epoxy, but the reverse situation is true for the fatigue strength. A PEEK-based composite degrades more strongly and the fatigue strength is almost independent of the matrix materials. This is shown in figure F2.0.20 by the S—N curves for crossplies with epoxy, PEEK and PC matrices. The crossply with a PC-matrix is an example of a weak fibre-matrix bond.

Figure F2.0.20. S—N curves of carbon fibre crossplies under tensile threshold fatigue loading at 77 K (Hübner and Hartwig 1996).

(d) Shear threshold fatigue on carbon fibre cross plies (0°° , 90°° ) The shear behaviour of crossplies is dominated by the matrix. In figure F2.0.21 shear fatigue curves of EP-based crossplies are shown. The static and fatigue strengths of the crossply are equal to those of the 0° and 90° layers, since shear loading is the same. The shear modulus Gx y degrades during shear fatigue cycling which is directly correlated to the increased formation of microcracks. The results are shown in figure F2.0.22 for crossplies with two different epoxy matrices (Hübner and Hartwig 1996).

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Figure F2.0.21. Shear fatigue on (0°, 90°) carbon crossplies at 77 K (Hübner and Hartwig 1996).

Figure F2.0.22. Relative degradation of the shear modulus during shear fatigue cycling at 77 K (maximum shear stress t = 100 MPa).

(e) Tensile threshold fatigue on carbon fibre ± 45°° composites The tensile properties of ±45° composites are determined mainly by the matrix and its bond strength to the fibres. The fatigue behaviour for PEEK and epoxy-based angleplies are shown in figure F2.0.23. F2.0.8.2 Glass fibre composites Investigations on the fatigue behaviour of glass concentrates on crossplies (e.g. G10CR, G11CR with woven laminates), on multi-axial loading and, in particular, on the combination of the in-plane shear and compression in the thickness direction are of importance in low-temperature technology (Nishijima et al 1995, Reed et al 1995). Stress analysis by the Tsai-Hill criterion is given in equation (F2.0.16) (Schutz and Fabian 1996). The Tsai-Hill criterion describes combined loading. Considering only compression and shear, both properties are correlated by elliptical failure envelopes. Typical results are plotted in figure F2.0.24 as well for the static case and after 105 or 106 load cycles (Reed et al 1995). The material investigated is an S-glass cloth crossply with an epoxy matrix. Compression is applied in the thickness direction.

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Figure F2.0.23. S-N curves for ±45° carbon fibre composites at tensile threshold fatigue loading at 77 K (Hübner and Hartwig 1995).

Figure F2.0.24. Static strengths and fatigue strengths of fiexibilized DGEBA epoxy/S-2 glass at 76 K under combined shear/compressive loading at 105 and 106 cycles to failure.

(a) Degradation of moduli of glass crossplies Biaxial cycling reduces strongly the Young’s modulus E and the shear modulus G (Wang and Khim 1981). Investigations have been performed on G10 glass cloth/epoxy under combined in-plane shear ∆τ and in-plane tension ∆σ. The measurements were done on tubes under tension-torsion fatigue. Typical results on the degradations of E and G are given in figures F2.0.25 and F2.0.26 (Wang and Khim 1981).

F2.0.8.3 Ceramic fibre composites (a) Tensile threshold fatigue on ceramic fibre UD composites As already mentioned, ceramic fibre composites exhibit a poor fatigue behaviour even with an epoxy matrix. Stress and strain life diagrams are plotted in figures F2.0.27(a) and F2.0.27(b) respectively. The ratio of σD /σU T is of the order of only 30% for tensile threshold cycling.

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Properties of fibre composites

Figure F2.0.25. Degradation of glass crossplies by different cyclic axial stresses ∆σ on tensile modulus E (fill) (a) and on shear modulus G (b) in biaxial fatigue at 77 K. The shear stress amplitude ∆τ was kept constant.

Figure F2.0.26. Degradation of glass crossplies by different cyclic torsional stresses ∆τ on tensile modulus E (fill) (a) and on shear modulus G (b) in biaxial fatigue at 77 K. The tensile stress amplitude ∆σ was kept constant.

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Fatigue behaviour

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Figure F2.0.27. (a) Stress life diagrams of UD ceramic fibre composites under tensile threshold fatigue loading at 77 K. (PESU: polyethersulphone). (b) Strain life diagrams of UD ceramic fibre composites under tensile threshold fatigue loading at 77 K.

(b) Tensile threshold cycling on ceramic fibre-ee poxy crossplies (fabric) Stress and strain life diagrams are plotted in figures F2.0.28 for ceramic fibre crossplies. When comparing both figures, the crossplies again exhibit a lower static and fatigue strength than expected from the number of load-bearing 0° layers. The reason is the waviness of the cloth. The relative fatigue ratio σD /σU T is about 30%.

(c) Degradation of moduli of ceramic composites By comparing the stress and strain life diagrams in figures F2.0.28(a) and F2.0.28(b), respectively, it can be seen that Young’s modulus of UD ceramic composites does not degrade by fatigue cycling. For crossplies the degradation is about 20%.

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Figure F2.0.28. (a) Stress life diagrams of ceramic fibre crossplies under tensile threshold fatigue loading at 77 K. (b) Strain life diagrams of ceramic fibre crossplies under tensile threshold fatigue loading at 77 K.

F2.0.9 Thermal expansion The integral thermal expansion is defined by the coefficient of thermal expansion α and the temperature difference relative to room temperature

For most materials α is positive and ∆L/L becomes negative on cooling. In figures F2.0.29 the integral thermal expansion ∆L/L and the expansion coefficient α are shown for several composites. The directly measured quantity is ∆L/L versus T ; the coefficient α is derived from these. Thermal expansion behaviour of fibre composites can be varied within a large range by the type and arrangement of fibres. Even for a single fibre type a great variability can be achieved by different fibre

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Thermal expansion

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Figure F2.0.29. Integral thermal expansion ∆L/L and α versus T; (a) for UD glass and (b) ∆L/L for UD ceramic and glass (Hartwig 1988, Okada el al 1991).

orientations. This is shown in figures F2.0.30 and F2.0.31 for the most common mono-angleply fibre composites. ∆L/Lx and αx depend strongly on the fibre angles ±ω ). The coefficient αx is due to room temperature and ∆L/Lx is the relative contraction when cooling from 293 K to 77 K (cooling to 4.2 K adds 10% to 15%). Glass fibre and ceramic exhibit only positive values of α , and ∆L/L is negative on cooling. For carbon and Kevlar negative values of a exist below ω ≈ ±45° and ∆L/Lx becomes positive on cooling. This is an unusual property. The small values at ω = 0° arise from the fibres (intrinsic negative expansion in the fibre direction). The larger negative values arise from thermally induced shear forces between +ω and -ω laminates. A maximum of negative thermal expansion exists roughly at ±ω = 30°. The principle has been explained in figures F2.0.6 and F2.0.7. This behaviour is specific to laminated fibre composites. The relatively large negative expansion coefficient at ±ω ≈ 30° is about half that of steel. It cannot be explained by the principles of solid-state physics; it is a result of the lamination theory. The magnitude of the negative expansion coefficient αx (e.g. at ±ω = 30° ) is especially large for anisotropic fibres with a large ratio of parallel to transverse stiffnesses E|| /E⊥. This is pronounced for Kevlar. The composite materials with a negative expansion can be used for compensating undesired positive expansion of other materials. An example will be given at the end of this chapter. At about ±ω ≈ 45° thermal expansion is zero in the x and y directions. For several applications a nonexpanding material is desired, especially in constructions which are exposed to large temperature variations. At angles ω > 45° the values of ∆L/L and α increase strongly. The large thermal expansion of the

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Properties of fibre composites

Figure F2.030. Coefficient of thermal expansion αx at room temperature (a) and ∆L/Lx (b) from 293 K to 77 K versus fibre angle ±ω . These values apply only to one direction (x-coordinate).

polymer matrix dominates the expansion behaviour. For carbon and Kevlar, however, the large transverse expansion of fibres contributes. As already mentioned, carbon and Kevlar fibres are anisotropic. They show a small negative thermal expansion coefficient α in the fibre direction, but a very large positive one in the transverse direction. The transverse expansion of Kevlar is larger than that of many isotropic polymers. A different situation is true for glass fibre and ceramic. They exhibit a small positive and isotropic coefficient α. Therefore, with the same matrix they have a smaller transverse expansion than carbon and Kevlar. In the fibre direction composites with glass or ceramic exhibit a small but positive expansion which cannot be overcompensated by the internal shear forces responsible for the negative expansion. The transverse expansion (ω = 90°) for UD composites, and in the thickness direction of crossplies is influenced by the matrix expansion. PEEK, most epoxies or PA12 are low expansion polymers: ∆L/L (293 K → 4 K) ≈ -0.8% to -1.1%; for PC: ∆L/L ≈ -1.5%. The transverse expansions of UD composites are: • • • •

for E-glass crossplies: DL/L ≈ -0.4% to -0.6% for A12O3 crossplies: DL/L ≈ -0.7% to -0.8% for carbon crossplies: DL/L ≈ -0.6% to -0.8% for Kevlar crossplies: DL/L ≈ -1.1% to -1.2%

The higher values apply to PC-based composites; the lower ones to EP- or PEEK-based composites. A special case of angleplies are crossplies (0°, 90°). Some data of in-plane and through-thickness expansions are given in figure F2.0.32. The expansion behaviour in the thickness direction is nearly that of UD composites in the 90° direction. The thermal expansion of crossplies is not uniform in-plane. The integral expansion at different angles of measurement are shown in figure F2.0.33.

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Thermal expansion

Figure F2.031. Thermal expansion: (b) carbon/PEEK (60 vol.%).

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∆L/L 0 versus T of mono-angleplies:

(a) E-glass/EP (70 vol.%);

Figure F2.0.32. Integral thermal expansions ∆L/L in-plane and in the thickness direction for crossplies: (a) G10CR and G11CR; (b) carbon (Hartwig el al 1991, Hartwig and Schwarz 1990, Nakahara et al 1996).

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Properties of fibre composites Table F2.0.22 Integral themal expansion ∆L/L (293 K → 4.2 K) of UD and 0/90 composities*.

Figure F2.0.33. Integral thermal expansion of crossplies in different measuring directions.

A nearly uniform expansion in-plane can be achieved by applying more fibre angles: ±ω1 , ±ω2 It is, for example, possible to construct a composite with zero expansion in every direction of the plate. This can be done by arranging UD carbon laminates every 30°. (a) Parameters that influence thermal expansion Thermal cycling influences the thermal expansion of planar carbon fibre composites (Hartwig and Hübner 1995). For filament wound tubes this effect is less pronounced. Mechanical preloading of planar carbon fibre angleplies (CRP) specimens leads to an irreversible deformation of the fibre angle ω and a change of thermal expansion. The effect, however, is small for ±30° composites, but relatively large for ±45° (see figure F2.0.30). When tubes are used, no creep or deformation is found (Hartwig and Hübner 1995). Geometry also influences the thermal expansion of composites. It is different for plates and tubes. For tubes there is an additional coupling of radial and azimuthal components, which leads to a lower value of |α|. The dependence on planar and on cylindrical geometries of carbon composites is shown in figure F2.0.34 for a plate, a tube and half a tube (Hartwig 1995). For tubes the (negative) coefficient of thermal expansion is smaller than for plates. A half cut tube is more similar to a plate. Support elements expand strongly on cooling. There are various conditions where different thermal expansions of components lead to undesired features on cooling. In figure F2.0.35 an example is shown for an inductive energy storage ring. The radial forces of the superconducting coil shrinks on cooling and the support between the coil and the warm environment becomes loosened if the support elements do not have a negative thermal expansion.

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Figure F2.0.34. Thermal expansion of ±30° carbon angleplies of different shapes: (a) integral thermal expansion; (b) coefficient α.

Figure F2.0.35. Support elements of an inductive energy storage ring with a superconducting (SC) coil. RT—room temperature.

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Properties of fibre composites

Another example is the support elements of superconducting-accelerator magnets and the cold tube for the beam line (figure F2.0.36). The outer and the cold, inner tubes should be kept concentric when cooled.

Figure F2.0.36. Superconducting-accelerator magnet with a cold and a warm tube. RT—room temperature.

Principle and systems (Hartwig 1995). Carbon and Kevlar fibres exhibit a negative thermal expansion. For composites with ±30° fibre angles a reasonable negative thermal expansion coefficient has been found; about half that of steel, but of a negative sign. This value, however, is too small for practical applications. A great enhancement can be achieved by a system of serial arrangement in opposite directions of ±30° CRP and a metal with a large thermal expansion. The principle is sketched in figure F2.0.37(a). An aluminium piece with a large positive thermal expansion rests on a CRP piece and carries another CRP component. The technical implementation is by tubes or tube elements. A single system consisting of tubes is shown in figure F2.0.37(b). An even larger negative expansion is achieved with a double system, as shown in figure F2.0.37(c). With single or double systems it is possible to reach thermal expansion coefficients of three or five times that of steel, but of negative sign. The integral thermal expansions of the components and the systems are shown in table F2.0.23.

Table F2.0.23. The integral themal expansions.

The stiffness of a single system is roughly half that of the CRP material, if the aluminium piece is made thick enough. The compressive strength σc o m of those systems is rather high ( ≥ 280 MPa) and they can be used as support elements. (If one sacrifices part of the negative thermal expansion by using, for example, CRP tubes with ±10° fibre angle, then a compressive strength up to 700 MPa is feasible (Hartwig et al 1994).) The fatigue strength is rather high for carbon composites. For ±30° tubes compressive fatigue strength σc o m ≈ 160 MPa at 77 K. The thermal conductivity χ of the system is determined by the CRP tubes. For a single system the value of χ is about half that of a CRP tube, related to the length L of the system. However, the ratio χ/E, which is the decisive parameter for a support element, is nearly constant. (The aluminium piece is taken as a thermal short circuit.) The ratio χ/σc o m is even half that of the CRP tube for a single system and a quarter of that for the double system.

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Thermal conductivity

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Figure F2.0.37. Top, principle; middle, single system; bottom, double system.

F2.0.10 Thermal conductivity Thermal insulation is an important factor when applying fibre composites in low-temperature technology. The main applications are in support elements, struts and straps for superconducting- magnet systems and transportation vessels of liquified gases (Giesy 1996, Hartwig 1995, Hirokawa et al 1991). The general results are as follows. (i) Carbon fibre composites exhibit a very low thermal conductivity c but only at very low temperatures. (ii) Carbon fibre composites exhibit a strong temperature dependence. A drastic decrease already starts below 120 K which is a consequence of the high Debye temperature of graphite. (iii) Fibre arrangement and fibre type do not influence thermal conductivity χ appreciably below 15 K. (iv) UD composites with glass fibre, Kevlar and ceramic fibres show a low thermal conductivity over a large temperature range. The temperature dependence is weak above 15 K. (v) For many applications the ratios χ/E or χ/σ are the decisive parameters. They are lowest over a large temperature range for ceramic fibre composites. A survey is given in figure F2.0.38 for UD composites. (a) Carbon fibre composites A specific feature of carbon fibre composites is their strong temperature dependence below room temperature. Compared with other composites, a strong decrease is already apparent at quite a high temperature. This feature is partly due to the very high Debye temperature Θ of graphite. Thermal conductivity is proportional to the specific heat, the low-temperature dependence of which is given generally

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Properties of fibre composites

Figure F2.038. Thermal conductivity of UD composites in the fibre direction (Hartwig 1994b).

by (T/Θ)3 for crystals and (T/Θ)2 for amorphous solids. A mixture of both is true for carbon fibres. For large values of Θ the drop of specific heat starts at higher temperatures. Graphite is a semiconductor, and it can be assumed that electronic components contribute to thermal conductivity. At lower temperatures they get frozen and thermal conductivity is reduced.

(b) Influence of fibre arrangement and fibre type Thermal conductivity depends on the fibre arrangement and on the fibre type. This feature is especially pronounced for carbon and Kevlar fibre composites since those fibres themselves behave anisotropically. The principle will be explained for carbon fibre composites. When considering UD carbon fibre composites, thermal conductivities in the directions parallel and transverse to the fibre differ considerably at least at high temperatures. As is obvious from figure F2.0.39, this anisotropic behaviour ceases at low temperatures and below 4 K UD carbon fibre composites behave quite isotropically. The reason for this is that at low temperatures only long wavelength phonons are activated and, similar to optics, a spatial resolution is no longer possible. Below 4 K thermal conductivity is rather insensitive to the fibre arrangement. Furthermore, it is almost independent of the fibre type. At room temperature composites with high-modulus carbon fibres, for example, exhibit higher values than those with high-tensile fibres. This is, however, no longer true at low temperatures, where phonon wavelengths are too large for resolving the fibre structure. Thermal conductivities of other UD composites parallel and transverse to the fibre direction are shown in figures F2.0.40 (Takeno et al 1986) Materials in common use are glass cloth crossplies G10CR (E-glass) and G11CR (boron-free glass). Their in-plane and through-thickness conductivities are plotted in figure F2.0.41 (Kasen et al 1980)

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Figure F2.039. Thermal conductivity of carbon fibre composites with different types and arrangements of fibres (Hartwig 1994).

Figure F2.0.40. Thermal conductivity of UD composites: (a) parallel and (b) transverse.

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Figure F2.0.41. The in-plane and through-thickness thermal conductivities of glass cloth crossplies.

(c) Thermal conductivity per strength or stiffness For many applications thermal conductivity is not the only decisive parameter. Certain levels of strength σ or stiffness E are required, e.g. for support elements. For those applications the ratios χ/σ or χ/E determine the capability of thermal insulation. In figure F2.0.42 the ratios χ/E of several UD composites are plotted versus temperature. As expected, the lowest ratio is true for carbon fibre composites but only at low temperatures. Glass fibre composites show reasonably low ratios over a large temperature range. Even lower values are exhibited by ceramic fibre composites over a large temperature range (Takeno et al 1986). They are the best choice for thermal insulating elements of high stiffness.

Figure F2.0.42. The thermal conductivity of UD composites per stiffness.

F2.0.11 Specific heat Heating a body of a mass m by an energy Q leads to a material specific temperature rise ∆T

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The coefficient C relating ∆Q to ∆T is called the specific heat. At low temperatures, and especially for low expanding composites, there is nearly no difference of specific heat at constant pressure or volume. The specific heat of composites can be calculated from its components by a linear mixing rule

where Vf is the fibre volume, f the fibre and m the matrix. The specific heat at low temperatures (< 100 K) is quite similar for all amorphous polymers; epoxy resins, in particular, are nearly independent of their chemical compositions. This is shown in figure F2.0.43 for several epoxy resins below 100 K (Hartwig 1994).

Figure F2.0.43. The specific heats of several epoxy resins versus temperature.

(a) Temperature dependence At very low temperatures (<5 K), phonon wavelength is large enough to propagate in three dimensions and the specific heat obeys the Debye law

At elevated temperatures phonon wavelength decreases, and vibrations between graphite layers or between polymer chains are no longer possible. In graphite (carbon fibres) phonon activation is then restricted to the two-dimensional graphite layers, and the specific heat obeys a quadratic dependence

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Properties of fibre composites

For Kevlar (aligned aramide chains) and generally for polymers at elevated temperatures (T > 60 K) phonons are restricted to linear chains, and

A roughly linear temperature dependence of the specific heat has been found for most polymers above 60 K and well below the glass transition (Hartwig 1994). Deviations occur because of the intrinsic bending stiffnesses of graphite layers and the polymer backbones. For most polymers the relation C ∼ T n with n ≈ 0.8 to 0.9 is more realistic (see figure F2.0.43). The polymer matrix mainly dominates the specific heat of fibre composites. As seen from figure F2.0.44 specific heats of fibre composites above 60 K obey a roughly linear dependence, as do composites with glass and carbon fibres (despite equation (F2.0.21)). Carbon and glass composites have nearly the same specific heat when related to the mass.

Figure F2.0.44. Specific heat (related to the mass) of crossplies/epoxy versus temperature: p, O—E-glass (Okada et al 1990); ×—carbon (Collings and Smith 1978).

The influence of the polymer matrix is shown in figure F2.0.45. The specific heats of glass-reinforced composites show a relatively small difference between matrix types and again an almost linear dependence on the temperature. The specific heats (related to the mass) are higher than for most metals. Teflon and polyester matrix composites have the highest specific heats. The epoxy, silicone and phenolic matrix composites are bunched on the lower side of the group. The specific heat, related to mass, is lower for composites than for the epoxy matrix. When relating specific heat to the volume, matrix and glass composite are nearly equal, as shown in figure F2.0.46. (b) Thermal diffusivity The time period for establishing a thermal equilibrium is controlled by the thermal diffusivity a

where χ is the thermal conductivity, c the specific heat and ρ the density. The time needed to equalize a temperature difference ∆T in a body can be calculated by the diffusivity a when the local temperature distribution is known. An example of the diffusivity is given in figure F2.0.47 for glass crossplies.

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Figure F2.0.45. Specific heat of E-glass with several polymeric matrices versus temperature (Kasen 1975). Matrices: 1—epoxies 2—phenolics; 3—polyester; 4—silicones; 5—phenyl silanes; 6—Teflon; 7—polybenzimidazole; 8—phenyl formaldehydes.

Figure F2.0.46. Specific heat (related to the volume) versus temperature, for glass crossplies and the epoxy matrix (Khalil and Han 1982).

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Figure F2.0.47. Thermal diffusivity versus temperature (Khalil and Han 1982).

It can be seen that a increases strongly at low temperatures, since c drops more strongly with temperature than χ. This means that the temperature equalizes much faster at low temperatures. F2.0.12 Dielectric properties and breakdown voltage The dielectric loss factor tan δe is determined mainly by the polymeric matrix of a fibre composite and is related to the time (frequency) and the temperature dependences. Only organic fibres, such as Kevlar, are dissipative themselves. Dissipation arises from dielectrically induced movement of polar groups in the polymer. This is analogous to the viscoelastic processes which cause mechanical damping. Loss factor data of pure polymers can be used for comparing those of composites. As an example a typical loss tangent curve for epoxy resins is shown in figure F2.0.48.

Figure F2.0.48. A typical dielectric loss tangent curve for epoxy resins (Ciba-Geigy) versus temperature at 50 Hz.

The loss factor is nearly independent of frequency at very low temperatures. The time constant is so large that fast molecular movements cannot be activated. At high frequencies, however, dielectric dissipation may lead to heating and temperature rise. Data on the dielectric loss tangent are given in

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figure F2.0.49 for some crossplies (G-11 CR and G-10 CR with epoxy matrix and Spaulrad with polyimide) (Nishijima et al 1986). The loss factor decreases strongly with temperature. Some extrapolations can be made by using figure F2.0.48. The dielectric constant depends on both the glass fibre and matrix. Similar to the loss tangent it depends on the time (frequency). The temperature dependence of crossplies in the thickness direction is shown in figure F2.0.49.

Figure F2.0.49. The dielectric loss tangent and dielectric constant versus temperature for glass crossplies in the thickness direction.

The dielectric strength is defined by the breakdown voltage per thickness. The disadvantage of this definition is a nonlinear dependence on thickness. Thin foils have a higher dielectric strength than thick plates. Kapton-H foils (25 µm) have a dielectric strength of about 400 kV mm−1 , whereas for polymer plates (1 mm) the value is lower by about one order of magnitude. Schutz et al (1995) propose a dependence of breakdown voltage V on thickness d

where k is a constant, specific to the material. For glass crossplies, of about 0.5 to 1 mm in thickness, the dielectric strength typically is 60 to 80 kV mm−1 for d.c. voltage at 77 K. This can be used as a guideline (Schultz et al 1995). The dielectric strength is almost independent of temperature. However, there is a dependency on the time profile of the voltage. The values of the d.c. voltage are somewhat different those of the a.c. voltage. For several materials there is a time lag between the application of the voltage and breakdown. The electrical resistivity of polymer fibre composites is lowest for carbon. It can be important to know the electrical resistivity in the vicinity of pulsed magnets which cause eddy currents. The resistivity of a carbon composite is almost independent of temperature and ranges from 10−4 to 10−5 Ω m (Hurley and Coltman 1984). The resistivity versus temperature is shown in figure F2.0.50 for high tensile carbon fibre/epoxy composites (which give a similar result to UD and crossplied composites) (Pinheiro et al 1987).

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Figure F2.0.50. The electrical resistivity of high tensile carbon fibre/epoxy composites versus temperature.

F2.0.13 Gas permeability An important design parameter of nonmetallic cryostats or vessels made of fibre composites is the gas permeability (Callaghan 1991, Evans and Morgan 1990, Okada et al 1 990). Lightweight construction is required for transport vessels in aeroplanes or vehicles. Usually glass fibre is sufficient, but for high-pressure vessels carbon fibre is used. The most important gases are helium for cooling and hydrogen or natural gases (methane) as fuel for propulsion. Permeation is thought to occur in three steps: (i) solution at the pressure side surface (ii) diffusion through the material and (iii) desorption at the low-pressure side. Solubility S and diffusivity D determine gas permeation in a different way for homogeneous or heterogeneous materials. The gas leakage rate Q through a material of thickness d and area A, driven by a pressure difference ∆p is given by

where P(T) is the permeation coefficient which depends on the temperature by an Arrhenius-type equation

where R is the gas constant, P0 a constant factor and EP the barrier against permeation. If the permeation barrier EP is known, an extrapolation from room temperature to any temperature is possible by equation (F2.0.25). This equation is valid for homogeneous and heterogeneous materials (e.g. fibre composites). The slopes of the Arrhenius plots (In P ∼ EP(1/T )) are proportional to the barrier EP (Hartwig and Humpenöder). Typical Arrhenius plots of the permeability P are shown in figures F2.0.51-F2.0.53 for glass fibre and carbon fibre crossplies with an epoxy matrix. Figure F2.0.51 illustrates the dependence on fibres and fibre volume; figures F2.0.52 and F2.0.53 show the dependence on the permeating gases. Most permeation occurs via the matrix as the fibres are relatively impermeable. The permeability P depends exponentially on the fibre volume V (Hartwig and Humpenöder 1997)

where Pm and Pf are the permeabilities of the matrix and the fibre material respectively. The diffusion in composites is no longer unidirectional, and the effective diffusion path in the matrix is drastically enhanced

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Figure F2.0.51. Helium permeabilities of carbon and glass crossplies, and the epoxy matrix.

Figure F2.0.52. Permeabilities of glass crossplies with different gases.

with the fibre content (zig-zag path). In figure F2.0.54 the permeability is plotted versus the glass fibre volume (Hartwig and Humpenöder 1997). The following results can be extracted from figures F2.0.51-F2.0.54. (i)

The permeabilities of glass fibre and carbon are very similar if the fibre content is similar and the same matrix is used. (ii) The permeations of helium and hydrogen gas are similar; the permeability of the large methane molecules is low and the barrier EP is large. (iii) The barriers EP of glass/EP crossplies (fibre volume 56%) are given by: EP = 20 kJ mol−1 with helium EP = 26 kJ mol−1 with hydrogen EP = 40 kJ mol−1 with methane. (iv) The permeability depends strongly on the fibre volume.

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Figure F2.0.53. Permeabilities of carbon crossplies with different gases.

Figure F2.0.54. Permeability of glass fibre versus the fibre volume V.

From Figures F2.0.51-F2.0.54 it is obvious that below 200 K typical fibre composites are nearly impermeable (P < 10−14 m2 s−1 ). Thermal and fatigue cycling, however, might induce microcracks and an increase in P. This applies to mobile storage vessels and cryostats for pulsed magnets. Investigations (Hartwig and Humpenöder 1997), however, revealed that thermal cycling has nearly no effect, and fatigue loading at cryogenic temperatures increases P by less than 30%. Disdier et al (1997) investigated the effect of surface cracks, voids and cracks in the thickness direction induced by different types of loading. No remarkable effect has been found. Organic or metallic liners and surface layers have been applied for the sake of reducing the permeability P. Kapton foils give no effect. Aluminium foils delaminate in composites under thermal and fatigue loading, except for cases where a special surface treatment is applied. Good adhesion has been found for tin foils (Hartwig and Humpenöder 1997). A thin tin foil (100 µm) can be laminated within a fibre composite. It is difficult to give recommendations for surface layers. Special surface treatment is necessary.

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F2.0.14 Radiation damage The resistance of fibre composites against irradiation is determined mainly by the polymeric matrix. Inorganic fibres are less sensitive, except for E-glass which reacts with neutrons because of its boron content; R-glass is quite stable. Polymeric fibres, such as Kevlar, however, are more sensitive to irradiation. Carbon is intermediate. It is a general rule that mechanical properties degrade first, followed by dielectric and thermal properties. Usually the admissible dose of irradiation is defined by a degradation of properties to half of their original values. The admissible dose depends on the source of irradiation (neutrons, γ-rays, electrons). The ionizing effect is different for these sources and depends, in addition, on their energies. Conversion formulae have been derived for calculating an effective dose which is representative for radiation damage behaviour. The damage processes, however, might be different for γ-rays, electrons and neutrons, the latter being able to enter nuclear reactions. However, ionization results from all sources. Intercalibrating dosimetry is well developed (Humer et al 1994, 1996). The absorbed dose is measured in gray (1 Gy = 100 rad). The temperatures of irradiation and measurement are other parameters which might influence the radiation damage. Irradiation and subsequent measurement both at low temperatures yield values different from those with an intermediate warm-up. Values for the latter case are considered to be the more critical ones, since warming to higher temperatures induces regeneration of free radicals or formation of other chemical compositions or crosslinking (Egusa 1990). Hydrogen atoms, knocked off by irradiation, form a gas which expands on heating and damages at least the polymeric matrix or the interface (Evans et al 1995). Resistive matrixes are those with a low hydrogen content and a stable chemical structure, such as aromatic rings. Polyimides are some of the most stable polymers. The admissible dose is of the order of 108 Gy. In combination with fibres even higher doses are tolerable. Most irradiation measurements revealed that the irradiation temperature and an intermediate warm-up are less sensitive to composites than to pure polymers (Egusa 1990, Humer et al 1990). (a) Degradation of mechanical properties Radiation damage of mechanical properties in the past was tested mainly on flexural strength. Long-beam flexural tests are fibre dominated as long as the matrix is not damaged by irradiation. In the case of matrix damage, interlaminar shear failure occurs even for long-beam tests. A suggested explanation of this is given in figure F2.0.55 for glass crossplies with a non-resistant epoxy matrix. The flexural strength degrades strongly for two-dimensionally reinforced plastics (2D-GFRP). An additional reinforcement in the thickness direction (3D-GFRP) increases interlaminar shear strength and nearly no degradation occurs at the same dose levels (Nishijima et al 1990). More data on radiation damage are given in the references (Benzinger 1983, Humer et al 1995, Reed et al 1995). Reed et al (1995) concentrated on neutron and γ irradiation at 4 K of glass composites with epoxy TGDDM/DDS (tetra-glycidyl-diamino-diphenyl-methane and diamino-diphenyl-methane). The general result of several investigations is that even matrix-dominated mechanical properties of composites are able to withstand a dose up to 108 Gy. There is a sequence of the best matrix materials as shown in table F2.0.24. Fibre reinforcement leads to higher admissable doses: (degration to half of the original value). A survey is given in table F2.0.25. In most cases ultimate tensile strength (in-plane) degrades less than the ultimate tensile strain. The modulus is in between. The compressive strength (in-plane) is similar to the tensile strength. The compressive modulus is almost insensitive to irradiation. The fatigue behaviour has been investigated on glass fibre crossplies (G11) at a dose of 106 Gy under tensile threshold cycling. An increase has been observed by γ irradiation for up to about 2000 load cycles, probably due to a higher radiation-induced crosslink density of the epoxy matrix. At 2 × 105 load cycles about 20% of the fatigue endurance limit degrades after a dose of 106 Gy. This result is similar for 77 K

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Figure F2.0.55. Degradation of nominal flexural strength of glass crossplies reinforced in two or three dimensions (Nishijima et al 1990).

Table F2.0.24. The best matrix materials.

Table F2.0.25. Admissable doses for laminates (fabrics) (Benzinger 1983).

and room temperature ( Kornkund et al 1983). More data on degradation of flexural or shear strengths of glass cloth crossplies are given in figures F2.0.56 and F2.0.57. The matrices are epoxy (TGDD/DDS) and PI (Kerimid). Flexural strength of composites is higherwith an EP matrix in the original state, but degradation is stronger than for a PI matrix.

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Figure F2.0.56. Ultimate flexural strength of glass fibre cloth versus absorbed dose in the matrix (Egusa 1990).

Figure F2.0.57. Ultimate shear strength of glass cloth in the 45° direction versus absorbed dose in the matrix (Egusa 1990).

The compressive strength of glass cloth in the thickness direction is not very sensitive to irradiation as shown in figure F2.0.58. An additional reinforcement in the thickness direction of composites ZI-003; 005 yields higher values, at least at room temperature.

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Figure F2.0.58. Radiation damage at room temperature of the compressive strength in the thickness direction for glass crossplies, G10CR, G11CR and 3D-composites ZI-003 and ZI-005 (Nishijima 1990).

(b) Dielectric properties Dielectric losses occur in the matrix. The degradation of the dielectric loss factor tan δ of an epoxy matrix by a radiation dose of 107 Gy is less than 30% and even lower for a fibre composite (Jäckel el al 1994). The epoxy resin is based on bisphenol A and cured with an aromatic amine. The dielectric constant (permittivity) is again sensitive to irradiation because of the matrix. At 100 K the permittivity of an epoxy matrix drops from 3.4 to 1.8 by a γ-radiation dose of 107 Gy. The radiation damage of a fibre composite is lower by more than a factor of two related to the matrix (60% fibre volume). The dielectric strength of glass (G10 and G11) degrades from 23 to 10 kV mm−1 by irradiation of 108 Gy. Higher values are achieved with glass/PI crossplies. Their values decrease from about 70 kV mm−1 by about 10% after irradiation with 8 × 107 Gy gamma rays and 3 × 1022 neutrons m−2. It should be mentioned that a pure Kapton 100 HA foil exhibits a much higher dielectric strength (about 400 kV mm−1 ) than and a similar percentage of degradation to the composite with a PI matrix. The reason is that thin foils are much better barriers than thicker ones. There is no linear relation to thickness (Reed et al 1995). (c) Thermal properties The thermal conductivity and specific heat is only slightly influenced by irradiation. Data are available only on the epoxy matrix which is the most sensitive component. At an electron irradiation of about 107 Gy the degradation of both properties is less than 30% (Jäckel et al 1994). References Section F2.0.1 Hartwig G and Knaak S 1984 Fibre-epoxy composites at low temperatures Cryogenics 24 639 Section F2.0.2 Bansemir H and Haider O 1991 Basic material data and structural analysis of fiber composites for space application Cryogenics 31 298

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References

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Section F2.0.3 Davidge R W 1979 Mechanical Behavior of Ceramics (Cambridge: Cambridge University Press) p 111 Donnet J-B and Bansel RoopChand 1990 Carbon Fibers 2nd edn pp 1-143; (New York: Dekker) From International Fiber Science and Technology, see also vol 1-7 Iwasaki Y, Yasuda I, Hirokawa T, Noma K, Nishijima S and Okada T 1991 Three dimensional fabric reinforced plastics for cryogenic use Cryogenics 31 261 Section F2.0.4 Hartwig G 1994 Polymer Properties at Room and Cryogenic Temperatures (New York: Plenum) Section F2.0.6 Chawla K K 1987 Composite Materials (Berlin: Springer) p 177 Engineered Handbook Composites 1987 vol 1 (Metal Park, OH: American Society of Metals) Hancox N L 1975 J. Mater. Sci. 10 234 Piggott M R 1980 Load bearing fiber composites (Elmsford, NY: Pergamon) p 83 Rosen B W 1965 Fiber Composite Materials (ASTM) p 58 Section F2.0.7 Ahlborn K 1991 Cryogenic mechanical response of carbon fibre reinforced plastics with thermoplastic matrices to quasi-static loads Cryogenics 31 252 Becker H 1990 Problems of cryogenic interlaminar shear strength testing Adv. Cryog. Eng. (Mater.) B 36 827 Evans D, Johnson I, Jones H and Hughes D D 1990 Shear testing of composites at low temperatures Adv. Cryog. Eng. (Mater.) B 36 819 Fabian P E, Reed R P, Schutz J B and Bauer-McDaniel T S 1995 Shear/compression properties of candidate ITER insulations systems at low temperatures Cryogenics 35 689 Fujczak R R 1978 Torsional fatigue behavior of graphite epoxy cylinders Proc. 2nd. Int. Conf. Composite Materials p 635 Hartwig G and Ahlborn K 1990 Fiber composites with thermoplastic matrices Adv. Cryog. Eng. (Mater.) 36 909 Hartwig G and Pannkoke K 1992 Fatigue behavior of UD fiber composites at cryogenic temperatures Adv. Cryog. Eng. (Mater.) B 38 453 Hübner R and Hartwig G 1996 Fatigue of crossply carbon fiber composites at low temperatures Adv. Cryog. Eng. (Mater.) A 42 233 Kasen M B 1981 Cryogenic Properties of filamentary-reinforced composites Cryogenics 21 323 Kasen M B 1990 Current status of interlaminar shear testing of composite materials at cryogenic temperatures Adv. Cryog. Eng. (Mater.) B 36 787 Kritz R D and McColskey J D 1990 Mechanical properties of alumina-PEEK UD composite: compression, shear and tension Adv. Cryog. Eng. (Mater.) B 36 921 Okada T and Nishijima S 1990 Investigation of interlaminar shear behavior of organic composites at low temperatures Adv. Cryog. Eng. (Mater.) B 36 811 Pannkoke K and Wagner H J 1991 Fatigue properties of UD carbon fiber composites at cryogenic temperatures Cryogenics 31 217 Puck A 1967 Zur Beanspruchung und Verformung von GFK-Mehrschichtverbund-Bauelementen, 1. (Grundlagen Kunststoffe 57) (Hauser) p 284 Reed R P and Golda M 1994 Cryogenic properties of UD composites Cryogenics 34 909 Walsh R P, McColskey J D and Reed R P 1995 Low-temperature properties of a unidirectionally reinforced epoxy fibreglass composite Cryogenics 35 723 Section F2.0.8 Ahlborn K 1988 Fatigue behavior of carbon fiber reinforced plastics at cryogenic temperatures Cryogenics 28 267 Ahlborn K 1991 Durability of carbon fiber reinforced plastics with thermoplastic matrices under cyclic mechanical and cyclic thermal loads at cryogenic temperatures Cryogenics 31 257 Hartwig G and Hübner R 1995 Thermal and fatigue cycling of fiber composites Cryogenics 35 675 Hartwig G and Pannkoke K 1992 Fatigue behavior of UD fiber composites at cryogenic temperatures Adv. Cryog. Eng. (Mater.) B 38 453

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Hübner R and Hartwig G 1996 Fatigue of crossply carbon fiber composites at low temperatures Adv. Cryog. Eng. (Mater.) A 42 233 Nishijima S, Ueno S, Okada T and Niwa K 1995 Stress analysis in shear/compression tests Cryogenics 35 681 Pannkoke K 1994 Static and fatigue properties of UD carbon and fiber composites at 77 K Adv. Cryog. Eng. (Mater.) B 40 1025 Reed R P, Fabian P E and Bauer-McDaniel T S 1995 Shear/compressive fatigue of insulation systems at low temperatures Cryogenics 35 685 Schutz J B and Fabian P E 1996 Failure criteria for low-temperature irradiated organic composite insulation systems Adv. Cryog. Eng. (Mater.) A 42A 73 Wang S S and Khim E S 1981 Degradation of fiber reinforced composite materials at cryogenic temperatures; multiaxial fatigue Adv. Cryog. Eng. 28 201 Section F2.0.9 Hartwig G 1988 Thermal expansion of fibre composites Cryogenics 28 255 Hartwig G 1995 Support elements with extremely strong negative thermal expansion Cryogenics 35 717 Hartwig G, Endres K and Haider O 1994 Support elements with negative thermal expansion Adv. Cryog. Eng. (Mater.) B 40 1107 Hartwig G and Hübner R 1995 Thermal and fatigue cycling of fiber composites Cryogenics 35 675 Hartwig G and Schwarz G 1990 Thermal expansion of polymers and fiber composites at low temperatures Adv. Cryog. Eng. 36 1007 Hartwig G, Krahn F and Schwarz G 1991 Thermal expansion of carbon fiber composites with thermoplastic matrices Cryogenics 31 244 Nakahara S, Tujita T, Sugihara K, Nishijima S, Takeno M and Okada T 1996 Two-dimensional thermal contraction of composites Adv. Cryog. Eng. 32 209 Okada T, Nishijima S, Takahata K and Yamomoto J 1991 Research and development of insulating materials for large helical device Cryogenics 31 307 Section F2.0.10 Giesy R K 1996 Composite support structures for cryogenic systems Adv. Cryog. Eng. (Mater.) A 42 257 Hartwig G 1994 Status and future of fiber composites Adv. Cryog. Eng. B 40 961 Hartwig G 1995 Support elements with extremely strong negative thermal expansion Cryogenics 35 717 Hirokawa T, Yasuda J, Iwasaki Y, Norna K, Nishijima S and Okada T 1991 Design of support straps with advanced composites for cryogenic applications Cryogenics 31 288 Kasen M B, McDonald G R, Beekman D H Jr and Schramm R E 1980 Mechanical electrical and thermal characterization of G10Cr and G11Cr glass cloth/epoxy laminates between RT and 4 K Adv. Cryog. Eng. 26 235 Okada T, Rugaiganisa B and Nishijima S 1990 Data Base Adv. Cryog. Eng. (Mater.) B 36 1027 Takeno M, Nishijima S, Okada T, Fujioka K, Tsuchida Y and Kuraoka Y 1986 Thermal and mechanical properties of advanced composite materials at low temperatures Adv. Cryog. Eng. (Mater.) 32 217 Section F2.0.11 Collings E W and Smith R D 1978 Specific heats of some cryogenic structural materials—composites Adv. Cryog. Eng. 24 290 Kasen B 1975 Glass reinforced composites Cryogenics 15 327 Khalil A and Han K S 1982 Mechanical and thermal properties of glass-fiber reinforced composites at cryogenic temperatures Adv. Cryog. Eng. (Mater.) 28 243 Okada T, Rugaiganisa B and Nishijima S 1990 Data Base Adv. Cryog. Eng. (Mater.) B 36 1027 Section F2.0.12 Hartwig G 1994 Polymer Properties at Room and Cryogenic Temperatures (New York: Plenum) Hurley G F and Coltman R R Jr 1984 J. Nucl. Mater. 123 1327 Nishijima S, Okada T and Hagihara T 1986 Thermostimulated current and dielectric loss in composite materials Adv. Cryog. Eng. (Mater.) B 32 187 Pinheiro F, Radcliffe D J and Rosenberg H M 1987 Proc. 7. Int. Cryogenic Engineering Conf. (IPC Science and Technology Press)

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Further reading

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Schutz J B, Fabian P E, Hazelton C S, Bauer-Daniel T S and Reed R P 1995 Effect of cryogenic irradiation on electrical strength of candidate ITER insulation materials Cryogenics 35 759 Section F2.0.13 Callaghan M T 1991 Use of resin composites for cryogenic tankage Cryogenics 31 282 Disdier S, Rey J M, Pailler P and Bunsell A R 1997 Helium permeation in composite materials for cryogenic applications Cryogenics at press Evans D and Morgan J T 1990 The permeability of composite materials to hydrogen and helium gas Adv. Cryog. Eng. (Mater.) 34 11 Hartwig G and Humpenöder J 1997 Gas permeation through polymers and fiber composites at low temperatures Cryogenics at press Okada T, Nishijima S, Fujioka K and Kuraoka Y 1990 Gas Permeation and Performance of a FRP Cryostat Adv. Cryog. Eng. (Mater.) 34 17 Section F2.0.14 Benzinger J R 1983 The manufacture and properties of radiation resistant laminates Adv. Cryog. Eng. (Mater.) B 28 231 Egusa S 1990 Irradiation effects on and degradation mechanisms of the mechanical properties of polymer matrix composites at low temperatures Adv. Cryog. Eng. (Mater.) 36 861–8 Evans D, Reed R P and Hazelton C S 1995 Fundamental aspects of irradiation of plastic materials at low temperatures Cryogenics 35 755 Humer H, Schönbacher H, Szeless B, Tarlet M and Weber H W 1996 CERN Report 96-05 4 July Humer K, Weber H W, Tschegg E K, Egusa S, Birtcher R C and Gerstenberg H 1994 Tensile strength of fibre-reinforced plastics at 77 K irradiated by various radiation sources J. Nucl. (Mater.) B 212–215 849-53 Humer K, Weber H W, Tschegg E K, Egusa S, Birtcher R C and Gerstenberg H 1990 Tensile and shear fracture behavior of fiber reinforced plastics at 77 K irradiated by various radiation sources Adv. Cryog. Eng. (Mater.) B 40 1015 Humer K, Weber H W, Tschegg E K, Egusa S, Birtcher R C, Gerstenberg H and Goshchititsky B N 1995 Low-temperature tensile and fracture mechanical strength in mode I and mode II of fibre reinforced plastics following various irradiation conditions Proc. 18th Symp. on Fusion Technology (Karlsruhe, 1994) Fusion Technol. 2 973–6 Jäckel M, Leucke U, Jahn K, Fitzke F and Hegenbarth E 1994 Thermal and dielectric properties of epoxy resins at low temperatures after irradiation Adv. Cryog. Eng. (Mater.) 40 1153 Kornkund B, Conway J C Jr, Queeny R A and Diethom W 1983 Effect of radiation and cryogenic temperatures on the fatigue resistance of G11 glass cloth/epoxy laminates J. Nucl. Mater. 115 197 Nishijima S, Nishiura T, Okada T, Hirokawa T, Yasuda J and Iwasaki Y 1990 Development of radiation resistant composite materials for fusion magnets Adv. Cryog. Eng. (Mater.) B 36 877 Reed R, Fabian P, Bauer-McDaniel T, Hazelton C and Munshi N 1995 Effect of neutron/gamma irradiation at 4 K on shear and compressive properties of insulation Cryogenics 35 739

Further reading Section F2.0.1 Hartwig G 1995 Status and future of fiber composites Adv. Cryog. Eng. (Mater.) B 40 961 Section F2.0.4 Ward I M 1983 Mechanical Properties of Solid Polymers 2nd edn (New York: Wiley) Section F2.0.13 Rogers W A, Butitz R S and Alpert D 1954 Diffusion coefficient, solubility, and permeability for helium in glass J. Appl. Phys. 25 Vieth W R 1991 Diffusion in and Through Polymers (Hanser)

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F3 Electrical resistivity B Seeber

F3.0.1 Introduction The electrical resistivity, ρ, is an important physical property of materials employed for applications in superconductivity. For instance, current leads for superconducting magnets must have a low ρ otherwise there will be an undesirable increase of heat dissipation when operating at the nominal current. A com promise has to be found between the generation of heat (due to a flowing current) and the thermal conductivity of the lead (see also chapter D10). The metallic matrix of superconducting multifilamentary wires should be of a low-resistivity material in order to prevent hot spots in the event of a quench occurring in a superconducting magnet. For a.c. applications, and fast transients, the electrical resistivity is at the origin of eddy current heating and additional mechanical forces. The purpose of this chapter is to give an introduction to the electrical resistivity of metallic materials used for superconductivity in the range between room temperature and the temperature of liquid helium. Factors that influence the electrical resistivity, such as the impurity content of pure metals or magnetic fields etc, are discussed. The resistivity of pure metals and of alloys is compiled in appendices A and B respectively.

F3.0.2 Basic considerations F3.0.2.1 Temperature dependence of the resistivity The electrical resistivity of metals with impurity atoms can be described by

where ρL is the resistivity due to the collision of conduction electrons with the atoms of the lattice. Because the amplitude of vibrations of these atoms (phonons) depends on the temperature, ρ L will go to zero when the temperature goes to zero. ρi is the resistivity resulting from the collision of conduction electrons with impurity atoms which are disturbing the periodicity of the lattice. Supposing a small concentration of impurity atoms, ρ i is independent of temperature. This is also known as Matthiessen’s rule. When

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decreasing the temperature, T, ρL goes to zero as T 5 and, below approximately 20 K, the electrical resistivity is dominated by ρ i which is then the residual resistivity. The ratio ρ(273 K)/ρ(4.2 K) is the residual resistivity ratio (RRR), an important parameter in the design of superconductive applications. In the case of a superconductor, the denominator has to be taken at a temperature slightly above the critical temperature. ρ L is also called intrinsic resistivity because it is characteristic for a material. The intrinsic resistivity of a metal can be found by measuring the temperature dependence down to temperatures where ρ does not depend on temperature. Subtracting this residual resistivity from ρ gives the intrinsic resistivity. The latter is tabulated in table F3.0.1 for copper, aluminium and silver (Fickett 1982, Matula 1979).

Table F3.0.1. Intrinsinc resistivity for copper, aluminum and silver.

ρ L depends on the temperature and two cases have to be distinguished

θ is a characteristic temperature and is called the Debye temperature (e.g. copper θ ∼ 330 K, aluminium θ ∼ 400 K). According to Grüneisen, ρ L can be calculated from the semi-empirical equation

where C is a constant. In figure F3.0.1 ρ/ρ(θ ) is calculated as a function of T/θ and compared with experimental data (Bardeen 1940).

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In reality agreement with theory is not always so good and deviations from Matthiessen’s rule, as well as from the T 5 behaviour at low temperatures, are frequently observed. The message is that, if required, precise data should be taken from measurements and theory can offer only qualitative behaviour and trends.

F3.0.2.2 Anisotropy The anisotropy of metals and alloys has to be considered when texturing is present. Texturing is a preferential crystallographic orientation (elongation) of grains and comes from deformation processes like extrusion, rolling and drawing. If the material has a noncubic crystal structure, an anisotropy of the resistivity can be expected. Because most metals employed in engineering have a cubic crystal structure, anisotropy is not a very important issue. Noncubic metals are for example Be, Mg, Zn, Hg, Ga, In, Sn, Ti etc (Meaden 1965).

F3.0.2.3 Crystallographic phase transitions Metals, but in particular alloys, have phase transitions at specific temperatures. At constant chemical composition of a material, different phases (crystal structures) exist at different temperatures. Phase transitions do not take place only by changing the temperature, they may also be initiated by pressure, strain or specific impurities. They mostly occur above room temperature. However, by appropriate quenching, high-temperature phases may also exist at room temperature and below. Phase transitions can also take place at low temperatures. For instance, Nb3Sn transforms upon cooling at about 20 K from a cubic (A15) crystal structure to a tetragonal one. Because the distortion of the crystal lattice is small.

Figure F3.0.1. Relative resistivity versus reduced temperature calculated according to Grüneisen. θ is the Debye temperature. Experimental data are included for: Au (θ = 175 K), Cu (θ = 333 K), Al (θ = 395 K) and Ni (θ = 472 K). Reproduced from Bardeen (1940) by permission of the American Institute of Physics.

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Figure F3.0.2. The electrical resistivity of β brass versus temperature. For different degrees of deformation, the transition temperature is shifted. Note the important hysteresis which is related to the formation (upon cooling) or dissolution (upon warming) of the martensitic phase (after Hummel 1968).

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physical parameters are not influenced in an important way. Another, more striking, example is β brass. Upon cooling a phase transition takes place and the transition temperature depends strongly on the degree of deformation (figure F3.0.2). Significant hysteresis arises from the gradual formation and dissolution of a second phase (martensite phase) upon cooling and warming respectively.

F3.0.2.4 Magnetoresistivity Pure metals have a magnetoresistivity, which means that the resistivity increases in a magnetic field. This is only observed at low temperatures (no collisions with phonons) and, depending on the purity of the metal, field-induced resistivity can be two orders of magnitude higher than the zero-field resistivity. The magnetoresistivity depends also on the relative orientation of the magnetic field and the direction of current flow. The transverse magnetoresistivity increases usually as the square of the applied field whereas the longitudinal magnetoresistivity saturates. At higher fields the transverse resistivity either continues to grow with B 2 or B, or saturates. Theory can only give a qualitative picture of the situation. When a magnetic field is switched on, the conduction electrons in a pure metal are subject to a magnetic force which makes them move in circular or helical orbits. The higher the field the smaller the radius of these orbits. The ratio l/r, where l is the mean free path of conduction electrons and r is the radius of the orbit, determines the resistivity. It can be shown that

where H is the magnetic field.

This equation is known as Kohler’s rule (Kohler 1938). The function f is characteristic for a metal and depends on the relative orientation of field and current. In single crystals, f depends also on the orientation of the crystal. Magnetoresistivity is often graphically presented as a so-called Kohler plot; this is a log-log plot of ∆ρ/ρ versus B(ρ(273 K)/ ρ(T) which should give a straight line. Sometimes, when different metals are compared, ρ(273 K)/ ρ(T) is replaced by ρ(θ)/ρ(T) with θ the Debye temperature. With the exception of aluminium, most metals in a transverse field show nearly straight lines in the Kohler plot and an example is given in figure F3.0.3 for copper. Other metals are summarized in appendix C for transverse and longitudinal fields. In single crystals the magnetoresistivity shows a strong anisotropy. The measurement of this anisotropy has commonly been used to obtain information regarding the Fermi surface of a metal (Chambers 1961). At a first glance this anisotropy may be only of academic interest, but it is not when dealing with textured materials. As already mentioned, texturing occurs when materials are deformed, e.g. by extrusion, rolling or drawing.

F3.0.2.5 Size effect As long as the mean free path of conduction electrons is of the order of 10 –7 m, which is the case for metals at room temperature, collisions with the surface are negligible. Of course, there are always electrons hitting the surface of the metal but their quantity is not sufficient with respect to the rest in order to contribute to the electrical resistivity. Upon cooling, the mean free path of conduction electrons increases and can reach values of the order of millimetres in pure metals. If the metal has a similar size, almost all conduction electrons can be scattered at the surface and a new term in Matthiessen’s rule has to be added

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Figure F3.0.3. Kohler plot for the transverse magnetoresistivity of copper. Reproduced from Fickett (1982) by permission.

where ρ s is the resistivity from surface scattering of electrons. At a first approximation (which works well in many cases) one can quantify the increase of resistivity in a thin wire due to surface scattering by (Nordheim 1934)

where ρb is the resistivity without surface scattering (bulk value), l b is the mean free path of conduction electrons in the bulk material and d is the diameter of the wire. An estimate of the order of magnitude of the mean free path of conduction electrons at 4.2 K can be obtained by ρ b lb = 6.6 × 10− 12 Ω cm 2 and 6.0 × 10−12 Ω cm2 for copper and aluminum, respectively (Brändli and Olsen 1969). In superconducting multifilamentary wires the spacing between the filaments can be small so at low temperatures the size effect may be observed in the matrix with high RRR. A typical example is Nb—Ti wires with a copper matrix (Cavalloni et al 1983). In the case of superconductors, one has also to take into account the presence of magnetic fields. There may be situations where the resistivity is reduced by applying a magnetic field. This can be explained by the fact that electrons move in helical orbits parallel to the field. If the field is sufficiently strong the electrons no longer hit the surface and the resistivity, due to the size effect, goes to zero. For instance, in a thin wire in a longitudinal field the current flows in the direction of the wire axis and the conduction electrons spiral around the same axis. If the radius of the electrons’ orbit is significantly smaller then the wire radius ρs is zero. The situation is more complex in a transverse field; here ρs is also reduced but does not go exactly to zero (MacDonald and Sarginson 1950). A detailed discussion of the size effect can be found in the article by Brandli and Olsen (1969).

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F3.0.2.6 Defect-induced resistivity (stress, fatigue) When a metal is mechanically deformed, defects are induced and the resistivity increases. As long as the material stays in its elastic regime, resistivity changes are reversible. In contrast, after a plastic deformation the resistivity increase is not reversible. Defects causing a resistivity increase are mainly deviations from the periodicity of the crystal lattice, such as vacancies, interstitials or dislocations, and can also be introduced by other means such as multiple thermal cycling through phase transitions or by irradiation with highly energetic particles. In general, the situation is rather complex because many factors can influence the resistivity. Depending on the application, defects should be introduced at the lowest possible temperature because recovery of the resistivity may already start at 15 K. As examples one could mention copper irradiated with 1 MeV electrons at 4.2 K or cold-deformed aluminium where the first recovery stage is observed at 15 K (Corbett et al 1959a, b) and at ∼50 K (Reed 1972) respectively. This means also that the resistivity at room temperature is not very greatly influenced by defects (Broom 1952). How the resistivity changes with the reduction of the cross-sectional area by wire drawing at different temperatures is illustrated for copper in figure F3.0.4 (Broom 1952). The lower the temperature of deformation the higher is the increase of resistivity. There is an empirical relation between the increase of resistivity ∆ρ and the applied strain ε .

where C is a constant and n is a material-specific exponent. In figure F3.0.5 a log—log plot of ∆ρ versus ε at 4.2 K is shown for two different aluminium qualities which have been strained up to about 1% at 4.2 K (Seeber et al 1997). Note that the n exponent is 1.0 and 1.2 for 4N0 and 4N8 aluminium respectively.

Figure F3.0.4. The increase of resistivity of copper as a function of deformation. The indicated temperatures are the temperatures for wire drawing. Reproduced from Broom (1952).

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Figure F3.0.5. The increase of the low-temperature (4.2 K) resistivity of two aluminium qualities as a function of uniaxial strain. The 4N0 (99.990%) aluminium is less stress sensitive then the higher purity 4N8 (99.998%) aluminium (Seeber et al 1997).

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The higher quality (4N8) aluminium is more strain sensitive than the less pure (4N0) one. In the literature one finds values of n for copper and aluminium between n = 1.4 to 1.8 and n = 1 to 1.3 respectively (Fickett 1982 and references therein). When cyclic stress is applied in the plastic regime of a material the resistivity increases as a function of the number of cycles. There is some confusion in the literature and one has first always to find out how stress cycles have been applied. Because of its importance this should be explained further. An important stress mode is the situation in a superconducting magnet (figure F3.0.6). When a magnet is charged to its maximum field, the electromagnetic forces also reach a maximum. If the design of the magnet is correct, these forces must be below a value which causes irreversible damage to the magnet. In the case of Nb—Ti, and also for Nb3Sn, at these forces the stabilizer may be deformed plastically (which is allowed up to a certain limit) and the resistivity of the latter increases. If the magnet goes back to zero field, there is a remaining plastic deformation and the stabilization of the magnet is reduced for further runs. As a consequence, the fatigue behaviour of materials used for stabilization must be measured as illustrated in figure F3.0.6. In the literature one also finds fatigue cycles where the yielded material is forced to go back to zero strain or even to negative strain (compressive strain). In such a case the material is alternately put under tensile and compressive strain and the increase of resistivity is faster than in the cycle shown in figure F3.0.6. As an example, the study of fatigue effects in an Nb—Ti multifilamentary superconductor is briefly discussed (Ekin 1978). An Nb—Ti wire with 180 filaments and 0.53 mm × 0.68 mm cross-section was uniaxially strained with a cyclic strain ∆ε (figure F3.0.6) between 0.53% and 0.88% until failure occurred (up to 105 fatigue cycles). Because the critical current Ic , did not show any fatigue influence but the resistivity of the copper matrix increased substantially, it could be shown that the limiting factor for failure is the matrix and not the superconductor. The situation is illustrated in figure F3.0.7 where the RRR is plotted versus the number of fatigue cycles for different cyclic strains. The original RRR of 70 is reduced

Figure F3.0.6. Stress-strain curve illustrating fatiguc cycles of a superconducting wire in a magnet. This mode has a constant cyclic stress.

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Figure F3.0.7. The RRR R(295 K)/R(9 K) of a multifilamentary Nb—Ti wire with a cross-section of 0.53 mm × 0.68 mm and with a copper-to-superconductor ratio α = 1.8. ∆ε is the peak-to-peak cyclic strain. Reproduced from Ekin (1978) by permission of Plenum Press.

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to 50 after 105 cycles with a peak-to-peak cyclic strain of 0.53%. With a cyclic strain of 0.88% only 103 cycles were possible before failure and the RRR was reduced to 20. A reduced RRR means also less stabilization of the wire and an increasing risk for damaging the magnet when a quench occurs. F3.0.3 Measurement of resistivity The simplest way to measure the electrical resistivity is to send a known d.c. current through a geometrically well defined sample and to measure the voltage drop. The resistivity can be obtained from

where V is the voltage, I is the current, A is the cross-sectional area of the sample and l is the distance between the voltage taps. The experimental situation resembles that of a critical current measurement (section B7.3). Because voltages are small, in the range of microvolts, care must be taken that no dis turbing noise is picked up (twisted leads) and thermally induced voltages are canceled out (change of current polarity). A convenient method is to ramp the sample current and to measure simultaneously the voltage. The current-voltage (I—V) curve is a straight line and the slope is the resistance of the sample. The data may be smoothed by a mean least-squares fit and accuracies < 1% can be obtained. If the whole system is in thermal equilibrium there are no time-dependent thermally induced voltages and any offset will stay constant during the ramping of the current. Because the slope is used for the determination of the resistivity, the always present voltage offset can be neglected. No cancellation of thermal voltages by changing the current polarity is necessary. Another possible way of measuring the resistivity is to induce a current into the cylindrical sample and to measure the decay of this current. The method, also known as the eddy current decay (ECD) method, is particularly suited to measure changes of the resistivity. The theoretical basis for this kind of measurement has been given by Bean (1959). The technique requires a small d.c. field applied to the sample. After switching off this field, eddy currents are induced and the decay is monitored, either with pickup coils or with Hall probes (Hartwig 1992). The obvious advantage is that no contacts are required and quite thick samples can be measured. If the temperature has to be changed, a possibility is to immerse the sample in a temperature bath of liquefied gases and slushes (solid—liquid mixture) (Fickett 1982). Maybe more convenient is to put the sample into a liquid-helium storage dewar and to position it slightly above the liquid. With an appropriate heater, temperatures up to room temperature can be obtained. A more elegant solution is the employment of a variable temperature insert (VTI) in a helium cryostat (see chapter D8). Samples should be mounted so that no strain is applied upon cooling. If the measurement is carried out in a magnetic field, the Lorentz force can also strain the sample. By an appropriate construction of the sample holder, Lorentz forces are taken over by a sample support structure and do not influence the measurement. Current contacts should be made ideally by soldering. However, pressure contacts can also be used where soldering is difficult or where the sample should not be heated. Voltage taps can be fixed by a conducting glue or, more simply, by silver paint. Regarding the selection of an appropriate temperature sensor, the reader is referred to chapter E4 of this handbook.

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Appendix A Resistivity of pure metals

Figure F3.0.A1. The resistivity of copper calculated from the Grüneisen function with ρ (273 K) = 1.553 × 10– 8 Ω m and θ = 330 K. The data for oxygen-free copper have been taken from Hust and Giarratano (1974) Reproduced from Fickett (1982) by permission.

Figure F3.0.A2. The resistivity of aluminium calculated from the Gruneisen function with ρ (273 K) = 2.428 × 10– 8 Ω m and θ = 400 K. The data for A11100 have been taken from Clark and Sparks (1973). Reproduced from Fickett (1982) by permission.

Figure F3.0.A3. The resistivity of silver. Data taken from Matula (1979).

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Electrical resistivity Table F3.0.A1. Resistivity of selected metals at the ice point (273 K).

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Appendix B Resistivity of alloys

Figure F3.0.B1. The resistivity of α brass, an alloy with copper and zinc for 300 K and 4.2 K. Data taken from Henry and Schröder (1963). Reproduced from Fickett (1982) by permission.

Figure F3.0.B2. The resistivity of the Cu-Ni alloy system for different temperatures (Iguchi and Udagawa 1975, Krupowski and Detlaas 1928, Legvold et al 1974, Svensson 1936). Reproduced from Fickett (1982) by permission.

Figure F3.0.B3. The resistivity of the Ag—Pd alloy system for 291 K and 4 K (Murani 1974, Svensson 1932). Reproduced from Fickett (1982) by permission.

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Electrical resistivity Table F3.0.B1. Electrical resistivity of alloys at low temperatures. Data from Clark et al (1970).

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Resistivity of alloys

1079 Table F3.0.B1. Continued.

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Electrical resistivity Table F3.0.B2. Low-temperature properties of resistance wires.

Appendix C Magnetoresistivity

Figure F3.0.C1. Kohler plot for the transverse magnetoresistivity of different metals. Note that the resistivity ratio is referred to the Debye temperature θ (Lüthi 1960).

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Figure F3.0.C2. Kohler plot for the longitudinal magnetoresistivity of different metals. Note that the resistivity ratio is referred to the Debye temperature θ (Lüthi 1960).

References

1081 Table F3.0.C1. Debye temperatures of different metals used in figures F3.0.C1 and F3.0.C2.

References Babic E, Ocko M and Rizzuto C 1975 The impurity resistivity of ZnFe alloys J. Low Temp. Phys. 21 243–55 Bardeen J 1940 Electrical conductivity of metals J. Appl. Phys. 11 88–111 Bean C P, DeBlois R W and Nesbitt L B 1959 Eddy-current method for measuring the resisitivity of metals J. Appl. Phys. 30 1976–80 Boato G, Bugo M and Rizzuto C 1966 The effect of transition-metal impurities on the residual resistivity of Al, Zn, In and Sn Nuovo Cimento B 45 227–40 Brändli G and Olsen J L 1969 Size effects in electron transport in metals Mater. Sci. Eng. 4 61–83 Broom T 1952 The effect of deformation on the electrical resistivity of cold worked metals and alloys Proc. Phys. Soc. B 65 871–81 Cavalloni C, Kwasnitza K and Monnier R 1983 Size effect of the longitudinal resistivity of multifilamentary superconducting wires Appl. Phys. Lett. 42 734–6 Chambers R G 1961 Magnetoresistance The Fermi Surface ed W A Harrison and M B Webb (New York: Wiley) pp 100–24 Cimberli M R, Michi V, Mori F, Rizzuto C, Siri A and Vaccarone R 1976 A search for low temperature stan dard resistance alloys Proc. 6th Int. Cryogenic Engineering Conf. (Guildford: IPC Science and Technology Press) pp 190–3 Clark A F, Childs G E and Wallace G H 1970 Electrical resistivity of some engineering alloys at low temperatures Cryogenics 10 295–305 Clark A F and Sparks L L 1973 The low temperature electrical resistivity of various purities of aluminum Bull. Am. Phys. Soc. 18 310 Cook J G, Van der Meer M P and Laubitz M J 1972 Thermal and electrical conductivity of sodium from 50 to 360 K Can. J. Phys. 50 1386–401 Corbett J W, Smith R B and Walker R M 1959a Recovery of electron irradiated copper. I. Close pair recovery Phys. Rev. 114 1452–9 —1959b Recovery of electron irradiated copper. II. Interstitial migration Phys. Rev. 114 1460–72 Ekin J W 1978 Fatigue and stress effects in NbTi and Nb3Sn multifilamentary superconductors Adv. Cryogen. Eng. 24 306–16 Feldman R, Talley L, Rojeski M, Vold T and Woollam J A 1977 Upper limit for magnetoresistance in silicon bronze and phosphor bronze wires Cryogenics 17 31–2 Fickett F R 1971 Aluminum 1. A review of resistive mechanisms in aluminum Cryogenics 11 349–67 —1972 A preliminary investigation of the behavior of high purity copper in high magnetic fields International Copper Research Association Report 186B

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—1982 Electrical properties of materials and their measurement at low temperatures National Bureau of Standards Technical Note 1053 Furukawa G T, Reilly M L and Saba W G 1964 Electrical resistance of wires of low temperature coefficient of resistance useful in calorimetry (10–380 K) Rev. Sci. Instrum. 43 113–4 Giauque W F, Lyon D N, Hornung E W and Hopkins T E 1962 Calorimetric determination of isothermal entropy changes in high magnetic fields at low temperatures. CoSo4.7H2O J. Chem. Phys. 37 1446–52 Hall L A 1968 Survey of electrical resistivity measurements on 16 pure metals in the temperature range 0 to 273 K National Bureau of Standards Technical Note 365 —1970 Survey of electrical pure metals in the temperature range 0 to 273 K National Bureau of Standards Technical Note 365–1 Hartwig K T, McDonald L C and Zou H 1992 Recent developments with the eddy current decay method for resistivity measurements Adv. Cryogen. Eng. Mater. 38 1169–75 Hedgcock F T and Muto Y 1964 Low-temperature magnetoresistance in magnesium and aluminum containing small concentrations of manganese or iron Phys. Rev. 134 1593–99 Henry W G and Schröder P A 1963 The low temperature resistivities and thermopowers of α-phase copper-zinc alloys Can. J. Phys. 41 1076–93 Hummel R E, Koger J W and Pasupathi V 1968 The effect of deformation on the martensitic transformation of beta brass Trans. AIME 242 249–53 Hust J G 1972 Electrical resistance ratios of Evanohm heater wire at low temperatures Rev. Sci. Instrum. 43 1387–8 Hust J G and Giarratano P J 1974 Thermal conductivity Semi-Annual Report on Material Research in Support of Superconducting Machinery NBSIR 74–394 (Gaithersburg, MD: National Bureau of Standards) pp 1–36 —1975 Thermal conductivity and electrical resistivity standard reference materials: electrolytic iron SRM’s 734 and 797, from 4 to 1000 K National Bureau of Standards Special Publication 260–50 Iguchi E and Udagawa U 1975 Analysis of Young’s modulus in the polycrystalline Cu-Ni alloy with the theory for Nordheim’s rule J. Phys. F: Met. Phys. 5 214–26 Keil D, Merbold U and Diel J 1974 Influence of annealing treatments on residual resistivity and the field dependence of the electrical resistance of niobium at 4.2 K Appl. Phys. 3 217–21 (in German) Kohler M 1938 Zur magnetischen Widerstandsánderung reiner Metalle Ann. Phys., Lpz. 32 211–8 Krupowski A and DeHaas W J 1928 The properties of the Ni-Cu alloys at low temperatures Commun. Leidena 194 1–12 Laubitz M J, Matsumura T and Kelly P J 1976 Transport properties of the ferromagnetic metals. II. Nickel Can. J. Phys. 54 92–102 Legvold S, Peterson D T, Burgardt P, Hofer R J, Lundell B, Vyrostek T A and Gartner H 1974 Residual resistivity and aging-clustering effects of Cu-rich Cu-Ni alloys Phys. Rev. B 9 2386–9 Lerner E and Daunt J G 1964 Magnetoresistance effect in advance and Evanohm wires at low temperatures in fields up to 50 kG Rev. Sci. Instrum. 35 1069–70 Lüthi B 1960 Widerstandsánderung von Metallen in hohen Magnetfeldem Helv. Phys. Acta 33 161–82 MacDonald D K Z and Sarginson K 1950 Size effect variation of the electrical conductivity Proc. R. Soc. A 203 223–40 Matula R A 1979 Electrical resistivity of copper, gold, palladium and silver J. Phys. Chem. Ref. Data 8 1147–298 Meaden G T 1965 Electrical Resistance of Metals (New York: Plenum) Murani A P 1974 Localized enhancement effects in Pd-Ag alloys Phys. Rev. Lett. 33 91–4 Nordheim L 1934 Die Theorie der thermoelektrischen Effekte Actual. Sci. Ind. 131 1–23 Reed R P 1972 Aluminium 2. A review of deformation properties of high purity aluminium and dilute aluminium alloys Cryogenics 12 259–91 Reich R 1966 Electric and superconducting properties of metals with different purities Mem. Sci. Rev. Met. 63 21–58 Seeber B, Erbüke L, Flükiger R and Horvath I L 1997 RRR-measurements of high purity aluminum under static and dynamic mechanical stress Proc. Int. Conf on Magnet Tech. MT15 (Beijing, 1997) to be published Svensson B 1932 Magnetic susceptibility and electrical resistivity of the polycrystalline (alloy) series Pd-Ag and Pd-Cu Ann. Phys., Lpz. 14 699–711 (in German) —1936 Ferromagnetic resistivity increase of copper-nickel alloys Ann. Phys., Lpz. 25 263–71 Woollam J A 1970 Evanohm resistance to 18 T (180 kG) at 4.2 K Rev. Sci. Instnum. 41 284–5

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F4 Materials at low temperatures: thermal conductivity B Seeber and G K White

F4.0.1 Introduction The thermal conductivity of materials at low temperatures is quite an important subject if, for instance, one is interested in minimizing heat leaks in cryogenic installations in order to reduce the boil-off of cryogens. By appropriate measures, where the conduction of heat is an important factor, cryostats may stay cold for a year or more without refilling with liquid helium. Low thermal conductivity is observed in alloys and in many nonmetallic materials. In contrast, there may be situations where a high thermal conductivity at low temperatures is required, e.g. to reduce temperature gradients; here high-purity metals or sapphire are suitable. This chapter gives an introduction to thermal conductivity in solids at low temperatures. It should provide the reader with some basic mechanisms concerning the conduction of heat and a minimum of knowledge regarding measurement techniques. In an appendix data are compiled for a number of materials of technological importance. F4.0.2 Conduction heat flow The conduction heat flow into a cryogenic installation by thermal conductivity can be expressed for a small temperature difference as follows

where Q• is the heat flow in watts, λ the thermal conductivity in W m–1 K–1, A the cross-sectional area in m2 and L the length in metres of the considered conductor. ∆T is the temperature difference between both ends of the conductor. However, for a usual temperature difference (e.g. 300 K and 4.2 K) the total conduction heat flow must be calculated by using thermal conductivity integrals. This is required because the thermal conductivity λ depends strongly on the temperature

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where θι = ∫ ΤιΤο λ dΤ is the thermal conductivity integral (Garwin 1956). If the cross-sectional area is not constant over the length of the conductor, the prefactor A/L must be replaced by ∫x2x1 dx /A. In the following, a brief introduction to the physics of thermal conductivity in metals, in nonmetallic materials and in superconductors is given. The interested reader, who wishes to go into detail, is referred to the further reading list at the end of the chapter. F4.0.3 Basic mechanisms As in the case of electrical conductivity, the thermal conductivity is the sum of contributions from different carriers, e.g. electrons, phonons, magnetic excitations etc. This can be expressed by the following equation

The subscript i denotes the type of carrier, C i is the specific heat, vi is the velocity and l i is the mean free path. The main question is what carriers are active and dominate in different materials. For instance, the most important carriers for the transport of heat in a metal are conduction electrons. In a nonmetallic material, however, there are no conduction electrons and heat is conducted by phonons. Depending on the temperature, the scattering processes involved with conduction electrons are different from those involved with phonons, yielding a strongly nonlinear behaviour of λ at low temperatures. From a qualitative point of view these mechanisms are well understood, although quantitative calculations are difficult. F4.0.3.1 Metals The thermal conductivity of a material can essentially be described by the following equation

where λe and λ p h are the thermal conductivities of electrons and phonons, respectively. In metals which are good conductors, λe » λp h so that the phonon term can be neglected. Then the conductivity can be expressed as

where W r is the thermal conductivity due to scattering of electrons by impurities, dislocations etc and is related to the residual electrical resistivity ρr (see also chapter F3 and Hust and Sparks 1973) by the Wiedemann—Franz—Lorenz law

Wi is the thermal resistivity due to scattering of electrons by lattice waves (phonons). At low temperatures, say T < θ/5 (where θ is a Debye characteristic temperature, Wi ≈ αT 2 where α is a constant specific to each material. Thus at low temperatures

According to this relation one expects a linear dependence of λ with T at low temperatures. At higher temperatures the T 2 term is dominant which yields a decrease of λ with increasing temperature. The situation is illustrated schematically in figure F4.0.1. Note that there is a peak in the thermal conductivity. On the left side of the peak λ increases linearly with temperature due to the scattering of conduction

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Figure F4.0.1. Thermal conductivity versus temperature for a metal. The carriers responsible for the transport of heat are electrons. At low temperatures λ ∝ T due to the scattering of conduction electrons at impurities. For a high-purity metal with high residual resistivity ratio, λ is high (λ1 ). For a less pure metal λ. is reduced (λ2 ). With increasing temperature, conduction electrons start to be scattered by phonons and l decreases as 1/ T 2 and becomes nearly constant at high temperatures (T ≥ θ/2).

electrons by impurities. This corresponds to the case of a (constant) residual electrical resistivity. For a given metal, the batch with λ1 has a lower residual resistivity than the one with λ2. With increasing tem perature, conduction electrons are scattered by phonons and the α T 2 term is responsible for the decrease in λ. The Wiedemann—Franz—Lorenz law relates the electrical conductivity of a metal to the thermal conductivity in a simple way (Lorenz 1881, Wiedemann and Franz 1853)

where λ is the thermal conductivity, σ the electrical conductivity, T the temperature, kB Boltzmann’s constant (1.38 × 10 –23 J K–1 ), e is the electrical charge of the electron (1.6 × 10–19A s) and L 0 is the Lorenz number. This allows an estimate of the thermal conductivity when the electrical conductivity is known. At low temperatures σ = 1/ρr is constant and λ varies linearly with temperature ( ρr is the residual resistivity). At higher temperatures, T > θ , Wi ≈ ρ i /LT ≈ constant and the Lorenz number, L , approaches the theoretical Sommerfeld value L 0 for many metals (see e.g. Hust and Sparks 1973 or Landolt—Börnstein 1991). However, between these limiting cases, λ does not follow the behaviour of σ . This can be described by a temperature-dependent Lorenz number (Hust and Sparks 1973). As shown in figure F4.0.2, where λ/σ T is plotted against T/θ, the cleaner the material (high residual resistivity ratio) the more L deviates from L0 . Table F4.0.1 shows some approximate values for the Debye temperature, based on the heat capacity observed near room temperature. They are generally different from the very low-temperature values of θ (see chapter F5 and compilation by Gschneidner 1964). If a magnetic field is applied at low temperatures (left side of the thermal conductivity peak), the residual resistivity increases as does the thermal resistivity. The situation is similar to that for the electrical resistivity. If the magnetic field is longitudinal (in the direction of current or heat flow) the thermal

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Figure F4.0.2. The temperature-dependent Lorenz ratio λ/σ T as a function of the reduced temperature T/θ where θ is the Debye temperature (after Hust and Sparks 1973).

Table F4.0.1. Debye temperatures of different metals.

resistivity saturates as a function of field, as is the case for the electrical resistivity. In contrast, in a transverse field the thermal resistivity has a B 2 dependence at low fields which changes at higher fields to either a linear B dependence or to saturation. Experimental data are sparse so in particular cases one has to estimate the thermal resistivity with the help of the Wiedemann—Franz—Lorenz law or carry out the measurement. Note that the Lorenz number itself is field dependent (Sondheimer and Wilson 1947, Wilson 1953). F4.0.3.2 Nonmetallic materials In nonmetallic materials there are no conduction electrons to carry heat and the main carriers are phonons. Thus one has to take into account the interaction of phonons with boundaries (at low temperatures), with defects (intermediate temperatures) and with other phonons (high temperatures). The situation can be described qualitatively with the help of equation (F4.0.3). First, with respect to other variables, the mean phonon velocity υ can be considered to be temperature independent. Because at low temperatures phonons with high frequencies are frozen out, one has to deal with low-frequency phonons with a long wavelength. This means that the mean free path of these phonons can be several millimetres long and λ depends of the boundaries of the sample; in the case of a single crystal, this is the shape and the size of the

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crystal. In polycrystalline materials the boundaries limiting thermal conductivity can be grain boundaries, precipitations, porosity etc. The other term influencing λ is the specific heat of the lattice which varies approximately as T 3 at the lowest temperatures i.e. where T ≤ θ/10 and becomes roughly constant for T > θ. At the lowest temperatures where boundary scattering is dominant, the mean free path is constant and therefore λ increases with temprature roughly as T 3 until dislocations, point defects (including isotopes) etc limit the conductivity (figure F4.0.3). At temperatures above the maximum, anharmonic interactions between the phonons (or lattice waves) cause λ to decrease. Here λ may fall exponentially because of the changing probability of effective (Umklapp) scattering of one phonon by another. The maximum at the peak and the exponential fall depend partly on the size of the sample and the presence or absence of point defects (including isotopes). A further increase in temperature produces increasing phonon-phonon interaction and the mean free path varies as l ∝ 1/T. Because in this region the specific heat is nearly constant, λ also varies as approximately 1/T.

Figure F4.0.3. Thermal conductivity versus temperature for a nonmetallic material. The carriers responsible for the transport of heat are phonons. At very low temperatures λ is governed by the T3 dependence of the specific heat and the scattering of long-wavelength phonons on boundaries (size of the crystal, grain boundaries in a polycrystalline material etc). The scattering at boundaries is also reflected in the height of the peak. The size of the crystal is greater in the case of λ1 than in the case of λ2 . At medium temperatures λ is determined by phonon scattering at defects and at high temperatures by phonon-phonon interactions.

F4.0.3.3 Superconductors In a superconductor one has to distinguish between the normal state (T > Tc ) and the superconducting state (T < Tc ). In the normal state of a metallic superconductor the behaviour of the thermal conductivity is that of a metal which means that the transport of heat is governed by conduction electrons. When the temperature goes below the critical temperature of the superconductor, conduction electrons form Cooper pairs and are no longer available as carriers for the thermal conductivity. Because Cooper pairs cannot transport heat, and the number of conduction electrons is strongly reduced, the electronic thermal conductivity of a superconductor below Tc goes exponentially to zero. However, the phonon component λp h remains and varies as T 3 as T → 0. So at the lowest temperatures in the superconducting state λs ≈ λp h ∝ T 3, while in the normal state λ n ≈ λe ∝ T. This difference is useful for the construction of thermal switches. For example, wires of Pb, Sn, Al are used as switches at temperatures below 1 K with

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conductivity ratios λn /λs of 1000 or more. One can switch between the superconducting and the normal state by applying a small magnetic field (see e.g. Berman 1976, p 168). Because phonons continue to conduct heat in the superconducting state and because they suffer less scattering by electrons as T falls below Tc , there may be a peak in heat conductivity below Tc due to λ p . This occurs in metallic superconductors such as Pb and Pb—Bi alloys where Tc is not too small comh pared with θ (Berman 1976, p 167). The heat conductivity in high-Tc cuprate superconductors behaves more like that of disordered ceramics than of metals. Their conductivities above Tc are largely (≥ 80%) due to phonons, at least for polycrystalline and sintered samples. In good single crystals, the measured conductivity λ a b in the ab (CuO) plane indicates that λ e and λp h are comparable above Tc . Below Tc , λ increases significantly to a maximum near Tc /2. For a single crystal of YBa2Cu3O7– x , this in crease may be by a factor of 2 to 3, from say 10 W mK–1 at 92 K to 25 W mK− 1 at 40–50 K. This rise has aroused considerable interest as to whether it is entirely due to the phonons being scattered less by the charge carriers in the superconducting state (Cooper pairs) or whether the charge carriers behave as quasi-particles (normal electrons) and are partly responsible for additional conduction (Uher 1992). F4.0.4 Measurement of the thermal conductivity The simplest method to measure thermal conductivity is the linear heat flow method based on equation (F4.0.1). A schematic set-up for such a measurement is shown in figure F4.0.4. A sample with a uniform cross-sectional area (e.g. cylindrical) is attached to an electrical heater which generates a well defined flow of heat Q• into the sample. At the other end, the sample is connected to a heat sink. After thermal equilibrium (steady state) is reached the measured temperature difference directly gives the thermal conductivity

Care has to be taken that the whole generated heat goes into the sample and the isothermals are planes perpendicular to the length of the sample. The most important source of errors is heat which leaves the sample by radiation and/or by conduction through attached leads (thermometers, heater), so that the isothermals are no longer planes. This is normally not a real problem for measurements below 100 K

Figure F4.0.4. The measurement of thermal conductivity by the linear heat flow method.

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Metals and alloys

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when the sample is thermally isolated from its surroundings by a vacuum. For temperatures up to room temperature, radiation losses are estimated either by using different L/A ratios and subsequent correction of the data or by a radiation shield with a temperature gradient similar to that of the sample. The temperature distribution of the shield is controlled by separate heaters. Finally, the measured temperature gradient, ( T1 – T2 ), should be small compared with T because of the strong temperature dependence of λ. The method is well suited for good conductors and for temperatures below ~300 K. An overall accuracy of about 1% can be achieved. For poor thermal conductors the above-described method is unsuitable because of prohibitively long time constants for thermal equilibrium and the large temperature gradients involved. For these reasons one is interested in working with as short a sample as possible and a length-to-diameter ratio of less then one is common. A standard method is the so-called guarded-hot-plate method. The method is described in ASTM C177 (1997) and schematically shown in figure F4.0.5. The sample, consisting of two equivalent parts, is fixed on both sides of a guarded heater. The other sides of the sample are connected to a heat sink and the whole is surrounded by a thermally insulating material. The heat sinks can be controlled, depending on the investigated temperature range, by cryogens. The guarded heater is adjusted by two independent power supplies so that there is no radial heat flow. Then λ can be measured according to equation (F4.0.1) where Q• is the heat flow provided by the main heater, A is the cross-sectional area of the heater and half that of the gap between the main heater and the guard, L is the total thickness of both samples and (T1 – T2 ) is the temperature gradient across the thickness of one sample. For temperatures above ambient, non-steady-state methods have been developed. They involve the full differential equation for heat flow and essentially measure the thermal diffusivity. To extract λ, the specific heat and the density of the material must be known. Because these methods are less relevant for low temperatures, they are not further discussed. A quite detailed description can be found in the book by Parrott and Stuckes (1975) and references therein.

Figure F4.0.5. The measurement of the thermal conductivity by the guarded-hot-plate method.

Appendix A Metals and alloys In the following figures thermal conductivity data have been compiled for metals and alloys of technological importance. Note the general sensitivity to impurity content at low temperatures, which is manifested by the residual resistivity ratio (RRR). This is explicitly shown in figures F4.0.A1–F4.0.A3 for copper, aluminium

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Figure F4.0.A1. The thermal conductivity of copper for different RRRs. Data from the copper data sheet of the International Copper Research Association.

Figure F4.0.A2. The thermal conductivity of aluminium for different RRRs. The RRRs have been calculated from ρ (273 K) = 2.428 µ Ω cm and the reported residual resistivities. Data from Touloukian et al (1970a).

Figure F4.0.A3. The thermal conductivity of silver for different RRRs. Data from Ho et al (1974).

and silver respectively. In figure F4.0.A4 other metals are summarized and their purities are defined by their electrical residual resistivity, as well as by the calculated RRR. The thermal conductivity of alloys is much less sensitive to impurities and the given data in figure F4.0.A5 do not vary much with the impurity content. There is a small difference in λ between the stainless steels AISI 304 and AISI 347, but this is not shown. The behaviour of other stainless steels is similar.

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Figure F4.0.A4. Examples of thermal conductivity curves for different metals, the effective purities being given by the values of residual electrical resistivity or RRR = ρ2 7 3 K/ρr . Typical errors are in the range of ±10%. Data from Touloukian et al (1970a).

Figure F4.0.A5. The thermal conductivity for different alloys. Typical errors are in the range of ±10%. Stainless steel (AISI 304, AISI 347), manganin (84 Cu, 12 Mn, 4 Ni, Cu + 2 Be ( ρr , = 5.54 µΩ cm)) data from Touloukian (1970a). Al 6063 (0.6 Mg, 0.4 Si), Cu + 30 Zn (brass), Cu + 30 Ni, constantan, inconel data from White (1987).

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Appendix B Nonmetallic materials In figure F4.0.B1 the thermal conductivity of sapphire (Al2O3 ) is depicted because of its high value for a nonmetallic material. The data have been obtained from a high-purity sapphire single crystal with the heat flow at 60 ° to the hexagonal axis. Note that the height of the peak in λ depends on the size of the crys tal, as shown schematically in figure F4.0.3. Other nonmetallic and amorphous materials are summarized in figure F4.0.B2. As in alloys, λ does not depend very much on the impurity content.

Figure F4.0.B1. The thermal conductivity of a (nonmetallic) high-purity sapphire single crystal with the heat flow at 60° to the hexagonal axis (Touloukian 1970b).

Figure F4.O.B2. The thermal conductivity of a few nonmetallic and amorphous materials: quartz glass, nylon and Plexiglas (Berman 1979); G10 (Kasen et al 1980) and Teflon (White 1987).

References ASTM 1997 Standard test method for steady-state heat flux measurements and thermal transmission properties by means of the guarded-hot-plate apparatus C177-97 pp 1–22 Berman R 1976 and 1979 Thermal Conduction in Solids (Oxford: Clarendon) Childs G E, Erichs L J and Powell R L 1973 Thermal Conductivity of Solids at Room Temperature and Below (National Bureau of Standards (NBS) Monograph 131) (Gaithersburg, MD: National Bureau of Standards) Garwin R L 1956 Calculation of heat flow in a medium the conductivity of which varies with temperature Rev. Sci. Instrum. 27 826–8 Gschneidner K A 1964 Physical properties and interrelationships of metallic and semimetallic elements Solid State Physics vol 16 (New York: Academic) pp 275–426 Ho C Y, Powell R W and Liley P E 1972 Thermal conductivity of the elements: a comprehensive review J Phys. Chem. Ref. Data 1 279–421 Hust J G and Sparks L L 1973 Lorenz ratios of technically important alloys Natl Bureau Standards Tech. Note 634 Kasen M B, McDonald G R, Beekman D H Jr and Schramm R E 1980 Mechanical electrical and thermal characterization of G10CR and G11CR glass cloth/epoxy laminates between room temperature and 4 K Adv. Cryogen. Eng. 26 235–44 Klemens P G 1958 Thermal conductivity and lattice vibrational modes Solid State Physics vol 7 (New York: Academic) pp 1–98 Klemens P G and Williams R K 1986 Thermal conductivity of metals and alloy Int. Met. Rev. 31 197–215 Landolt-Börnstein 1991 Thermal Conductivity of Pure Metals and Alloys Group III, vol 15c, ed O Madelung and G K White (Berlin: Springer)

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Further reading

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Lorenz L 1881 über das Leitvermgen der Metalle für Wärme and Elektrizitat Ann. Phys., Lpz. 13 422–46 Sondheimer E H and Wilson A H 1947 The theory of the magnetoresistance effect in metals Proc. R. Soc. A 190 435–55 Touloukian Y S, Powell R W, Ho C Y and Klemens P G 1970a Thermophysical Properties of Matter, Vol. 1: Thermal Conductivity—Metallic Elements and Alloys (New York: IFI-Plenum) Touloukian Y S, Powell R W, Ho C Y and Klemens P G 1970b Thermophysical Properties of Matter, Vol. 2: Thermal Conductivity—Nonmetallic Solids (New York: IFI-Plenum) Uher C T 1992 Thermal conductivity of high temperature superconductors Physical Properties of High Temperature Superconductors vol 3, ed D M Ginsberg (Singapore: World Scientific) pp 159–284 Wiedemann G and Franz R 1853 Über die Wärmeleitungsfähigkeit der Metalle Ann. Phys., Lpz. 2 497–531

Further reading Parrott J E and Stuckes D 1975 Thermal Conductivity of Solids (London: Pion) Tye R P 1969 Thermal Conductivity vol 1 and 2 (London: Academic) White G K 1987 Experimental Techniques in Low-Temperature Physics 3rd edn (Oxford: Clarendon) Wilwon A H 1965 The Thoery of Metals 2nd edn (Cambridge: Cambridge University Press)

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F5 Specific heat G K White

F5.0.1 Introduction The specific heat or heat capacity of a material is denoted by c (per unit mass) or C (per mole) and is the quantity of energy needed to change the temperature of the material by one degree. The calorie as a unit of heat will still be familiar to many and was defined as the quantity of heat or energy needed to warm up a gram of water by one degree Celsius. With the introduction of the International System of Units (SI) the calorie has been officially superseded by the joule and 1 calorie = 4.184 joule. At normal and higher temperatures, values of specific heat for a fixed number of atoms (e.g. for one gram atom of N = 6.02 × 1023 atoms) are rather similar for most solids whether they are copper or rocksalt or a niobium—titanium alloy. The original observation that C ≈ 25 J g-at–1 K–1 for many solids at ambient temperature was made by Dulong and Petit in 1819. The heat capacity per unit volume also does not differ by more than a factor of two for most solids because the volume per atom is roughly the same. The similarity of specific heat values at high temperature arises because of the internal energy content and does not apply to many other physical properties which may vary by an order of magnitude. In the low-temperature limit as T approaches zero, the specific heat must also approach zero but can vary greatly from one material to another in the low-temperature region above zero. The rate of approach to zero has important consequences in cryogenic design as the heat content and thermal diffusivity (ratio of thermal conductivity to heat capacity) are important in designing a cooling system. For example, the quantity of liquid helium required to cool a kilogram of copper is many times less than that required to cool a kilogram of lead and many times more than needed to cool a kilogram of alumina. F5.0.2 Theory F5.0.2.1 Lattice specific heat Most of the thermal energy U of a solid is associated with the vibrations of the atoms and in the case of molecules this may include rotations. The specific heat at constant volume, CV = dU/dT, measures the energy needed to excite these vibrations. At sufficiently high temperatures, all the modes of vibration are excited and CV approaches the Dulong and Petit value of 3Nk = 3 R per N vibrating atoms where k is Boltzman’s constant and R = 8.314 J K–1 mol–1 is the gas constant.

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Specific heat

At normal and lower temperatures, Cv may be represented fairly well by the Debye function

The derivation of this formula is based on Debye’s elastic continuum model of a solid for which the number of modes of vibration in a frequency interval v , v + dv is given by

V is the volume of the solid and υl , υt , are wave propagation velocities for longitudinal and transverse waves respectively. The maximum frequency vm a x is given by the normalizing condition that in a volume V containing N atoms, there are only 3N modes

and the Debye characteristic temperature

Tabulations of the Debye integral appear in many standard texts such as those by Gopal (1966), Cezairliyan (1988) and Furukawa et al (1972). The integral reduces to CV → 3R = 24.94 J mol−1 K−1 for T > ΘD and to CV = 1943.8(T/ΘD )3 J mol−1 K−1 for T ≤ ΘD /25. In practice, a solid does not behave as an elastic continuum except in the low-frequency (very lowtemperature) limit. The true frequency spectrum of lattice modes is more complex than the Debye model with various peaks corresponding to different transverse and longitudinal acoustic vibrations as well as possible optical modes. Generally the transverse modes have lower frequencies than longitudinal modes and therefore are more heavily weighted at low temperatures. At the lowest temperatures, the so-called Debye T 3 region, the value of ΘD is usually denoted by Θ0 and should be the same as that calculated from the measured ultrasonic velocities. Note that the ‘characteristic’ temperature, Θ, does not have a single unique value for a given material. This is a frequent source of confusion: different values are appropriate to the particular properties which they seek to describe. They represent different weighted averages over the frequency spectrum, whichever is appropriate to describing that property. For example, different values of Θ are necessary to describe CV at low temperatures, CV at high temperatures, the entropy function S(T), the x-ray Debye—Waller factor, the Bloch—Gruneisen equation for electrical resistivity, etc. Values of ΘD calculated from experimental specific heat data (figure F5.0.1) illustrate the extent to which the Debye model departs from reality. For open (low coordination number) structures such as zincblende, departures are very marked at temperatures in the range from Θ/20 to Θ/10 (e.g. Sn in figure F5.0.1). Turning briefly to the effect of impurities and strain, it is important to note that these should not have much effect on the lattice energy of the crystal and therefore on specific heat except in a few circumstances. The exceptions include some magnetic impurities such as manganese, chromium and iron which can have a significant effect on the specific heat at low temperatures where lattice energy is small. Generally speaking, impurities have a much greater effect on transport properties than on static properties because they intefere with the regular pattern of the crystal lattice and cause additional scattering of charge carriers. F5.0.2.2 Electronic specific heat In metals the free electrons contribute to the specific heat. Their contribution, Ce , is proportional to T, at least at low temperatures, and is denoted by Ce = γ T. This T term is easily distinguished from the

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Theory

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Figure F5.0.1. Variation of ΘD with temperature for some solids (after White 1979).

more rapidly changing T 3 lattice component at low enough temperatures. γ is a measure of the electron density of states at the Fermi surface. Measurements show that γ varies from about 0.6 mJ mol−1 K−2 for copper, silver and gold to ∼10 mJ units for some transition metals and more than one hundred millijoules for heavy-fermion (or heavy-electron) metals such as UPt3 , CeCu6 , etc. We shall see below (section F5.0.4) that in superconducting materials the electron contribution is expected to decrease below the transition temperature as the charge carriers ‘condense’ into the ground state. At normal temperatures Ce is small relative to the lattice term, Cl . F5.0.2.3 Superconductors The superconducting transition (in zero magnetic field) is a second-order process; there is no latent heat but there is a change in the temperature derivative, dS/dT, of the entropy and an associated λ-shaped anomaly in specific heat. The charge carriers ‘condense’ gradually on cooling below Tc to a ground state such that at T = 0 they have zero entropy and zero specific heat. In a superconductor the lattice spectrum and associated specific heat do not differ measurably from their values in the normal state. This is reflected in the very small difference in dimensions and elastic moduli between the normal and superconducting states—of the order of parts per million. The Bardeen—Cooper—Schrieffer (BCS) theory attributes superconductivity to the coupling of electron pairs produced by electron-phonon interaction (e.g. Phillips 1971). The thermodynamics of the superconducting transition depends on the difference between the electronic specific heat in the normal state, Ce n , and that in the superconducting state, Ce s . The theory indicates that at Tc there is a second-order

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Specific heat

transition with change in Ce of

There is an energy gap at the Fermi surface which increases from 2∆(T) = 0 at Tc to a maximum value of 2∆(0) = 3.52kTc at T = 0. The value of 3.52kT is that for the weak-coupling limit. The pairing process and energy gap leads to an exponential decay in Ce s given approximately by

over limited temperature intervals. In so-called type I superconductors (which includes most of the elemental superconductors such as Pb, Sn, In, Ta), a magnetic field exceeding the critical field Hc completely penetrates the solid and destroys the superconductivity in a first-order transition with latent heat and small discontinuity in volume. However, most alloys are type II in which partial field penetration begins at a lower critical field Hc l but there remains a ‘lattice’ of superconducting and normal regions within the solid until the field exceeds the upper critical field Hc 2. There are consequences for the magnetization and thermodynamics which are discussed by Hake (1969). In section F6.0.4.4 there is a discussion of the behaviour of the expansion coefficient at Tc . For a second-order phase transition, we expect a discontinuity in the volume coefficient of thermal expansion, β, related to that in the specific heat by the Ehrenfest equation (see e.g. Shoenberg 1952)

Measurements of these discontinuities can give useful information about the pressure dependence of Tc which may be difficult to measure directly for some materials. In particular for anisotropic crystals, the differences in linear expansion coefficient, αa, b, c, along the a, b, c axes can give values for the uniaxial stress derivatives of Tc . F5.0.3 Experimental methods The specific heat is usually measured at constant pressure and denoted by CP = dH/dT, where enthalpy H = U + PV. CP is larger than CV by an amount equal to the external work required to expand the solid during heating. The differences between CP and CV depend on the thermal expansion coefficient (β) and compressibility (χ) and are negligible at low temperatures but may reach a few per cent at room temperature

where V is the molar volume. Various methods of measuring specific heat, suited to different temperature regimes and materials, are described by Anderson (1988), Martin (1988), Westrum (1988) etc in Ho and Cezairliyan (1988). The most common methods used at low temperatures are variants of the adiabatic calorimeter pioneered by Nernst early in the century. A schematic diagram in figure F5.0.2 illustrates the salient features. They include good thermal isolation of the sample which is supported in high vacuum by low thermal conductivity ‘threads’ (e.g. nylon) and has a small ‘addendum’ comprising thermometer T and electric heater H which are thermally anchored to the sample. There may be an additional vacuum enclosure (or radiation shield) to improve the therma isolation. This shield can be controlled at temperatures close to that of the sample. Measurements are made by supplying a heat pulse Q through the electrical heater to the sample sufficient to raise the temperature by an amount ∆T which is large enough to be measured accurately (error of < 1%) but small compared with the temperature of the sample. Usually ∆T is 1% or 2% of T. The efficiency of the isolation and thermal contact to heater and thermometer are judged by the stability or ‘drift’ rate

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Experimental methods

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Figure F5.0.2. Schematic diagram of an adiabatic calorimeter for low temperatures. H and T are heater and thermometer respectively and are thermally anchored to the sample.

of the sample temperature before and after the heat pulse. With suitable thermometry (e.g. germanium or platinum resistance thermometers, thermocouples) values of CP = Q/∆T should have errors of order of 1% or less. A test of the reliability is to measure a sample of high purity (99.999%) copper and compare it with ‘standard’ data (e.g. Marsh 1987 or Martin 1987). An alternative to this method of adiabatic calorimetry by intermittent heating is the ‘continuous heating’ method in which a fixed heating rate is used and the temperature is recorded at regular intervals. The specific heat is then obtained by differentiating the heating curve. This method lends itself to automation but for high accuracy depends on isothermal conditions within the sample-heater-thermometer assembly. That is, the heat transfer must be adequate to achieve thermal equilibrium at a rate much faster than the heating rate. These conditions may be met at low temperatures but are difficult to achieve at normal temperatures where larger values of specific heat mean longer relaxation times. Also radiation transfer can introduce serious errors unless thermal shields are very well controlled. Transient methods which are more suited to very small samples, thin films and ultra-low temperatures have become more popular with the advent of high-speed recording and computation. They include (i)

the temperature-wave method in which a sinusoidal heat source is applied to a rod-shaped sample—the temperatures measured at two points give values of both specific heat and thermal conductivity; (ii) pulse methods for which the time-rate of decay of temperature can give the specific heat (e.g. Anderson 1988, Martin 1988).

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Specific heat

Superconductors pose no special problems for calorimetry except in the region close to Tc where Cp may be changing rapidly with T and high-resolution thermometry is needed. Detailed study of orderdisorder transitions generally requires very sensitive thermometers with microdegree resolution. F5.0.4 Observations F5.0.4.1 Behaviour of nonsuperconducting materials Figure F5.0.3 shows the variation of CV with temperature of an insulating crystal and of three common metals. The insulating crystal, alumina or sapphire, has a relatively high characteristic temperature, ΘD (∼1000 K) so that the modes of vibration of the crystal are not all excited until T exceeds 1000 K above which CV approaches the Dulong and Petit value of 5 × 24.94 = 125 J mol−1 K−1 l well above 1000 K. The value measured directly (at constant pressure), CP , exceeds CV for sapphire by less than 1% at room temperature and about 10% at 2000 K.

Figure F5.0.3. Plot of CV (T) at high temperatures for an insulating crystal (alumina) and for copper, tungsten and molybdenum.

The metals in figure F5.0.3 have Debye temperatures between 300 and 400 K. For the two transition elements, Mo and W, the contributions of the electrons increase CV significantly above the Dulong and Petit value at temperatures above 1000 K. At low enough temperatures, it is usual to separate out the electronic term by plotting C/T against T 2 so that the intercept on the y axis (see data for copper in figure F5.0.4) gives the coefficient of the electronic term, Ce = γT while the slope gives the coefficient of the T 3 or lattice term, i.e. Cl = n × 1943.7(T/Θ0 )3 where n is the number of atoms per formula unit, namely n = 1 for copper and n = 2 for NaCl. Obviously for the nonmetal, NaCl, the intercept on the y axis is at 0. Since the electronic coefficient γ is a measure of the electron density of states at the Fermi surface, N(EF ), changes of the Fermi surface with alloying will change the electronic specific heat. As we move along rows of the periodic table, γ or N(EF ) oscillates with change in the electron/atom ratio (e.g. Heiniger et al 1966). The lattice energy and Debye ΘD are not very sensitive to small changes in composition unless there is a phase change or significant density change involved. The density and elastic moduli are good guides

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Observations

Figure F5.0.4. Plot of C/T versus T (NaCl).

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2

at low temperatures for a typical metal (copper) and an insulating crystal

to the Debye temperature. Anderson (1963) gives a simple method of calculating a Debye temperature (Θ0 ) from the mean ultrasonic velocity

where h is Planck’s constant, N is Avogadro’s number, ρ is density, M is the molecular weight and q is the number of atoms in the molecule. The average sound velocity υm for an isotropic or polycrystalline sample approximates to the weighted mean of the transverse and longitudinal wave velocities, υt and υl

The importance of being able to estimate the Debye temperature is that we can then estimate the rate of change of specific heat, internal energy and enthalpy at low temperatures, all of which are important factors in cryogenic design, whether for estimating cooling times or required amounts of refrigerant. F5.0.4.2 Superconducting metals Figure F5.0.5 illustrates a typical behaviour of a superconducting metal, vanadium, for which Tc , = 5.3 K. The critical magnetic field needed to destroy superconductivity for metallic elements is ≤ 0.1 T so that the specific heat in the normal state, Cn (shown in the figure), is easily measured (in a magnetic field) as well as Cs (in zero field). Figure F5.0.6 shows data for an alloy system of Ti—Mo which has been studied extensively (see Collings 1986a, vol 1, p 316). Less extensive measurements on other important alloys of niobium, titanium, etc are discussed by Collings (1986a, vol 1, ch 8). The difference in specific heat between normal and superconducting states is of fundamental interest in determining energy gaps, entropy change and an appropriate theoretical model rather than being of practical concern. However, the total specific heat is of practical concern in determining cooling costs (see the example below) and can be of importance in stabilizing the temperature of superconducting coils (e.g. see discussion by Collings 1986b, p 221).

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Specific heat

Figure F5.0.5. Variation of specific heat for vanadium in normal (Cn ) and superconducting (Cs ) states (after Gopal 1966).

Figure F5.0.6. Low-temperature specific heat data for quenched Ti-Mo alloys plotted as C/T versus T2 (Collings 1986a, p 317).

As an example of cooling requirements, consider cooling a 10 kg magnet coil of Ti-Nb (50:50 composition by weight) from 300 to 78 K with liquid nitrogen and from 78 to 4 K with helium. For an accurate calculation we need to know the difference in enthalpy or integrated CP over the relevant temperature ranges. For an approximation, we can use the difference in internal energy U obtained from integrating the Debye function for CV . Tabulations of U (Debye)/3RT as function of Θ/T are given by Gopal (1966), Furukawa et al (1972), White (1979), etc.

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In this example, 10 kg of Ti-Nb is equivalent roughly to 100 mol of Ti plus 55 mol of Nb. If we assume that ΘD ~ 300 K, then at T = 300 K the Debye tables give U/3RT = 0.674 per mole, and at 78 K U/3RT = 0.195 per mole, therefore the difference in energy is given by (0.674 – 0.195) × 3 × 8.31 × (300 – 78) × 155 = 0.41 MJ. The latent heat of liquid nitrogen is 0.16 MJ per litre so that if only the latent heat were used for cooling, this would need about 2.5 1 of nitrogen. For further cooling to 4 K, the energy to be removed from the coil is about 0.195 × 3 × 8.31 × 74 × 155 = 55 kJ. The latent heat of liquid helium is 2.6 kJ 1−1 therefore about 20 1 of helium would be needed if only latent heat were used. In practice, by careful cooling, the specific heat of the evaporating helium gas can be used and this amounts to about 5.2 J g−1 K−1 corresponding to 650 J K−1 for gas produced by 1 1 of liquid or nearly 50 kJ for the temperature interval of 74 K. So total usage of liquid helium might be reduced to 2 1 or less. F5.0.4.3 High-Tc superconductors In the superconducting metals discussed above, Tc << ΘD , so that the electronic specific heat is comparable with or greater than the lattice term below Tc . The result is that the discontinuity in CP is very noticeable. By contrast, for the high-Tc materials Ce is small compared with the lattice term so that the ‘bump’ or ‘cusp’ (see figure F5.0.7 after Roulin et al 1996) is relatively small, e.g. 2% or 3% in the cuprates. It follows that the shape of the ‘ordering’ peak is more difficult to determine for the high-Tc materials than for say, single crystals of tin or indium. Also the very high critical fields preclude measurement of the normal-state electronic specific heat at low temperature and hence direct determination of γ. Indirect methods of measuring γ include the comparison of superconducting and nonsuperconducting samples by differential calorimetry or analysis of high-temperature data. These have given values of γ for YBCO ‘123’ of 10 to 20 mJ mol-1 K-1 (see Loram et al 1994 or the review of Phillips et al 1992). From a practical viewpoint, the specific heat at temperatures above 50 or 60 K can be represented as for most other ceramic oxides by a Debye function with appropriate choice of Θ. For the ‘123’ YBCO,

Figure F5.0.7. Specific heat of a YBa2Cu3O6.93 crystal near to Tc in a magnetic field B parallel to the c axis (after Roulin et al 1996).

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Specific heat

ΘD ≈ 500 K above about 100 K compared with Θ0 ~ 420 K at low temperatures. Of some fundamental interest is the specific heat at liquid helium temperatures because of the apparent existence (at least in the YBCO family) of a linear or T term which is unexpected in a completely superconducting sample as T → 0. It has been variously attributed to tunnelling states (as in a glass), a small nonsuperconducting component and an intrinsic contribution of the interaction process. It is partially masked by a magnetic contribution from the presence of impurity compounds such as barium cuprate or Cu2+ ions (e.g. Phillips et al 1992). Most recent work suggests for this linear term (at liquid helium temperatures) in YBCO a ‘…floor value of about 1 mJ mol– 1 K– 2…’ (Junod 1996). It may be smaller or nonexistent in the BSSCO materials. F5.0.5 Data sources There appear to be no comprehensive data compilations on superconductors per se. Volume 4 of the Thermophysical Properties of Matter Series ( Touloukian and Buyco 1970 ) gives tables of original results and their sources for the metallic elements and many alloys but not for the A15 compounds or many important Ti alloys. Volume 1 of Collings’ (1986a) two-volume series on applied superconductivity of the titanium alloys devotes a chapter to low-temperature specific heats including Ti—Mo, Ti—V, Ti—Nb, etc. The review by Phillips (1971) includes a compilation of γ and Θ0 values for metallic elements and a discussion of electronic specific heat in normal and superconducting states but not the alloys or compounds. Corrucini and Gniewek (1960) compiled data for technical solids used in cryogenics including some alloys, glasses, resins, solders and metallic elements. Figure F5.0.8 compares some such solids below room temperature in terms of specific heat per unit mass. Further data on technical solids at liquid helium temperatures may be found in books on cryogenic techniques (e.g. White 1979). Useful data are also distributed through the many volumes of Advances in Cryogenic Engineering (New York: Plenum) which cover the biennial meetings of the International Cryogenic Materials Conference

Figure F5.0.8. Specific heat per gram for some cryostat materials including polyethylene (PET) and glass-reinforced epoxy laminate (G10), an 18:8 stainless steel (SS), vitreous silica and constantan (60Cu-40Ni).

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References

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(ICMC) and the International Cryogenic Engineering Conference: e.g. volume 40 (ed F Fickett and R P Reed) covers the ICMC (Materials) section of the 1993 conference held in Albuquerque, New Mexico. Likwise the journal Cryogenics (Butterworth—Heinemann) includes data and conference proceedings in areas of applied superconductivity, cryoelectronics and low-temperature engineering. For high-Tc superconductors, reviews by Fischer et al (1988), Fisher et al (1988), Junod (1990, 1996) and Phillips et al (1992) give many references to measurements of CP on La1.85Sr0.15CuO4 , YBa2Cu3O7-δ , Bi—Ca—Sr—Cu oxides, Tl—Ca—Ba—Cu oxides, other members of the RBa2Cu3O7 family (where R = rare earth other than Y) and the cubic noncuprate, (Ba1-χKχ )BiO3 . The reviews include tables or graphs showing values of Tc , γ (0), ∆C(Tc )/Tc , Θ0 . References Anderson A C 1988 Calorimetry below 0.3 K Specific Heat of Solids eds C Y Ho and A Cezairliyan (New York: Hemisphere) Anderson O L 1963 A simplified method for calculating the Debye temperature from elastic constants J. Phys. Chem. Solids 24 909–17 Cezairliyan A 1988 Theory of specific heat of solids Specific Heat of Solids eds C Y Ho and A Cezairliyan (New York: Hemisphere) Collings E W 1986a Applied Superconductivity, Metallurgy, and Physics of Titanium Alloys, vol 1, Fundamentals (New York: Plenum) Collings E W 1986b Applied Superconductivity, Metallurgy, and Physics of Titanium Alloys, vol 2, Applications (New York: Plenum) Corruccini R J and Gniewek J J 1960 Specific Heats and Enthalpies of Technical Solids at Low Temperatures, (NBS Monograph 21) (Washington, DC: US Government Printing Office) Fischer H E, Watson S K and Cahill D G 1988 Specific heat, thermal conductivity and electrical resistivity of high temperature superconductors Comments Condens. Matter Phys. B 14 65–127 Fisher R A, Gordon J E and Phillips N E 1988 Specific heat of the high-Tc oxide superconductors J. Supercond. 1 231–94 Furukawa G T, Douglas T B and Pearlman N 1972 Heat capacities American Institute of Physics Handbook 3rd edn, ed D E Gray (New York: McGraw-Hill) ch 4e Gopal E S R 1966 Specific heats at Low Temperatures (New York: Plenum) Hake R R 1969 Thermodynamics of type-I and type-II superconductors J. Appl. Phys. 40 5148–60 Heiniger F, Bucher E and Muller J 1966 Low temperature specific heat of transition metals and alloys Phys. Kondens. Materie 5 243–80 Ho C Y and Cezairliyan A (eds) 1988 Specific Heat of Solids (New York: Hemisphere) Junod A 1990 Specific heat of high temperature superconductors Physical Properties of High Temperature Superconductors II ed D Ginsberg (Singapore: World Scientific) pp 13–121 Junod A 1996 Specific heat of high temperature superconductors in high magnetic fields Studies of High Temperature Superconductors vol 18, ed A V Narlikar (New York: Nova) Loram J W, Mirza K A, Cooper J R, Liang W Y and Wade J M 1994 Electronic specific heat of YBa2Cu3O6+x J. Supercond. 7 243–9 Marsh K N (ed) 1987 Recommended Reference Materials for the Realisation of Physicochemical Properties (Oxford: Blackwell) pp 236–8 Martin D L 1987 ‘Tray’ type calorimeter for the 15–300 K temperature range Rev. Sci. Instrum. 58 639–46 Martin D L 1988 Calorimetry from 0.3 to 30 K Specific Heat of Solids ed C Y Ho and A Cezairliyan (New York: Hemisphere) Phillips N E 1971 Low-temperature heat capacity of metals Crit. Rev. Solid State Sci. 2 467–553 Phillips N E, Fisher R A and Gordon J E 1992 The specific heat of high-Tc superconductors Prog. Low-Temp. Phys. 13 267–357 Roulin M, Junod A and Walker E 1996 Scaling behaviour of the derivatives of the specific heat of YBa2Cu3O6.93 at the superconducting transition up to 16 Tesla Physica C 260 257–72

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Shoenberg D 1952 Superconductivity (Cambridge: Cambridge University Press) ch 3 Touloukian Y S and Buyco E H 1970 Specific Heat-Metallic Elements and Alloys (New York: Plenum) Westrum E F Jr 1988 Calorimetry from 5 to 300 K Specific Heat of Solids ed C Y Ho and A Cezairliyan (New York: Hemisphere) White G K 1979 Experimental Techniques in Low Temperature Physics (Oxford: Clarendon) ch 11

Further reading For background reading about the specific heat of solids there are many standard university texts devoted to solid state physics which treat the crystal lattice, electron energy states, superconductivity in an easily accessible fashion: these include the following. Ashcroft N W and Mermin N D 1976 Solid State Physics (New York: Holt-Saunders) Ho C Y and Cezairliyan A (eds) 1988 Specific Heat of Solids (New York: Hemisphere). This volume is one of a series from the Purdue University Data Center and includes chapters on calorimetric methods suited to various temperature ranges written by leading practitioners, e.g. A C Anderson, D L Martin, E J Westrum Jr, D Ditmars, etc. Kittel C 1976 Introduction to Solid State Physics e.g. 5th edn (New York: Wiley) Lynton E A 1964 Superconductivity (London: Methuen) is a handy pocket-size book to complement Shoenberg below. It is more theoretical but brief and very readable. Rosenberg H M 1978 The Solid State (Oxford: Clarendon) Shoenberg D 1952 Superconductivity (Cambridge: Cambridge University Press) is still a masterly account of the macroscopic properties, thermodynamics and magnetic properties including the intermediate state of metallic superconductors.

More recent books include the following. Collings E W 1985, 1986 Applied Superconductivity, Metallurgy, and Physics of Titanium Alloys, vol 1, Fundamentals and vol 2, Applications (New York: Plenum) cover the principles and applications of the titanium alloy superconductors in nearly 1500 pages, a magnum opus. There is yet no equivalent work on the high-temperature superconductors: they have not yet reached the stage of application of the titanium alloys. Ginsberg D M (ed) 1989, 1990, 1992 Physical Properties of High Temperature Superconductors vols I, II and III (Singapore: World Scientific) is a continuing series which attracts experienced practitioners to review the progress of research on the high-Tc system. Phillips N E, Fisher R A and Gordon J E 1992 The specific heat of high-Tc superconductors Prog. Low-Temp. Phys. 13 267–357 is a recent review of the specific heat of high-temperature superconductors and complements the earlier reviews by Fisher et al (1988) and Junod et al (1990) listed above. Rose-Innes A C and Roderick E H 1994 Introduction to Superconductivity 2nd edn (Oxford: Pergamon) Tinkham M 1975 Introduction to Superconductivity (New York: McGraw-Hill) (reprint edn 1980, Malabar, FL: Krieger)

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F6 Thermal expansion G K White

F6.0.1 Introduction A thermal property which manifests itself in everyday life is thermal expansion, perhaps in the sad sight of a cracked glass vessel on a hot stove or the ‘onion skin’ weathering of large granite boulders exposed to extremes of cold nights and hot days or the buckling of train rails in summer; it appears also in more useful ways such as bimetallic strips used for temperature control, shrink-fitting metal cylinders together or just warming up a screw-top in hot water to loosen it. The intuitive feeling that heating produces expansion is generally true because heating excites vibrations of the atomic lattice and increases vibrational amplitudes so that the average interatomic distance usually increases. However, this is not always the case, particularly at low temperatures where in some solids warming produces a contraction. Such is the case when transverse modes of vibration dominate (as in a guitar string) and produce a ‘cross-contraction’ effect. Examples of cross-contraction occur in vitreous silica and many diamond-structure crystals which contract on warming, at least over a significant range of low temperatures. A negative expansion coefficient may also arise when some magnetic forces called magnetostriction are dominant. By careful design, materials can be produced which have almost zero expansion over a convenient temperature range. The development of low-expansion iron-nickel alloys (e.g. Invar) at the beginning of the 20th century for clocks was very important for accurate timekeeping. More recently glasses and glassceramics have been produced with near-zero expansion at room temperature for use in large telescope mirrors. Of course the expansion coefficient of all materials (and their specific heat) approaches zero as the temperature T approaches absolute zero as required by the third law of thermodynamics. An important technical problem in cryogenics is to tailor composite materials that combine mechanical strength and stability with suitable expansion and thermal conductivity properties. A family that conforms to these criteria are the graphite-fibre- (and silica-fibre-) reinforced polymers used in lightweight airframes. The design is complicated by the large anisotropy of expansion of the graphite fibres which have positive expansion along one axis and negative expansion normal to this axis. In large superconducting appliances such as big magnets, compatibility of expansion between the component parts is necessary to avoid distortion, strain and failure. Similarly, in small superconducting devices made from thin films and substrates the same criterion must apply or performance may be degraded.

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F6.0.2 Theory The volume coefficient of thermal expansion β (at constant pressure P ) is usually denoted by

where V is the volume. For cubic and isotropic solids β is obtained from the coefficient of linear expansion, α , by

where l is the length. More generally, for anisotropic solids there are two or three principal coefficients, αj corresponding to the principal crystallographic directions: two for the axially symmetric structures (hexagonal, tetragonal, trigonal) and three for those of lower symmetry. Early in the 1900s, Grüneisen measured the expansion of many solids at temperatures down to 90 K and observed that the temperature dependences of α and the specific heat, C, were similar. The ratio of β or α to C has become closely identified with his name and is now known as the Gruneisen γ in the dimensionless form

where CP and CV are the respective specific heats at constant pressure and volume of a mole of volume V; χS and χT are respectively the adiabatic and isothermal values of the compressibility. The measurements by Gruneisen showed that γ had values between 1 and 3 for many solids and was fairly constant with temperature for a given solid. The early microscopic theories of Mie in 1903 and Gruneisen in 1912 (see e.g. Barron et al 1980 or Born and Huang 1954) assumed an Einstein model with atoms having a single vibrational frequency, v , and led to a parameter γ = – d In v/d ln V which reflected the anharmonicity of the potential well within which the atoms vibrate. In a real crystal there are many modes of vibration (frequency vi ) with different individual values of γi and the thermal γ (of equation (F6.0.1)) will be the average

weighted by their contributions, Ci , to the specific heat. The weighting factor or importance of different modes will change with temperature but we may expect γ to be constant at higher temperatures where all modes are fully excited. Some thermodynamic relations between β, χ, T, entropy S and pressure P are

and

The sign of dS/dV determines the sign of the expansion coefficient: in most crystals the entropy (or disorder) decreases under pressure as the volume is reduced and so dS/dV and β are positive. However, the sign is negative at low temperatures in a few important solids arising from the effect of transverse modes of vibration, examples being many crystals of open structure such as the tetrahedrally bonded zincblende family. In metals there are other contributions to the free energy and entropy and therefore to the specific heat and thermal expansion from electrons, magnetic spins, etc. The electronic contribution, βe , is governed by the volume dependence of the electronic density of states at the Fermi surface, dN (EF )/dE, just as the

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specific heat component, Ce , is a measure of the density of states, N(EF ). We can define an electronic Grüneisen parameter, γe , by

which for an ideal free electron gas would have a value of 3-2 . Likewise any magnetic contribution, βm , derives from the volume dependence of the spin interaction energy (see Barron et al 1980, p 642). Anisotropy is most obvious in layered materials like graphite or in chain-like polymers but is important in many crystals including the noncubic perovskite family to which many high-Tc superconductors belong. The simplest examples of anisotropy occur in the axially symmetric crystals (e.g. hexagonal) for which there are two principal coefficients of expansion. In such cases

and

where a and c denote lengths perpendicular and parallel, respectively, to the main crystal axis. The principal coefficients are related to principal Gruneisen coefficients and elastic moduli; for example, to the elastic compliances, si j

Note that the cross-compliance s13 is negative so that a large expansion along one direction can produce a contraction in a direction normal to this. Superconductivity is a cooperative ordering process within the electron gas or the charge carriers. Since the ordering is reflected in the free energy and entropy, there should be differences in volume, length, expansion coefficient (as well as specific heat) beween the normal (n) and superconducting (s) states. Shoenberg (1952) derives the thermodynamic relations for the volume and length differences for type I superconductors

and

where σi denotes a uniaxial stress in direction i. The second terms on the right-hand sides of the equations are the small magnetostrictive contributions. Differentiation of these equations leads to differences in the expansion coefficients, and at T = Tc to the Ehrenfest relation

These relations show that thermal expansion data are a useful alternative to high-pressure data and may be easier to measure and a more reliable guide to the pressure dependence of Tc (i.e. dTc /dP) than direct measurements of pressure or uniaxial stress. In type II superconductors, field penetration begins at a relatively low critical field, Hc 1 , and flux entry increases with applied field until the material becomes entirely normal at the upper critical field, Hc 2 . There is no discontinuity in length at Hc 1 or Hc 2 but a change in slope of l( H ) (see Hake 1969 for discussion of thermodynamics of type II superconductors).

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F6.0.3 Experimental methods At low temperatures, the expansion coefficient changes rapidly with temperature in much the same fashion as the specific heat. For most solids β and CP roughly follow the Debye approximation with values of ΘD varying from near 90 K for lead to about 400 K for iron and 1000 K for alumina. It follows that at liquid helium temperatures, β or α will be very small and demand very sensitive methods for their measurement. Dimensional changes can be measured by methods ranging from x-rays and neutrons (as internal probes) to contacting methods (push-rod dilatometers) and noncontacting methods (optical, capacitative, inductive). Their sensitivities vary from micrometres to picometres. The linear coefficient is usually measured as an average α = ( 1/l )(∆l/∆T ) from the length change ∆l over a finite temperature interval ∆T. At ambient temperatures, α ∼ 10−5 K−1 for most solids so that if ∆T ∼ 5 K, a resolution of ∆l/l ∼ 5 × 10−7 will give α to 1%. With l ∼ 10 mm, this requires a measurement sensitivity of 5 nm. Conventional x-ray and neutron diffraction methods can resolve changes in lattice spacings of ∆a/a ∼ 10−5. This does not allow accurate determination of α over a small temperature interval but with intervals of 100 K or so, average coefficients can be determined to a level of 10−7 K−1. They are a convenient way of determining the differences in expansion (anisotropy) in noncubic solids without the need for large single crystals. Optical interferometry has a long history in measuring expansion with the Fizeau interferometer dating back to the 1860s. Detection limits lie in the range of 1/100 of a fringe corresponding to ∆l/l ≤ 10−7. More recently Fabry—Pérot interferometers with stabilized laser light sources have increased the precision to ∆l/l ∼ 10− 9. These optical methods have the merit of being absolute compared with the relative nature of push-rod dilatometry. Optical levers of various forms have been used to detect length changes of less than 0.1 nm (1 Å) but most have mechanical couplings which are sensitive to vibration and result in hysteresis effects which reduce their performance. Many are reviewed by Barron et al (1980) (see also the books by Krishnan et al (1979), Touloukian et al (1975, 1977), Yates (1972) which compare various methods). The most common commercial measuring systems use push-rod dilatometers with an inductance device (linear variable differential transformer—LVDT) as the sensor for length change. They are differential, that is they measure the change in sample length relative to the tubular sheath, and use silica or alumina for construction of the sheath and push-rod, depending on the temperature range. The commercial versions can have sensitivities of the order of nanometres and using 10 or 20 mm long rods can give meaningful values of α to ±10−7 K−1 if calibrated with a suitable reference standard. Reference materials include silica, silicon, tungsten, copper and alumina (White 1993b). The most sensitive dilatometers are those custom made using capacitative or inductive sensors. Both are capable of detecting length changes of 10−12 m (0.01 Å) or less. They are most used for the low-temperature range where α is very small and maximum sensitivity is needed. The capacitance method has proved more popular because it is not sensitive to magnetic effects and high-sensitivity three-terminal capacitance bridges are readily available. Figure F6.0.1 shows a cryostat in which length changes in the sample are measured relative to a frame (copper) with a three-terminal parallel plate capacitor, similar to that developed by White in 1961 (see Barron et al 1980). Commercial bridges are now available capable of resolving 10−7 pF which with a 20 pF capacitance and 0.2 mm gap corresponds to a resolution of 10−12 m in gap change. F6.0.4 Observations (examples of behaviour patterns) F6.0.4.1 General Figure F6.0.2 shows the percentage change in length (contraction in these materials) which occurs on cooling from 293 K for some representative materials used in cryogenics. The curve denoted ‘E +

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Observations (examples of behaviour patterns)

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Figure F6.0.1. Capacitance cell for measuring the length change of a cylinder (2) relative to a frame (3). Capacitance C12 (between parallel surfaces of 1 and 2) is measured in a ratio-transformer bridge (e.g. Barron et al 1980).

F’ is for an epoxy resin containing 20 vol.% powder filler. The values for TiNb are for a technical (superconducting) alloy of Ti + 55 wt% Nb (see Collings 1986, p 374) while S.S. denotes two common 18:8 stainless steels, 304 and 316, used in cryogenic equipment. Table F6.0.1 gives numerical values for these and additional materials, also measured relative to the length at 293 K. Values for copper are representative of ultra-pure Cu, oxygen-free high conductivity and electrolytic Cu within the limits of measurement. Trace impurities have no significant effect above 20 K. The Al2024 is a commercial alloy of aluminium with 4 wt% Cu, 1.4 wt% Mg, 0.5 wt% Mn etc. Values given for Inconel are also applicable (within a few per cent) for other Ni-rich alloys containing Cr, Fe, etc, such as Inconels, Hastelloys. The values given for PET (polyethylene), YBCO ‘123’ (YBa2Cu3O7–δ ) and alumina are average values for polycrystalline material without preferred orientation. The single-crystal data for YBCO along the a, b and c axes are approximate and taken from graphs published by Meingast et al (1991). We have included in the table magnesium and aluminium oxides as representative of two insulating

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Figure F6.0.2. Linear thermal contraction, Dl/l, relative to 293 K for some representative solids. E + F denotes epoxy resin with 20 vol.% powder filler. S.S. is for 18:8 stainless steel.

substrate materials used for making thin-film superconducting devices. The tabulated data are based on the sources given below in section F6.0.5, particularly Clark (1968, 1983), Corruccini and Gniewek (1961), Collings (1986) and White (1979). The differences in length change are considerable, particularly among nonmetals. For close-packed metals the differences are less marked and arise largely from the differences in the compressibility whereas in insulating crystals the anharmonic factor, (dS/dV), can be most important. F6.0.4.2 Nonsuperconducting metals Figure F6.0.3 shows the temperature dependence of ∝ for some common cubic metals, the magnitudes reflecting roughly their relative ‘softness’ or compressibility. The shapes of α(T) are similar to the respective specific heat curves, CP(T), and values of the Grüneisen ratio γ are ∼2. Clark (1983) has collated values of (l2 9 3 – l0 )/l2 9 3 and α2 9 3 for a number of metals and alloys used in cryogenics. He shows that the ratio (∆l/l )/α2 9 3 lies in the relatively narrow range of 180 to 200 for these materials, which is a useful guide for cryogenic design. This similarity arises because α varies with T roughly as a Debye function over the temperature interval where most length change occurs. The ratio lies outside this narrow range for metals having very different values of ΘD such as lead (ΘD ≈ 90 K (or beryllium ΘD ≈ 900 K)). Like the specific heat data (see section F5.0.4), values at low enough temperatures (≤ ΘD /25) can be well represented by an expression of the form AT + BT 3, the respective terms corresponding to electron

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Table F6.0.1. Linear Thermal contractions, ∆l / l × 104, relative to 293 K.

Figure F6.0.3. Temperature dependence of linear expansivity, α , for metals of different compressibilities, χ (see White 1993c).

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and lattice contributions. Magnetostrictive contributions play a strong role at low temperatures in a few elements such as manganese and chromium, some rare earths and some alloys of the Fe/Ni family (e.g. Invar), some stainless steels and manganese alloys. Some of the rare-earth—Fe and rare-earth—Mn mixtures display giant magnetostrictive effects at more normal temperatures. Another group with ‘anomalous’ behaviour (at low T) are the heavy-electron metals such as CeCu6 , UPt3 , etc (de Visser et al 1989). Single crystals of noncubic metals can show large anisotropy of linear expansion, particularly at low temperatures. Common examples from the hexagonal close-packed (hcp) system are cadmium and zinc for which the axial ratio c/a is approximately 1.85 compared with the ideal packing value of ( -83)1/2(=1.633). This ‘stretching’ along the hexagonal axis increases the elastic compliance and the expansion coefficient in this direction while reducing them in the basal plane. Figure F6.0.4 illustrates for cadmium the large positive values of α|| and the smaller negative values of α⊥ (at low T) resulting from the cross-compliance term (s13 ) in equation (F6.0.2). The individual γ values for these close-packed metals do not differ very much and indeed the volume coefficient, β = 2α⊥ + α|| , is positive and varies with T much as for copper. For the tetragonal elements indium and tin, the anisotropy is also very marked while for the hcp elements titanium and zirconium it is less so.

Figure F6.0.4. Principal linear expansivities α|| and α⊥ for Cd. Reproduced from a review by Munn (1969) by permission.

F6.0.4.3 Nonmetallic solids (nonsuperconductors) The expansion behaviour of cubic nonmetals reflects partly their compressibility (like metals discussed above) but also is strongly influenced by the closeness of packing or coordination number (CN). The close-packed rare gas solids (CN = 12) have Grüneisen ratios which are similar to those of copper or those of aluminium and do not change with temperature by more than 10 or 20%. By contrast the crystalline materials of more open structures such as rocksalt (CN = 6) and zincblende (CN = 4) show the effect of low-frequency transverse acoustic modes of vibration, which at low temperatures may have a dominant influence on thermal properties and produce a negative volume expansion coefficient (and negative value of γ ), the so-called ‘guitar string’ effect (Barron et al 1980).

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Anisotropy in the nonmetals is most noticeable in the chain-like crystals of tellurium or polyethylene (and other polymers) which have weak van der Waals forces between chains and strong covalent bonding along the chains: this can lead to large positive expansion normal to the chain direction (soft directions) and contraction along the chains (hard direction). The reverse situation applies to layer structures such as graphite or boron nitride which expand normally to the basal plane and contract within the plane. We shall see evidence of this in some of the high-Tc crystals discussed below. F6.0.4.4 Superconducting metals Figure F6.0.5 illustrates for tantalum, niobium and vanadium the typically small differences in length (∆l/l ∼ 10– 7 ) which are observed in changing from normal to superconducting states. The changes reflect the small magnitude of the electronic contribution to expansion and are not sufficiently large to have any practical significance, i.e. cause serious stresses. However, they are of fundamental interest because of their relation to the strain dependence of Tc and to the understanding of the mechanisms of superconductivity. For most type I superconductors, ls — ln is positive, and at T = Tc βn — βs is positive (except in vanadium). Therefore since Cn — Cs is negative (see section F5.0.2) the Ehrenfest relation leads to the fact that dTc /dP is usually negative, i.e. pressure usually lowers Tc . However, in vanadium and many type II superconductors, βn — βs is negative. Compare

Figure F6.0.5. Linear thermal expansions of Ta, V and Nb rods (l0 = 51 mm) in normal (z) and superconducting states (O) (arbitrary zero). Broken lines show the normal state below Tc (White 1962).

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Figure F6.0.6. Electronic coefficients, αe , for (a) Zr + 71 at.% Nb and (b) Nb (after Simpson and Smith 1978).

Figure F6.0.6(a) showing the change in the electronic coefficient at Tc for Zr + 71 at.% Nb and F6.0.6(b) showing that for pure Nb. For the alloy dTc/dP is positive while for Nb, it is negative. F6.0.4.5 High-Tc superconductors Most measurements of expansion for the high-Tc cuprate ceramics have been made on polycrystalline or sintered samples which did not show any strong preferred orientation. Values of α(T) are similar in form to those of many other ceramics including MgO, ZrO2 , BaTiO3 , etc (see the review by White (1993a) and figure F6.0.7). Differences at low temperatures between different cuprates largely reflect the differences in ΘD . The spread of 10 to 20% in values shown in figure F6.0.7 for the YBCO (‘123’) probably arises from differences in porosity, oxygen content and small degrees of preferred orientation. Near Tc , the average linear coefficient exhibits a ‘cusp’ which is roughly λ shaped and ∼2% in height, similar to that shown by the specific heat (see section F5.0.4.3). For the YBCO ‘123’ samples, these lead (via the Ehrenfest relation) to dTc /dP ≈ 0.6 K GPa–1 and d ln Tc /d ln V ∼ 1 assuming bulk modulus B ≈ 120 GPa. Most of the superconducting cuprates have a perovskite structure with orthorhombic symmetry and are quite anisotropic in single-crystal form. Untwinned single crystals of YBCO ‘123’ have been measured and show the behaviour illustrated in figure F6.0.8(a), namely maximum expansion up the c axis and smallest along the b axis which is the CuO chain direction (Meingast et al 1991). The accompanying figure F6.0.8(b) amplifies the behaviour near to Tc Along the c axis there is no clear sign of a ‘cusp’ at Tc and therefore stress along this axis has little effect on Tc . Along the a and b axes, the ‘signs’ of the anomaly are opposite so that uniaxial stresses have opposite effects on Tc , the first decreasing and the second increasing Tc . F6.0.5 Data sources There have been no tabulations of expansion data devoted entirely to superconductors. Volume 12 of the Thermophysical Properties of Matter Series (Touloukian et al 1975) gives tables of original data for

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References

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Figure F6.0.7. α (T) for five polycrystalline samples of YBCO (‘123’) (θ0 ≈ 430 K) and MgO (θ0 ≈ 950 K).

Figure F6.0.8. Linear coefficients for YBa2Cu3O7 along principal axes: (a) shows α values and (b) shows expanded views of ∆α near Tc (after Meingast et al 1991).

length changes and linear expansivities for the metallic elements and many alloys while volume 13 of the same series (Touloukian et al 1977) gives similar data for nonmetallic solids. They do not cover A15 compounds or many important titanium alloys or the high-Tc compounds. An earlier useful compilation by Corruccini and Gniewek (1961) tabulates data for ‘technical solids’ at low temperatures. These include the common metallic elements, steels, silica, Pyrex, MgO and many plastics. Collings’ (1986) book on applied superconductivity includes a chapter on thermal expansion and gives tables of length changes, ∆l/l, from 300 down to 4 K for many Ti-Nb alloys, copper, titanium, aluminium, stainless steels, technical Ti-based alloys and some superconducting coil composites e.g. fiberglass—epoxy. Clark et al (1981) have tabulated length changes, ∆l/l, from 293 to 4 K for superconducting magnet materials ranging from copper (0.32% contraction), Nb3Sn wire (0.21%), V3Ga wire (0.22%), CuSn bronze

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Thermal expansion

(0.37%), NEMA G10 fibreglass—epoxy laminate (0.24% in the parallel direction and 0.705% in the normal direction) to a NbTi—fibreglass coil composite (0.03 and 0.05%). Clark (1968) also gives data over the 293 to 4 K range for a number of commercial alloys based on aluminium, nickel and iron and other materials used in superconducting magnets. Further data on polymers, graphite-fibre-reinforced polymers and other fibre-reinforced polymers can be found in the proceedings of various ICMC (International Cryogenic Materials Conference) meetings devoted to nonmetals and composites at low temperatures published by Plenum in 1978, 1980 and 1984 (ed G Hartwig and D Evans) and more recently published as special issues of the journal Cryogenics e.g. vol 28 (1988), vol 31 (1991) and vol 35 (1995)). References Barron T H K, Collins J G and White G K 1980 Thermal expansion of solids at low temperatures Adv. Phys. 29 609–730 Born M and Huang K 1954 Dynamical Theory of Crystal Lattices (Oxford: Clarendon) Clark A F 1968 Low temperature thermal expansion of some metallic alloys Cryogenics 8 282–9 Clark A F 1983 Thermal expansion Materials at Low Temperatures ed R P Reed and A F Clark (Metals Park, OH: American Society of Metals) ch 3 Clark A F, Fujii G and Ranney M A 1981 The thermal expansion of several materials for superconducting magnets IEEE Trans. Magn. MAG-17 2316–9 Collings E W 1986 Applied Superconductivity, Metallurgy, and Physics of Titanium Alloys vol 1, Fundamentals (New York: Plenum) Corrucini R J and Gniewek J J 1961 Thermal Expansion of Technical Solids at Low Temperatures (NBS Monograph 29) (Washington, DC: US Government Printing Office) de Visser A, Franse J J M and Flouquet J 1989 Magneto-volume effects in some selected heavy-fermion compounds Physica B 161 324–32 Hake R R 1969 Thermodynamics of type-I and type-II superconductors J. Appl. Phys. 40 5148–60 Krishnan R S, Srinivasan R and Devanarayanan S 1979 Thermal Expansion of Crystals (Oxford: Pergamon) Meingast C, Kraut O, Wolf T, Wühl H, Erb A and Muller-Vogt G 1991 Large a—b anisotropy of the expansivity anomaly at Tc in untwinned YBa2Cu3O7– δ Phys. Rev. Lett. 67 1634–7 Munn R W 1969 The thermal expansion of axial metals Adv. Phys. 18 515–43 Shoenberg D 1952 Superconductivity (Cambridge: Cambridge University Press) ch 3 Simpson M A and Smith T F 1978 Thermal expansion in the superconducting state J. Low Temp. Phys. 32 57–65 Touloukian Y S and Buyco E H 1970 Specific Heat—Metallic Elements and Alloys (New York: Plenum) Touloukian Y S, Kirby R K, Taylor R E and Lee T Y R 1975 Thermal Expansion—Metallic Elements and Alloys (New York: Plenum) Touloukian Y S, Kirby R K, Taylor R E and Lee T Y R 1977 Thermal Expansion—Nonmetallic Solids (New York: Plenum) White G K 1962 Thermal expansion of vanadium, niobium and tantalum at low temperatures Cryogenics 2 292–6 White G K 1979 Experimental Techniques in Low Temperature Physics (Oxford: Clarendon) ch 11 White G K 1993a Thermal expansion and Grüneisen parameters of high Tc superconductors Studies of High Temperature Superconductors vol 9, ed A Narlikar (New York: Nova) pp 121–47 White G K 1993b Reference materials for thermal expansion: certified or not Thermochim, Acta 218 83–99 White G K 1993c Solids: thermal expansion and contraction Contemp. Phys. 34 193–204 Yates B 1972 Thermal Expansion (New York: Plenum)

Reading list Most standard texts on solid-state physics do not devote much space to thermal expansion compared with their treatment of specific heat so for background reading on the theory and measurement of thermal expansion the book by Krishnan et al (1979) and the review by Barron et al (1980) are recommended.

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References

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A forthcoming book on Thermal Expansion (edited by R E Taylor of Purdue University and published by CINDAS at Purdue University) will contain useful chapters on theory, measurement methods and reference materials and will be a companion volume to that on Specific Heat of Solids (Ho C Y and Cezairliyan A (eds) (New York: Hemisphere)) mentioned in chapter F5. The chapter by Clark (1983) in Materials at Low Temperatures gives useful background as well as data on superconductors, composites and reference materials. Another report by Clark et al (1980) also discusses modelling of composites and refers to other reports on the effect of strain (arising from expansion mismatch) on critical currents in magnets. The biannual volumes of Advances in Cryogenic Engineering covering the combined International Cryogenic Materials and International Cryogenic Engineering Conferences (New York: Plenum), particularly the ICMC volumes, are a good source of recent research on superconductors and their applications. For high-Tc materials a review by White (1993a) covers most thermal expansion measurements published up to the end of 1990. Since then more measurements have been made on untwinned single crystals, particularly of YBCO (e.g. Meingast et al 1991).

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F7 Dielectric properties

J Gerhold

F7.0.1 Introduction Any field of electromagnetic technology is based on current flow and on voltage stress. Power can only be managed via the Poynting vector P = E × H where E is the electrical field and H is the magnetic field. P acts in an insulating space, not within a conductor. This highlights the importance of dielectric materials. There are power applications of superconductors where the need for electrical insulation is selfevident, e.g. in transmission lines, superconductor magnetic energy storage, transformers etc. Beyond power applications, any d.c. magnet has to be charged and discharged. This means power has to be fed in and has to be extracted. Finally, any signal processing manages small energies, i.e. power pulses. Voltages may be low in this case but electrical field strengths can be >108 V m–1 in extremely thin insulating sheets. Insulating solids have to meet various needs. (i) (ii) (iii) (iv) (v)

They must hold conductors in position, often under high mechanical stress. Moving can be very harmful to superconductors, e.g. in a magnet. Insulation performance, i.e. resistivity, dielectric strength, aging, must be adequate. Dielectric losses are a key point in a.c. applications. Heat conductivity must be adequate to remove losses to a cooled surface. Solid insulators must be compatible with cryogens or with high vacuum. Thermal contraction must be at an acceptable level. Manufacturing expenditure must be reasonable.

Figure F7.0.1 illustrates the use of insulating solids in a superconducting device. Solids are present within the cold space at temperature Tc , but are used in addition to bridge the gap to ambient temperatures as feedthrough insulators. In any case, solids must be seen within the particular environment. Dielectric phenomena often are extrinsically controlled. F7.0.2 Intrinsic properties Some of the dielectric phenomena can be solely described by intrinsic properties. Resistivity and polarization do not suffer from degradation, but may favour premature breakdown or surface flashover.

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Figure F7.0.1. Insulating solids in a superconducting device. Reproduced from Hara (1988) by permission of the Cryogenic Society of Japan.

F7.0.2.1 Resistivity Although no net free charges exist in an ideal insulator, some ions or electrons may move across an imperfect lattice. The corresponding conduction currents often decrease drastically at low temperatures, e.g. with exp( – b/T) where b is a materials constant and T the absolute temperature (Sillars 1973). In general the currents decrease with time which indicates a kind of long-term polarization. A high electrical stress may inject additional electrons. An exponential current—voltage (I—V) relationship is often found at ambient temperatures. Space charges can build up in the solid, thereby modifying the internal stress considerably. Very little is known about resistivity and space-charge build-up in the cryogenic temperature regime. Figure F7.0.2 indicates the course of resisistivity ρ versus temperature for a polyester varnish. Results of similar order have been obtained for glass-fibre reinforced epoxies for instance. However, ρ > 1018 Ω m has often been claimed for temperatures < 50 K (Fallou 1975). In fact, such a high resistivity cannot be measured. Nevertheless this does not mean that charge moving is completely out of the question. The situation is analogous to a superconducting wire where a critical current has to be defined by a resistivity criterion for instance, without a distinct limit for genuine superconductivity. Therefore, long-term spacecharge build-up in a cold insulator cannot be excluded completely, and degradation or aging may be encountered in d.c. applications. Time constants τρ = ε0εr ρ can be extremely long (ε0 is the permittivity of free space; εr the relative permittivity). Surface conduction which often gives rise to severe problems at ambient temperatures is of less concern in the cryogenic domain. The causal humidity traces are completely frozen, and ions are immobilized. However, long-term d.c. stress may again accumulate surface charges which can impair the original surface voltage gradient considerably. This impairs surface flashover for instance. Impurity particles in a cryogen may be the primary source of charge accumulation.

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Figure F7.0.2. Resistivity of polyester varnish versus temperature. From the Joint Institute for Nuclear Research, Dubna 1974, P8-7663.

F7.0.2.2 Permittivity and dielectric losses Movement of bound charges in a solid insulator is restricted. Polarization is the net outcome of the charge displacements caused when an external field is switched on. The first kind of polarization, i.e. the relative movement of electrons and nuclei of atoms, occurs in all materials. It is an atomic phenomenon and not influenced by the temperature at all. A relative permittivity of about two is typical for this mechanism. An ionic or partly ionic crystal may show an additional displacement of positive and negative ions (the second kind of polarization). Again, temperature is often of little concern. The permittivity range is near five or more. Many ceramics belong to this class. Various common plastic materials embody dipolar groups which become oriented under electrical stress. Thermal agitation impedes orientation so this third kind of polarization increases with 1/T unless a glass—rubber transition freezes the dipoles. Permittivities may be high when polar dipoles are involved. However, there is a strong frequency influence (Sillars 1973). Finally, the long-term space-charge build-up as mentioned already may be classified as a fourth kind of polarization. A very general picture of permittivity and dielectric loss characteristics can be given by means of a complex permittivity εr = ε′r − jε″r. Hence

where εr s and εr i are the low-frequency and high-frequency limit of the permittivity, respectively, ω is the angular frequency and τ is the relaxation time. τ increases at low temperatures. The dielectric dissipation factor is defined by tan δ = ε″r /ε′r. The dielectric losses per unit volume Pυ are to be found from

Er m s is the root mean square (rms) electrical field; the product εr tan δ indicates the dielectric loss number. This simple picture needs no detailed information about the actual dipolar source. The only prerequisite is a linear increase of polarization with stressing field. The typical course for a solid with one

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Dielectric properties

Figure F7.0.3. Principal course of complex permittivity: (a, b) distinct relaxation time constant; (c, d) time constant scattered around the central value with a distribution as shown by the shaded area in (e) (after Sillars 1973).

distinct dipole type is illustrated in figure F7.0.3(a) and F7.0.3(b). The ε″r peak shifts to lower frequencies when the insulator is cooled down. Scattered relaxation times widen the peak region considerably, see figure F7.0.3(c)—(e). Unfortunately, contour maps such as those shown in figure F7.0.4 giving full information are scarce. Only sectional graphs are available in general. Figures F7.0.5–F7.0.7 show data for two common plastics, and for Kraft paper. Polyester varnish as well as other plastics have been measured at power frequencies only, see figures F7.0.8 and F7.0.9. It is obvious that loss peaks can occur at cryogenic temperatures in the power frequency domain. All these materials embody various kinds of dipole. Additives and treatment may

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Intrinsic properties

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Figure F7.0.4. Relief of permittivity and dielectric loss factor versus frequency and temperature (in degrees Celsius); polyethylene tetraphtalate. Reproduced from Sillars (1973) by permission of IEE.

have a strong influence on the losses at very low temperatures. A distinct correlation between dielectric losses and mechanical damping has been found. This is illustrated for an epoxy in figure F7.0.10. Glasses and ceramics may be of special interest for the low-temperature regime. Permittivities often are high, and dielectric losses can considerably increase near absolute zero, see figure F7.0.11. Note that a crystalline quartz shows a loss increase only when impurity ions are involved, see figure F7.0.12. Extremely low dielectric losses are of extreme importance in the domain of microwave applications. Sapphire is one promising substrate candidate for high critical temperature superconducting films. Measuring techniques rely on dielectric resonators, by using the TE011 mode for instance (Krupka et al 1993). The permittivity of single-crystal sapphire at 4 K is near 9.3 and tan δ is below 4 × 10–10 at 2.7 GHz. However, dielectric measurements are troublesome at low temperatures. Careful shielding by guard

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Figure F7.0.5. Permittivity (a) and dielectric loss Figure factor (b) versus temperature in polyethylene at various frequencies: ◊— 1 kHz; O— 2.34 kHz; S—5.1 kHz; T— 10.3 kHz; U—21.5 kHz; „—44 kHz. Reproduced from Allan and Kuffel (1968) by permission of IEE.

Dielectric properties

Figure F7.0.6. Permittivity (a) and dielectric loss factor (b) versus temperature in Nylon 11 at various frequencies: ♦— static; •—47 Hz; „—229 Hz; O—2.34 kHz; U— 21.5 kHz. Reproduced from Allan and Kuffel (1968) by permission of IEE.

electrodes and elimination of any discharges must be guaranteed when using a bridge technique, e.g. the Schering bridge. Figure F7.0.13 shows the electrode system for measuring a plastic film sample. The full arrangement can easily be immersed into a cryogenic fluid. Most materials have been tested at distinct temperatures and power frequencies. Figure F7.0.14(a) gives an overview at 4.2 K; an 80 K survey is indicated in figure F7.0.14(b). Nonpolar materials should be preferred for a.c. power systems. Table F7.0.1 lists materials which have been investigated for use in superconducting a.c. power cables (Forsyth 1991).

F7.0.2.3 Dielectric strength A discharge-free arrangement is the first prerequisite when searching for near intrinsic dielectric breakdown, even though a true intrinsic strength can never be measured. Stress level as well as stress duration for instance may have an influence (Sillars 1973). As an order of magnitude, 109 V m–1(=1000 kV mm–1 ) can be assumed for most insulators. Temperature has little effect between 300 K and 4 K. Figure F7.0.15 shows a representative test sample (McKeown type). Some significant results for polymer films (∼20 µm) are indicated. These extremely high strength values are considerably affected by extrinsic effects, as will be discussed in the next section.

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Extrinsic phenomena

Figure F7.0.7. Permittivity (a) and dielectric loss factor (b) versus temperature in dry Kraft paper at various: ♦—static; •—47 Hz; „—229 Hz; O—2.34 kHz; U—21.5 kHz. Reproduced from Allan and Kuffel (1968) by permission of IEE.

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Figure F7.0.8. Permittivity (a) and dielectric loss factor (b) versus temperature in polyester varnish at a frequency frequencies of 50 Hz. From the Joint Institute for Nuclear Research, Dubna 1974, P8-7663.

F7.0.3 Extrinsic phenomena Environmental conditions often have a strong influence on the dielectric behaviour of solids. This is distinctly true for the dielectric strength, and for dielectric losses when partial discharges are involved. Only the most important effects are highlighted here; the reader is referred to more specific textbooks and review papers for detailed discussions (Fallou 1975, Gerhold 1992, Sillars 1973).

F7.0.3.1 Dielectric strength No real solid is perfect. Conducting inclusions as well as voids distort the stressing field locally. Distortion also may arise at solid—electrode interfaces. Conducting inclusions often lead to treeing, which is a well known effect at ambient temperatures. Treeing has also been observed at cryogenic temperatures, for instance in polyethylene. A test sample configuration is shown in figure F7.0.16. Electrode adjacent voids in the sample are eliminated by a matching cooling down procedure. Table F7.0.2 indicates tree starting voltages, which are higher at 77 K than at ambient temperatures. Following modern design philosophies, any treeing initiation damages the solid. Final breakdown will follow due to premature aging. Thus, treeing must be eliminated in practical insulations. No detectable discharges may be allowed in tests of devices. There is no self-healing of damaged solids at cryogenic temperatures.

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1128 Dielectric properties

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Observations (examples of behaviour patterns)

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Figure F7.0.9. Permittivity (a) and dielectric loss factor (b) of various insulators at a frequency of 75 Hz: •— polyimide; ♦—silicone-bonded SAMICA; „—polypropylene; ▲—polytetrafuorethylene. Reproduced from Chant (1967) by permission of Elsevier Science Ltd.

Figure F7.0.10. Dielectric loss factor (a), and mechanical loss factor (b), in epoxy (Cy221/Hy979) (after Hartwig and Schwarz 1984).

Voids are the second source of degradation. Apart from manufacturing defects, cracking during cooldown is a possibility with bulk solids. Glass-fibre-reinforced epoxies are a well known example. The internal void field strength can be assumed to be between ES and εrEs where ES is the field strength in the bulk solid (Sillars 1973). A crack in a glass epoxy insulator at 4.2 K may cause a void

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Figure F7.0.11. Permittivity and dielectric loss factor Figure in fused silica: (a) tand at 1 kHz; (b) tanδ at 2 kHz; (c) permittivity at 1 kHz; (d) permittivity at 2 kHz (after McCammon and Work 1965).

Dielectric properties

Figure F7.0.12. Dielectric loss in crystalline quartz at a frequency of 32 kHz: (a) pure crystal; (b) with Na+ ions; (c) with K+ ions (after Stevels and Volger 1962).

Figure F7.0.13. Test sample for dielectric measurements. Reproduced from Mizuno et al (1991) by permission of IEEE.

strength up to 5Es , see figure F7.0.14. A partial discharge becomes likely whenever the voltage across a void of depth δυ , U = εr Esδυ , exceeds the Paschen voltage of the filling fluid (see chapter D2). Fluid states may range from the liquid down to a near vacuum condition. The result is arbitrary. Thus, Paschen minimum conditions have often been cited as safe criteria. Then, a maximum admissible void depth δυ can be defined from the Paschen minimum voltage. Partial discharges cause aging of the solid. Aging has often been investigated by introducing artificial voids. A typical sample arrangement is shown in figure F7.0.17. Partial discharge inception and extinction is illustrated in figure F7.0.18. The filling fluid is supercritical helium (discharge level 1 pC). Qualitatively similar effects have been found with liquid nitrogen (LN2 ) as the filling fluid.

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Figure F7.0.14. Permittivity and dielectric loss factors of various insulating materials at 50 Hz: (a) temperature 4.2 K; (b) temperature 80 K.

TableF7.0.2. Tree starting voltage in polyethylene (from Kosaki et al 1977).

Figure F7.0.19 illustrates the adverse effect of repeated voltage pulses on insulator life. Aging at cryogenic temperatures has often been found to be more distinct than at ambient temperatures; the exponent n in the life equation

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Figure F7.0.15. Test sample and intrinsic d.c. breakdown field strength data for polymers: (a) polymethylmethacrylate (PMMA); (b) polyvinylchloride-acetate; (c) polystyrene; (d) low-density polyethylene; (e) polyisobutylene. Reproduced from Nagao et al (1991) by permission of M Nagao.

Figure F7.0.16. Test sample for treeing tests. Reproduced from Kosaki et al (1977) by permission of IEEE.

Figure 17.0.17. Test sample for partial discharge measurements; artificial voids in a sheet package: (a) mid-void; (b) electrode adjacent void.

where t means the time to failure, can have a value of up to 100 (Bulinski and Densley 1980). Table F7.0.3 gives the results of a survey for various materials impregnated with LN2 . Aging may be reduced

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Figure F7.0.18. Partial discharge inception and extinction stress versus void depth for polyethylene sheets with mid-void and supercritical helium as the filling fluid: — inception; - - - extinction. Reproduced from Weedy et al (1982) by permission of IEEE.

Figure F7.0.19. The life of LN2 -impregnated layered insulation packages with mid-voids under impulse voltage stress: — 250/2500 ms pulse; – – –1.2/50 ms pulse (after Bulinski and Densley 1980).

considerably by limiting the stress below the partial discharge level. A general indication of the lifetime characteristics without detectable discharges is given in figure F7.0.20, which has been measured with wrapped insulation samples which simulate a superconducting cable dielectric.

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Dielectric properties Table F7.0.3. Life exponent n for lapped tape systems (from Weddy and Swingler 1979).

Figure F7.0.20. Lifetime of lapped polymer tape insulation with supercritical helium as the filling fluid (1011 cycles equals 60 years at 50 Hz) (from Forsyth 1982).

F7.0.3.2 Interface phenomena Interfaces also are a critical source of degradation. Interfaces may occur between different solids, or on solid surfaces exposed to vacuum or to a cryogenic fluid. It is noteworthy that the voltage strength along such an interface is lower than the strength of the wetting fluid or of an equivalent vacuum gap. Figure F7.0.21 illustrates a typical situation. The flashover voltage increases much less than linearly with the gap length. Interface performance is normally assessed by examining the covering fluid. Figure F7.0.22 illustrates the mutual interdependences of flashover degradation. Obviously, the state of the fluid as well as the type of stressing voltage are decisive. Flashover in vacuum also has been studied, but the results are often somewhat erratic. Cooldown of

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Survey on field calculations

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Figure F7.0.21. Flashover voltage versus gap length in normal boiling LN2 at a frequency of 50 Hz: 1—gap without spacer; 2, 3—typical spacer configurations (low-density polyethylene). Reproduced from Bobo and Thoris (1975) by permission of Centre Francois du Copyright.

Figure F7.0.22. Relative flashover voltages along low-density polyethylene spacers covered by a cryogenic fluid; a.c. voltage and 1/50 µs pulse voltage. Reproduced from Bobo and Thoris (1975) by permission of Centre Francois du Copyright.

a solid may give some improvement compared with ambient temperature conditions. In summary, flashover along cryogenic solids shows a qualitatively similar behaviour to that found under ambient temperature conditions. There is a distinct ‘total-voltage’ effect—see figure F7.0.21—which means that the behaviour is nonlinear. To control a low voltage at a high local interface strength may present no problems at all. To insulate a high voltage may be impossible, even at a very low stress, unless sophisticated means such as forced subdivision of the total voltage or a forced local stress control are taken.

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Figure F7.0.23. Dielectric loss factor and partial discharges at a frequency of 50 Hz: (a) dielectric loss factor; (b) number of discharge pulses (300–800 pC) per second. Reproduced from Fallou and Breteau (1975) by permission of Centre Francais du Copyright.

F7.0.3.3 Dielectric losses The dielectric losses of a solid insulation can also be severely affected by extrinsic effects. Partial discharges have been identified as the releasing mechanism. Figure F7.0.23 may highlight the correlation. The sample was a layered polyethylene sample with artificial voids immersed in liquid helium. Pressurizing the impregnating fluid shifts the loss increase to higher stress levels. F7.0.4 Summary A broad variety of solid insulators ranging from normal plastics to expensive substrates such as sapphire can be offered in the cryogenic domain. Without exception, the intrinsic properties of these materials are perfectly adequate. However, some important functions are dominated by external effects, i.e. breakdown, flashover, partial discharges and aging. Extrinsic control involves a nonlinear performance. Total voltage effects are found at a lower voltage level than in conventional electrical techniques. The cryogenic temperature regime seems to fit into low- or medium-voltage systems with high stress rather than into very high-voltage systems. References Allan R N and Kuffel E 1968 Dielectric losses in solids at cryogenic temperatures Proc. IEE 115 432–40 Bobo J C and Thoris J 1975 Proprietes dielectriques de structures isolantes aux basses temperatures Rev. Gen. Electr. 84 758–63 Bulinski A and Densley J 1980 The impulse characteristics of electrical insulation operating at cryogenic temperatures

IEEE Trans. Electr. Insul. EI-15 89–96

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References

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Bulinski A, Densley J, Sudarshan T S 1980 The ageing of electrical insulation at cryogenic temperatures IEEE Trans. Electr. Insul 15 83–8 Chant M J 1967 Dielectric properties of some insulating materials over the temperature range 4.2–300 K Cryogenics 7 351–4 Fallou B 1975 A review of the main properties of electrical insulating materials used at cryogenic temperatures Proc. 5th Int. Conf on Magnet Technology (Rome, 1975) p 664 Fallou B and Breteau J P 1975 Comportement dielectrique sous haute tension des structures rubanées impregnées de fluides cryogeniques Rev. Gen Electr. 84 748–57 Forsyth E B 1982 Test results of a.c. superconducting cables IEEE Trans. Power Apparatus Syst. PAS-101 2049–55 Forsyth E B 1991 The dielectric insulation of superconducting power cables Proc. IEEE 79 31–40 Gerhold J 1992 Electrical insulation in superconducting power systems IEEE Electr. Insul. Mag. 8 14–20 Hara M 1989 Electrical insulators in superconducting apparatus Cryogen. Eng. Japan 24 72–81 Hartwig G and Schwarz G 1984 Correlation of dielectric and mechanical damping Advances in Cryogenic Engineering vol 30 (New York: Plenum) pp 61–70 Kosaki M, Shimizu N and Horii K 1977 Treeing of polyethylene at 77 K IEEE Trans. Electr. Insul. EI-12 40–5 Krupka J, Klinger M, Kuhn M, Baranyak A, Stiller M, Hinken J and Modelski J 1993 Surface resistance measurements of HTS films by means of sapphire dielectric resonators IEEE Trans. Appl. Supercond. AS-3 3043–8 Krähenbühl F, Bernstein B, Danikas M, Densley J, Kadotani K, Kahle M, Kosaki M, Mitsui H, Nagao M, Smit J and Tanaka T 1994 Properties of electrical insulating materials at cryogenic temperatures: a literature review IEEE Elect. Insul. Mag. 10 10–22 Mathes K N 1963 Electrical insulation at cryogenic temperatures Electro-Technology 72–7 McCammon R D and Work R N 1965 Measurement of the dielectric properties and thermal expansion of polymers from ambient to liquid helium temperatures Rev. Sci. Instrum. 36 1169–73 Mizuno Y, Ohe T, Nomizu K, Muneyasu H, Nagao M, Kosaki M, Shimizu N and Horii K 1991 Evaluation of ethylene propylene rubber as an electrical insulation materials of superconducting cable Proc. 3rd Int. Conf on Properties and Applications of Dielectric Materials (3rd ICPADM) (Tokyo, 1991) pp 317–21 Nagao M, Kosaki M, Mizuno Y and Ono M 1991 Distinctive electrical breakdown of polar polymeric films in cryogenic temperature region Proc. 7th Int. Symp. on High Voltage Engineering (Dresden, 1991) pp 139–42 Phillips W A 1970 Low temperature dielectric relaxation in polyethylene and related hydrocarbon polymers Proc. R. Soc. A 319 565–81 Sillars R W 1973 Electrical Insulating Materials and their Application (Stevenage: Peregrinus) Stevels J M and Volger J 1962 Further experimental investigations on the dielectric losses of quartz crystals in relation to their imperfections Philips Res. Rep. 17 283–314 Weedy B M, Shaikh S and Swingler S G 1982 Partial discharges in cavities in insulation impregnated with supercritical helium IEEE Trans. Electr. Insul. EI-17 46–52 Weedy B M and Swingler S G 1979 Life expectancy of liquid-nitrogen taped cable insulation IEEE Trans. Electr. Insul. EI-14 222–8

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F8 Thermoelectric effects of superconductors

A B Kaiser and C Uher

F8.0.1 Introduction Thermoelectricity, representing the interplay between the electric and thermal gradients in conductors, is widely utilized in important technological applications. Examples are thermocouple temperature sensors, which measure temperature differences in terms of thermoelectric voltage, and thermoelectric energy conversion devices, which convert heat energy to electrical energy or vice versa. In thermoelectric coolers, high-temperature superconductors have an advantage as passive elements, as we explain in this chapter. High-temperature superconductors also find application as the thermoelectric reference material in thermocouples, since they make it possible to determine accurately the absolute thermoelectric power of a single material up to the transition temperature of the superconductor. Thermoelectric effects can be very sensitive probes of the nature of electronic states and interactions in conductors, and are therefore a standard tool in the investigation of the properties of new conducting materials. In the case of hightemperature superconductors, the thermoelectric power in the normal state shows systematic behaviour related to the transition temperature Tc and the carrier density, and so can be helpful in optimizing Tc Particularly significant are thermomagnetic/thermoelectric effects in the mixed state of high-temperature superconductors, which have proved to be a very useful probe of the motion of the magnetic flux lines, a key factor in dissipation of supercurrents. F8.0.2 General characteristics of thermopower (Seebeck coefficient) The thermoelectric power S (or thermopower or Seebeck coefficient) determines the interaction between electric and thermal currents in a conductor. It is defined by S = ∆V/∆T, where ∆V is the voltage arising (in the absence of an applied electric field) across a sample with a temperature difference of ∆T between its ends, or equivalently by the relation E = S∇T, where E is the electric field created in the sample (Blatt et al 1976). A typical arrangement for the practical measurement of thermopower is shown in figure F8.0.1. The temperature difference across the sample is established by the heater (e.g. a small resistor) at one end, and thermal and potential differences are measured at two points along the length of the sample. Temperatures are measured by the copper—constantan thermocouples with legs 1 and 2, the reference junctions being kept at a fixed temperature. ∆V can be measured using the legs of the thermocouples or separate fine-gauge copper wires. In fact, it can be seen that what is actually measured is the thermopower difference between two materials, the sample and the copper or other material used

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Figure F8.0.1. Typical experimental set-up for the measurement of the thermopower using copper-constantan thermocouples (numbers 1 and 2 designate copper and constantan wires respectively).

for voltage leads (we discuss the determination of the absolute thermopower in the section on applications below). Details of techniques in thermoelectric measurements are given in chapter 3 of the book by Blatt et al (1976). Two other related thermoelectric effects are also described in the standard texts. In the Peltier effect, heat is absorbed when a current flows from one conductor (A) to another (B) of different material and evolved when the current flows in the opposite direction from B to A. This effect forms the basis for thermoelectric coolers, an important technical application discussed later. The other effect is the Thomson effect, in which heat is evolved or absorbed as a current flows in a single conductor with a temperature gradient; this effect is quite separate, and additional to, the normal Joule heating, changing from absorption to evolution of heat on reversal of the current direction. The sign and magnitude of both these effects can be determined from a knowledge of the thermopower S(T) (Blatt et al 1976). The basic physical origin of thermopower lies in the asymmetry of the diffusion of electrons and holes down the temperature gradient, which results in the accumulation of charge at one end of the sample as a result solely of the temperature gradient. If the charge carriers are predominantly holes, there tends to be an accumulation of positive charge at the cooler end, in which case the thermopower is defined to be positive, and vice versa for electrons. This ‘diffusion thermopower’ Sd can be expressed as

where εaυ , is the average energy of the electrons (relative to the chemical potential) weighted by their contribution to the conductivity, e is the (negative) electronic charge and kB is Boltzmann’s constant. Thus TSd is sometimes thought of as the ‘transport’ specific heat εa υ /e per unit charge and Sd as the entropy per unit charge. It is clear from equation (F8.0.1) that if the average charge carrier energy εa υ , equals the thermal energy kBT, the thermopower equals kB /e, which has the value 86 µV K−1. In crystalline semiconductors, conduction occurs predominantly near the bottom of the conduction band at energy εc , relative to the chemical potential. Hence the diffusion thermopower Sd is large (since usually εc » kBT ) and varies approximately as T −1 (since εc , is temperature independent), as shown in the sketch in figure F8.0.2. For p-type semiconductors, the thermopower will be positive (εa υ negative), and for n-type semiconductors it will be negative, in accordance with the type of carrier present. For metals, on the other hand, conduction is in the vicinity of the Fermi level, so εa υ , and thermopower are small (usually less than 10 µV K−1 at room temperature). The asymmetry of diffusion increases approximately linearly with temperature, since electron states within ±kBT of the chemical potential participate in the conduction, making Sd approximately linear in temperature (figure F8.0.2). Even in simple metals, however, this linearity is not really valid owing to mass enhancement by the electron—phonon

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Figure F8.0.2. A sketch of the thermopower behaviour for different conduction mechanisms: metallic conduction (including the electron—phonon nonlinearity), crystalline semiconductor band conduction and variable-range hopping (VRH).

interaction, which gives a characteristic change in slope of thermopower well below the Debye temperature TD for the phonons. In addition, in crystalline metals there is usually a large ‘phonon drag’ thermopower Sg arising from the effect of the ‘disequilibrium’ in a thermal gradient of the phonon distribution, which causes a corresponding disequilibrium in the electron distribution with which the phonons interact. Typically, peaks of a few microvolts per kelvin are seen well below TD . This peak is suppressed with increasing disorder (e.g. Kaiser 1987), which limits the phonon mean free paths, so that it is essentially absent in amorphous metals. For materials with localized states in which conduction is by variable-range hopping (VRH), the thermopower tends to be of a magnitude intermediate between that of metals and crystalline semiconductors and is expected to vary as T1/2, although such simple behaviour is rarely seen experimentally. For conduction in very narrow bands, thermopower at high temperatures where all states participate in conduction is approximately temperature independent (see Kaiser and Uher (1991) for references). F8.0.3 Thermopower of superconductors in the normal phase F8.0.3.1 Conventional, A15 and Chevrel-phase superconductors The thermopower of conventional superconductors consisting of metals and their alloys is consistent with the general thermopower behaviour for metals, as would be expected, except that it is zero in the superconducting state, as discussed below. For example, the thermopower of niobium (Tc ∼ 9 K), shown in figure F8.0.3, shows an apparent phonon drag peak around 75 K, which then decays as temperature increases. The thermopower of the A15 compound Nb3Sn (Tc ∼ 18 K) is also reported to have a peak, near 60 K (Ruan and Lin 1986). For V3Si (Tc ∼ 17 K) shown in figure F8.0.3, and V3Ga (Tc ∼ 16.5 K), which has a similar thermopower, the behaviour is somewhat unusual in that the thermopower does not extrapolate to zero as temperature decreases. The technologically important superconductor Nb—Ti (Tc ∼ 9 K) does have a thermopower extrapolating to zero (figure F8.0.3), but unfortunately data above 90 K do not appear to be available. In general, for the case of good crystalline metals, superconducting or not, the apparent presence of large phonon drag contributions and more than one source of scattering, as well

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Figure F8.0.3. The thermopower of niobium (Nb) (Carter et al 1970), Nb—Ti (Uher 1987) and the A15 superconductor V3Si (Sarachik et al 1963).

the great sensitivity of thermopower to details of the electronic structure, makes it difficult to explain thermopower quantitatively in terms of theoretical models. However, in the case of disordered materials, the situation is much simpler. The scattering is often dominated at all temperatures by a single mechanism (elastic disorder scattering) and, as mentioned above, phonon drag is suppressed by the disorder. Thus the observed thermopower is usually approximately linear in temperature as in the standard metallic model, although characteristic changes of slope often occur around 50 K. These changes in slope can largely be explained as due to mass enhancement at low temperatures by a factor (1 + λ), where λ is the electron—phonon coupling constant, as in the better known enhancement of the specific heat. In thermopower (unlike specific heat), it is posssible to see the decay of this electron—phonon enhancement as temperature increases, i.e. the reduction in the slope of the thermopower temperature dependence. Including the electron—phonon enhancement effect, the standard expression for metallic diffusion thermopower becomes

where XbT is the bare thermopower (often taken as linear), λs(T) is the electron—phonon enhancement for thermopower, which dies away as temperature increases, and a is a constant approximately equal to unity if other electron—phonon effects are small (Kaiser 1987). Clearly, in superconductors in which the electron—phonon interaction is responsible for the superconductivity (i.e. λ is large), the enhancement of low-temperature thermopower, and hence the change in slope, should be significant. Experimentally, this is found to be the case, as illustrated in figure F8.0.4(a) for a Chevrel-phase superconductor in which disorder scattering is large owing to the partial substitution of sulphur. The fitted value aλs(0) ∼ 0.6 for the Chevrel superconductor is comparable to the value of λ calculated from Tc using the usual McMillan formula, although a direct correspondence is not expected owing to the possible presence of other electron—phonon effects (i.e. a ≠ 1), especially when the thermopower is small (Kaiser 1987). In addition, the scale of the temperature dependence is consistent with the phonon energies in these materials.

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Figure F8.0.4. Measured thermopower (crosses) of (a) the Chevrel-phase superconductor Cu1.8Mo6S5Te3 with Tc ∼ 5 K (Rao et al 1984) and (b) the fullerene superconductor K3C60 with Tc ∼ 19 K (Inabe et al 1992), fitted to theoretical expressions for metallic diffusion thermopower (full lines) (Kaiser 1987, Kaiser et al 1995). The broken line in (a) shows linear behaviour.

F8.0.3.2 Fullerene superconductors One of the most exciting discoveries of a new material in recent years is that of fullerene (‘buckyball’), C60 , which with its numerous modifications has since been the focus of extensive research. Unexpectedly, it was found that when C60 was doped with alkali metals, superconductors were formed with Tc up to 31 K (Murphy et al 1993), higher than any other superconductors except for the cuprates. It is thought that the electron—phonon interaction involving intraball phonon modes is responsible for this superconductivity. In agreement with this, the thermopower of the so-called ‘fulleride’ crystals K3C60 and Rb3C60 shows a similar behaviour to that for the disordered metal and Chevrel-phase compounds, i.e. roughly proportional to temperature but with a change of slope near 50 K. Since the bare thermopower of the fullerides is negative, the enhancement is also negative. The rather pronounced changes in slope seen near 50 K for K3C60 (Tc ∼ 19 K) (figure F8.0.4(b)) and Rb3C60 (Tc ∼ 29 K) (Inabe et al 1992) indicate a strong electron— phonon coupling, which is obviously consistent with an electron—phonon mechanism for the superconductivity. The fulleride superconductors are good candidates for the observation of the electron— phonon change in slope, since their large residual resistivity and orientational disorder indicate the presence of the necessary strong disorder scattering to suppress other effects. F8.0.3.3 Organic and heavy-fermion superconductors Two new classes of superconductor are the recently discovered organic and heavy-fermion superconductors, which both have interesting novel features but appear to be of lesser significance for technology, partly because of their relatively low Tc values. For the organic superconductors, the highest Tc at ambient pressure is reported as Tc ∼ 12 K for κ-(ET)2Cu[N(CN)2 ]Br, a charge-transfer salt derived from the electron donor molecule BEDT-TTF or ET (Williams et al 1991). This material has a layered structure, with a lower resistivity parallel to the layers than perpendicular. The resistivity parallel to the layers is still large (approximately 20 mΩ cm at room temperature) and has an unusual temperature dependence

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featuring a pronounced peak near 100 K. The thermopower (Bondarenko et al 1995) is also larger than typical for metals (up to about 25 µΩ cm) and shows a peak or minimum around 100–150 K, depending on the direction within the two-dimensional (2D) layer. Examples of heavy-fermion superconductors are UPt3 (Tc = 0.5 K) and CeCuSi2 (Tc ∼ 0.6 K) (Stewart 1984). In these materials, it is the highly correlated f electrons with high effective mass that participate in the superconductivity, and there are suggestions that the nature of the superconductivity is unconventional. The resistivity and thermopower do not follow the usual metallic behaviour, with sharp low-temperature thermopower maxima for CeCuSi2 and UPt3 (Steglich et al 1985). F8.0.3.4 Perovskite high-temperature superconductors The recent dramatic increase of interest in superconductivity is of course due to the discovery of cuprate superconductors with a perovskite structure with Tc up to 134 K for HgBa2Ca2Cu3O8 −δ , or even higher under pressure. The thermopower of the cuprate superconductors, while of metallic magnitude in samples with high Tc , is more complex than that of the more conventional superconductors discussed above. It nevertheless shows a surprisingly systematic pattern that is closely linked to the occurrence of superconductivity and the value of Tc , and therefore requires an explanation in terms of the electronic interactions related to the superconductivity. This pattern is illustrated in figures F8.0.5 and F8.0.6 by data for the La214, Hg1223 and Tl1201 cuprate systems; data for Bi-based cuprates and YBa2Cu3O7– δ are similar, although anisotropy in the latter is discussed further below.

Figure F8.0.5. (a) Thermopower of La2 – xSrxCuO4 – δ for Sr concentrations x = 0.2 and 0.25 (Uher et al 1987). (b) Thermopower of HgBa2Ca2Cu3O8 – δ samples a—g from the same batch after annealing at approximately 500°C for increasing times (Subramaniam et al 1995a).

The thermopower pattern is shown most fully in the Tl1201 superconductor series with relatively low Tc , which has the advantage that the normal-state thermopower can be investigated down to lower temperatures. In this series, substituting La3+ ions for Sr2+ ions reduces the hole concentration in the CuO2 planes. The variation of Tc is shown in figure F8.0.6(a). For small La concentrations x, the superconductor is in the ‘overdoped’ regime, with zero or small Tc, and as holes are removed, by increasing x , Tc first increases to its maximum value (‘optimal doping’), and then decreases again in the ‘underdoped’ regime where the hole concentration is small. The thermopower S of the polycrystalline superconductors in figure F8.0.6(b) is essentially that for the direction parallel to the CuO2 planes, owing to the very small conductivity perpendicular to the planes.

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Figure F8.0.6. (a) Transition temperature Tc and (b) thermopower temperature dependence as a function of doping for the Tl1201 cuprate superconductor series Tl0.5Pb0.5Sr2 – xLaxCuO5 . The numbers beside the thermopower data sets are the La concentrations x (Subramaniam et al 1994a).

The temperature dependence of thermopower for the overdoped samples (x < 0.4) is somewhat similar to that of the conventional disordered metal, Chevrel-phase and fullerene superconductors discussed above, i.e. S is approximately proportional to T above Tc, perhaps with some change of slope. This is also the regime where the Hall coefficient and resistivity show conventional behaviour. As the hole concentration is reduced, the thermopower data tend to show a temperature-independent upward shift, retaining a similar negative slope. Hence, although still approximately linear, the thermopower is no longer proportional to temperature but instead extrapolates to positive values as T → 0. For optimal doping (x = 0.4), the room-temperature thermopower is near zero. As hole concentration decreases further to reach the underdoped regime (x > 0.4) where Tc decreases, the thermopower becomes positive at all temperatures below room temperature, showing a further upward shift. Ultimately, electron states become localized, the resistivity temperature dependence becomes nonmetallic, thermopower becomes very large and roughly temperature independent and superconductivity is suppressed. Thus the key unusual feature of the thermopower of the cuprate superconductors is the approximately temperature-independent upward shift from conventional metallic behaviour in the overdoped limit, especially for the Bi-, Tl- and Hg-based series. We note that the room-temperature thermopower is often near zero when the hole concentration is such that Tc is at its maximum value for a given series, which can be useful as a test for optimal doping (Obertelli et al 1992). For example, the thermopower for the highest-Tc Hg1223 superconductor in figure F8.0.5(b) failed to show the usual change to negative values of S at room temperature even after extensive annealing in oxygen, suggesting that it remained slightly underdoped. The cuprate superconductor YBa2Cu3O7 – δ is rather different from most of the other cuprate superconductors in having a large contribution to conductivity from CuO chains as well as from the usual CuO2 planes. This is revealed by the anisotropy in the a and b crystal directions which are both parallel to the CuO2 planes: the conductivity is higher in the b direction parallel to the CuO chains. The thermopower in the a direction (figure F8.0.7(a)) shows a negative slope as a function of temperature, like the other cuprate superconductors (but with a reduction in slope above 150 K). In the b direction, however, to which the chains contribute strongly, the thermopower shows both positive and negative slopes. Small differences in oxygen vacancies are probably the main cause of the differing thermopowers for different samples.

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Figure F8.0.7. (a) The thermopower of untwinned YBa2Cu3O7 – δ crystals in the a and b directions parallel to the CuO2 planes (Subramaniam et al 1995b), with additional data (labelled L) from Lowe et al (1991) and (labelled C) from Cohn et al (1992). (b) The thermopower of YBa2Cu3O7–d crystals in the c direction perpendicular to the CuO2 planes (data from: +–Sera et al (1988) and ∆–Wang and Ong (1988)).

We also show in figure F8.0.7(b) the thermopower Sc for YBa2Cu3O7 – δ in the c direction perpendicular to the CuO2 planes. These data indicate that the thermopower in this direction is more closely proportional to T than for the in-plane directions, but owing to the high anisotropy Sc is difficult to measure and results vary. Nevertheless, c-axis behaviour similar to that in figure F8.0.7(b) has also been seen in single crystals of other cuprate superconductors. An interesting special case of a cuprate superconductor is the ‘electron-doped’ material Nd2 – CexCuO4 (maximum Tc ∼ 25 K), in which the formation of Ce4+ is expected to dope the CuO2 planes x with electrons instead of holes. This is confirmed by the observation that this material has a negative Hall coefficient and large negative thermopower for low Ce doping levels. However, as the doping level increases, the thermopower magnitude becomes small, and for the high-Tc regime the thermopower pattern appears to be surprisingly similar to that shown in figure F8.0.3 for hole doping, i.e. a negative slope with a change in sign as temperature increases (Xu et al 1992). In this case, there is an upward rather than downward shift of the thermopower—temperature plot as doping is increased. The similarity shows that the electronic states of the electron-doped material are very similar to those of the hole-doped materials for the superconducting compositions, emphasizing the universality of the thermopower pattern and the need for a common explanation. This systematic pattern provides a test for theories seeking to account for the properties of the cuprates. It is in fact remarkably similar to that seen in NbN superconductors progressively disordered by irradiation (Kaiser et al 1995, Siebold and Ziemann 1993), but the thermopower in the cuprates is an order of magnitude larger. At present, there is no consensus regarding the origin of the cuprate behaviour, although a number of theories have been advanced. Some of these are developments of conventional models involving the electron—lattice interaction, such as phonon drag, extremely large electron-phonon

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renormalizations due to anharmonic phonons and very narrow band features such as van Hove singularities; others involve more exotic concepts such as spin polarons and marginal Fermi liquids, although few quantitative predictions are available for these models. A survey of several different theories, and of detailed thermopower data for the various families of cuprate superconductors, is given by Kaiser and Uher (1991). Another interesting case is that of the bismuthate superconductors Ba1 – xKxBiO3 , which have some similarities with the cuprate superconductors, being perovskite oxide superconductors with a low carrier density near a metal—insulator transition. However, there are also dramatic differences: Ba1 – xKxBiO3 has an isotropic cubic instead of a 2D plane perovskite structure, and no magnetic ions (its normal-state suceptibility is diamagnetic). Hence similar behaviour to the cuprate superconductors would suggest that 2D structure and magnetism play no role in this behaviour. In fact, the thermopower of Ba1 – xKxBiO3 does show some similarities to the standard cuprate pattern mentioned above, namely a small magnitude and a negative sign for the thermopower temperature dependence above Tc , sometimes with changes of sign as temperature increases (Subramaniam et al 1994b). However, the thermopower of Ba1 – xKxBiO3 appears to show a flattening or upturn above 150 K. F8.0.4 Thermoelectric effects in the superconducting phase In the superconducting phase, any current density Jn of normal-state carriers in response to an applied thermal gradient is opposed by a counter-flow Js of supercurrent without the need for the usual thermoelectric voltage (Van Harlingen 1982). Thus thermopower is expected to be zero in superconductors, except for small effects when the crystal is anisotropic and ∇T is not applied along one of the principal axes. As illustrated by the data for various superconductors in zero field shown in figures F8.0.3–F8.0.7, the thermopower is indeed observed to drop to zero as temperature is reduced below Tc . (Note that when magnetic flux penetrates into a superconductor to produce the mixed phase, the thermopower and other thermoelectric coefficients are not zero, as we discuss in the next section.) However, even in superconductors with total current J = Jn + Js = 0, some thermoelectric effects arising from the normal-state current Jn may be detectable (Van Harlingen 1982). For example, in an isolated superconductor, a circulating current pattern is set up, with normal current being transformed into supercurrent (and vice versa) at the ends of the sample. A quasiparticle charge imbalance near the end of superconducting Al films has been detected (Mamin et al 1984) in the presence of a thermal gradient, and the inferred value of the thermoelectric current density Jn in the superconducting state was consistent with the thermopower measured in the normal state. Turning now to effects near Tc , it might be expected that the effect of superconducting fluctuations in the normal state above Tc would be to produce a reduction in thermopower magnitude as a precursor to the zero value of thermopower in the superconducting state. This is often seen, but the behaviour in other cases appears to be more complex and not well understood. F8.0.5 Thermomagnetic effects in the mixed phase F8.0.5.1 Introduction Studies of the thermomagnetic/thermoelectric properties in the mixed state of superconductors have gained considerable impetus over the past few years and proved to be a very useful probe of the dynamics of vortices and other parameters characterizing the superconducting state, so we discuss these effects at some length. In conventional superconductors, thermomagnetic effects are typically small and occur over a very narrow temperature range (Huebener 1979), but in the highly anisotropic structure of CuO2 perovskite superconductors, the effects grow to be readily measurable over a wide range of temperatures below the superconducting transition temperature Tc (Freimuth 1992). While the more familiar transport parameters

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such as the electrical resistivity and zero-field thermopower vanish upon entry into the superconducting domain, thermomagnetic phenomena in the mixed state continue to provide useful information and are a direct experimental probe of the carrier and vortex dynamics. Before we describe the various thermomagnetic coefficients and discuss their behaviour, we comment in this section on the structure of flux lines and define several relevant parameters which are essential for understanding the role of vortices in the transport properties. For the vast majority of practically viable superconductors, including all high-temperature perovskites, the true Meissner state (with magnetic flux excluded from the interior of the superconductor) exists only at low magnetic fields. In these type II superconductors, for magnetic fields H greater than the lower critical field Hc 1 , the magnetic flux starts to penetrate into the bulk of a superconductor in the form of microscopic filaments called flux lines or vortices, each containing a fundamental quantum of magnetic flux, Φ0 = 2.07 × 10–15 Wb; the cores of these filaments are in the normal state. With increasing magnetic field the density of vortices increases and they interact to set up a triangular vortex lattice (Abrikosov lattice) with lattice spacing a0 ~ (Φ /H) - gradually squeezing out the superconducting phase. At the upper critical field, Hc 2 , the overlap between the vortices leads to a collapse of the superconducting phase, the material becomes normal and the field penetrates completely. It is this vortex state of a superconductor for fields Hc 1 < H < Hc 2 (the mixed state) that is the regime of prime interest to us from the point of view of the thermomagnetic/thermoelectric properties of superconductors. One can distinguish between type I and type II superconductors based on the value of the Ginzburg—Landau parameter κGL defined as κGL = λ/ξ. Here λ stands for the penetration depth which characterizes the length scale over which the magnetic field penetrates into a superconductor and ξ is the coherence length which can be viewed as the range over which superconducting correlations extend or, in other words, the ‘size’ of the Cooper pair. Materials for which κGL < 1/p2 are type I superconductors while those for which κGL < 1/p2 are type II superconductors. Since all hightemperature superconductors possess a very short coherence length (∼10 Å) and a substantial penetration depth (∼ 1000 Å), they have a Ginzburg—Landau parameter of the order of 100 and are therefore extreme type II superconductors. A schematic picture of an isolated flux line is shown in figure F8.0.8. The normal core of the flux line is roughly delineated by a region between – ξ and +ξ. The density of the superconducting electrons ns and the superconducting order parameter ∆ vanish in the centre of the flux line and reach their full equilibrium values at a distance of the order of the coherence length. The magnetic field H(r) peaks at the centre of the flux line and decays approximately exponentially beyond a radius of the order of the penetration depth. The shaded region indicates quasiparticle excitations which are localized within the vortex core. In 1

0

2

Figure F8.0.8. The structure of an isolated flux line indicating the variation of the density ns of superconducting electrons and the magnetic field H(r) as a function of distance from the centre of the flux line. Excitations bound within the core of the flux line are indicated by shading.

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addition, normal excitations with energy above ∆ which extend throughout the bulk of the superconductor may also be present, but will eventually die out as the temperature decreases. All experiments probing flux-line transport in high-temperature superconductors have, so far, been performed at fields far below the upper critical field Hc 2 and thus the vortex cores were kept well separated. From the above description it is not surprising that the flux lines are frequently pictured as rigid tubes of normal phase of radius ξ threading the superconducting phase. Such a view, often referred to as the London model, is quite adequate for most of the conventional type II superconductors with a modestly large Ginzburg—Landau parameter. However, in high-temperature perovskites, because of their highly anisotropic (layered) structure, the flux lines are less robust. In fact, when crossing over into a quasi-2D realm of superconductivity, the shape and form of the flux lines can dramatically change. The model based on rigid tubes gives way in the Lawrence—Doniach approach to a Josephson-coupled stack of superconducting sheets of thickness d spaced a distance s apart. In the case of extreme anisotropy this leads to a chain of 2D pancake vortices each with a circular current essentially confined to an individual superconducting layer and connected by Josephson vortices rather like ‘beads on a necklace’ (Clem 1991). Chains of pancake vortices lack rigidity and thermal fluctuations can displace the pancakes in individual layers creating a zig-zagged string. With enough thermal energy the string can ‘evaporate’ into a gas of individual 2D vortices (for a superconducting structure consisting of identical layers, this temperature coincides with the temperature (Kosterlitz and Thouless 1973) at which vortex—antivortex unbinding occurs in 2D superconducting films). However, because the electromagnetic response of a stack of 2D pancake vortices is qualitatively similar to that of a three-dimensional (3D) vortex line, we adhere in our description of the flux-line transport to a model depicting vortices as straight lines, i.e. as an Abrikosov vortex lattice. Although we are primarily interested in the response of a superconductor in the mixed state to the thermal force, it is instructive and also essential to understand how the vortex state reacts to the transport current. For this reason we include a brief discussion of flux-flow resistance. The most prominent feature of type II superconductors is their ability to carry large transport current in the presence of a magnetic field provided flux lines are pinned. The critical current density is given by the balance of two opposing forces acting on the flux lines: the pinning force due to spatial variations of the condensation energy and the Lorentz force exerted by the transport current. Whenever flux lines are in motion, energy is dissipated and resistance arises. One distinguishes between two regimes of dissipation: flux creep where the pinning force dominates and flux flow where the Lorentz force dominates. Let us assume that flux lines are free to move. A flow of transport current Jx directed along the x axis exerts a Lorentz force on a unit length of the flux line given by

In the configuration of figure F8.0.9 this force points down along the. negative y axis. Here Φ0 is the flux quantum associated with a vortex oriented along the z axis. Being free to move, the flux lines will traverse the sample in the direction of the Lorentz force. Under steady-state conditions, new flux lines continuously enter at the upper edge and leave upon reaching the lower edge of the specimen. It is important to keep in mind that the flux lines are being dragged through a lattice of positive ions constituting the sample and, as such, their motion is hindered by some relaxation mechanism and they eventually acquire velocity υL , y transverse to the transport current Jx . The force responsible for damping the vortex motion is a viscous drag force f η = —ηυL , y where η is the viscous drag coefficient. The motion of free vortices is governed by the equation

The resulting motion is referred to as flux flow or the viscous flow of vortices.

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Figure F8.0.9. Transport current, thermal gradient and magnetic field assignments with respect to the Cartesian coordinate system.

It is frequently stated that the flux lines moving with velocity υL ‘induce’ the macroscopic electric field

where B is the magnetic induction, B = n Φ0 , and n is the density of vortices. At a glance it looks as if the electric field E is a consequence of Faraday’s law of induction, but this is not the case. The misconception stems from the assumption that the vortices are individually indestructible. In reality, as the flux lines enter on one side of the specimen and leave on the other, the total magnetic flux in the measuring circuit remains unchanged during the flux motion and thus Faraday induction cannot be the source of the electric field. The actual origin of the field E is in the quantum-mechanical phase slip resulting from the motion of vortices and given by the Josephson relations. The essential point is that a potential gradient and hence resistance is possible in a superconductor, but only in association with a time derivative of Ψ. Although this can occur by acceleration of supercurrent, it is more often effected by the flow of flux lines, since the passage of one vortex between two points requires a change of 2π in the relative phase of Ψ. According to Josephson, this leads to the voltage

where ∆γ is the phase difference. Assuming the flux lines move with the average (constant) velocity υL , the rate of phase slippage is proportional to υL and equation (F8.0.5) follows from equation (F8.0.6). Combining equations (F8.0.3)–(F8.0.5) and assigning the respective directions as shown in figure F8.0.9, the electric field in the x direction becomes Ex = Φ0BZ /η) Jx (which implies the appearance of electrical resistivity called the flux-flow resistivity

At magnetic field equal to Hc 2 the material becomes normal and the resistivity is that of the normal metal, ρn

Eliminating the coefficient η between equations (F8.0.7) and (F8.0.8) we finally obtain

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The proportionality on the right-hand side of equation (F8.0.9) follows from the well known relation for the upper critical field, Hc 2 ∝ 1/ξ2, and from the expression for the vortex lattice parameter a0 = (Φ0 /B)1/2 . It basically states that the flux-flow resistivity is proportional to the fraction of normal volume occupied by the vortex cores in a unit cell of the flux-line lattice. The flux-flow resistivity is thus clearly associated with the motion of the flux lines and reflects the bulk properties of a type II superconductor. Apart from the phase boundary Hc (T) which delineates the superconducting domain from the normal phase, there is another important line, called the irreversibility line Ti r r , which is the boundary between regions within the superconducting domain where magnetization is reversible (at T > Ti r r (H)), and where irreversibility (hysteretic behaviour) sets in at lower temperatures. In conventional superconductors it is difficult to distinguish the irreversibility line from the phase boundary of superconductivity. In contrast, high-Tc perovskites display an unusually large region of reversibility. The correct interpretation of the irreversibility line is a contested issue and the proposals include conventional flux creep, various glass phase transitions and melting of the flux-line lattice. Discussion of these fascinating topics is beyond the scope of this chapter; for our purposes it suffices to note that, because of their highly anisotropic structure, short coherence length and vortex pinning energies comparable to thermal energies, the dynamics of vortex motion are dramatically altered as the system crosses the irreversibility line. The plethora of unusual transport properties observed in high-temperature superconductors has its origin in a free, unimpeded motion of vortices above the irreversibility line where flux pinning is ineffective. F8.0.5.2 The Ettingshausen effect In the previous section we considered the effect of the transport current on the flux-line lattice and we arrived at the important result that the motion of flux lines leads to dissipation and therefore to resistance. We now consider another important aspect associated with the motion of flux lines—the flow of entropy. Since the core of a flux line is normal, it has higher entropy density than the surrounding superconducting phase. Thus, a flux line in motion represents the transport of entropy. We again refer to figure F8.0.9 where the transport current Jx exerts a Lorentz force on the flux line given by equation (F8.0.3) and the flux lines move in the negative y direction. As discussed before, the flow of flux lines is sustained by their generation at one edge of the specimen and their annihilation at the opposite edge. Since flux lines carry entropy, this also means that there is absorption of heat at the first edge and liberation of heat at the other edge. This leads to a thermal gradient along the direction of flux flow as shown in figure F8.0.9. The Ettingshausen coefficient ∈ is defined in terms of this thermal gradient ∇yT by

which is proportional to the transport entropy of the flux lines. Thermodynamics requires that the transport entropy is zero at T = 0. This should not be surprising since at and near absolute zero no quasiparticle excitations exist within the vortex core due to a gap in the excitation spectrum. Unlike the caloric entropy which goes into the normal entropy, the transport entropy is also expected to vanish at Tc . The reason is simple: the flux lines have taken over the entire space and the sample has become normal. Thus, one expects a maximum for the transport entropy (and for the measured transverse thermal gradient ∇yT ) somewhere between absolute zero and Tc . Very few measurements of the Ettingshausen coefficient have been made. Using a sensitive measuring system based on a transformer-coupled lock-in amplifier which enhanced the small transverse voltage generated by a differential constantan— chromel thermocouple, Palstra et al (1990) were able to detect the transverse thermal gradient ∇yT in a single crystal of YBa2Cu3O7 – δ (see figure F8.0.10). Measurements were performed in a magnetic field of up to 12 T oriented perpendicular to the CuO2 planes. The gradient ∇yT is detected only in the range of temperatures where the flux-flow resistivity is significant. Typical

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Figure F8.0.10. The transverse thermal gradient ∇yT induced by the transport current Jx and the magnetic field Bz for YBa2Cu3O7 – δ . The inset indicates the lead and sample configuration. Reproduced from Palstra et al (1990) by permission.

values of the transverse thermal gradient are a few kelvin per metre. An interesting observation is the persistence of the transverse thermal gradient at temperatures which exceed Tc by as much as 15 K. The authors note that this may be due to a pronounced fluctuation effect. F8.0.5.3 The Peltier effect In response to an applied electric field Ex , the motion of charges giving the electric current density Jx also produces a heat current density qx , even in the absence of a temperature gradient. This effect governed by the Peltier coefficient Π, which is defined as the ratio qx /Jx subject to the condition of zero thermal gradient along the x direction. In zero magnetic field and in the Meissner state the Peltier effect clearly disappears because the Cooper pairs do not carry entropy. However, in the mixed state of a superconductor, the motion of vortices resulting from their interaction with the transport current Jx generates the electric field Ex . There is a normal-state component of the electric current Jn driven by the field Ex and with this component is associated the flow of Peltier heat across the sample. Evaluating the Peltier heat current associated with the normal-state current density Jn flowing parallel to the x axis in the cores of the moving vortices, Logvenov et al (1991) showed that the Peltier coefficient in the mixed state is given by

where Πn is the Peltier coefficient in the normal state. Equation (F8.0.11) indicates that the Peltier effect in the mixed state arises from the flux-line motion and is proportional to the flux-flow resistivity ρf . Thus, as temperature is lowered in the mixed state, the Peltier coefficient Π decreases from its value Πn in the normal state in the same way as ρf . For the same reason that the flux-flow resistivity curves are dramatically broadened by the magnetic field, we expect the transition region of the Peltier effect to be broadened also. Because of the much wider range of reversibility, it should be easier to measure the Peltier heat in the high-temperature superconductors than in conventional superconductors. However, a word of caution

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is in order: because the above model assumes rigid flux lines with a core diameter ξ which form a regular lattice, questions arise as to whether a much less robust vortex structure (pancake-like vortices) of some of the highly anisotropic high-Tc perovskites would influence the outcome of the experiment. At this stage only one set of data is available (Logvenov et al 1992) obtained on a single crystal of TI2Ba2CaCu2Ox . Although contributions other than those from the superconductor were present, the observed Peltier signal indicated a broadening of the transition region in an applied magnetic field consistent with the theory as represented by equation (F8.0.11). F8.0.5.4 Thermal force In the previous sections the driving force responsible for the motion of flux lines was the Lorentz force arising from interaction of the vortices with the transport current. Whenever the flux lines were free to move or once the transport current exceeded the critical current density determined by the strength of vortex pinning, the flux lines were set in motion perpendicular to the transport current. Instead of the Lorentz force, we now consider another kind of driving force—the thermal force Ft h given by

where ∇x T is the imposed thermal gradient and SΦ is the transport entropy per unit length of flux line. The thermal force arises for the following reason: because of its dependence on temperature, the penetration length is larger at the hot end of the sample than at the cold end. This results in stronger repulsion of flux lines near the hot end and therefore their motion down the temperature gradient. An alternative explanation is based on the Onsager reciprocity argument: since the moving vortex carries with it the entropy, the temperature gradient develops along the direction of the vortex motion which compensates the entropy flow by ordinary heat conduction. Conversely, an applied temperature gradient leads to the thermal force per unit length of flux line given by equation (F8.0.12). The action of the thermal force on vortices in the presence of the applied magnetic field affects the following three transport effects: the Righi—Leduc effect (thermal Hall conductivity), the Nernst effect and the Seebeck effect. The Righi—Leduc effect is the exact thermal analogue of the Hall effect, i.e. a heat current qx along the x axis in the presence of magnetic field in the z direction gives rise to a transverse (along the y axis) temperature gradient ∇yT given by

where RL is the Righi—Leduc coefficient. Recent measurements (Krishana et al 1995) indicate that the Righi—Leduc effect in high-Tc superconductors arises from quasiparticle dynamics and that quasiparticles make a contribution to in-plane thermal conductivity comparable to that of phonons. This is discussed in a review of thermal conductivity in a magnetic field (Uher 1996; see also Uher 1992). F8.0.5.5 The Nernst effect Perhaps the most frequently studied thermomagnetic effect in the mixed state of superconductors is the Nernst effect. This is also the effect which unequivocally confirms the main premise that most of the thermoelectric and thermomagnetic transport effects in superconductors arise as a consequence of flux-line motion. As a distinctive and unambiguous signature of the flux-line motion, the Nernst effect reveals the field and temperature range where pinning is absent or very weak and the flux lines move freely. An additional point to note about the Nernst effect is its very small normal-state value. Thus, the appearance of the Nernst signal in the mixed state is a rather striking feature which contrasts sharply with the small signal measured in the normal state. In conventional superconductors, the Nernst effect was observed

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first by Otter and Solomon (1966). The first reports describing measurements of the Nernst effect in the high-temperature superconductors appeared early in 1990 (e.g. Galffy et al 1990) and much activity has occurred since (e.g. Ri et al 1994). The Nernst effect refers to generation of the transverse electric field due to a longitudinal thermal gradient in a perpendicular magnetic field. Placing the thermal gradient in the negative x direction, the thermal force Ft h acts on the flux lines in the positive x direction and, provided the flux lines are free to move, they will be set in motion down the thermal gradient (see figure F8.0.9). As in the case of the flux-flow resistivity, the flux-line motion is hindered by a viscous drag force fη and equation (F8.0.4) becomes

The same argument which led to equation (F8.0.5) here gives rise to the electric field E except that now this field is directed along the y axis. Thus, from equation (F8.0.14) and using Ey = –υΦ, x B z one obtains

The appearance of the transverse electric field Ey is an unmistakable sign of the Nernst effect. It is indeed difficult to imagine a situation whereby the transverse electric field Ey is caused by anything but the quantum-mechanical phase slip associated with the motion of flux lines in the x direction. The actual Nernst coefficient is defined as

which, after substituting from equation (F8.0.15), yields

The last equality in the above relations follows from a substitution for the viscous drag coefficient η from equation (F8.0.7). In making this substitution one must keep in mind that the coefficient η in equation (F8.0.7) refers to the viscous drag of vortices in the y direction while the viscous drag coefficient introduced in equation (F8.0.15) reflects damping of vortices along the x axis. This may be an insignificant detail as far as conventional superconductors are concerned but it is an important issue for some of the high-Tc perovskites which display significant in-plane anisotropy. In this latter case, one must make sure that the appropriate η is used, i.e. that the Lorentz force and the thermal force in the actual experiments are acting along the same crystallographic direction. Rearranging equation (F8.0.17), the transport entropy becomes

Numerical estimates of the transport entropy based on different experiments frequently disagree by a wide margin. From the experimental point of view, it is essential that the experimental conditions reflect the reality of what one actually measures. For instance, if in addition to the flux-flow resistivity ρf there is another resistive process with the resistivity ρA acting in series with ρf , the overall resistivity ρ will be larger than the flux-flow resistivity and the viscous drag coefficient η will be smaller. This, in turn, will lead to a decrease in the transport entropy according to

Thus, any resistive mechanism which has nothing to do with the flow of vortices but limits the flow of the transport current will cause a reduction in Sφ .

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Two important points should be mentioned here which potentially influence the Nernst effect in highTc superconductors but which are usually inconsequential for the Nernst effect in conventional superconductors. Both issues arise as a consequence of the high anisotropy of CuO2 perovskites. In the introduction to this section we noted that anisotropy has a dramatic influence on the flux-line structure and that for the most anisotropic high-Tc , superconductors (Bi and Tl compounds) the concept of rigid flux tubes needs to be replaced by 2D pancake-like vortices centred on individual superconducting planes and connected by Josephson vortices. In this context, it is important to realize that, because Josephson vortices lack the normal core, the thermal force on such a vortex is zero and thus Josephson vortices do not contribute to the Nernst effect. Likewise, even if realized, the vortex—antivortex unbinding above the Kosterlitz— Thouless temperature is not going to contribute to the Nernst effect in spite of the fact that it is an important resistive mechanism. Very simply, under the action of the Lorentz force the vortex and the antivortex move in opposite directions and, having opposite orientation of magnetic flux, they will generate a resistive voltage. In contrast, under the influence of the thermal force the vortex and the antivortex both move down the thermal gradient and the electric fields they generate will cancel each other. As already mentioned, the Nernst effect has been studied frequently and measurements have been made on samples of various structural forms representing major families of high-Tc superconductors. Figure F8.0.11(a) shows the Nernst effect (Ey /∇xT) together with the resistivity for a c-axis-oriented film

Figure F8.0.11. The increase of resistivity and normalized Nernst electric field as magnetic field (parallel to the c-axis direction) is increased for: (a) c-axis-oriented YBa2Cu3O7 – δ film; (b) c-axis-oriented Bi2Sr2CaCu2O8+x film. Reproduced from Ri et al (1994) by permission.

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of YBa2Cu3O7 – δ with a transition temperature of 90 K (Ri et al 1994). The data illustrate the general trend seen in all forms of CuO2-plane superconductors. The resistive transition broadens dramatically in a magnetic field, the broadening arising mainly from a strongly reduced temperature at which the resistivity eventually becomes zero. We note that a nonvanishing Nernst signal is detected at the same temperatures at which the resistivity is broadened, i.e. in the flux-flow regime. Starting from the high-temperature side, the Nernst effect is very small in the normal state, starts to rise some 10–15 K above the superconducting transition temperature and reaches a peak value of a few microvolts per kelvin. The peak is strongly field dependent and shifts to lower temperatures with increasing magnetic field strength. The Nernst effect becomes immeasurably small at temperatures where the resistivity vanishes. The disappearance of the Nernst signal represents the onset of pinning and the flux-flow motion of vortices ceases to exist. To illustrate the influence of anisotropy, figure F8.0.11(b) shows the data of the same authors but this time for a c-axis-oriented film of Bi2Sr2CaCu2O8+x (zero-field Tc of 85 K), a material which has a far more anisotropic structure than YBa2Cu3O7 – δ . This enhanced anisotropy of Bi2Sr2CaCu2O8+x and the related weaker pinning is clearly reflected in the data where one observes a much broader regime of flux-flow resistivity and a correspondingly wider temperature range over which the Nernst effect is measurable. The larger anisotropy of Bi2Sr2CaCu2O8+x also results in a wider fluctuation regime, i.e. significant Nernst signals extend to considerably higher temperatures above Tc than in the case of YBa2Cu3O7 – δ. In general, because of the extremely short coherence length and high transition temperature, superconducting fluctuations play an important role in the dynamical behaviour of high-temperature superconductors in a temperature range above the transition temperature Tc . In systems with reduced dimensionality, such as Bi- and TI-based perovskites, the contribution of fluctuation effects is further enhanced. The range of temperatures over which the Nemst effect and other thermomagnetic/thermoelectric transport phenomena are measurable could be extended to lower temperatures, i.e. the regime of fluxflow vortex motion could be expanded into the domain of pinning, if a driving force in excess of the pinning force fp is supplied. For instance, one can attempt to drive a sufficiently large transport current, Jx > Jc , through the sample so that the Lorentz force exceeds the average pinning force. It is not so easy to extend the range of flux-flow resistivity in thermally driven experiments such as the Nernst effect because the temperature gradients generated by small resistive heaters are limited (typically of the order of 100–1000 K m–1 ) and the resulting thermal force is a small fraction of the critical current density. To overcome this limitation, the use of pulsed lasers has proved very successful. By focusing a laser beam on the surface of a superconducting film, it is possible to generate thermal gradients in excess of 107 K m–1 and thus thermally driven flux motion could be studied over an extremely wide range of driving forces (Zenner et al 1994). This, in turn, has opened the way to investigate all three regimes of flux motion: thermally assisted flux flow, flux creep and viscous flux flow as well as the crossover between them as a function of the laser pulse intensity. At high laser fluences, corresponding to viscous flux flow, the Nernst voltage is proportional to the fluence; deviations from this proportionality signal the onset of pinning. At low fluences, in the thermally assisted flux-flow regime, the variation of Nernst voltage with temperature yielded an activation energy of order 500 K for flux hopping, which indicates the scale of the dominant pinning mechanism in the Tl—Ba—Ca—Cu—O film investigated. F8.0.5.6 The Seebeck effect (magnetothermopower) As discussed above, the Seebeck effect manifests itself as the longitudinal electric field Ex = S∇xT proportional to the longitudinal temperature gradient. For a superconductor in the mixed state, the flux lines (along the z direction as in figure F8.0.9) are subject to a thermal force Ft h pointing along the positive x direction. Assuming they are free to move, they will drift down the thermal gradient. However, the electric field that arises is not directed along the x axis (the Seebeck field) but along the y axis (the Nernst field). Thus, unless the flux lines somehow acquire a nonzero velocity component along the y axis, a

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longitudinal (along the x axis) field associated with their motion down the thermal gradient cannot arise, and therefore the Seebeck effect due to vortex motion is zero. Even if the flux lines do not move perfectly parallel to the thermal force, but rather at an angle θυ , this should yield a contribution to the Seebeck effect of at most S = QBZ tan θυ where Q is the Nernst coefficient. If the Hall angle θυ is very small, typically tan θυ ≈ 10– 3, the vortex contribution to the Seebeck effect is negligible. In conventional superconductors the Seebeck effect in the mixed state has never been detected. In highTc superconductors the situation appears to be quite different: one observes a robust Seebeck signal that does not vanish until well below Tc and the magnitude of the Seebeck effect rivals or exceeds that of the Nernst effect. The question is: what causes this large signal? An interesting model that is now widely accepted was proposed by Huebener et al (1990), who invoked the counterflow model mentioned in section F8.0.4 and assumed its validity even in the mixed state. Its important feature is the postulate that only the superfluid current component Js interacts with the vortices. Through a Lorentz-like force Js × Φ0 the vortices acquire a y component of velocity and, via equation (F8.0.5), a longitudinal (Seebeck) field arises (see figure F8.0.12). It should be kept in mind that, while the Seebeck voltage is the result of the phase slip due to the motion of vortices along the y axis, the actual origin of the effect rests with the quasiparticle motion down the thermal gradient that gives rise to an equal and opposite superfluid current and this, in turn, interacts with the vortices. It is interesting to note that, in accord with experiment, reversal of the magnetic field leaves the Seebeck voltage unchanged. The two-fluid model leads to a simple formula for the Seebeck coefficient in the mixed state given by

where ρf (T) is the flux-flow resistivity and ρn(T) and Sn(T) are the normal-state values of the resistivity and of the Seebeck coefficient. In fact, any two-component model with the thermopower of one of the channels greatly exceeding that of the other will lead to this kind of formula. Note that this formula is analogous to that (equation (F8.0.10)) for the Peltier effect, which is the thermal inverse of the Seebeck effect. Equation (F8.0.20) provides a theoretical foundation for the experimental fact that broadening in the thermopower transition follows remarkably closely the corresponding broadening in the flux-flow resistivity. Figure F8.0.13 shows an example of the close resemblance between the behaviour of the Seebeck coefficient and that of the resistivity.

Figure F8.0.12. The transverse component of the flux-line motion resulting from the interaction of flux lines with the counterflow supercurrent Js .

While the counterflow model of Huebener et al (1990) is certainly applicable far away from the fluxline centres, close to the vortex core the supercurrent is redistributed and the boundary conditions are important. However, calculations indicate that equation (F8.0.20) still holds. In addition, there are only small Hall effect contributions associated with the angles θυ and θQ P between the direction of the thermal force and the directions of the flux-line and quasiparticle motion respectively. There are nevertheless some aspects of the Seebeck effect that are not yet fully understood. For example, the wide range of temperatures over which the Seebeck effect is observed is surprising. One would normally expect a rapid decrease in the density of unbound quasiparticles below Tc . Their apparent persistence to very low temperatures may be

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Figure F8.0.13. The temperature dependence of the reduced Seebeck coefficient Sf /Sn (open circles) in YBa2Cu3O7 – δ compared with that of the reduced magnetoresistance ρf /ρn (full curves). The curves for different magnetic fields are shifted by 0.2 along the y axis. Reproduced from Hohn et al (1991) by permission.

another sign of the peculiar nature of the quasiparticle spectrum and may even provide additional support for the proposed d-wave pairing mechanism.

F8.0.6 Thermoelectric applications of high-temperature superconductors Thermoelectricity has important technological applications such as thermoelectric energy conversion devices (Beeforth and Goldsmid 1970, Harman and Honig 1967) and thermocouple temperature sensors (Kinzie 1973). In this section we outline two applications where high-temperature superconductors have significant advantages.

F8.0.6.1 Determination of absolute thermoelectric power It is possible to use the fact that the thermopower of a superconductor is zero (in zero magnetic field) to measure the absolute thermopower of a single material up to the transition temperature of the superconductor (Uher 1987). The importance of this application arises from the difficulty of determining the thermopower of a single element by other means. A typical experimental set-up to measure thermopower consists of a thermoelectric circuit formed from the material under investigation and a reference material. In the configuration shown in figure F8.0.14, the thermoelectric power of the couple SAB is (Blatt et al 1976)

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Figure F8.0.14. basic thermocouple circuit.

Clearly, if we choose the material A so that it has zero absolute thermopower (SA = 0), the thermopower of the couple is equal to the absolute thermopower of material B. The sign of the thermopower SB is also determined by this method, being positive if the hotter junction is at higher potential, and vice versa. If normal metals are used as a reference material (Seebeck probes), they need to be calibrated and their absolute thermopower as a function of temperature (and perhaps also of a magnetic field) well established. This is a difficult task to accomplish. Materials that come immediately to mind, such as copper, gold, aluminium and perhaps even platinum (all having good ductility to make wires) have a rather complicated, nonmonotonic variation of the thermopower with temperature that is strongly influenced by even minute (few parts per million) traces of impurity. Consequently, it is very difficult to standardize thermoelectric properties of such wires, and variations occur not just between different manufacturers but even between different spools originating from the same source. Although several valiant efforts have been made to prepare alloys with a nearly zero thermopower and very small temperature dependence, concerns regarding reproducibility of the material remain and cannot be avoided. It was recognized a long time ago that for work below room temperature Pb has useful properties as a possible thermoelectric reference material, since it not only has a relatively small thermopower but also is substantially insensitive to trace amounts of impurity elements. The first thermopower scale based on Pb was assembled by Christian and coworkers in 1958 and relied on the data dating back to the early 1930s. A great majority of the existing thennopower data, including extensive investigations in the particularly busy period of the 1960s, use this scale as a reference. An important modification of the thermoelectric scale was undertaken by Roberts (1977) who determined the absolute thermopower of lead by direct measurement of its Thomson heat. Significant errors were found in the original scale amounting to as much as 0.3 µV K–1 above 20 K. Errors of this magnitude are, of course, inevitably present in all the thermopower data published prior to 1977 and are often propagated even into more recent tabular compilations. While Pb itself would seem to be a good choice for the Seebeck probe, its direct use is limited by the difficulties of making sufficiently thin wires. Since the thermopower is frequently studied simultaneously with the thermal conductivity, heat loss along the relatively thick Pb leads presents a serious drawback. Experimental conditions are made even more difficult when one wants to apply an external magnetic field. In that case the rather large magnetothermopower of Pb complicates the data reduction. An ideal reference material should have zero thermopower and be immune to external stimuli such as magnetic field. Superconductors substantially fulfil these requirements. Until 1987, the temperature range over which the intrinsically zero thermopower of a superconductor could be utilized was limited to about 18 K, the transition temperature of Nb3Sn tapes or composite wires. The discovery of high-Tc superconductors greatly expanded the range of zero thermopower and opened new avenues in the field of precision metrology. With now readily available Hg-based perovskites, absolute thermopower measurements can be routinely extended to temperatures as high as 134 K (provided that any magnetic field present is sufficiently small so there is no significant contribution from the Seebeck effect in the mixed

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state). Although high-Tc superconducting perovskites are not yet available in the form of thin flexible wires to be used directly as the Seebeck probe, their bulk form provides a very convenient material for a rapid and precise determination of the absolute thermopower of any section of wire arranged in the thermoelectric circuit of figure F8.0.14 where the segment cd represents a high-Tc material. This concept was first realized by Uher (1987) in his measurements of absolute thermopower of several different wires. F8.0.6.2 Thermoelectric coolers An important application of thermoelectricity is the generation of electricity from heat and its inverse process, thermoelectric refrigeration (Goldsmid 1986). In each case, the efficiency or the coefficient of performance with which such energy conversion can be accomplished is expressed in terms of the temperatures of the hot and cold junctions of the thermocouple and the material quantity known as the figure of merit Z. The figure of merit is defined in terms of the three transport parameters of the two thermocouple branches, electrical conductivity (σ), thermal conductivity (κ) and the Seebeck coefficient (S), as

where the parameters for two branches are indicated by subscripts p (a material with positive thermopower) and n (usually with negative thermopower). Although, in principle, the figure of merit refers to the entire couple, it is frequently convenient to use the figure of merit of a single thermoelectric material, z. In analogy to equation (F8.0.22), this is defined as z = S 2σ/κ. In general, large values of the figure of merit are obtained in semiconductors and certain semimetals. In predicting the cooling power of the thermoelectric refrigerator, it is often assumed that the resulting figure of merit of a thermocouple is equal to the average of the figures of merit of its constituent branches. This assumption works well for materials that are used in the fabrication of thermoelectric couples designed to operate at or near ambient temperature. In this case, the material of choice is typically Bi2Te3 and its alloys with Sb2Te3 or Bi2Se3. It so happens that Bi2Te3-based thermocouples deteriorate rapidly with decreasing temperature and, in the temperature range of liquid nitrogen, by far the best materials are the alloys of bismuth with antimony. A significant drawback that hampers the practical use of thermoelectric couples based on Bi—Sb alloys is the fact that the n-type branch of the couple has far superior thermoelectric parameters to any material that might come into consideration for the p-type branch. It is in this context that Goldsmid et al (1988) proposed the concept of a passive thermoelement and suggested high-Tc superconductors for this purpose. The idea is clever and simple: rather than compromise the figure of merit of the couple by using a ptype semiconductor or semimetal with inferior thermoelectric properties, it might be advantageous to substitute a high-Tc superconductor for the p-type branch for operation in the liquid-nitrogen temperature range (see figure F8.0.15). Even though the thermopower of a superconductor is zero, i.e. it contributes nothing to the Seebeck term (Sp — Sn ), the ratio κp /σp is essentially zero (because of the infinite electrical conductivity and finite thermal conductivity dominated by phonons) and the superconductor does not detract from the (high) performance of the n-type branch. The superconductor thus serves as a passive thermoelectric branch. An important consideration in any practical use of this concept is the current density needed to maximize the thermoelectric cooling. For instance, Goldsmid et al (1988) estimate that for a convenient length of thermoelements of 10 mm and the typical transport parameters of Bi—Sb, to achieve maximum cooling effect the superconducting leg would have to withstand a current density of the order of Jc ∼ 120 A cm−2 . While this value may be beyond the current-carrying capacity of sintered high-Tc materials at 77 K, significant advances achieved with preferred grain orientation lead to an enhancement of Jc well above 104 A cm−2, considerably brightening the prospect of the passive thermoelement use.

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Figure F8.0.15. superconductor.

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A low-temperature thermoelectric cooler utilizing a passive branch made from a high-Tc ,

Figure F8.0.16. Temperature of the cold junction as a function of applied current. The curve indicates the fit to the data with coefficients a and b given in the text. Reproduced from Fee (1993) by permission.

Furthermore, a very substantial gain in the figure of merit of Bi—Sb can be achieved on application of a modest transverse magnetic field of the order of 0.2–0.3 T. The concept of the hybrid thermoelectric cooler utilizing a high-temperature superconductor in place of the p-type leg has recently been realized by Fee (1993). The author made a Peltier cooler using sintered YBa2Cu3O7 – δ (YBCO) with a transition temperature of 89 K and a critical current density of 208 A cm– 2. This thermoelectrically neutral leg was combined with an n-type leg made from crystalline Bi0.835Sb0.165 , the c axis of which was oriented at an angle of 70° from the long axis of the sample. The dimensions of the two legs comprising the thermocouple bridge were: 12.5 × 3 × 2 mm3 for YBCO and 12.5 × 6 × 2 mm3 for Bi—Sb. To facilitate low contact resistance, silver pads of 2 mm diameter and 0.2 mm thickness were bonded to the ends of the YBCO bar by pressing to 4 × 107 Pa in the die before the superconductor was sintered and annealed. A bridge-type thermocouple structure similar to that shown in figure F8.0.15 was assembled with the aid of copper plates soldered to the YBCO and Bi—Sb legs using indium solder. Operated in a vacuum better than 10– 5 Torr (~ 133 × 10−5 N m-2) with the heat sink held at 79 K, the couple developed a maximum temperature drop of 5.35 K at a current of 4.75 A (see figure F8.0.16). This implies a figure of merit of the couple Z = 2 × 10– 3 K– 1. The temperature drop developed by a Peltier cooler is a function of the current and consists of two

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terms: a linear term representing Peltier cooling and a quadratic term giving Joule heating. In the case of a cooler with a neutral leg, we have ∆T = aI + bI2 where a = SnT1 /Kj and b = Rj /2K j , Sn , stands for the thermopower of the active leg, T1 is the temperature of the cold junction, R j is the resistance of the couple, K j its thermal conductance and I is the applied current. The fit to the data of figure F8.0.16 yields a = −2.251 K A–1 and b = 0.2368 K A– 2. For a prototype device, the performance of the cooler was encouraging. From the perspective of superconductivity, it was especially important to prove that the YBCO leg of the couple was not a limiting factor of performance. Nonetheless, there are several obvious improvements which could, in principle, triple the maximum temperature drop one could achieve. At its optimum operating current of 4.35 A, the YBCO leg still had more than twice the critical current capacity. Thus, there are wide margins for improvement left which could significantly enhance the performance of the device. One can certainly reduce the cross-section of the YBCO leg to reduce the movement of heat from the sink into the couple. In particular, by using melt-processed YBCO with enhanced critical current density, one could make the superconducting leg with quite a small cross- section. Even better, by using Bi2Sr2CaCu2O8+x which has a higher Tc , larger critical current density and withstands small magnetic fields better than YBCO, one could dramatically enhance the operation by applying a magnetic field to increase the figure of merit of the active Bi—Sb leg. It is also useful to note that the Bi—Sb was not used in its most favourable configuration which would be realized if the current were passed along the trigonal direction instead of 70° away from it. With optimized materials and in a modest magnetic field, a temperature drop in excess of 16 K should be possible at liquid- nitrogen operation. What remains to be done is to develop a multistage Peltier cooler, the operation of which extends to sufficiently low temperatures below the Tc of a suitable high-temperature superconductor, so that one could incorporate a hybrid thermocouple in its top stage and thus extend the operational range of the device still further. Discovering a high-temperature superconductor with a record-high transition temperature would, of course, aid in the design of such a multistage Peltier cooler. F8.0.7 Conclusion We have outlined in this chapter a number of ways in which thermoelectric/thermomagnetic effects in superconductors have been utilized. The main uses might be summarized as follows. (i) (ii) (iii)

(iv)

(v) (vi)

A systematic pattern of thermopower behaviour that is intimately related to the carrier density and the value of Tc is observed in the cuprate superconductors; explaining this pattern provides a test for theories of the superconducting mechanism. A room-temperature thermopower near zero can be used as an approximate indicator of optimal doping for maximum Tc in bismuth-, thallium-, mercury- and yttrium-based cuprate superconductors. A strong increase in the Nernst and Ettingshausen effects in the mixed state of high-temperature superconductors (compared with that in the normal state) demonstrates the motion of flux lines: in the Nernst effect the flux-line motion due to the thermal force generates a transverse electric field and in its inverse, the Ettingshausen effect, the flux motion due to the Lorentz force leads to a transverse temperature gradient. The decrease of the Seebeck coefficient (and of its inverse the Peltier coefficient) to zero below Tc is broadened in the presence of a magnetic field in the same way as the flux-flow resistivity. The persistence of a nonzero Seebeck effect in the mixed state indicates the existence of unbound quasiparticles down to relatively low temperatures. High-temperature superconductors can be used as a reference material to enable the determination of absolute thermopower. High-temperature superconductors can be used as passive elements in thermoelectric coolers.

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Acknowledgments Part of the work described here was supported by the New Zealand Foundation for Research, Science and Technology (ABK) and by ONR grant N00014-96-1-0181 (CU). References Beeforth T H and Goldsmid H J 1970 Physics of Solid State Devices (London: Pion) (an introduction to the physical principles of thermoelectric energy conversion devices) Blatt F J, Schroeder P A, Foiles C L and Greig D 1976 Thermoelectric Power of Metals (New York: Plenum) (gives an introduction to thermopower and describes measurement techniques in detail) Bondarenko V A, Petrosov R A, Tanatar M A, Yefanov V S, Ogenko V M, Kushch N D and Yagubskii E B 1995 Synth. Metals 70 955 Carter R, Davidson A and Schroeder P A 1970 J. Phys. Chem. Solids 31 2374 Clem J R 1991 Phys. Rev. B 43 7837 Cohn J L, Skelton E F, Wolf S A and Liu J Z 1992 Phys. Rev. B 45 13 140 Fee M G 1993 Appl. Phys. Lett. 62 1161 Freimuth A 1992 Selected Topics in Superconductivity, Frontiers in Solid State Science vol 1, ed L C Gupta and M S Multani (Singapore: World Scientific) (reviews early work on thermomagnetic effects in high temperature superconductors) Galffy M, Freimuth A and Murek U 1990 Phys. Rev. B 41 11 029 Goldsmid H J 1986 Electronic Refrigeration (London: Pion) (a detailed account of the physical principles of thermoelectric refrigeration and design criteria for efficient thermoelectric coolers) Goldsmid H J, Gapinathan K K, Matthews D N, Taylor K N R and Baird C A 1988 J. Phys. D: AppL Phys. 21 344 Harman T C and Honig J M 1967 Thermoelectric and Themmomagnetic Effects and Applications (New York: McGrawHill) Hohn C, Galffy M, Dascoulidou A, Freimuth A, Soltner H and Poppe U 1991 Z Phys. B 85 161 Huebener R P 1979 Magnetic Flux Structures in Superconductors (Berlin: Springer) (reviews the thermomagnetic properties of conventional superconductors) Huebener R P, Ustinov A V and Kaplunenko V K 1990 Phys. Rev. B 42 4831 Inabe T, Ogata H, Maruyama Y, Achiba Y, Suzuki S, Kikuchi K and Ikemoto I 1992 Phys. Rev. Lett. 69 3797 Kaiser A B 1987 Phys. Rev. B 35 4677 Kaiser A B, Subramaniam C K, Ruck B and Paranthaman M 1995 Synth. Met. 71 1583 Kaiser A B and Uher C 1991 Studies of High Temperature Superconductors vol 7, ed A V Narlikar (New York: Nova) p 353 (gives a detailed review of normal-state thermopower in high-temperature superconductors) Kinzie P A 1973 Thermocouple Temperature Measurements (New York: Wiley) Kosterlitz J M and Thouless D J 1973 J. Phys. C: Solid State Phys. 6 1181 Krishana K, Harris J M and Ong N P 1995 Phys. Rev. Lett. 75 3529 Logvenov G Yu, Ryazanov V V, Ustinov A V and Huebener R P 1991 Physica C 175 179 Logvenov G Yu, Hartmann M and Huebener R P 1992 Phys. Rev. B 46 11 102 Lowe A J, Regan S and Howson M A 1991 Phys. Rev. B 44 9757 Mamin H J, Clarke J and van Harlingen D J 1984 Phys. Rev. B 29 3881 Murphy D W, Rosseinsky M J, Fleming R M, Tycko R, Ramirez A P, Haddon R C, Siegrist T, Dabbagh G, Tully J C and Walstedt R E 1993 The Fullerenes ed H W Kroto, J E Fischer and D E Cox (Oxford: Pergamon) p 151 (reviews fullerene superconductors) Obertelli S D, Cooper J R and Tallon J L 1992 Phys. Rev. B 46 14 928 Otter F A and Solomon P R 1966 Phys. Rev. Lett. 16 681 Palstra T T M, Batlogg B, Schneemeyer L F and Waszczak J V 1990 Phys. Rev. Lett. 64 3090 Rao V V, Rangarajan G and Srinivasan R 1984 J. Phys. F: Met. Phys. 14 973 Ri H-C, Gross R, Gollnik F, Beck A, Huebener R P, Wagner P and Adrian H 1994 Phys. Rev. B 50 3312 Roberts R B 1977 Phil. Mag. 36 91 Ruan Y and Lin P 1986 Acta Phys. Temp. Humilis Sin. 7 210

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Sarachik M P, Smith G E and Wernick J H 1963 Can. J. Phys. 41 1542 Sera M, Shamoto S and Sato M 1988 Solid State Commun. 68 649 Siebold Th and Ziemann P 1993 Solid State Commun. 87 269 Steglich F, Rauchscwalbe U, Gottwick U, Mayer H M, Spam G, Grewe N, Poppe U and Franse J J M 1985 J. Appl. Phys. 57 3054 Stewart G R 1984 Rev. Mod. Phys. 56 755 (reviews heavy-fermion systems) Subramaniam C K, Rao C V N, Kaiser A B, Trodahl H J, Mawdsley A, Flower N E and Tallon J L 1994a Supercond. Sci. Technol. 7 30 Subramaniam C K, Kaiser A B and Tang H Y 1994b Physica C 230 184 Subramaniam C K, Paranthaman M and Kaiser A B 1995a Phys. Rev. B 51 1330 Subramaniam C K, Trodahl H J, Kaiser A B and Ruck B J 1995b Phys. Rev. B 51 3116 Uher C 1987 J. Appl. Phys. 62 4636 Uher C 1992 Physical Properties of High Temperature Superconductors vol III, ed D M Ginsberg (Singapore: World Scientific) p 159 (reviews thermal conductivity of high-temperature superconductors) Uher C 1996 Proc. 10th Anniversary High Temperature Superconductor Workshop (Singapore: World Scientific) p 304 (reviews the thermal conductivity of high-temperature superconductors in a magnetic field) Van Harlingen D J 1982 Physica B 109/110 1710 (reviews thermoelectric effects in the superconducting state) Wang Z Z and Ong N P 1988 Phys. Rev. B 38 7160 Williams J M, Schultz A J, Geiser U, Carlson K D, Kini A M, Wang H H, Kwok W-K, Whangbo M-H and Schirner J E 1991 Science 252 1501 (reviews organic superconductors) Xu X-Q, Hagen S J, Jiang W, Peng J L, Li Z Y and Greene R L 1992 Phys. Rev. B 45 7356 Zeuner S, Prettl W, Renk K F and Lengfellner H 1994 Phys. Rev. B 49 9080

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PART G PRESENT APPLICATIONS OF SUPERCONDUCTIVITY

Copyright © 1998 IOP Publishing Ltd

G1 Ultra-high-field magnets for research applications

N Kerley

G1.0.1 Introduction The discovery of type II superconductors such as Nb-Ti and Nb3Sn has revolutionized the production and use of intense magnetic fields. Over the last three decades the use of superconducting magnets has become widespread in many research laboratories throughout the world. Laboratories with access to liquid helium can conveniently operate many different types of superconducting magnet that generate fields of up to over 20 T. However, even after over 30 years of development only two superconducting materials are in common use today. Magnets up to 10 T are usually made entirely from Nb-Ti. Magnets operating in the range from 10 T to 20 T are of duplex construction where the inner high-field solenoid is made from Nb3Sn and the outer section from Nb-Ti. The high cost and difficulty of processing Nb3Sn mean that its use is always minimized in any given magnet. The superconducting performance of both materials improves significantly when the temperature is reduced from 4.2 K to 2.2 K. Nb-Ti improves by typically 20% and Nb3Sn by around 10%. For maximum performance of the magnet at minimum cost, much use is made of these enhanced properties—most research magnets built today will have one field rating when run at 4.2 K and a higher rating when cooled to 2.2 K. Typically a magnet designed to give 9 T at 4.2 K, and constructed only from Nb-Ti, will operate up to 11 T when cooled to 2.2 K. An 18 T magnet whose performance at 4.2 K is limited by the performance of the central Nb3Sn section will run to 20 T at 2.2 K. The design of modern research magnets is almost entirely dominated by the performance of superconducting wires. Earlier limitations of premature quenching and coil damage caused by factors such as excessive training, internal stresses and energy dissipation have to a large extent been overcome. In addition the performance of Nb-Ti superconductor has been almost fully optimized, although there is continuing and significant evolution in the design, production and performance of Nb3Sn wires. Ternary additions of small amounts of tantalum or titanium to Nb3Sn have significantly improved its performance at the highest fields. This is very significant because quite small improvements in the performance of superconducting wires can have a dramatic effect on the design, overall size, weight and cost of building a magnet with a given performance, particularly at the highest available fields. Copyright © 1998 IOP Publishing Ltd

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The manufacture of both superconducting wires and magnets for research has largely been industrialized since the late 1960s. This has enabled superconducting magnets to be produced in significant numbers and has led to reductions in costs and improvements in performance and reliability. Within industry the design and manufacture of magnets has become rather an exact science and an optimized process that has been carefully refined to minimize costs. Between 1970 and 1995 the maximum magnetic field available for research magnets had steadily increased from below 10 T to over 20 T as the technology evolved. In a number of central research laboratories, hybrid magnets are used. Hybrids have superconducting outer coils generating fields in the range 10–14 T with high-power water-cooled resistive inner coils requiring power inputs of typically 10–20 MW. With such systems steady magnetic fields beyond 35 T can be generated. The relative ease of obtaining and using superconducting magnets has resulted in their widespread use around the world, for a very wide range of experiments. Research magnets are frequently used in conjunction with special sample cooling systems which include variable temperature inserts (1.2–300 K), helium-3 inserts (0.3–70 K) and dilution refrigerators (0.004–20 K) (Reinders et al 1987). Much of this equipment nowadays is automated and operated under computer control for ease and economy of use through maximized operation. In many cases samples can be inserted or changed through a top-loading arrangement that allows the magnet to remain at its operating temperature. Many fundamental and important discoveries in condensed matter physics such as the quantum Hall effect (von Klitzing et al 1980) have been made with equipment that provides low temperatures combined with high magnetic fields providing a high B/T ratio. The quantum Hall effect has an important practical use in that it is now used to determine experimentally the internationally accepted value of the standard ohm to an accuracy of 1 in 108 that depends only on fundamental quantum constants (Hartland 1992). The critical differences between superconducting magnets and earlier water-cooled electromagnets are their ability to generate extremely strong, uniform, stable and quiet magnetic fields using electrical power available in any laboratory. The low-power requirement and improved field quality are a direct consequence of superconductivity itself and it has led directly to widespread applications of superconducting magnets in many scientific disciplines both within and outside pure physics. The first large-scale application of superconducting magnets outside pure physics was in nuclear magnetic resonance (NMR) spectroscopy—an analytical technique that is now widely used in biochemical and pharmaceutical applications. Superconducting magnets for this application were developed in the early 1970s. The usefulness of NMR depends directly on the quality of the signal from the sample which is directly related to the strength, uniformity and stability of the magnetic field over the sample. The remarkable field strength, homogeneity, persistence and complete freedom from vibration can only be achieved using superconducting technology. The highest field standard NMR magnet currently available operates at 800 MHz (proton frequency) which corresponds to a field of 18.8 T. For further information on NMR methods the reader is referred to chapter G2.1. Another significant application of superconducting magnets with widespread benefit to humanity is that of whole body magnetic resonance imaging (MRI). This has become a routine medical diagnostic technique since the early 1980s and is in widespread use throughout the world. Standard MRI magnets operate in the 1.5–2 T range although new demands related to spectroscopy on living subjects and evoked response work on the brain are pushing towards the use of higher fields up to 5 T and in some special cases even up to 12 T. Industrial applications of superconducting magnets are still in their relative infancy although some uses have emerged. The use of superconducting magnets in the kaolin industry where high-gradient magnetic separators are used to purify and whiten the product is now established on an industrial scale. For further information on magnetic separation the reader is referred to chapter G6. Other applications of superconucting magnets beyond the research laboratory are set to increase as processes are developed and easier and more industrially acceptable methods of keeping magnets at their operating temperature are developed.

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Many person years of development have gone into superconducting magnets which are now a remarkable combination of condensed matter physics and subtle mechanical engineering. In recent years much more has become known about how superconducting magnets really work in practice. In the majority of examples today, enough is known about the specialized design and engineering needed to permit a desired special magnet to be engineered and built at the first attempt. Continuing improvements in both constructional materials and available superconductors means that yet higher performance superconducting magnets can be expected in the future. G1.0.2 Scope Magnets for research applications are normally specified by the magnetic field, bore size, field uniformity over a given volume and stability in time. Additional parameters may include field sweep rate, field modulation frequency and amplitude. The homogeneity of the central field of a typical general research magnet is in the range 1 in 103 to 1 in 104 over a 10 mm diameter spherical volume (DSV). Homogeneities of 1 in 105 to 1 in 106 are typical for magnets used for ‘solid-state’ NMR where resonances naturally are broadened by the nuclear environment. For high-resolution NMR spectroscopy, usually in liquid samples, natural linewidths are extremely narrow and homogeneities of 1 in 108 or better are required for optimum resolution and results. Common experimental requirements determine the two basic magnet types in common use—solenoids and split pairs (see Figure G1.0.1).

Figure G1.0.1. A solenoid and a split pair.

The simplest and most commonly used geometry is a solenoid which is the most efficient and cost effective solution for producing a magnetic field. To give an idea of scale, many high-field solenoids for research purposes have typical bore sizes of 40 to 50 mm in diameter and lengths and outside diameters of 200 to 300 mm. For experiments requiring a homogeneity of 1 in 103 over 10 mm DSV a simple solenoid geometry of adequate length is sufficient. For homogeneities of 1 in 104 over 10 mm DSV or better, compensation Copyright © 1998 IOP Publishing Ltd

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coils are added to minimize the amount of superconductor required in the design. To modify the shape of the field, improve the homogeneity or add a known field gradient, additional coils are fitted to the basic windings. The next most common type of magnet is a split pair. Depending on the design of the cryostat and the orientation of the magnet, transverse access can be provided to the sample or samples may be inserted into a transverse field. Transverse access to the sample may be required for the transmission of optical neutrons or other beams. Sample access transverse to the magnet may be needed for magnetic anisotropy measurements. Additional parameters needed to define a split pair include the size, shape and number of transverse access ports. Split pairs are more difficult to design and construct than solenoids. There are several extra design and engineering factors for split pairs that limit maximum available fields to around 9 T and 15 T for Nb—Ti and Nb3Sn magnets respectively. The size of the split access is usually minimized as it has a major effect on the overall size and cost of the magnet. The basic winding geometry is not as efficient at producing magnetic field as a solenoid partly because there is an absence of windings in the most efficient field-generating position. There are also higher peak fields in the windings relative to the central field than on a solenoid. Split pairs generate large attractive forces between the coils that frequently exceed many tonnes even on quite small magnets. A combination of high pressures and a tendency for the windings to slide against the fixed former as they expand under Lorentz forces makes split pairs particularly difficult. G1.0.2.1 Unusual geometries Some experiments require unusual geometries of magnets and a few examples are given for illustration. (a) Asymmetric split pairs Asymmetric split pairs have been built to allow experiments involving polarized neutrons. To retain beam polarization it is necessary to ensure the neutrons do not pass through a region of zero field. By building an asymmetric coil the zero field point (actually a circle within the split) can be moved just outside the path of the incoming neutron beam. As the polarized beam enters the magnet the neutron spins precess around the field lines and ensure polarization is maintained up to the target (see Figure G1.0.2).

Figure G1.0.2. An asymmetric split coil.

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(b) Four coil geometry For some neutron-scattering experiments a horizontal magnetic field is required but without any superconducting material in the path of the incoming or outgoing neutron beam. A novel way of achieving this has been developed using a ‘four-coil’ design. When oriented as shown in Figure G1.0.3, the ‘stray’ fields of the four solenoids combine to generate a horizontal field of up to about 4 T with moderate homogeneity (typically 5% over the sample).

Figure G1.0.3. Four-coil geometry

Figure G1.0.4. A vector rotate magnet.

(c) Vector rotate magnet Some experiments require the possibility of being able to rotate or tilt the magnetic field vector from the vertical towards the horizontal. This can be achieved by fitting a solenoid with a vertical field around a split pair with a horizontal field (see Figure G1.0.4). Use of two power supplies allows the field vector to be accurately positioned at the required angle but the large forces and torque involved limit fields to around 4 T. Other special geometries are sometimes required to produce particular field shapes, field gradients and null-field regions. Modern finite-element analysis and design techniques allow accurate predictions of magnetic fields and forces. G1.0.3 Design The designing of superconducting magnets is a complex task. No attempt is made fully to describe this here but indications of some of the important factors are given in this section. G1.0.3.1 Simple solenoids Once the basic requirements of magnet bore size, field strength and field uniformity are defined, the shape of the coils can be designed very precisely. Axial fields can be calculated analytically, but off-axis fields can only be determined using complex numerical integration. Although magnets can be designed using hand calculations and graphical aids (Wilson 1983), for reasons of accuracy and speed computer programs are used in practice. The measured field and homogeneity produced by the magnet normally corresponds Copyright © 1998 IOP Publishing Ltd

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closely to the predicted performance at the 1 in 103 level. The as-wound data, recorded during winding, and the effects of thermal contraction can be taken into account. Superconducting wire manufacturers normally supply measured short-sample data for a given batch of wire to ensure it complies with or exceeds the minimum standard performance. The short-sample properties of superconducting wires as a function of magnetic field are well known and extremely repeatable from batch to batch. At the design stage it is necessary to ensure the operating current for any given wire and that local magnetic field at no point exceeds the critical level. There must be a typical margin of safety of 5–10% on short-sample performance everywhere throughout the magnet. The windings within a given magnet are normally graded by wire diameter to minimize the size, weight, cost and stored energy of the magnet. The thickest wire is nearest the bore where the magnet field on the windings is highest. Successively smaller wires are used further out in the coil as the peak field on the windings reduces. For user convenience all sections are connected in series and the same current is used in all sections of the magnet. A typical magnet will have wires with standard diameters of 1.1, 0.85, 0.75, 0.7, 0.6, 0.5 and 0.4 mm within the winding. Several kilometres of wire are used in the winding of even a modest magnet. Wires much smaller than these are not easy to wind and so are not often used. Exceptions to this may include small magnets where only a few amperes are available for operation such as for small magnets used on satellites for space research. Typical on-board current sources are only a few amperes and under these circumstances magnets for fields up to about 4 T are wound using wires as thin as 0.1 mm. With such magnets even layer winding may not be achievable or even necessary. The maximum field that can be produced economically using an Nb–Ti superconductor in a solenoid operating at 4.2 K is between 9 and 10 T. In practice 10 T is only economic for small magnets with a bore size below 40 mm. Beyond this field and bore size, it is necessary to use an Nb3Sn superconductor in the inner part of the magnet. This material is typically an order of magnitude more expensive than Nb—Ti. It is also much more difficult to wind into a magnet because of its extremely brittle nature. For reasons of cost a ‘duplex’ design is employed with the central coils made of a minimum amount of Nb3Sn superconductor, surrounded by coils made from Nb—Ti as shown in Figure G1.0.5. For convenience of use, the two magnet sections are designed and constructed to operate together in series from one power supply. The additional complexity of using Nb3Sn adds significantly to the cost of the magnet but enables fields of up to over 21 T to be reached.

Figure G1.0.5. A duplex solenoid showing graded windings.

By reducing the temperature from 4.2 K to around 2.2 K the current-carrying performance of Nb—Ti and Nb3Sn can be increased by about 20% and 10% respectively. Providing the performance of the magnet is only limited by the short-sample performance of the superconductor and not some other structural or stored energy limitation, the maximum field of the complete magnet may be increased substantially by operating it at 2.2 K. Typically a 9 T Nb—Ti magnet will generate 11 T at reduced temperature and an 18 T magnet whose field is limited by the Nb3Sn will enhance that to 20 T. Magnets can be operated continuously at reduced temperature in a simple cryostat using a ‘lambda-point refrigerator’. This device Copyright © 1998 IOP Publishing Ltd

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enables the lower part of a 4.2 K helium bath to be run continuously at 2.2 K by pumping helium through a needle valve and heat exchanger and allowing convection of the helium in the bath below the heat exchanger to cool the magnet. For further information on magnet operation below 4.2 K the reader is referred to chapters D8 and D9. G1.0.3.2 Stresses Superconducting high-field magnets have to be constructed in a way that prevents any movement of wire or individual turns to avoid quenching. It is necessary to impregnate all the windings with waxes or resins to achieve this. The use of modern materials such as epoxy resins within the windings means high field research magnets can usually operate reliably up to the short-sample performance of the wire. High forces are generated within superconducting magnets during operation. The direction of the Lorentz forces are such that the coil tends to burst radially and also to compress axially. In some magnets the bursting force locally can approach the tensile yield stress of copper. Single-turn hoop stresses in excess of 320 MPa can be accommodated locally within the windings. In larger magnets it is necessary that integrated stresses on sections do not exceed that of the yield strength of copper. In practice the design stress on individual turns within a section may exceed this but support from windings further out where the bursting forces are reversed usually prevents difficulties. During quenching transient stresses may occur which exceed the static designed value by a significant amount through self-transformer action. The structure of the windings must be sufficiently robust to prevent damage occurring during these brief periods of excess current. G1.0.3.3 Homogeneity The winding geometry of a magnet is determined by the required field profile and homogeneity. An ‘optimum’ solenoid that has a minimum weight of wire for a given field will produce a homogeneity of about 1 in 103 over a 10 mm DSV. For higher homogeneity, windings are normally fitted with compensation coils in order to improve the homogeneity with minimum use of superconducting wire (see Figure G1.0.6).

Figure G1.0.6. A compensated solenoid.

The design process for high homogeneities is simplified if the axial field in the central region of the magnet is thought about in terms of a series expansion of components, starting at the centre of the magnet. Over a short axial distance the field strength varies with axial distance Z. The field change can be fitted in powers of Z in terms of increasing order. These terms are known as the Z0 or field term, the Z2 term where the component of field varies with Z2, and so on for Z4, Z6 etc. Odd order terms such as Z1, Z3 are antisymmetric and represent types of gradient field which are zero for a symmetric magnet. Copyright © 1998 IOP Publishing Ltd

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Clearly for a simple solenoid the field is highest at the centre of the magnet and falls off with distance along the axis—a simple solenoid has a negative Z2 component. By adding compensation coils which produce an equal and opposite Z2 term the second-order variation can be cancelled over a given central region. By ensuring that Z4 and higher orders are cancelled a magnet of any homogeneity can be designed by a combination of making the magnet longer and adding compensation coils as appropriate. For the highest homogeneity applications such as NMR, magnet designs are corrected up to sixth or even eighth order. The off-axis homogeneity also needs to be determined to an appropriate level for many applications. For further information on NMR techniques the reader is referred to chapter G2.1. In designing a magnet one must consider practicalities. The design must not be so sensitive to the actual position of real coils that correct placement cannot be achieved in practice, allowing for initial positioning, thermal contraction and possible movement. The design must not be over-sensitive to the effect of individual turns (which are necessarily integers) such that the required homogeneity cannot be achieved. G1.0.3.4 Cancellation coils Some superconducting magnets require cancellation coils in order to provide a region of low field (see Figure G1.0.7). The low field may be needed for a variety of experimental purposes such as the positioning of a field sensitive device like a temperature sensor, a superconducting quantum interference device (SQUID) or heat switch. Cancellation coils can be produced which typically maintain a defined region at below about 3 mT, for all values of the central field. They consist of reverse energized coils that are normally run in series with the main solenoid. If even lower fields are required additional shielding can then be used. A superconducting niobium cylinder that uses the Meissner effect can be used to produce extremely low fields and can enable sensitive devices such as SQUIDS to be used in close proximity to the magnet. The cancellation coil can introduce a field gradient back at the centre of the main magnet. This gradient is usually minimized by making the main magnet compensation coils slightly asymmetric. Invariably there are large repulsive forces between the magnet and cancellation coil and a robust mechanical structure is used to ensure the coils do not move significantly or separate in use.

Figure G1.0.7. A solenoid with cancellation coil and support structure.

G1.0.3.5 Stored energy During use a superconducting magnet stores energy. The energy E (joules) depends on the inductance of the magnet L (henries) and the current I (amperes) and is given by the equation E = ½ L I 2 . Depending on the magnet size and field, the stored energy for research magnets can vary from a few tens of kilojoules to several megajoules. There is always a possibility that a superconducting magnet can quench unexpectedly due to a low helium level or some other factor. Copyright © 1998 IOP Publishing Ltd

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The magnet support system must be designed to be completely safe during a quench to protect both the operator and the equipment. Without adequate protection even quite small superconducting magnets can generate dangerously high voltages and currents. Most of the energy released during a quench eventually appears as heat emerging from the magnet over a period of a few tens of seconds. This causes the liquid helium within the cryostat to be evaporated quickly. Adequate gas exit ports fitted with one-way valves must be provided to avoid the build-up of dangerous pressures and prevent the ingress of air following a quench. For complete safety the emergency vent valve should be of sufficient size that it can cope with a simultaneous magnet quench and full cryostat vacuum failure. During a quench, significant voltages can appear across the magnet terminals and also across different sections of the magnet. These voltages can be estimated from the operating current of the magnet and the normal-state d.c. resistance of the coil. Simple calculations show that to keep voltages below a few hundred volts it is necessary to add a protection tap approximately every 2 km throughout the windings. To ensure optimum safety, the energy-absorbing protection circuit is best fitted close to the magnet, within the cryostat. This ensures that even if the current causes the power supply to fail or to become disconnected the quench energy can still be safely absorbed. A typical protection circuit is shown schematically in Figure G1.0.8. The circuit is designed so that during normal energization the diodes remain insulating which prevents currents flowing through the resistive elements and eliminates energization losses and reduces helium consumption. Providing there is no persistent mode switch, the use of diodes also ensures close correspondence between the current and field for swept field applications.

Figure G1.0.8. A typical resistor and resistor diode magnet protection circuit.

Each section of the protection circuit should be capable of carrying the full operating current of the magnet to avoid being damaged during a quench. In some extreme cases a ‘nonpropagating’ quench can occur where only one section of the magnet quenches. This can cause almost all the energy of the magnet to be dissipated in just one of the protection sections, and is potentially a damaging situation. Ideally each section of the protection circuit should have a sufficient rating to absorb the stored energy from the entire magnet. Copyright © 1998 IOP Publishing Ltd

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It is normal to try to ensure that nonpropagating quenches are minimized at the design stage. Full propagation of a quench throughout the magnet is desirable and is normally ensured by close magnetic coupling of all sections. During a quench all sections of the system experience rapidly reducing fields that are alone sufficient to quench all the windings. In a compact solenoid close coupling is normally ensured automatically by the proximity of all the windings. However, in split pairs, cancellation coils, four-coil magnets or other geometries, close coupling is not always automatically achieved. To assist with the propagation of quenches, remote parts of the coil can be connected in series with a close-coupled part and protected in a single protection unit. In this way quenches can be fully propagated automatically and potentially damaging nonpropagating quenches can normally be avoided. Superconducting magnet power supplies are usually fitted with energy-absorbing circuits in order to cope with quenches and limit the voltages at the magnet terminals. Protection from the power supply should always be regarded as secondary back-up, and used in addition to the local magnet protection circuit within the cryostat. G1.0.3.6 Persistent mode For many applications the field uniformity and field stability as a function of time are closely related. A guide to this is that the field variations across the sample determined by the magnet homogeneity should be similar to the field variations over the sample caused by field decay of the magnet over the length of time of the experiment. This means in practice that the higher the homogeneity the lower should be the decay of the magnet. In theory superconducting magnets may be constructed so that the field in persistent mode is absolutely stable in time because of the zero resistance in the superconducting circuit. In practice it is not straightforward to design or construct such perfect magnets due to practical limitations in the superconducting materials and joints. The level of persistence achieved in magnets is determined largely at the design stage. Choice of superconductor, proximity of the superconductor operating point relative to short-sample performance and type and position of joints between lengths of superconductor all have an impact on the decay. Magnets wound with single-core Nb—Ti can have truly superconducting joints made by ‘cold welding’ the superconducting cores together and so can be fully persistent in operation. Magnets built in this way for NMR have produced steady fields that are stable to better than 1 in 109 measured over a period of more than 10 years. The majority of superconducting magnets today are built from filamentary superconductors where more sophisticated jointing procedures are used. Details of zero-resistance joints between successive lengths of Nb—Ti or joints to Nb—Sn are proprietary industrial processes but all joints are sensitive to magnetic fields which means that they have to be placed in low-field regions usually at one end of the magnet as shown in Figure G1.0.9. To improve the performance of joints further it is necessary to screen them from the magnetic field with local shields using the Meissner effect. Shields can be made from tubes of Nb—Sn or high-temperature superconductor with a closed end. An alternative technique is to use small reverse-energized coils wound around the joints. These shielded joints have an effect on the homogeneity of the central field of the magnet at the 1 in 105 level and beyond and need to be taken into account at the design stage. G1.0.4 Practical magnets Successful magnets need to operate reliably to maximum field without quenching. The magnet must also be capable of surviving an accidental quench without degradation or damage. The low heat capacity

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Figure G1.0.9. A magnet with joints. Courtesy of Oxford Instruments.

of materials at low temperatures means that only small amounts of energy (typically a few microjoules) released locally within the windings during energization can initiate a quench. Local movement of any of the superconducting wire by just a few micrometres under the large forces can release enough frictional energy to create a small normal region in the wire. This region rapidly spreads throughout the windings usually in less than 1 s, causing the whole magnet to quench. Quenches can also be started by the initiation or propagation of a crack in the material that is impregnated into the windings during construction, specifically to prevent wire movement. The final assembly and interconnection of a magnet must be very carefully performed and are critical to the success of the magnet. Newly built magnets often quench prematurely or ‘train’ when they are first energized. A well-constructed magnet will train fully in just a few large steps and will retain this training during subsequent operation and use over many years. Because of the complexity of the winding structure one might expect no training or virtually indefinite training. Exactly why a few initial training quenches are sometimes but not always required is not entirely clear. G1.0.4.1 Winding Although the windings may be designed to provide any desired level of field homogeneity, real windings can only be constructed with a precision determined by practical limitations. Coil winding is a skill practised by trained staff who often make the production of ‘perfect’ layer wound coils look straightforward. Windings possess a natural pitch or helix and crossover effects occur on successive layers. This means that windings cannot be made perfectly round. Wire sizes and insulation thickness may vary along the length of the conductor and complete integral numbers of turns of successive layers may not fit precisely into the winding mandrel. End effects caused by layer changes may build up. In practice magnets with a homogeneity in the range of about 1 in 103 or 104 over 10 mm DSV can be designed and wound in a relatively straightforward manner. To ensure a tight, well-compacted winding, coils are wound on a mandrel or coil former with significant tension, typically 10 to 20% of the yield stress of the wire. It is beneficial for winding machines to possess an ‘active’ tensioner that allows winding tension to be maintained during winding and also unwinding if turns need to be repositioned. Copyright © 1998 IOP Publishing Ltd

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The winding mandrel needs to be sufficiently rigid to avoid the bore tube stretching or the end flanges deflecting during winding. Dural end flanges 2 to 3 cm thick are adequate for winding laboratory-scale magnets and one flange is normally provided with a radial slot to allow start and end leads to be led out. The presence of short circuits or any potential shorts within a magnet is extremely serious for the production of reliable magnets. Short circuits can lead to the magnet possessing abnormally long settling time constants after sweeping the field. In some cases a short circuit can initiate a premature quench through transformer action within the magnet during energization. This can cause the current in the shorted section to ‘run ahead’ of the magnet current and hence exceed the short-sample limit. A quench in a coil with a short circuit is likely to cause damage in the form of a local or extended burn-out. Shorts can occur for a variety of reasons. Insulation methods and production quality assurance have improved the reliability of insulated wires. Superconducting wires are normally tested for inclusions during manufacture using eddy current testing methods. In spite of cleaning precautions, conducting metallic inclusions in the windings are the most common cause of problems. The winding process can build up static charges that attract contaminants from the environment. On the basis that prevention is better than cure, most coil winding is now performed in a semi-clean area, away from sources of metallic particulates such as a machine shop. Nb—Ti wire is normally supplied insulated with enamel or Formvar. The resistance of windings is normally carefully monitored during manufacture to ensure short circuits do not occur or develop during the winding, impregnation or test process (see Figure G1.0.10). Winding mandrels are insulated with plastics such as Mylar, Kapton or PTFE. In this way coils can be checked for short circuits to ground at both low- and intermediate-voltage levels. Use of low-energy, high-impedance 500 V test sources for this procedure minimizes the chances of a voltage breakdown causing further damage through arcing.

Figure G1.0.10. Winding a superconducting solenoid. Courtesy of Oxford Instruments.

The winding procedure for Nb3Sn is quite different from that used for Nb—Ti. Because the superconductor is extremely brittle it cannot be wound into coil form without severe degradation in performance. To overcome this the wire is supplied in a flexible unreacted ‘green’ state. After winding the entire coil is heated to around 700°C for several days in vacuum or low-pressure inert gas to form the superconductor by means of an in situ solid-state diffusion process. This procedure is known as ‘wind Copyright © 1998 IOP Publishing Ltd

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and react’ and is a nonreversible process. All the materials used in the construction of the coil must withstand the reaction temperature. Stainless steel and copper are used for the winding former. The former is normally insulated using glass-fibre sheet and ceramic materials and the wire is insulated with glass braid. The coil terminations typically use ceramic beads for insulation. After winding, the coil is extremely vulnerable to damage and must be handled with great care until it is vacuum impregnated. Even after impregnation the coil terminations are vulnerable to damage and must be supported against Lorentz and eddy current forces. G1.0.4.2 Impregnation To stop superconducting magnets from quenching, the windings need to be impregnated to prevent the turns from moving during operation. There are two fundamental approaches to this requirement. The turns need to either be held very firmly (strong bonds which do not break) or very weakly (weak bonds which all break but with insufficient energy to quench the magnet). Strong-bond magnets use various types of sophisticated resin for impregnation whereas weak-bond magnets typically use paraffin wax. The choice of impregnant depends on the application. Broadly speaking magnets for NMR that tend to be less highly stressed and only energized a few times before being left at steady field for extended periods can be wax impregnated. The wax has no significant strength and is used merely as a filler to stop turns of the coil from moving. When used within its limits, wax potting is extremely successful and is cheap and easy to perform. If a coil wound from Nb—Ti fails it can be unwound and the wire reclaimed, but where certain stress limits are exceeded wax impregnated coils tend to fail and further degrade with repeated quenching. With wax potted magnets, only fairly slow field sweeping is possible. It is necessary to allow all the weak bonds to break at an acceptable rate otherwise the energy releases sum together and cause premature quenching. Paraffin wax tends to powder during cooling and use which allows helium to impregnate the windings. Although the presence of helium can add to the thermal stability of the windings the dielectric strength of cold helium gas is extremely low. This can lead to the possibility of inter-turn or inter-layer burn-outs during quenching especially if the wire insulation has been damaged during winding or termination of the coil. For highly stressed magnets for general research applications it is necessary to use resin impregnation for overall strength of the windings. Most resin methods are proprietary to industrial manufactures because of the necessary investment in time, effort and money required to develop successful processes and materials. Some general indications of requirements for success are described. The impregnation needs to be perfect, without voids, bubbles or cracks. The winding former is provided with holes or slots to allow complete impregnation of all the turns. Before the resin is introduced the magnet must be evacuated and degassed to a medium vacuum (<10−1 mbar i.e. 0.1 Pa) to avoid problems with trapped air. Once the resin has been allowed to soak into the magnet, the chamber can be let up to atmospheric pressure to finally force the resin into all parts of the coil. If a separate winding or potting mandrel is used, it must be designed so that it can be removed after impregnation and treated with release agent to avoid sticking. Although resin potting results in a stronger structure than wax potting, it is also more brittle and susceptible to cracking during cooling or magnet operation. The initiation and propagation of cracks are a major source of premature quenching in superconducting magnets. Areas particularly liable to cracking such as resin-rich areas that might occur near magnet terminations must either be avoided or fibre-filled to prevent or suppress cracking. Some modern resins are intrinsically resistant to cracking during thermal cycling and are used for impregnation, producing magnets that demonstrate little or no training.

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G1.0.4.3 Joints For basic research magnets, adequate low-resistance joints between separate wires can be formed by twisting and soldering about 1 m of superconducting wire. The joints are normally wound up and put into small brass cups at one end of the magnet. For added security Woods metal is usually run into the cup to prevent wire movement. This method is adequate for magnets with a homogeneity of 1 in 103 over 10 mm DSV where a field decay of 1 in 104 per hour is acceptable. For better joints a few centimetres of copper surrounding the superconducting filaments is etched away using nitric acid. The filaments are plaited, knotted and tinned using indium and an ultrasonic soldering iron, before being put into joint cups and filled with Woods metal. Placing these joints in fields of less than 1 T results in a magnet with a decay in the region of a few parts per million per hour. Using special proprietary jointing methods decays of 1 in 109 per hour or better can be achieved in magnets for high-resolution NMR. G1.0.4.4 Superconducting switches A unique feature of superconducting magnets is their ability to operate in persistent mode because the windings can be made with zero resistance and finite inductance. This means that once the magnet has been energized and the start and end terminals connected using a superconducting switch the current will continue to circulate and the magnetic field will remain essentially constant. This is known as persistent mode operation. The switch consists of a special wire with superconducting filaments but with a matrix made from a resistive material such as cupro-nickel. The normal-state resistance of the switch is typically in the range 10 to 50 Ω. The switch is usually wound into a noninductive coil and a small heater is fitted. The whole assembly is surrounded with an insulating filled resin. This provides a thermal insulation layer that allows the heater to warm the superconductor of the switch above its transition temperature without excessive heat passing into the helium bath. The switch can be opened or closed in just a few seconds. In practice a superconducting switch is permanently connected across the terminals of a magnet as shown in Figure G1.0.8. During energization of the magnet the switch is ‘opened’ using a small heater that quenches the superconductor in the switch, causing it to have a resistance of a few ohms. The magnet can then be energized using a power supply. Most of the current from the power supply flows through the magnet while sweeping. Once the current sweep has stopped and the inductive voltage reduced to zero, the switch is closed by turning off the heater and allowing the superconductor in the switch to cool. The current from the power supply can be reduced to zero while the magnet continues to run in persistent mode. For many general research applications the magnet is left connected to the power supply, but for some special applications such as NMR and MRI, the magnet is left running in persistent mode, and the power supply and even the current leads into the cryostat can be disconnected and withdrawn. Removing the current leads eliminates a major source of heat into cryostats designed for very low helium consumption. Very economic operation of the cryostat can be achieved, and providing the magnet is kept at its operating temperature, it will continue to generate a steady magnetic field and can be regarded as a permanent magnet. G1.0.4.5 Interfaces Many magnets for research applications such as split pairs and solenoids with cancellation coils have loaded interfaces between magnet coils and supporting structural formers. These interfaces can be a source of premature quenching through frictional effects. Typically the windings will be tend to enlarge and press on to the former under Lorentz forces as the magnet is energized while the former will react passively to the load. Pressures as high as 50 MPa can be generated within a large split pair magnet operating

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to 15 T. Any relative motion that happens discontinuously through possible stick-slip effects can cause enough frictional heating to initiate a quench (Kensley and Iwasa 1980). The use of Teflon (PTFE) sheets on these loaded interfaces can have a remarkably beneficial effect because of the low coefficient of friction and benign creep properties that occur even at low temperatures. Most laboratory magnets are precooled initially with liquid nitrogen. When nitrogen is removed from the cryostat it is important that the magnet does not trap residual nitrogen. Interfaces should be designed to allow drainage as residual frozen nitrogen can initiate premature quenching presumably through either preventing correct cooling of the magnet or by localized cracking. G1.0.4.6 Shims The term ‘shim’ comes from early water-cooled electromagnets that were developed for NMR. These magnets were fitted with thin annular layers of iron on the pole tips in order to obtain the desired level of homogeneity. However, not only was the shimming for homogeneity difficult, these magnets relied on power supplies for continuous energization that could only be stabilized to about 1 in 105 thus placing severe limits on the strength, quality and stability of achievable fields. Superconductivity provided a method of producing fields that were uniform and highly stable in time. For homogeneities beyond 1 in 104 over 10 mm DSV or larger volume, ‘shim coils’ are used. These correct for manufacturing imperfections as well as diamagnetic and ferromagnetic effects that originate from the presence of a superconductor and a metallic cryostat respectively. Superconducting shims can operate either in series with the magnet (series shims) or can be independently operated and fitted with their own superconducting switches (independent shims) and power supplies. Series shims are used on magnets up to around 1 in 105 over 10 mm DSV and have the advantage that they provide reasonable field correction at all values of magnet operating current. For finer shimming to the 1 in 106 level and beyond, independent shims are used. Because of inevitable and unavoidable coupling, magnets with independent superconducting shims normally operate at a fixed field. Shims fall into two basic types. The simplest correct for field errors along the central axis of the magnet and are known as axial shims. The second type compensates for errors in the off-axis magnetic field as measured in a circle around the axis at the mid-plane of the magnet. The first axial shim is the Z1 that corrects for a linear gradient error produced by the magnet and is illustrated in Figure G1.0.11. It consists of two coils in a split pair configuration but with one coil reversed.

Figure G1.0.11. A solenoid with an axial Z1 shim.

A Z1 shim is fairly straightforward to use as it produces no net field at the centre of the magnet. In an ideal case it does not couple significantly with the magnet since it is antisymmetric and the mutual inductance is in principle zero, although this is hard to achieve in practice. To begin shimming an axial Copyright © 1998 IOP Publishing Ltd

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field plot of the bare magnet is performed using an NMR probe. The effect of an independently energized Z1 shim can be measured and adjusted exactly to oppose the error produced by the basic magnet. After the test the number of turns can be adjusted to allow the shim to run in series with the magnet, at the same operating current. This is known as a ‘series Z1 shim’. This type of shim tends not to couple strongly with the main magnet and may require a special quench protection to avoid being damaged during a quench. A similar procedure is used to correct any Z2 error of the main magnet. The shim consists of two coils in a split-pair configuration but with the coil separation greater (or less) than the ideal Helmholtz configuration (which by definition gives a zero second-order contribution). A simple Z2 shim (see Figure G1.0.12) is somewhat more difficult to trim and correct in practice than a Z1 shim since it produces field at its centre and also couples inductively with the main magnet causing overall field shifts. Once a shim has been trimmed and connected in series with the main magnet it should operate without difficulty. A similar procedure is used to shim out higher-order axial field errors until the desired axial field homogeneity is achieved.

Figure G1.0.12. A solenoid with an axial Z2 shim.

Off-axis shims are not as straightforward to describe. They correct for field errors as plotted on a circle at the centre plane of the magnet. The diameter of the plotting circle is normally the same as the diameter of the spherical volume used to define the homogeneity of the magnet. The field is essentially an axial field (although NMR techniques measure total field not just axial components) so the error being measured is effectively the variation of field around a circle. It is not a field variation in the radial direction. X and Y shims have a somewhat confusing name. For a perfect solenoid this off-axis circular field plot is a constant, but for real magnets first-order errors show as a sinusoidal variation around the plotting circle. An X shim is a three-dimensional shape with four rectangular coils placed on the surface of a cylinder as shown in Figure G1.0.13. The direction of the currents in the four coils is indicated in the figure. The arrangement produces an approximately sinusoidally varying field in the Z direction around a small circle on the mid-plane. During shimming, a circular field plot of the magnet at the radius of interest is measured to determine the strength and direction of the field errors. The strength and direction of the X shim is then determined by replotting the system with the shim energized and then subtracting the effect of the magnet. After the test the magnet is series shimmed by rotating the shim to the correct position. The number of turns required for the shim to operate at the correct strength at full magnet current is also adjusted. The ability to rotate the shim to line up with the magnet error avoids the need for both an X and Y series shim. In practice shimming a magnet is a complex and somewhat inexact science. The first difficulty is that the homogeneity of the magnet as built is not known until it is tested and measured. Fortunately the errors that occur are fairly predictable in a general way. For high-homogeneity magnets up to about 1 Copyright © 1998 IOP Publishing Ltd

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Figure G1.0.13. An X shim.

in 105 over a 10 mm DSV only first- and second-order axial and first-order off-axis shims are normally necessary. For larger volumes or higher homogeneity higher-order shims are required. Magnets for high-resolution NMR applications are fitted with ten or more independent superconducting shims. The shims are designed to have minimum coupling interaction with the main magnet to minimize changes of the central field. They are constructed with their own superconducting switches and are used in independent persistent mode operation. Extremely fine shimming of an NMR magnet is possible with such an arrangement and is one of the unique benefits that superconductivity has brought to the NMR user community. The final shimming of NMR magnets is often performed using low-power copper room-temperature shim coils which are continuously energized from stable power supplies.

G1.0.4.7 Flux jumping A phenomenon known as flux jumping is observed in some types of superconducting magnet. The effect is characterized by sudden small changes in the field that are apparently rather random, fast, backward step field changes of typically 2 mT against the direction of field sweep. Flux jumps occur only while the field is being swept and also usually only in the field range below 2 T. The effect of flux jumping can be reduced by subdividing the superconductor in the wire down into small filaments. Reducing the inductive coupling between filaments further reduces the effects of flux jumping. Reduced coupling can be achieved by twisting the filaments during wire manufacture. In practice Nb–Ti superconductors with filament diameters below about 100 µm, and a twist pitch of a few centimetres, do not show measurable flux jumping. For Nb3Sn the filaments need to be smaller than 10 µm to avoid flux jumping. The first-generation Copyright © 1998 IOP Publishing Ltd

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filamentary Nb3Sn produced since the early 1970s has several thousand discrete filaments. These fine filaments, necessary to allow successful wire production using the bronze-route solid-state diffusion process, do not exhibit flux jumping. Some more recent high-performance Nb3Sn wires produced since the mid-1980s by the ‘internal tin’ and the ‘jelly roll’ processes have fewer relatively large filaments (~100 µm diameter) which tend to coalesce during reaction. Magnets made from these wires exhibit quite marked flux jumping when operated in the range 0–2 T. Flux jumping can be a problem for some types of sensitive measurement particularly those which involved a pickup coil surrounding the sample for measuring induced voltages such as in a vibrating sample magnetometer (VSM). A compromise has to be made between magnet size, cost and performance and in such cases the more expensive bronze-route wire is used even though it results in a somewhat larger, more costly magnet. G1.0.4.8 Nonlinear effects For most practical work the strength of the field generated by a magnet is best determined from measuring the current. For a modern high-field magnet built from filamentary superconducting wire the field will be known from the current to better than about 10 mT over the entire range of operation. This is sufficient for most practical purposes. Some applications of magnets need to achieve linear correspondence between power supply current and magnetic field during sweeping. Instruments such as a VSM rely on achieving this. It is necessary to ensure that all the current from the power supply passes through the magnet. Parallel current paths must be avoided by using a magnet protection circuit with diodes to avoid current leakage paths and a superconducting switch should not be fitted. An incidental advantage with this arrangement for swept magnetic field applications is that considerable heat dissipation that otherwise would be generated in the superconducting switch and protection circuit is eliminated, minimizing the consumption of liquid helium. There are heating effects associated with sweeping magnets such as eddy current and hysteresis losses in the superconducting wire. These heat sources may dominate the helium consumption of an instrument such as a VSM where the field is swept continuously and can be much greater than the losses associated with the current leads. Superconducting magnets also demonstrate small nonlinear effects due to the hysteretic nature of many of the materials used in their construction, such as superconducting wire and magnetic materials nearby. Materials such as soft solder and ‘Woods metal’ (White 1959) often used in the manufacture of joints show strong diamagnetic effects which vary with applied field. Such materials, used in the joints of the magnet, are usually kept well away from the magnet (or placed symmetrically around the circumference of the magnet at the centre line) to minimize distortions of the magnetic field. There are some peculiar nonlinear effects believed to be associated with flux redistribution in type II superconductors that can cause the measured magnet inductance to change by 20-30% even though the measured field is essentially linear with current. The anomalous inductance seems to be associated with the Nb—Ti section of the magnet and at high fields the magnet inductance reverts to the calculated value. Flux jumping and trapping in the superconducting wire means that the residual field left in the magnet at zero current depends on its recent sweep history and appears somewhat hysteretic in nature. For a typical research magnet the residual field after sweeping would be in the range of –2 to –5 mT, i.e. negative compared with the direction of the last field sweep. G1.0.4.9 Current leads The magnet current leads inside a cryostat are often the largest source of heat leak into the helium. To minimize helium consumption and operating costs it is important to make use of the thermal properties

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of helium when optimizing their design. The current lead should have a sufficient surface area in order to make maximum use of the large amount of cooling available in the exhausting cold helium gas. Brass tubes of 1 cm diameter and with a wall thickness of 0.5 mm are adequate for up to 120 A operation. Gas cooling of the leads is enhanced by convection through the tubes providing they are left open at the top and bottom. Leaving the tubes open at the top and bottom also ensures that thermo-acoustic oscillations do not occur. Current leads tend to warm during use, which automatically helps to promote convection and increases the cooling. Because the latent heat of liquid helium is low it is necessary to avoid Joule heating where the lead enters the surface of the liquid helium. This can be achieved by ensuring that the lower part of the lead (up to just beyond the maximum helium level) is superconducting by shunting it with a piece of Nb—Ti superconductor. Optimized brass leads designed to operate up to 120 A can add as little as 70 cm3 h−1 to the helium consumption of a cryostat at zero current and approximately 200 cm3 h−1 at full current. Brass leads, in addition to being robust, are also fairly resistant to burning out through thermal runaway. To construct efficient current leads, materials with a poor thermal conductivity and high electrical conductivity are required. The availability of high-temperature superconductors has provided a means of designing even more efficient leads. High-temperature superconductor materials being ceramics have a very low thermal conductivity and are superconducting, but high-temperature superconductor leads have not yet come into widespread use in liquid-helium bath cryostats for several practical reasons. Firstly it is inconvenient to provide a positive thermal anchor at or below a denned temperature at the upper end of the lead where a transition from a normal-metal lead is made. Secondly the brittle nature of the high-temperature superconductors makes them susceptible to fracturing during repeated rapid cooling with liquid nitrogen or helium. High-temperature superconducting current leads have found an important application in fully refrigerated magnet systems where they can be gently cooled by conduction. For further information on current leads the reader is referred to chapter D10. G1.0.4.10 Refrigerated magnet systems The recent availability of a new generation of two-stage Gifford-McMahon (G-M) cryocoolers which operate to just below 4 K has provided a method of operating superconducting magnets independently of a supply of liquid helium. The performance of G-M cryocoolers has been improved by means of the use of rare-earth enhanced materials in the regenerators. They have sufficient cooling power (typically >0.5 W at 4.2 K) to cool a superconducting magnet to its operating temperature. The potential difficulty of constructing current leads with sufficiently low heat leak and dissipation when running has been solved by the use of high-temperature superconductor current leads—a technology entirely complementary to the enhanced G-M refrigerators. The first stage of the cryocooler operates typically below 40 K and can be used to provide both a thermal radiation shield for the magnet and also a convenient upper thermal anchor for high-temperature superconductor current leads. Research magnets up to beyond 10 T have been operated in this way. It provides the opportunity for laboratories without access to liquid helium to use magnetic fields. The availability of low-cost refrigeration has increased the possibility of superconducting magnets being used for wider industrial applications. G1.0.5 Testing G1.0.5.1 Initial testing A new magnet is normally prepared for first test by mounting it on a support system with current leads, switch heater and a protection circuit connected. Potential tap wires from each joint in the magnet are

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Figure G1.0.14. A test cryostat with room-temperature access. Courtesy of Oxford Instruments.

connected to a multichannel data-acquisition system. The test cryostat is normally a simple bucket dewar equipped with re-entrant tails that allow room-temperature access to the central region of the magnet (see Figure G1.0.14). The magnet is slowly precooled with liquid nitrogen to minimize thermal shocks. Fitting a flow restriction of about 1 cm in diameter to the exhaust port of the dewar is generally sufficient to limit the flow of exhausting gas and ensure a reasonable rate of precooling. The liquid nitrogen is blown out with care to ensure it is all removed. As a precaution the helium bath is pumped and back filled with helium gas. Sufficient liquid helium is transferred to ensure the magnet is covered. The first test procedure is to run the full magnet operating current through the superconducting switch to check the current leads and switch itself. The voltage across the leads at the top of the cryostat should stabilize in the range 1-2 V after about 15 min. The current is then reduced to zero, the superconducting switch opened and the magnet energized to a few amperes. The field produced in the centre of the magnet is checked against the predicted value to ensure the magnet has been correctly connected. The current through the magnet is then gradually increased. If a quench occurs, the current at which it happens is logged and the data acquisition checked to determine which section of the magnet initiated the quench. The voltages recorded by the data-acquisition system are a complex interplay of coupled inductive and resistive effects but the most important information is to locate which magnet section first indicated a resistive transition, indicating the initial source of the quench. After a quench the magnet is recooled and energization repeated. Modern magnets can be expected to train to maximum field in just a few large steps. Small steps or negative steps are usually indicative of a problem within the magnet which must be located and rectified before excessive amounts of liquid helium are wasted on pointless quenching. Sometimes magnets respond positively to a thermal cycle, which is indicative of a crack or other temperature-induced stress point. Usually poor training is a result of a structural problem within the magnet. Considerable experience and expertise is required to locate and determine the exact nature of any fault. Once the magnet has reached maximum field the magnet calibration constant is determined by measuring the central field using a calibrated Hall probe and an accurate current shunt. The next stage of testing depends on the function and homogeneity of the magnet. It is normal to ensure commercial research magnets will survive a quench from maximum field without damage by over-running beyond the Copyright © 1998 IOP Publishing Ltd

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maximum rated field to quench. Magnets would normally be run back to their maximum rated field before warming up. G1.0.5.2 Low-homogeneity magnets For magnets with a nominal homogeneity of 1 in 103 over 10 mm DSV an axial field plot is performed with a Hall probe (calibrated against NMR) to check that the magnet corresponds with its design and to find the magnetic centre. For magnets with special field profiles, such as those fitted with a cancellation coil that provides a low-field region at one end, extended axial field plots are performed over the region of interest. When testing and calibrating split pairs it is important to check that the true centre of the magnet is located as it normally occurs at a minimum in the axial field situated midway between two peaks. The measured magnetic centre of research magnets is normally clearly marked on the outside of the magnet to allow users to accurately place samples in the correct position. G1.0.5.3 High-homogeneity magnets Magnets with a homogeneity of 1 in 105 or higher need to be shimmed using the coils described earlier. It is normal to perform axial shimming before off-axis correction. Firstly the axial homogeneity of the bare magnet is measured every millimetre using an NMR plotting probe. Spreadsheet-based fitting programs are used to estimate the size of first- and second-order field errors. The separately energized axial shim coils are used to correct the axial field errors. Usually at this level of homogeneity it is desirable to run the shim coils in series with the basic magnet which means that the magnet remains shimmed at all values of operating current. From the measurements described it is possible to correct the strength of the shim coils by adjusting the number of turns and then connecting them in series with the magnet. Off-axis shimming involves a similar procedure. The NMR probe is mounted on the mid-plane but displaced off-axis by the radius of interest, normally 5 mm. The probe is moved in a circle and field measurements taken every 15°. The result of this plot for a carefully wound magnet approximates to a sine wave. The X shim is then energized and the plots repeated. By subtracting the effect of the magnet the strength and direction of the X shim are determined. By rotating the shim to be in antiphase with the magnet error and adjusting its strength the magnet may be series shimmed off-axis. With experience and practice magnets can be shimmed to 1 in 105 over 10 mm DSV in just one or two tests. Magnets with a higher or larger region of homogeneity need more extensive plotting and correction although the principles remain the same. Once a magnet has been series shimmed to 1 in 105, independent first-order axial and transverse shims can be used to shim it to better than 1 in 106. At this level of homogeneity it is easier to shim using half-height linewidth measurements with an NMR linewidth probe and a sample which is the size of the region of interest. Independent shims are supplied with their own persistent mode switches. After shimming it is normal to check that the magnet persistence is adequate and that the shims maintain the desired homogeneity and do not decay. G1.0.5.4 Modulation coils Some experiments require the use of small-sweep or field-modulation coils. Because of the relatively large inductance of superconducting magnets it is not possible to modulate the magnetic field directly even at low audio frequencies. Consequently for field-modulation purposes, separate coils are used. If the modulation coil couples too strongly to the main magnet its operation beyond a critical level will quench

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the main magnet through a.c. losses and eddy current heating (Reinders et al 1987). The higher the desired modulation frequency and amplitude the smaller the modulation coil needs to be. Modulation coils are an imprecise science and exact predictions are very difficult. Typically a magnet with a close-fitting superconducting modulation coil in the bore will probably be capable of a few tens of millitesla modulation at a few hundred hertz. Superconducting magnets fitted to cryostats with a room-temperature bore can have water-cooled modulation coils fitted in the bore. With this arrangement modulation of around ±100 mT at 1 kHz can be achieved. For higher frequencies in the kilohertz range very small, well cooled modulation coils mounted in close proximity to the sample must be used. G1.0.6 The future The maximum field available from a commercial, compact, laboratory-scale superconducting magnet is just over 20 T. This limit is set by the performance of the best available Nb3Sn superconducting wires. The structural engineering and quench energy management of such magnets is now sufficiently well understood not to be a main barrier to progress. So the possibility of higher-field superconducting magnets in the future depends critically on the availability of better superconductors. Some possibilities are considered. G1.0.6.1 High-temperature superconductors High-temperature superconducting materials are promising as high field inserts as their critical fields at 4.2 K are extremely high and probably over 100 T. The difficulty with high-temperature superconductor materials arises from low critical current when in the form of wires or tapes and their brittle nature. The anisotropic nature of the properties of the materials requires careful processing to ensure correct grain alignment. High-temperature superconductor coils have been used to generate fields of over 1.5 T when in a background field of 20 T but significant dissipation (hundreds of milliwatts) in the joints and windings is normal and persistent operation of high-temperature superconductor coils has only been achieved at around 0.15 T. Clearly further materials and processing development are required before high-temperature superconductors can be used to generate high magnetic fields. G1.0.6.2 Alternative A15 materials The standard A15 material Nb3Sn is very well developed although even today, some 30 years after it was first introduced, improved wires are still being developed. Nb3Sn has been successful because a convenient and optimized metallurgical process of formation via a solid-state reaction exists at around 700°C. Other potentially higher-performance A15 materials such as Nb3Al and Nb3Ge exist but because of the equilibrium metallurgical phase diagram, there is no solid-state diffusion process equivalent to that used in forming Nb3Sn. In the case of Nb3Al the stoichiometric phase necessary for high performance can only be formed at temperatures of around 1500°C. To retain pure Nb3Al in the A15 phase, the wire or tape has to be extremely rapidly cooled into a metastable state. This has been achieved by quenching the wire in a liquid-gallium bath. Recent research has increased the chances of finding a nonequilibrium reaction or annealing process at <1000°C. Since Nb3Al is not as brittle as Nb3Sn the possibility of using react and wind methods of construction of coils exists for this material. G1.0.6.3 Chevrel phase The Chevrel-phase material lead molybdenum sulphide (PMS) has a critical field of around 50 T at 4.2 K. As with high-temperature superconductor materials the problem with Chevrel-phase wires has been low

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References

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current densities achieved in wires. Recent work on powder optimizations and heat treatment including hot isostatic pressing (HIP) has resulted in significantly increased current densities that are comparable with Nb3Sn at fields over 20 T. The material is isotropic which eases the problems of processing. Further work on this material may yield suitable materials for use as high-field inserts in superconducting magnets. For further information concerning Chevrel phases, the reader is referred to chapter B9. G1.0.6.4 Pole tips The use of field-enhancing pole tips in superconducting magnets can result in significantly increased fields. Poles can be made from materials with a high saturation field such as dysprosium or holmium. For a pole tip diameter of 2 mm and a separation of 1 mm, typically an extra 4 T field can be generated in a small volume but only with modest homogeneity. The small experimental space available with pole tips means that new ways of making useful measurements need to be developed before they become widely used. References Hartland A 1992 Metrologia 29 175–90 Kensley R S and Iwasa Y 1980 Cryogenics 20 25 Reinders P, Springford M, Hilton P, Kerley N and Killoran N 1987 Cryogenics 27 689–92 von Klitzing K, Dorda G and Pepper M 1980 Phys. Rev. Lett. 45 494–7 White G K 1959 Experimental Techniques in Low Temperature Physics (Oxford: Clarendon) Wilson M N 1983 Superconducting Magnets (Oxford: Clarendon)

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G2.1 Nuclear magnetic resonance spectroscopy for chemical applications

W H Tschopp and D D Laukien

G2.1.1 Introduction Nuclear magnetic resonance (NMR) spectrometers are used to determine the structures of chemical compounds. They have been commercially available since the late 1950s and today are standard equipment in chemistry laboratories. NMR spectrometers consist mainly of a radiofrequency (RF) detector system (probehead), a highly sophisticated transmitter and receiver system, a computer system (since the late 1960s) and a magnet system for generating the highest possible field. In the beginning, these spectrometers made use of the continuous wave (CW) method. Using this technique, the NMR spectrum was acquired by sweeping the frequency. At that time, the simple resolution of the ethyl alcohol spectrum in its multiplets was considered a sensation. The magnetic field was produced with permanent magnets or with electromagnets that contained iron yokes. The highest fields were achieved with these electromagnets which produced a maximum field strength of approximately 2.3 T and were limited to this value by the saturation of the iron. To achieve the high stability necessary for high-resolution NMR, the magnetic field had to be stabilized carefully in several steps. In a first step, current stabilization, a power supply delivered a stabilized high current to the electromagnet. The second step, flux stabilization, used a pickup coil placed in the region of the air gap to detect field instabilities. This coil supplied a voltage proportional to the magnetic field variations per unit of time. By means of the voltage, a correction signal was derived, which in turn was fed into the current stabilizer to vary the current and produce a field change that would compensate for the field error. The third step was achieved with the well-known NMR lock circuit. The NMR lock signal was fed directly into the pickup coil and there simulated a field error. The flux stabilizer in turn tried to compensate for this error and in so doing compensated for the very small field errors detected by the NMR lock. In its entirety, this stabilization was complicated and difficult to adjust. To achieve the necessary basic homogeneity of the magnet the pole pieces had to be shaped appropriately. Problems often appeared as a result of saturation effects and of material inhomogeneities in the iron. This happened particularly when the air gap was small and the sample had to be placed close to the surface of the pole pieces. If the technology had remained as described above, NMR could never have achieved its current success. In general, one can say that the development of NMR was supported by two progressive technologies: the introduction of computer technology and the appearance of superconducting magnet systems. NMR spectroscopy is inherently a very insensitive method, but it provides on the other hand extremely precise information on the chemical structure of the compound. To overcome the inherent low sensitivity, Copyright © 1998 IOP Publishing Ltd

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new detection methods with increased sensitivity have always been of utmost importance. With the introduction of computers in NMR in the late 1960s, it became possible to improve the sensitivity of NMR spectroscopy considerably. Instead of detecting the individual NMR lines one after another by means of frequency sweeps, computers allowed excitation of the entire spin system by means of short RF pulses. The immediately following response signal was digitized and stored in the computer. By means of a digital Fourier transformation (FT) it was possible to reconstruct the desired spectrum. FT spectroscopy in NMR was born. Simultaneously exciting all NMR lines is more efficient, and as a consequence, the signal-to-noise ratio related to a given measuring time could be improved substantially. For the first time, it became possible to run efficient and elegant 13C spectroscopy, which is clearly of utmost importance in resolving the structures of organic compounds. Thanks to computers, FT spectroscopy, a variety of methods for exciting spin systems using composite pulses and multidimensional spectroscopy exist in NMR today. The second major development also took place in the late 1960s, when superconducting magnets for NMR appeared and led the way for higher field strengths, in excess of 2.3 T. As a consequence, the interpretation of spectra of large molecules became possible. Individual NMR signal groups were spread out further because of increased chemical shift. Undesirable second-order spin effects, which are dominant at lower fields, are much less prominent at higher fields (see figure G2.1.1). The use of higher fields also allowed a considerable increase in sensitivity. The combination led to the possibility of applying NMReven to very large molecules. As a result, NMR made substantial inroads into the fields of molecular biology and polymer chemistry. The generation of high fields with compact coils is not the only advantage of superconducting magnets. Their stability is excellent as well because persistent-mode operation is used. This means that the magnet has a superconducting switch across its two ends, so that the magnet current can flow in a closed circuit with no electrical connections to the outside. As a result, higher stability of the current and the field is achieved. In addition, the homogeneity of the magnetic field generated by a superconducting magnet is excellent, because of three provisions: first, through implementation of special regions within the winding package having half or zero current density (notch-sections) in order to homogenize the field; second, by precision winding procedures, and third, through superconducting current shims, also for homogenizing the field. In contrast, the maximum field of an iron magnet is limited to about 2.3 T and its homogeneity is strongly dependent on the homogeneity of the material used for the pole caps. Finding the best supplier for this pole-cap material has always been a major problem. In the early 1980s, NMRcontinued to find new applications in medicine with the development of novel magnetic resonance imaging systems. The concept of implementing precise gradient fields to encode NMR spins spatially made it possible to perceive an unambiguous relationship between the NMR signal and the spin distribution in the volume of interest. In this way, a noninvasive imaging of the human body slice by slice (tomography) became possible. Once again, computer technology and, more importantly, superconducting magnet and gradient design, were instrumental. By using large superconducting magnets in tomography, it became possible to profit from the advantage of higher fields because of the improvement of the signal-to-noise ratio in the NMR signal. This led to an improvement of the spatial image resolution and to the introduction of in vivo NMRspectroscopy (magnetic resonance spectroscopy). G2.1.2 General aspects of superconducting NMR magnets Superconducting magnets belong to the group of magnets that generate magnetic fields by means of current-carrying conductors. The following explanations will be restricted to high-resolution NMRmagnets as used in chemistry and biology for the structural analysis of molecules, and to magnets wound in a solenoidal

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Figure G2.1.1. NMRspectra of 1,7-mathano(12)annulen at different field strengths.

form because all modern NMR magnets are of this type. NMR magnets for imaging in medicine will not be treated here. Superconducting magnets offer two important advantages over resistive electromagnets: the current flowing in the superconductor does not cause any heating of the conductor itself, and the current density in the superconductor can be much higher than is possible with a resistive conductor such as copper. These two properties enable the design of compact magnets that can generate large magnetic fields. Superconductors also have disadvantages. They must be cooled to very low temperatures— preferably to 4.2 K, the temperature of liquid helium at ambient pressure. In addition, the current-carrying capacity in the superconducting state depends on the magnetic field. Once the critical temperature ( Tc ) and the critical field strength ( BC 2 m a x ) for the specific material are exceeded, superconductivity is not possible. Details of this behaviour are shown in figure G2.1.2. Curves 1-5 in figure G2.1.2(A) define a surface that is the boundary between the superconducting and the normal-conducting region. Superconductivity is possible only below this surface. The normal case in which the superconductor is cooled down to 4.2 K is shown hatched in figure G2.1.2(A) and is shown in more detail in figure G2.1.2(B). The hatched area represents a region where the temperature is constant and equal to the usual operating value of 4.2 K. Generating magnetic fields above 17-18 T leads to problems because of the inherent limitations of the superconducting materials. Today, the best commercially available superconductors for high-field magnets Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.2. The critical current density of a superconductor.

are made of the compound Nb3Sn or, better, ternary compounds like (NbTa) 3Sn. Unfortunately, the critical current densities of these materials at 4.2 K and above 18 T are rather low. As a result, coils wound with such materials require a large volume to produce the desired field and therefore large amounts of expensive superconducting wire are required. In order to compensate for part of this drawback, the coil current can be brought closer to the critical current of the wire, thereby achieving a higher current density in the wire. This, however, affects the persistent-mode operation and normally leads to a loss in field stability. A better way to avoid the above-mentioned drawback is to cool the coil below 4.2 K (e.g. 2 K) which then allows for higher critical current densities (figure G2.1.2). As an example, an (NbTa)3Sn superconductor at 2 K and 18.8 T has a current density which is about 67% higher than at 4.2 K. To achieve the lower temperatures, an active cooling unit which cools the liquid helium at 4.2 K down to the desired temperature is necessary. With this method, NMR magnets generating fields of 18 to 21 T are feasible at present. The highest field achieved today with a high-resolution NMR magnet is 18.8 T (introduced by Bruker, Germany, in 1995). High-resolution NMR magnets operate at a single field strength and remain at that field strength for many years. Because these magnets are used for NMR, their fields are normally expressed in terms of the NMR resonance frequency of the hydrogen nucleus (proton). A field of 2.35 T corresponds to a proton frequency of 100 MHz. The relation between field and proton frequency is linear so that all other proton frequencies can be derived from the above numerical values. Therefore a field of 18.8 T corresponds to a proton frequency of 18.8 × 100 MHz/2.35 = 800 MHz. Today field strengths range from 4.7 to 18.8 T (200 MHz to 800 MHz) with a typical minimum room-temperature bore of 52 mm diameter. The highest available field for persistent NMR magnets is 18.8 T (800 MHz) and will certainly increase in coming years. The magnetic energy Em stored in a magnet containing no iron is given by

From equation (G2.1.1) we can see that the stored energy depends on the field strength B and on the size or volume of the magnet. The above-mentioned NMR magnets have stored magnetic energies ranging from 0.018 to about 5.5 MJ for the corresponding fields from 4.7 to 18.8 T. For comparison, 1 MJ is equivalent to the energy required to boil 333 1 of liquid helium, or 6.7 1 of liquid nitrogen, or 0.43 1 of water. Because the stored magnetic energy is closely related to the volume of the magnet, it is also closely related to the costs of the wire. In figure G2.1.3 the sizes of NMR magnets designed for fields between Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.3. A comparison of size between NMRmagnets with a 5 cm room-temperature bore designed for magnetic fields from 4.7 to 23.5 T.

4.7 and 23.5 T are shown. From that it is obvious that the conductor requirements for higher fields do not increase proportionally with the field strength; they increase more rapidly. The magnetic field has to have a long-term stability of better than 10−8 h−1 and the field homogeneity has to be approximately 2 × 10−10 within a measurement volume of 0.2 cm3. The homogeneity requirements are achieved by using specially designed coils which generate a flat field plateau, using superconducting shims and a room-temperature shim system as well. Although laboratory magnets with field strengths of over 21 T have already been built, the world record for NMR magnets is much lower, 18.8 T. One may ask why this is the case. Laboratory magnets need only produce a high magnetic field; field homogeneity is much less important. This means that the solenoid coil can be fairly short and the winding package can be designed without any regions with reduced current density (notches). The electromagnetic forces, as well as their influence on the superconductor, are therefore much lower and coil design is significantly simplified. On the other hand, the requirements for field homogeneity in NMR magnets demand a much more complicated coil design (see figure G2.1.4). The coil needs a larger axial length and an inhomogenous current density distribution in the axial direction (notches) to achieve the required homogeneous field plateau (see figure G2.1.5). In addition the stability requirements for the field demand excellent persistent-mode operation which is only possible when the current of the magnet is not too close to the critical current of the wire. In order to fulfil these demands, NMR coils have to be designed much larger than a corresponding laboratory magnet for the same field strength. Table G2.1.1 shows in more detail a comparison between laboratory magnets and NMR magnets. G2.1.3 Construction of a superconducting NMR magnet A superconducting NMR magnet consists basically of a superconducting main coil and superconducting shim coils, both mounted inside a cryostat. In order to achieve the required high field homogeneity in NMR, a highly sophisticated room-temperature shim system must also be provided. We concentrate here on the superconducting magnet itself, and we will not go into further details concerning the room-temperature shim system. Figure G2.1.6 shows the basic design of a superconducting magnet system. Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.4. A comparison between a laboratory magnet and an NMRmagnet.

G2.1.3.1 The electrical circuit Figure G2.1.7 is a circuit diagram for a superconducting magnet system. To achieve a high basic field stability, superconducting NMR magnets should be operated in the persistent mode; i.e. in a superconducting, closed-circuit mode in which the beginning and the end of the superconducting main coil are connected by a superconducting main switch after the magnet has been energized to its nominal current value. The magnet current can then flow through the main switch in a closed circuit and no current leads to the outside are necessary. In a sense, the superconducting magnet behaves like a permanent magnet and does not require a constant supply of energy. The principle of operation for the superconducting switch is discussed later. To energize the main coil to its desired value, it is connected to the main power supply and the switch is opened (heater on). The power supply provides a constant voltage u0 to the main coil and thereby causes the current iM in the coil to increase linearly. LM is the inductance of the main coil

After the desired current has been reached, the voltage across the main coil is brought to zero by adjusting the voltage of the power supply. Now the main switch is closed by switching the heater off. The output Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.5. The axial field profile for a sixth-order NMR magnet compared with a laboratory magnet.

current of the power supply is slowly brought to zero, and the power supply is then disconnected from the coil. The magnet current then flows in a closed circuit consisting of the main coil and the switch. Diode 6 in figure G2.1.7 is a safety diode designed to protect the main switch from excessive voltage and to produce a return path for the current in the main coil when the main switch is ‘open’. This is important when the main coil quenches (a transition of the coil from the superconducting to the normal-conducting state) or when the power supply is faulty and not able to deliver the desired current. The dump resistors (5) are described in more detail later. Their job is to protect the main coil from damage during a quench. The wiring of the superconducting shim coils is similar to that of the main coil. The shim coils are bridged by superconducting switches, which also allow for persistent-mode operation. The individual shim coils and their switches are connected in series, and the beginning and the end of this network are connected to a shim power supply. This series connection allows current to be delivered to all shim coils by only two current leads. The principle of operation is as follows. Suppose that the current in the z shim has to be adjusted. All shim switches are closed. First, heating current is applied to the z shim switch. This opens the switch. The current from the shim power supply then flows across the switch of the z 2 shim to the z shim and from there across the x and y shim switches back to the power supply. The currents in the z 2 , x and y shims are not affected by this process because they are shortened by their own switches. After adjusting the desired current value in the z shim, the heating current is switched off and the z shim switch operates again in the superconducting state. The other shims can be adjusted similarly. Only the shim coil with an open switch (heater on) is affected. Copyright © 1998 IOP Publishing Ltd

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Strictly speaking, this last statement is not quite correct. The superconducting switch in the closed state has a small but nonzero inductance that must be considered. This inductance is connected in parallel to the small inductance of the shim coil. The current coming from the shim power supply is therefore divided between the switch and the shim coil according to the inverse values of their inductances. Most of the current flows through the shim switch because it has the much smaller inductance. A small amount, however, can flow though the shim coil to produce an unwanted gradient. To minimize this effect, the inductance of the switch must be much smaller than that of the shim coil. The current is supplied to the NMR magnet through removable current leads. Because the magnet operates in the persistent mode after being energized, these current leads can be removed afterwards, whereby an improved thermal isolation of the magnet is achieved. G2.1.3.2 The cryostat The cryostat provides thermal isolation of the superconducting magnet at 4.2 K from the surroundings at approximately 295 K. This is initially achieved by evacuating the inside of the cryostat (figure G2.1.6). However, heat from outside can still enter the cryostat in two ways: ( i ) by thermal conduction along the thin stainless steel tubes connected to the top of the helium and nitrogen container and along the thin adjustable rods that are used to centre the two containers and the radiation shield ( ii ) by thermal radiation between materials inside the cryostat that have large surface areas at different temperatures. Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.6. Mechanical construction of an NMRmagnet.

Efficient thermal isolation is achieved by using a nitrogen container filled with liquid nitrogen at a temperature of 77 K and located between the radiation shield and the outside shell. It can be considered as a thermal barrier that separates the outside shell from the helium container and therefore leads to lower thermal radiation. Because the nitrogen container is in thermal contact with the upper parts of the two stainless steel tubes of the helium container, the areas of thermal contact on the tubes are also cooled to 77 K. As a result, these tubes conduct less outside heat into the helium container. A radiation shield is placed between the nitrogen and the helium container to minimize thermal radiation between each of them. This radiation shield is also in thermal contact with the two stainless steel tubes of the helium container. In this way, the radiation shield and the tubes are cooled efficiently by the evaporating helium gas flowing through the tubes. This helium gas has a total cooling capacity which depends on the difference in enthalpy of the gas leaving the helium container at 4.2 K and the gas leaving the outside shell at about 295 K (room temperature). This difference is equal to 1519 J g−1 and can be divided into 384 J g−1 available between Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.7. Electrical circuit of an NMRmagnet.

4.2 and 77 K and 1135 J g −1 available between 77 and 295 K. One part of the 384 J g −1 is used for cooling the radiation shield, the remaining part for cooling the stainless steel tubes below the nitrogen thermal contact. The 1135 J g −1 are used for cooling the tubes between the nitrogen thermal contact and the outside shell. These design features, combined with an optimal choice for the geometry and for the materials used, minimize helium loss. Today NMR cryostats are available exhibiting helium hold times of more than 12 months and boil-off rates of approximately 10 cm3 liquid helium per hour. They have a reservoir of liquid helium of about 105 1 of which approximately 90 1 will evaporate between refilling. Cryostats for an operating temperature of only 2 K are built similarly to the cryostat described above. The main difference appears in the construction of the helium can. It normally consists of two compartments, a lower one at 2 K and an upper one at 4.2 K, separated by a membrane that acts as a thermal barrier. In addition, an active cryogenic cooling system (i.e. refrigerator) is needed for cooling the 2 K compartment. Such a cooling system normally operates in conjunction with a Joule-Thomson valve, which allows compressed helium gas to expand into the 2 K compartment and to generate there a cooling effect. Copyright © 1998 IOP Publishing Ltd

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G2.1.3.3 Superconducting wire The wires used today in NMR magnets are made of superconducting compounds containing two or more of the following elements: niobium (Nb), titanium (Ti), tin (Sn) and tantalum (Ta). Most superconducting NMR magnets are manufactured with NbTi separately or combined with NbsSn wire. For field strengths above 12 T, wire made out of a ternary compound, such as (NbTa)3Sn or (NbTi)3Sn, is commonly used. The amount of Ta or Ti in the Nb is very low for these ternary compounds, a few atomic per cent only. Figure G2.1.8 illustrates some typical cross-sectional views of these wires.

Figure G2.1.8. Cross-sectional views of typical NbTi and Nb3Sn superconducting wires used in NMRmagnets.

The coat, matrix or core of copper inside the wire helps in stabilizing the superconductor. Superconductors are inherently susceptible to local heating caused by friction or mechanical motion. This heat can lead to local ‘quenching’, in which the wire loses its superconductivity. In such a case, the copper has a stabilizing effect because it momentarily carries the current flowing in the superconductor and allows the superconductor to recover and return to its initial state. This prevents the superconductor from entering the irreversible resistive state with consequent quench, which would cause a total discharge of the coil. The Nb3Sn wire in figure G2.1.8 is shown in its unreacted state. The filaments consist of pure niobium and have only poor superconducting properties. The excellent superconducting properties are induced by a thermal reaction process at 700°C that lasts several days and transforms the niobium into crystalline Nb3Sn. This transformation is accomplished through a thermal diffusion process during which part of the tin contained in the bronze matrix diffuses into the niobium filament. This creates a crystalline layer of Nb3Sn on the surface of the niobium filaments which slowly grows towards the inside. Because of the bronze Copyright © 1998 IOP Publishing Ltd

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involved in this process, it is also called the bronze diffusion technique. The Nb3Sn wire consists of a copper core surrounded by a bronze matrix. The two are separated by a tantalum barrier to prevent a tin diffusion from the bronze matrix into the copper core. This is undesirable and would degrade considerably the electrical conductivity of the copper core and therefore reduce its stabilizing effect. (Nb,Ta)3Sn, (Nb,Ti)3Sn and Nb3Sn conductors are almost identical except that the filaments are made of NbTa, NbTi or pure niobium (Nb). To prevent the filaments from breaking during the winding process, the coil is wound before being reacted (‘wind-and-react technique’). Unreacted niobium filaments are ductile and therefore less likely to break. After the reaction, the filaments are extremely brittle because of the crystalline structure of Nb3Sn. If the coils were wound after reaction, some filaments could break and would cause severe drifting of the magnetic field. Clearly, this would be unacceptable for superconducting NMR magnets. Single-filament NbTi wires are commonly used for magnetic fields below 7 T and only for small, simple magnets. These wires are less stable than are NbTi wires consisting of multiple filaments, but they are less expensive and simpler to connect with a superconducting joint. In the range between 7 and 9.5 T, NbTi multifilament wire is used exclusively. Above 9.5 T a hybrid design is used, where the inner part of the coil is wound with NbSn material and the outer part with NbTi. For field regions between 9.5 and 12 T, Nb3Sn is used. For field regions greater than 12 T, the material of choice is (NbTa)3Sn or (NbTi)3Sn. The limits can be derived from the critical current density versus field curves of the respective materials (see figure G2.1.9).

Figure G2.1.9. Critical current densities of typical superconductors used in NMRmagnets.

Between 6 and 9.5 T, a range normally covered by NbTi, Nb3Sn has a higher critical current density than does NbTi, and it could be used as well. However, NbTi is less expensive, needs no thermal reaction process and is easier to wind. Use of Nb3Sn in this field range is therefore not economical. It is obvious that the critical current density of the wire must be as high as possible in order to achieve an economical and compact coil. However, it is not only the critical current density which is important. For NMR magnets which have to operate in the persistent mode, it is equally important that the residual Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.10. Specific resistance r of the superconducting wire.

resistance of the wire in the transition region between super- and normal-conductivity has a high rate of change. Figure G2.1.10 illustrates this region and shows also how the graph of the specific resistance can be approximated by a potential function of the nth power. n is called the n factor or index of the wire. It is an indicator for the persistent-mode performance of the wire and it should have a value greater than 30. With this value, experimental results have shown that the residual resistance at the operational point is low enough to achieve the required minimal field stability for NMR, which is about 1 × 10−8 h−1. G2.1.3.4 The main coil The superconducting NMR coil, which produces the main field, is normally called the main coil. Main coils are usually built with wire that has different thicknesses in different radial regions. The thinnest wire is used in the outermost parts of the coil and the thickest wire is used close to the centre of the coil. However, the same amount of current flows through both the thick and thin parts, producing a higher current density on the outside than on the inside. This leads to an inhomogeneous current density distribution in the radial direction throughout the coil. For a better understanding, it is necessary to look at the magnetic field distribution within the coil. Figure G2.1.11 shows the radial and axial magnetic field profiles. From the radial field profile, we see that the field within the coil decreases with increasing radius r, goes through zero and actually becomes slightly negative (reversed field direction) outside the coil. If we keep in mind that the current is the same throughout the coil and that the current-carrying capacity of the superconducting wire increases with decreasing magnetic field, we see that it is possible to use thinner wire on the outside than on the inside. By taking advantage of the higher current density in the thinner wire, it is possible to achieve a more compact coil. Such coils are called graded coils, and they are used for fields of 300 MHz and above. From figure G2.1.4(B) it can be seen that NMR coils have ‘notches’; regions within the winding layers that have lower current densities than do the remaining windings on the same layer. Notches are Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.11. Radial and axial field profiles of a superconducting NMR main coil.

typical for NMR coils and necessary for achieving the specified field homogeneity. The axial field profile of a coil without this notch would fall off too quickly with increasing axial distance z from the centre of the coil. To compensate for this, higher current density is needed in the ends than in the middle. Instead of increasing the current density in the ends, we also can decrease the current density in the middle. This is achieved with a notch construction. The current density distribution of an NMR coil is therefore inhomogenous not only in r (due to the grading), but also along the axis z (due to the notches). The two typical notches in NMR magnets are those with current density zero and those with current density 1/2. A notch with current density zero has no active wire and consists of either air or of the coil bobbin itself. A notch with current density 1/2 is built by using a superconducting wire parallel to a second, nonsuperconducting wire of the same dimensions (also called a dummy wire) during the winding process. This means that half the volume of the notch is filled with nonactive wire which results in a reduction of the current density to half of the original value. NMR coils are usually designed as sixth-order coils. To gain a better understanding of this, we look at the axial field profile B( z ) in the z direction and approximate the central region with a series expansion

z is zero at the magnetic centre of the coil. Because the coil is constructed symmetrically about the z axis, the axial field distribution has the same symmetry and the odd terms of the expansion in equation (G2.1.3) are zero. If we look at a Helmholtz coil, we find that a2 = 0, and the field decay is defined by the next higher nonzero term, which is a4z 4. Such a coil is called a fourth-order coil. NMR coils at least fulfil the condition a2 = a4 = 0 with the help of notches. The next higher nonzero term is therefore a6z 6, and so Copyright © 1998 IOP Publishing Ltd

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this coil is called a sixth—order coil. The axial field distribution for this coil is given by

A higher—order coil has a larger homogeneous field plateau but its field outside the plateau decays more rapidly. Figure G2.1.5 shows a typical axial field profile of a sixth—order coil. G2.1.3.5 Series correction coils Although great care is taken in manufacturing an NMR coil and its notches according to specifications, small errors that result from manufacturing tolerances cannot be avoided completely. A consequence of this is that the axial field profile can diverge by 10 to 100 ppm from the theoretical sixth—order field function. This calls for several corrections, one of which is the series correction, which is made by placing additional solenoidal coils on the outside of the cylindrical main coil and connecting them in series with the main coil. These coils can have varying numbers of loops and can generate fields parallel or antiparallel to the direction of the main field, depending on the desired correction. This series correction is designed so that some important field gradients generated by the main coil are compensated for as well as possible. Because of manufacturing tolerances we might find besides the even—order gradients (second and fourth) also odd—order gradients (first and third), which have all to be compensated. Because the main coil and the series correction coils are connected in series and carry the same current, we can define them as a new coil. The new coil has a much better homogeneity than does the coil without the added corrections. The improved homogeneity resulting from the series correction is virtually independent of the current in the magnet, i.e. of the generated magnetic field, and therefore need not be readjusted for every desired field strength. This is a special advantage of this kind of correction and does not apply to a further kind of correction, provided by the superconducting shim coils, described in the next section. G2.1.3.6 The superconducting shim system Although the introduction of the series correction improves homogeneity substantially, further corrections are necessary to achieve the requirements of high—resolution NMR. These corrections are accomplished by using superconducting shim coils. The adjustment of the currents in these coils is called the shimming process. To get a better understanding of the homogenizing process for an NMR magnet, it is necessary to apply some mathematics. The region of interest in an NMR magnet is defined by the measuring sample and the RF receiving coil, which is arranged on a cylindrical surface around the sample. Both are located at the magnetic centre (see figure G2.1.11) of the NMR magnet. It is important to note that the NMR frequency of a nucleus is directly dependent on the magnitude B of the magnet field at the location of the nucleus. Expressing the magnitude B with the z component and the radial component of the magnet field (see figure G2.1.12) we find

Because the sample tube containing the nuclei is located near the centre of the magnet, all field lines at that position are almost parallel to the z direction with only small deviations. This means that the second Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.12. Field B and its components Bz and Br at point P on the surface of a sphere with radius ρ .

term in the expansion of equation (G2.1.5) can be neglected and that the Bz component of the magnet field is the only component of interest. Bz can be expressed by an orthogonal expansion in spherical harmonics. This has the advantage that the expansion is composed of functions (spherical harmonics) defined on the surface of a sphere with the radius ρ. If the coefficients of these functions are minimized, e.g. by using shim coils, then all inhomogeneities inside the sphere are also minimized. Furthermore the functions are orthogonal, so that minimizing the coefficient of one function does not influence the minimizing process for the other functions. We can now define a cylindrical coordinate system with its centre exactly in the centre of the RF coil. The expansion in spherical harmonics is given by

It is convenient to use cylindrical coordinates, because normally, when field measurements are performed, the field points are chosen on the surface of a cylinder. The functions Fn m ( z , r ) cos( mϕ ) and Fn m ( z , r ) sin( mϕ ) are the associated Legendre functions. They represent the harmonics which are generated by the main coil (see table G2.1.2). n is the order and m the degree of the harmonics. To generate the above functions with the shim coils, they are composed of several subcoils connected in series and placed on the surface of a cylinder with axis z. These subcoils are wound with superconducting wire and have special geometric shapes. Each shim coil has its own superconducting switch connected Copyright © 1998 IOP Publishing Ltd

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Figure G2.1.13. Shim coils for generating z and x gradients.

in parallel, so that a persistent—mode operation is possible without any permanent connection to a power supply. It is an identical method of operation to the one already described for the main coil. Figure G2.1.13 shows the typical geometric layout for the z shim (n = 1, m = 0) and x shim (n = 1, m = 1). The Cartesian coordinate description of the gradients is normally used in NMR to name the shim coils. They are usually referred to as z shim, x shim, y shim, z 2 shim, etc. Because the z shim produces an on—axis gradient which typically is not dependent on the angle ϕ , it must have a cylindrical symmetry relative to the z axis and therefore consists of solenoidal— shaped coils. On the other hand, the x shim generates an off—axis gradient which typically is dependent on the angle ϕ and therefore cannot exhibit a cylindrical symmetry. The x shim is therefore composed of several saddle—shaped coils arranged symmetrically around the surface of a cylinder. The y shim is identical but displaced 90° in ϕ . Shimming is accomplished by adjusting the current in the individual shims and thus producing individual field components Fn m ( z , r )cos( mϕ ) and Fn m ( z , r )sin( mϕ ), which compensate for those produced by the magnet coil itself. The currents needed in the shim coils are proportional to the negative values of the coefficients An m and Bn m produced by the magnet coil. The shimming process is feasible only if the compensation of one gradient is not affected considerably by another. This condition is met if the gradients produced by the shims are defined by functions which are mathematically orthogonal to each other, and this is the case for the above— mentioned shim functions. NMR magnets usually have at least four to eight superconducting shim coils: n , m = 1,0: 2,0; 1,1; (total of four) or n , m = 1,0: 2,0; 1,1; 2,1; 2,2 (total of eight). The off—axis shim functions ( m > 0 ) always appear in pairs for individual values of n and m. These pairs have an identical geometric shape; however, they are rotated with respect to one another by 90° divided by the grade m ( ∆ϕ = 90°/m ). One example of such a pair is the x and y shims. The superconducting shim coils are placed symmetrically about the centre of the main coil and can be mounted either on the outside of the main coil, between two sections of the main coil or inside the innermost section of the main coil; placement on the outside is simplest. However, this may require shim coils with a larger number of ampere windings, i.e. either more windings or larger shim currents. This Copyright © 1998 IOP Publishing Ltd

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is especially the case for the second—order shims ( n ; m = 2,0; 2,1; 2,2 ) as their sensitivity decreases in proportion to 1/r02 ( r0 = radius of the cylinder containing the shim coils ). To achieve the ultimate resolution necessary for NMR spectra, an additional and very sophisticated room—temperature shim system is needed which creates up to 32 different shim functions with order numbers up to six and degree up to three. The adjustment of all these shims can be accomplished only automatically by special measuring methods under computer control. G2.1.3.7 Superconducting joints Superconducting NMR coils operate in the persistent mode with a superconducting switch across the beginning and the end of the main coil. It is important that the switch and all connections between the superconducting wires (the joints) of the NMR coil are superconducting. Every superconducting NMR coil has at least two joints, namely those that connect the beginning and the end of the coil to the superconducting switch. However, because NMR coils usually are built of several wire sections with different wire diameters, additional joints are necessary. Very little information is present in the literature about the techniques used in jointing different sections of superconducting wire. Because this part of the magnet design is probably the most important, manufacturers are secretive about their techniques. In principle, superconducting joints can be produced by one of three techniques: the wire ends can be connected directly by cold welding under high mechanical pressure; the wire ends can be connected by hot welding under high temperature; the wire ends can be connected indirectly by using a superconducting intermediate material. In the latter case the connection is made from one wire end to the intermediate material and from there to the other wire end. The highest—quality connections are usually achieved with the first method, although it cannot be used in all cases. Hot welding can lead to a degradation of the superconducting properties. The third method is used mainly for Nb3Sn wire but also for NbTi wire that contains a large number (500 or more) of filaments. Good superconducting joints have a resistance of 10–13 Ω or less. Such minute resistances can be measured by forming a superconducting loop and measuring the decay of an applied current. Knowing the inductance of the loop and the rate of decay then enables one to calculate the resistance. The current—carrying capacity of a superconducting joint decreases with increasing magnetic field. Superconducting joints are therefore placed as far away from the main coil as possible. This is of particular importance for the design of high—field magnets. G2.1.3.8 A superconducting switch A superconducting NMR magnet has a main switch connected in parallel to the main coil and several shim switches connected in parallel to their respective shim coils. These switches allow operation of the coils in the persistent mode. All of these switches consist of a special superconducting wire in close thermal contact with a thin wire that can heat the superconducting wire to above its critical temperature. The superconducting wire itself is the switch, and the heating wire acts as a means to operate the switch. Both wires are wound in a bifilar way on the same coil body. In this way, the field generated by each wire is compensated to zero. The two bifilar wires are wound such that they have a close thermal contact to each other. This gives a switch with minimal inductance and three additional advantages: the electromagnetic forces between the main coil and the switches are negligible; the switches generate only a very small field and therefore have practically no influence on the field homogeneity at the centre of the main coil; the switch in the closed state (heater off) represents only a very small inductor, so that the current of the shim connected in parallel is practically not influenced when the current of the other shim coils is adjusted. The superconducting wire (normally NbTi) used for the switches has a matrix consisting of a resistive material, such as CuNi, instead of copper. The electrical resistance of the matrix, which is a few ohms,

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defines the resistance of the switch in the open state (heater on). In the closed state (heater off) the switch is superconducting and represents a very small inductance. The principle of operation of the switch is as follows. In the open state, current flows through the heating wire, heats the superconducting wire to above its critical temperature and turns the superconductor into the normal— conducting state. The resistance of the switch is then a few ohms. This relatively small resistance, compared with the resistance of the superconducting coil itself which is zero, is sufficient to define the open state of the switch. In the closed state of the switch the heating wire has no current. Therefore the switch is superconducting and fulfils the necessary requirements for persistent—mode operation of the superconducting coil. G2.1.3.9 Dump resistors The dump resistors are shown in figure G2.1.7. They protect the main coil and the main switch during a quench, which can happen if a small portion of the coil experiences a local rise in temperature. Such an effect can result from a slight motion of the superconducting wire under the influence of the electromagnetic forces present in the winding package of an energized magnet. If this friction is enough to heat the superconductor to above its critical temperature, a local quench starts that can propagate quickly throughout the coil. The magnetic energy EM = (L M i 2M stored in the main coil will then be converted into heat. The total magnetic energy can amount to more than 1 MJ for a large NMR magnet. It is therefore necessary to control the dissipation of this energy to prevent damage to the coil. During normal operation an NMR magnet does not have any high—current leads to the outside, so the magnetic energy cannot be dissipated outside the helium container. Therefore, the heat dissipation must take place within the helium container. The object best suited to absorb this heat energy is the magnet itself, with its relatively large mass. As long as the heat can be evenly distributed over the whole mass of the magnet in a short time, the magnet will increase its temperature only to about 100 K, and no serious damage should be expected. This is the case if the quench rapidly propagates throughout the coil. However, if the propagation is not fast enough, most of the stored energy will be dissipated in the region of the winding package where the quench started. This can lead to a local overheating of the superconductor and can destroy the main coil. Dump resistors are used to prevent this. They are designed to provide protection against two potential problems during a quench: overheating of the main coil and excess voltages within the winding package of the main coil. Dump resistors are not designed to absorb all of the heat from a quench, but they help to distribute the quench as quickly as possible throughout the coil. In this way, the whole mass of the coil can absorb the heat energy. This process can be explained as follows. Imagine that the main coil is divided into two sections, each bridged by a low—resistance dump resistor. If a quench takes place in the first section, then the resistance of that section will increase and the current will decrease. At the same time, an increase in current will occur in the second section because the sections are magnetically coupled and the total magnetic flux of the main coil tends to stay constant (law of Lenz). This causes the current in the second section to rise above the critical current value and therefore induce a quench in that section. Now, the second section can also contribute to the absorption of heat. This process is possible only if different currents can exist simultaneously in different sections of the coil. Thanks to the dump resistors, this is possible. They have low resistance values ( typically 0.1 to 1 Ω ) and can therefore take over the difference in current between the coil sections. A further task of the dump resistors is to protect the main coil from excess voltage during a quench. This leads to sparking in the coil and eventually to its destruction. One could be tempted to consider the coil as already protected against excess voltage by the main switch, which acts as a short circuit across

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the two ends of the main coil. However, within the winding package two different kinds of voltage are generated, which have opposite polarities: the resistive voltage across the quenched wire and the inductive voltage across the whole wire. Both can be dangerously high within the winding package but compensate to zero or nearly zero across the whole coil or across a section with a dump resistor. Therefore, by increasing the number of dump resistors it is possible to keep the resistive and inductive voltages in the winding package low.

G2.1.4 Future NMR magnets The demand for ever higher magnetic fields from the NMR community, for example from the area of molecular biology, is strong and challenges the magnet manufacturers. The highest fields suitable for NMR are not easily reached and are limited by electrical properties of the wire and by large electromagnetic forces. With increasing field strength, the critical current values of the superconducting wire decrease and the design of compact magnets becomes more and more difficult. Furthermore, the operating field moves closer to the critical field strength ( Bc 2 ) so that the n factor (or index) decreases as well and can lead to long—term instabilities in the form of field drift. Another limiting factor, perhaps the most important one, is the large electromagnetic forces generated within the winding package of the main coil at high fields. Three approaches are currently available or under investigation for manufacturing superconducting wire that has improved electrical properties at higher fields. In a first approach, the electrical properties of the already available NbSn superconductors are being improved by using ternary or even quarternary materials such as (NbTa)3Sn, (NbTi)3Sn and (NbTaTi)3Sn. Although this development has led to a considerable increase of the critical field strength from 19 T to about 26 T at 4.2 K, these values are still too low for operating fields above 23 T. In a second approach, one can lower the operating temperature of the magnet. This increases the critical field and the critical current density of the wire and allows a higher n factor and consequently an improved stability of the field, and/or the achievement of higher operating fields. In a third approach, other superconducting materials are being studied, including Nb3Al (critical field Bc 2 = 30 T at 4.2 K ), Nb3Ge ( Bc 2 = 36 T at 4.2 K ), PbMo6S8 (the Chevrel phase material, Bc 2 = 55 T at 4.2 K ) and last but not least the new ceramic high—temperature superconductors such as YBa2Cu3Ox or (Bi, Pb)2Sr2Ca2Cu3Ox ( Bc 2 > 100 T at 4.2 K ). If the amount of interest and money invested in this new high—temperature superconductor technology is an indication of industrial applicability, it seems that the high—temperature superconductors have the best chance to win the race. However, the manufacturing of usable and economical wires made from these materials is difficult, and the results are not yet satisfactory. In addition, the higher quality demands for NMR applications complicate the problem considerably. Besides the electrical limits of the wire, the magnet designer will be confronted with one overwhelming problem-the control of the enormous electromagnetic forces at high fields. In most cases, the superconducting material itself will not have the necessary mechanical strength to cope with the electromagnetic forces. Therefore, superconducting magnets for high fields require a certain amount of a nonsuperconducting material with higher mechanical strength (copper, copper alloys, stainless steel, molybdenum, etc) to absorb the forces without damage and to protect the superconductor itself from exposure. We can expect that future superconducting NMR magnets for very high fields (above 25 T) will contain only a small amount of superconducting material and a larger amount of nonsuperconducting material for mechanical stability. This, however, will also reduce the overall critical current density in the winding package and lead to very large magnets. It is obvious that designers of superconducting NMR magnets will be confronted with a most challenging future, full of new and interesting problems.

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Further reading Brechna H 1973 Superconducting Magnet Systems (Berlin: Springer) Montgomery D B and Weggel R J 1969 Solenoid Magnet Design (Malabar, FL: Krieger) Pobell F 1992 Matter and Methods at Low Temperatures (Berlin: Springer) Reed R P and Clark A F 1983 Materials at Low Temperatures (Metals Park, OH: American Society for Metals) Richardson R C and Smith E N 1988 Experimental Techniques in Condensed Matter Physics at Low Temperatures (New York: Addison—Wesley) Wilson M N 1983 Superconducting Magnets (Oxford: Clarendon)

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G2.2 Magnetic resonance imaging and spectroscopy (medical applications) W H-G Mütter and D Höpfel

G2.2.1 Introduction In medicine the technique of magnetic resonance imaging (MRI) has become, in a period of only ten years, a standard diagnostic tool worldwide. The most important part of the whole MRI system is the magnet, and the breakthrough in MRI resulted chiefly from the development of large, strong and homogeneous superconducting magnets. Since high magnetic fields are desired for the resonance process, superconducting magnets have been developed of the size necessary to provide a very homogeneous magnetic field over a volume of interest (e.g. a part of a human body). Because MRI has been accepted very quickly as a valuable tool for medical diagnosis the market for these superconducting magnets has become the largest for applied superconductivity, apart from the research—oriented magnet systems for large accelerators. In addition, the medical—application field has the attractive advantage of being a relatively continuous one. MRI is based upon a resonance phenomenon, nuclear magnetic resonance (NMR) or magnetic resonance (MR), which will be explained in the next section. Historically, the idea of a nuclear magnetic moment was introduced in 1924 by Pauli, whereas the first resonance experiments by induced transitions on free atoms were performed in 1939 by Rabi. The first NMR experiments on liquids were performed in 1946 by Purcell and Bloch, both being honoured for their work in 1952 with the Nobel prize. The first MRI experiment (i.e. the acquisition of an MR image) was achieved by Lauterbur in 1973. For the acquisition of the data and the reconstruction of the image he used a technique which was already known from x—ray images, called backprojection. In 1975 Ernst introduced the two— dimensional (2D) MRI technique, which today represents the technique most often used in MRI because of the significant technical advantages it offers in comparison with backprojection reconstruction. In the last few years very attractive and promising new fields of research, such as rapid imaging techniques and functional imaging, have been developed especially for the medical applications of NMR. These techniques place very severe demands on the equipment. An example of the technical problems involved is the eddy currents in the metal tubes of the superconducting magnet induced by the rapid switching of strong magnetic field gradients. Active shielding of the magnetic field gradients is today the solution to this problem, the same technique as that employed for reducing the stray field, a method which is very important for the installation of an MRI magnet. Copyright © 1998 IOP Publishing Ltd

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G2.2.2 Basic concepts of NMR and MRI G2.2.2.1 The principles of NMR All atomic nuclei with an odd number of protons or neutrons possess an angular momentum, the spin I, and coupled with this a magnetic moment µ

where γ is the gyromagnetic ratio and h = h/2 π (h is Planck’s constant), e.g. for protons: γF = γ/2π = MHz T−1 (figure G2.2.1).

Figure G2.2.1. Rotating charge, spin I and magnetic moment.

The directions of the magnetic moments normally have a stochastic distribution. In an external magnetic field they tend to line up with the field direction like a compass in the magnetic field of the earth. However, the thermal fluctuations prevent all magnetic moments from lining up in the same direction as the external magnetic field. The distribution of the magnetic moments is determined by a Boltzmann distribution, dominated by the energy ratio µB/κT (κ is Boltzmann’s constant; k = 1.38 × 10−23 W s K−1). Since µB « κT there is nearly the same amount of magnetic moment parallel to the external magnetic field as there is in the opposite direction. The net magnetization is therefore essentially very low (i.e. NMR is inherently an insensitive method). To understand the NMR experiment the laws of quantum mechanics must be applied. They allow only certain directions of a magnetic moment in an external magnetic field, resulting in two distinct orientations for the proton. Due to the angular momentum the magnetic moments precess around the external field B0, as shown in figure G2.2.2. The precession frequency (angular frequency) is proportional to the field B0

This can be compared with the precessing of a torque around the axis of the gravitational force. The precession of all magnetic moments in a probe and the resulting macroscopic magnetization M are shown in figure G2.2.2. As is common in NMR, in this section the term magnetic field is used for B instead of the more correct terms magnetic flux density or magnetic induction. (a) NMR spectroscopy In equation (G2.2.2) the relation between the precession frequency of the magnetic moments and the magnetic field B0 is expressed. However, the electrons of the molecule screen the external magnetic field B0 at the location of a nucleus in that molecule. The local magnetic field is therefore dependent upon the chemical environment

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Figure G2.2.2. Precession of all the magnetic moments in an external magnetic field B 0 and the resulting total magnetization M.

where σ is the screening constant. Due to the local field the local processing frequency changes

Different chemical environments therefore result in different processing frequencies, which is the principle that forms the basis of magnetic resonance spectroscopy (MRS) (figure G2.2.3).

Figure G2.2.3. 31P Spectrum of a human forearm. The peaks of the different chemical compounds in the tissue can be clearly distinguished.

MRS is a standard technique used in the chemical industry to study compounds and the structure of molecules. In medicine in vivo MRS has not yet become an important tool in spite of the fact that it would be very interesting to study human metabolism locally with MRS. Even at high magnetic fields ( B ≥ 1.5 T ) it suffers mainly from a poor signal—to—noise ratio, thus demanding very long examination times. The inherent problem is that one tries to acquire a signal from a volume of the order of 1 cm3 , but the volume of the receiving coil covers, for example, the whole head. Thus the noise— producing volume is much larger than the volume from which the signal arises resulting in a poor signal—to—noise ratio. Copyright © 1998 IOP Publishing Ltd

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G2.2.2.2 Detection of the NMR signal The question is now: how can we detect the small magnetization M ? The easiest way to understand this is to visualize that we are in a coordinate system which precesses around B0 with the same frequency as the magnetic moments. In such a system we would not experience any force (compare this with the precession of the earth around the sun, which you also do not feel). If you now switch on a second magnetic field perpendicular to B0 (referred to as B1 below) in this rotating system, the magnetization vector will again precess around B1 due to the angular momentum characteristics of the nuclei. The precession frequency of the B1 field must have in the laboratory system the same frequency as the magnetic moments around B0 to have a steady influence on them (i.e. it is in resonance with the nuclei), hence the name nuclear magnetic resonance. The processing B1 field can be easily created by a radiofrequency field (RF field) of the proper frequency. The total movement of the magnetization M in the laboratory frame is a combination of the precession around B0 (which is of course always present, being produced by a superconducting magnet) and the precession around B1, the latter being an RF pulse switched off after a precession around B1 by an angular rotation of 90° (figure G2.2.4).

Figure G2.2.4. Movement of M in the laboratory frame under the influence of the static magnetic field B0 and the processing field B1 .

Figure G2.2.5. Decay of the induced NMR signal, the so—called ‘free—induction decay’ ( i.e. after the RF field is switched off ).

The precession of M around B0 in the x—y plane induces a voltage in an RF coil which is tuned to the resonance frequency. This signal can be detected after switching off the RF ( B1 ) field and is called the free—induction decay (FID). Due to relaxation effects the signal decays with time as shown in figure G2.2.5. The decay of the NMR signal is determined by two relaxation processes. After the excitation with the RF pulse the resulting magnetization M is in the x—y plane and relaxes back to thermal equilibrium. This requires a definite amount of time because the energy transmitted to the spin system from the RF pulse must be transferred to the lattice. Therefore this time is referred to as the spin lattice relaxation time T1. Additionally the interaction between magnetic moments leads to a decay of the induced signal because their phase coherence is lost by this interaction. The time required for the signal decay caused by the spin—spin interaction is called the spin—spin relaxation time T2. Inhomogeneity of the magnetic field has the same effect upon the signal decay, due to the different processing frequencies of the single magnetic moments at different locations (ω = γ B ), consequently the single magnetic moments have different phases in the x—y plane and their sum, the magnetization vector M , decays to zero (figure G2.2.6). One is not interested in the signal decay resulting from external magnetic field inhomogeneities. However, it is not possible technically to create an absolutely homogeneous magnetic field, but there is a clever method which may be used to overcome this problem. By flipping the magnetic moments after Copyright © 1998 IOP Publishing Ltd

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Figure G2.2.6. Decay of the magnetization vector M due to the temporal dephasing of the magnetic moments.

a certain time τ by 180° they will regain their phase coherence after 2τ, thus forming the so—called spin echo. This can be compared to runners with differing speeds. If they turn back after a time τ they will all be back at the starting point at the same time, provided they maintain their original speed (figure G2.2.7).

Figure G2.2.7. ‘Dephasing’ and ‘rephasing’ of runners with different speeds.

With this method, which is fundamental for MRI, it is possible to measure the signal decay established through the spin—spin interactions and T2. Both relaxation times T1 and T2 depend on the chemical environment and differ therefore between different types of tissue. This is very important for MRI because, as a result of the variable relaxation times, one acquires excellent contrast between different tissues, without the use of a contrast agent. G2.2.2.3 Imaging MRI means imaging with the NMR technique. For clinical MRI only the nucleus of hydrogen, or the proton, is used. With all other nuclei the signal is insufficient for MRI, mainly because of their low natural abundance. An interesting nucleus for clinical studies would be carbon. However, the carbon nucleus with the highest abundance, 12C, or has an even number of protons and neutrons, thus having no spin or magnetic moment respectively. The only possibility for NMR with carbon would be the 13C isotope, but with a natural abundance of only 1% the signal is far too small for imaging. Now we understand how the NMR signal is detected, but so far the measured signal stems from the whole probe. To reconstruct an image from these signals we must know which signal contribution stems from which part of the probe. For this purpose we use the simple equation ω = γ B (i.e. if we Copyright © 1998 IOP Publishing Ltd

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change the B field then the signal frequency will change in an analogous manner). The variation of the external magnetic field is achieved with linear magnetic field gradients which means that the processing frequencies change as a linear function of distance along the gradient, corresponding to a spatial encoding of the MR signal received (figure G2.2.8).

Figure G2.2.8. The principle of the spatial encoding of the MR signal.

A prerequisite for definite spatial encoding is that the linear change of the magnetic field gradient is at least ten times greater than the inhomogeneities of the external magnetic field B0 This leads to the following condition for the relative homogeneity of the magnetic field over the region to be imaged (in a whole—body magnet and with a sphere of 500 mm diameter): ∆B/B0 10−5 with the best available magnets ranging up to ∆B/B0 10−6. The spatial encoding of the signal in the second direction is done by phase—encoding the signal in this direction by changing the strength of the magnetic gradient with each projection step. Therefore after switching off the gradient the phase of the signal changes with every step depending on the location of the volume element (voxel). One obtains a 2D image by simply performing a 2D Fourier transformation (2D—FT). The 2D—FT is performed in two steps: first, line by line and then one row after the other. In the third direction one could essentially perform the same phase—encoding as in the second direction. However, this is normally too time consuming. Instead a slice selection is performed in this direction by using an RF pulse with a restricted bandwidth during the time a magnetic field gradient is present. From the whole spectrum of frequencies, corresponding to the various locations of the magnetic moments, only those spins with processing frequencies existing within the bandwidth of the pulse will be excited (figure G2.2.9). The modulation of the excitation pulse can only be performed in the time domain. To excite a rectangular profile in the frequency domain the profile in the time domain therefore must be the Fourier transform of a rectangle, which is a sin( t )/t function. The slice parameters can be easily changed: the slice orientation is achieved by superimposing magnetic field gradients, while the slice thickness can be altered either by varying the bandwidth of the excitation pulse or by changing the gradient strength. Normally the latter is done. Lastly, the slice location is determined simply by the frequency of the excitation pulse. The entire RF and magnetic field gradient pulse sequence typically resembles that shown in figure G2.2.10. Copyright © 1998 IOP Publishing Ltd

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Figure G2.2.9. A slice selection with an RF pulse of restricted bandwidth.

Figure G2.2.10. RF and magnetic field gradient pulses for a typical MRI experiment, where Gx is the magnetic gradient field in the x direction (in this case the so—called ‘read gradient’, i.e. the gradient field during which the signal is read out), Gy is the ‘phase—encoding gradient’ (above referred to as the second direction) and Gz is the ‘slice gradient’, i.e. the slice of the image is excited perpendicular to the z direction.

G2.2.2.4 Standard MRI applications A significant advantage of MRI is the high image contrast in different kinds of tissue. The image contrast is determined by the difference in luminance of adjacent picture elements (i.e. by the variable signal intensity). Compared to x—ray computer tomography (CT), where the contrast is determined only by absorption and scattering of the radiation in the various tissues, the strength of the NMR signal is influenced by several parameters: the density of the protons, the interaction of neighbouring magnetic moments (chemical environment) determining the relaxation time T1 and the diffusion and/or interaction of the molecules Copyright © 1998 IOP Publishing Ltd

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influencing the signal strength determined by the relaxation time T2. Flow effects can also occur (e.g. blood flow in vessels). All these influences must be considered in interpreting the image, but nevertheless it can provide valuable information concerning the status of the tissue. After considering such influences it follows logically that there exist several MRI sequences designed especially for studying specific states of tissue. The most important MRI applications will be discussed briefly in the text below. (a) Spin—echo sequence The pulse sequence shown in figure G2.2.10 remains the standard MRI technique. The reason for this is that this sequence is relatively insensitive to magnetic field inhomogeneities and that the signal intensity’s dependence upon the measuring parameters is well understood. The spin—echo pulse sequence can be easily extended to acquire multiple slices. In this way one uses the relaxation time T1 of the excited magnetic momenta for multiple excitations, or slices, with the same phase—encoding gradient before stepping to the next phase for each slice. In this manner, a set of eight to 20 slices can be obtained within 4–8 min (figure G2.2.11). The acquisition of multiple echo signals is also possible, the later echoes being more and more T2 weighted since long T2 values correspond to a long signal decay. Today in clinical practice there are certain spin—echo times used which provide distinct contrasts between tissues possessing different T2 values. Another technique uses each acquired echo signal for incrementing the phase—encoding gradient, and is called the RARE (rapid acquisition with relaxation enhancement) technique (Hennig et al 1986). By this method the total number of excitations of the magnetization is divided by the number of acquired echo signals (e.g. 256/8 = 32). Thus, the total acquisition time is reduced by the same factor. This sequence consequently has a twofold advantage: providing T2 -weighted images and possessing a short acquisition time (figure G2.2.12).

Figure G2.2.11. An example of a T1-weighted image: sagittal image of the neck, showing a disc prolusion, Courtesy of Bruker Medizintechnik.

Figure G2.2.12. A T2-weighted transversal image through the head with an acquisition matrix of 426 × 512 pixels using a ‘RARE’ sequence. Courtesy of Bruker Medizintechnik.

A disadvantage of these sequences is the acquisition time of 4–8 min typically needed to acquire the images. The patients must not move at all during this time if motion artifacts in the resultant images are to be avoided. For some patients this is very difficult. Therefore one of the main aims in MRI was, and still is, to reduce the acquisition time. Since the acquisition time is determined by the relaxation time T1, which in turn is fixed by the chemical compounds and cannot be changed, the only way to reduce the acquisition time is to acquire only a portion of the signal with each phase—encoding step (i.e. one uses only part of the magnetization which could relax in the time between two excitation pulses). The signal—to—noise ratio on the other hand Copyright © 1998 IOP Publishing Ltd

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is determined mainly by the strength of the magnetic field B0. Therefore all fast image sequences in MRI perform better at higher magnetic field strengths. One manner of shortening image acquisition time is to employ the RARE technique. This could be extended to the case where one excites only once and then uses 128 phase—encoded echoes to reconstruct the image. Thus only tissue with very long T2 values will contribute to the signal, (e.g. fluids in the body). Since the resulting images are similar to those provided by a well known technique in x—ray CT, this technique is called MR—myelography. A very common fast—imaging method is the gradient—echo sequence: the echo of the signal is in this case acquired through an inversion of the magnetic field gradient and not by an 180° RF pulse. Since the repetition time of this experiment is much faster than the relaxation time T1 , the excitation pulse is less than 90° to prevent signal saturation (which also would be caused by the refocusing 180° pulses). The disadvantage of this method is that the inhomogeneities of the magnetic field are not compensated and the signal therefore decays more rapidly. On the other hand this sequence is also more sensitive to susceptibility changes in the tissue, therefore creating additional contrasts in the image. The total acquisition time of an image acquired with the gradient echo sequence is typically in the range of 10 s, which is much shorter than that of a standard sequence. Figure G2.2.13 shows an example of an image in the abdomen where the patient has been asked to stop breathing during the acquisition to prevent motion artifacts.

Figure G2.2.13. A coronal slice through the body obtained using a gradient-echo sequence (total scan time is only 20 s). Courtesy of Bruker Medizintechnik.

If there is enough signal intensity available the gradient-echo sequence could be shortened to a time which is essentially limited only by the time required for switching the magnetic field grandients. In this manner total acquisition times in the range of 100 ms or even shorter are possible. Motion artifacts due to breathing can be eliminated with this technique and even heart images without electrocardiogram (ECG) triggering are possible. Another important MRI application using fast gradient-echo sequences is the acquisition of three-dimensional (3D) data sets, which in practice are not possible wiht the spin-echo sequence because of impractical imaging times. With a 3D-data set, imaging planes a certain range. Such surface ‘extracted’ images can be made to rotate in real time through all possible viewing angles. (b) MR angiography As already mentioned, the MR signal intensity is sensitive to the diffusion or to the motion of magnetic Copyright © 1998 IOP Publishing Ltd

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moments. On one hand this can yield undesired (motion) artifacts, yet on the other hand one can use this effect to obtain information concerning moving spins. There are several techniques involved in MR angiography which provide images of the vascular system. Contrary to the well established DSA method (digital subtraction angiography) using x—rays, MR angiography requires no contrast agents. The diagnostic results of the two techniques are different, DSA showing the absorption of the x—rays and MR angiography the local MR features of the moving spins. X—ray CT angiography has the advantage of superior spatial resolution and the ability to demonstrate dynamic processes (e.g. blood flow in the cardiac vessels). On the other hand MR angiography presents no significant health risk and is therefore preferred, if the resolution is sufficient. G2.2.3 MRI—instruments and safety aspects G2.2.3.1 Technical equipment In figure G2.2.14, a block diagram shows the essential components of an MRI system.

Figure G2.2.14. A block diagram of an MRI system.

The main part of the system and of the construction costs is represented by the magnet system which includes the shims and the gradient system. The technical aspects of these are discussed in section G2.2.4, therefore only a few remarks regarding MRI need be made here: as already discussed, for a high signal—to—noise ratio a high field strength is necessary. Of course higher field strengths create higher costs and contrast in the images will be reduced due to the longer relaxation times. Additionally, at very high field strengths the RF requirements regarding power are very large (increasing with the square of the field strength, e.g. at 2 T up to 15 kW) and the design of the RF coils becomes increasingly more complicated. In clinical MRI systems today the magnetic field strength ranges from 0.2 T to 2 T, and is typically between 0.5 and 1.5 T. Resistive or permanent magnet systems can provide up to 0.3 T whereas higher fields require superconducting magnets. The relative homogeneity requirements are ∆B/B0 ≤ 10−5 which can easily be fulfilled with modern superconducting magnets. The field stability of these magnets is also significantly better than that of Copyright © 1998 IOP Publishing Ltd

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resistive systems. Field strength and stability therefore are the main reasons why most of the MRI systems sold today include superconducting magnets. The screening of the large stray fields is an important point in the clinical environment and was achieved initially with iron. Today active screening is employed which involves a second coil built around the main coil for the purpose of screening it. The screening of stray fields will be discussed in more detail in section G2.2.6.1. This same technique of active screening is also used for the gradient systems in order to prevent the induction of eddy currents which causes severe problems with gradient switching. Of course the gradient strength of the inner coil is diminished by the screening coil but this effect can be normally compensated by using higher coil currents. The gradient strength for a whole—body system is typically 10 mT m-1 but can be higher for special applications such as echo planar imaging (EPI) (Mansfield and Chapman 1977). EPI involves a very fast MRI sequence with acquisition times down to 50 ms for the whole image. Switching times for the magnetic gradients of the order of 100 µs are necessary for this sequence. Therefore this method has only been one or two years in clinical environments, since the technical demands for switching high gradients in a 1 m bore magnet with a strength of 20 mT m−1 or higher in 100 µs are very high. Power supplies with several hundred volts and amperes are necessary for these applications. For smaller research systems, used for so—called MR microscopy gradient systems, up to 1000 mT m−1 are available. With respect to the RF system whole—body MRI systems require powerful linear transmitters with an RF power of up to 15 kW. The RF power is normally transmitted by the whole—body coil in order to achieve a homogeneous RF field. The RF coil receiving the MR signal can be the same as the transmitting coil. However, today one typically uses a separate receiving coil if whole—body scans are not being performed. The receiving coils are adapted to various parts of the body (head, shoulder, arm, leg, etc) to optimize the signal—to—noise ratio. Furthermore, it is easier to construct an RF probe for handling less than 1 W than one for handling several kilowatts. Solenoid—shaped coils are not usable with cylindrical magnets since the B0 field vector must be perpendicular to the B1 field vector of the RF coil. Today, so—called birdcage RF resonators are typically used for whole—body and head coils. Otherwise Helmholtz coils or saddle—shaped coils are employed. One advantage of new open magnets (C—type magnet, see section G2.2.6.3) is that solenoid coils can be used. A more sophisticated technique to improve the signal—to—noise ratio is a ‘quadrature’ coil which consists of two coils with their B1 vectors perpendicular to each other. Theoretically the gain in the signal—to— noise ratio is . This sort of RF coil has become standard equipment, especially with low—field systems. For processing images, sophisticated software packages are now available, including 3D data—processing and image display. For data acquisition, easy—to—handle software menus are commonly used in a clinical environment. G2.2.3.2 Actively shielded gradients The gradient coils are a major hardware component of an imaging system because they are necessary for the spatial encoding of the NMR information. Without gradients the instrument can only be used as an NMR spectrometer. The NMR—imaging process requires a change of the gradient fields and this switching has become more and more important for the newer fast—imaging methods. The stray fields of the gradient system interact with the metallic part of the cryostat of a superconducting magnet, or the pole pieces of an electromagnet, in such a way that the fields inside the volume of interest are changed on timescales appropriate for the different imaging processes. The time constants (Jehenson et al 1990) for the cryogenic structure are given by the different metallic shields inside the bore tube of the magnet and, to a smaller extent, also by the metallic parts of the superconductor and the outer parts of the cryostat. The physical parameters are the geometry of the parts (e.g. diameter, thickness and the resistivity of the material), leading to time constants of 0.1–100 ms.

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When the problem of eddy currents became obvious, optimization of the material and geometry were discussed to reduce this negative effect. The second generation of cryostats used fibreglass—reinforced material as bore tubes. With those nonconducting tubes the effect was drastically reduced but, since the requirements of NMR increased at the same time, the eddy current problem was not solved satisfactorily. Even for small gradient coils inside large-bore magnets, such as head coils in a whole—body system, for which the coupling to the conducting shield is relatively small due to the favourable geometry, the unwanted time dependence of the effective gradient field is still too large. The solution was found with actively shielded gradient coils (Turner 1988, 1993), which use the same principle of shielding as the basic concept for actively shielding the main magnet. A second current system, possessing opposite current with respect to the main gradient coil, is placed on a large radius in the case of cylindrical geometry. Figure G2.2.15 shows the field-line plot of an actively shielded gradient coil in order to illustrate the principle of active shielding. The contours, which are only shown in one quarter of the cross—section, are obtained by plotting the field lines at ten equidistant intervals. The polarities of the currents in the sub-coils are indicated in the inset. In the outer space the fields of the two—coil system cancel so that there is no interaction between the gradient coils and the cryostat. Inside the arrangement the difference between the two fields is still sufficient for effective MRI. These actively shielded gradient systems are optimized with respect to their gradient strength, linearity and minimal inductance. They are the major hardware component of the MRI system, besides the main magnet, and together with the necessary power supply they determine the essential specifications of the imaging process. It is interesting that the idea of actively shielded gradients is based upon the assumption that the coil is surrounded by a superconducting shield which leads to the disappearance of the gradient fields outside the double—coil system (Tuner 1988). The introduction of the actively shielded gradient systems has been the turning point for superconducting magnets since, due to the cryostat requirements, superconducting magnets could previously not be used for effective imaging. This is especially true for newer MRI applications such as EPI or chemical shift imaging.

Figure G2.2.15. An actively shielded gradient system. Copyright © 1998 IOP Publishing Ltd

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G2.2.3.3 Safety aspects All modern techniques used in medicine must fulfil strict safety requirements. The potential dangers of MRI are the magnetic field Bo, the fast switching of strong magnetic field gradients and the RF power, all of which will be discussed briefly in the following. The static magnetic field Bo can only act on magnetic parts in the body (e.g. implants consisting partially of ferromagnetic material) or on moving charges due to the Lorentz force. Severe danger is only known in those cases where patients are carrying a cardiac pacemaker or metallic surgical clips in very sensitive regions such as the brain. By moving the patient into the magnetic field eddy currents and magnetic fields are induced in these metallic implants which create a force contrary to the moving direction (Lenz’s law) and lead to a small movement of the implant, which may result in severe consequences if this should happen in the brain or other vital regions. These patients should not be examined with MRI. Otherwise no danger associated with the static magnetic field is known at present. Nevertheless limits for the field strength in a routine clinical environment exist, the highest field in most countries being 2 T. Patients with cardiac pacemakers are not allowed to enter a magnetic field higher than 0.5 mT. The magnetic gradient field induces large flux changes during the switching period which can induce voltages and currents in conducting media and in this manner potentially influence the conduction of nerves. Problems can theoretically arise in very fast sequences such as EPI; however, to date nothing of consequence has been reported in patient use. Limitations for the gradient switching differ in various countries but are all in the range of several tesla per second. The RF power absorbed in the body is described by the specific absorption rate (SAR)

SAR ∼ f 2B12tP /TR

f being the excitation frequency, B1 the magnetic amplitude of the RF field, tP the duration of the RF pulse and TR the relaxation time from the last RF pulse to the next excitation pulse. The power dissipated in the body by the RF pulse obviously grows with the square of the frequency or the magnetic field. The body is able to dissipate heat of the order of 1 W (kg body weight)−1 into the surrounding environment, depending on the isolation provided by clothes or by subcutaneous fat. Thus actual heating of the body occurs only if the power of the RF pulses exceeds that which the body can dissipate into the surrounding environment during the same time interval. However, at high fields this can lead to a limitation in the application of certain MRI methods. The SAR rates allowed in various countries are normally ≤1 W kg−1 for whole-body examinations and ≤5 W kg−1. for other regions of the body such as the extremities. G2.2.4 Superconducting magnets for MRI It has been made clear in the preceding sections that there are major prerequisites for MRI magnets such as field strength, homogeneity and stability. Moreover there are other properties of the magnet which are not mandatory for NMR but which nevertheless are important for routine application and commercial success. Examples for this second type of requirement are small stray fields, compact design with small overall weight and, of course, a minimum of cryogenic maintenance. G2.2.4.1 Basic concepts for superconducting NMR magnets The use of superconducting magnets for NMR requires additional features in comparison to solenoids for general research. The most important properties are very high homogeneity and excellent stability of the magnetic field. In order to obtain homogeneities where ∆B/B0 ≤ 10−7 in the volume of interest Copyright © 1998 IOP Publishing Ltd

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(VOI) NMR magnets consist of several sections arranged with rotational symmetry around one axis and symmetry around the mid-plane of the magnet. The basic homogeneity concepts for imaging magnets are very similar to those for NMR magnets used for spectroscopy (see section G2.1 and Laukien et al 1993). These are based on the expansion of the magnetic field of an axial symmetry loop in a series of the Legendre functions Pn(cosθ ) (Garrett 1967, Sauzade and Kan 1973)

As a result of the symmetry of a solenoid no tesseral components are present and further, due to the mirror plane, the odd components are zero. Therefore the expansion reduces to a simple Taylor expansion with even components

The cross-section of the magnet coil is structured in such a way that the even components of lower order of the different coil parts cancel each other in the centre of the coil. Normally the coils are named after the first order which is unequal to zero in the basic design. The simplest case is a coil of fourth order, the so-called Helmholtz coil, consisting of two loops with a radius R and a distance z = ±R/2 forming the central plane. The spectroscopy magnets normally have high fields which are generated by very thick coils while the MRI magnets with moderate fields consist of thin coils which are as short as possible for a given free access. This difference leads to the differences in the relative homogeneity, normally defined as deviation ∆B/B within a sphere (diameter spherical volume (DSV)). For NMR spectroscopy sixth-order coils can be used; for MRI magnets coils of 10th and 12th order are necessary in order to obtain the desired homogeneity in the VÓI.

Figure G2.2.16. Homogeneity of a 12th-order system.

The homogeneity region of a 12th-order system is shown in figure G2.2.16 for only one quarter of the cross-section of a six-coil magnet. The contours of equal field deviations ∆B/B0 = ±5 × 10−3, ±3 × 10−3 and ±10−3 in figure G2.2.16 are similar to the well-known Legendre functions because the Copyright © 1998 IOP Publishing Ltd

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dominating order is the 12th order. As seen in figure G2.2.16 the contours terminate at the end positions of the subcoils. The homogeneity along a given radius is not at all constant but varies with a certainoscillation character as shown in figure G2.2.17 for a radius of 250 mm. The zeros of this function are given by the zeros of the corresponding Legendre function (Abramowitz and Stegun 1965). These zeros are the positions of the 12 planes normally used for the field analysis of a magnet (a so-called 12-plane plot). Six positions of a 12-plane plot are shown in the centre of figure G2.2.16.

Figure G2.2.17. Field variation for a quadrant (DSV = 500 mm).

In addition to this high homogeneity the field must also have good temporal stability over the period of an experiment or a series of experiments. The drift should be less than 10−8 h−1. This stability is at least two orders of magnitude better than the very best of modern high-current power supplies. Consequently, a simple power supply mode cannot be used and a better method of running the magnet is required. This is the ‘persistent mode’ of operation and it necessitates a closed superconducting loop with perfect joints between different sections of the coil, existing together with a superconducting switch. Since the principle of this persistent mode is very basic to all NMR magnets it will be explained in more detail. Figure G2.2.18 shows a schematic diagram of the complete magnet with the major items indicated. The superconducting switch (3) consists of a length of superconducting wire surrounded by an electrical heater (4) and a thermal insulator. To energize the magnet the superconducting switch is first opened by operating the electrical heater and driving the superconductor into the normal state above the transition temperature. Current is then introduced down the ‘demountable lead’ (5) flowing mainly through the superconducting path, i.e. through the main coil (1). When the operating current is reached the superconducting switch is closed by allowing it to cool down to the temperature of the helium bath. When the switch closes and the current is reduced in the demountable lead, then the magnet current remains constant and the switch carries the difference. At zero current in the leads the switch carries all the magnet current and the leads can be removed leaving a superconducting loop with current flowing through it. In this state the main coil is connected to the outer world only by control leads for the helium-level meter and an emergency discharge unit. The heat load of the helium vessel is drastically reduced. Therefore after some hours the system will reach its minimal helium consumption. Despite the very high accuracy of machining the magnet formers and performing the winding process, Copyright © 1998 IOP Publishing Ltd

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Figure G2.2.18. Magnetic circuit (persistent mode).

the first field in the VOI, after production of the coil, will deviate from theoretical results by two to three orders of magnitude. These deviations are due to mechanical tolerances, small deviations from the theoretical number of turns, unknown thermal expansions, etc. The correction involves a special procedure and, in general, different hardware components. Spectrometer magnets use specially wound coils (so-called shim coils) which can be individually energized to shim the magnet (Romeo and Hoult 1984). These coils may be superconducting and built into the magnet itself or resistive and inserted inside the room-temperature bore of the magnet. The correction procedure starts with a theoretical concept based upon the same ideas as the basic design, but there is one major difference: the theoretical design does not include angle deviations, whereas the experimental results do have these deviations because there are corresponding errors in the real coil. The simplest theoretical analysis is based upon a point-by-point measurement of the field. An NMR coil is usually used as a sensor with a small sample having a diameter of a few millimetres. Seven or 12 measuring planes are common and the angle increments are 30° or 22.5° respectively. This set of measuring values is then analysed to produce the empirical coefficients of the spherical harmonics which are present in the measured field due to impurities or intrinsic design constants. This analysis is based on the expansion in spherical harmonics which is given by the following expression (Vlaardingerbroek 1996).

The basic idea of correction is the same for all NMR magnets (see Laukien and Tschopp 1993). A correction system now adds its influence to the original field and compensates the components present due to the field analyses explained above. This procedure in general needs several iteration steps in order to improve the homogeneity. With few exceptions neither superconducting nor resistive active-correction systems are used for MRI magnets. For routine field strengths the so-called passive shimming method is used, and is described in the following section. G2.2.4.2 Superconducting MRI magnets The very first magnets for MRI were strongly influenced by the technology of superconducting magnets for NMR spectroscopy (section G2.1) which was well established by the early 1980s. The obvious differences were given by the horizontal bore, the large bore diameter and the moderate field strength, in comparison with spectroscopy magnets. The systems had a typical bore diameter of 1 m, a DSV of some 50 cm, a Copyright © 1998 IOP Publishing Ltd

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refill time of ten days for liquid nitrogen and six weeks for liquid helium and a very large stray field. The rapid acceptance of MRI as a clinical method and the increasing number of instruments led to the development of special MRI magnets whose improvements are discussed below in more detail. A major step towards a magnet which is easy to use has been the realization of the active-shield concept. The stray-field problem, which will be discussed in more detail below, significantly restricts the application of superconducting magnets. The basic idea of an actively shielded magnet includes a second coil which has a larger radius than the main coil as well as a negative field contribution. The sums of the fields cancel each other outside the concentric arrangement, whereas in the centre the difference in the fields is still sufficient for the NMR experiments. The first actively shielded magnets had a moderate field strength of 0.5 T but subsequent development has extended the maximum field in commercial products to 2 T. The ratio of the diameters of the shielding coil and the main coil is 1.3 to 1.5. With this ratio the cryostat dimensions are changed only slightly in comparison with nonshielded magnets. Figure G2.2.19 shows the field-line plot of an actively shielded magnet in order to illustrate the principle of active shielding. The contours, which are only shown in one quarter of the cross-section, are obtained by plotting the field lines in ten equidistant intervals. The polarities of the currents in the sub-coils are indicated in the inset. The peak fields and the hoop stresses are in general much higher for actively shielded magnets at the same centre field. The forces and stresses have to be carefully analysed to realize a safe magnet design. The shielding requires a special quench-protection concept in order to ensure that, during a quench, the

Figure G 2.2.19. The field lines of an actively shielded magnet. Copyright © 1998 IOP Publishing Ltd

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currents in the two coils will decrease with the same time constant. Otherwise, the shielding effect will be eliminated during a quench. Another important feature, which had a great influence on the routine application of superconducting magnets for MRI, is passive shimming (Hoult and Lee 1985). This method uses small pieces of ferromagnetic material inside the bore for the correction procedure and has become the standard correction method for MRI magnets. The theory is based upon the same basic idea as active shimming (i.e. the description of the inhomogeneities by the coefficients of the spherical harmonics and the cancelling of those coefficients by the field contribution of the ferromagnetic pieces). The shimming procedure includes the application of an appropriate computer algorithm and optimizes the homogeneity in several iteration steps, where the results are very close to the theoretical limit. The ferromagnetic pieces are iron plates which are located on a cylindrical rail system inside the room-temperature bore. A finite number of fixed positions on this cylinder (e.g. ten positions in the axial direction and 24 in the radial direction) and the thickness of the iron plates is varied to optimize the homogeneity. After shimming a prototype of a new magnet it is possible to measure the homogeneity of the bare magnet and to analyse the shimming theoretically in the factory. If the outcome of this homogeneity analysis leads to reasonable theoretical shimming results, the shimming procedure is actually done at the customer site. This method is very economical; one has the advantage that environmental influences on the homogeneity at the customer site are included in the correction procedure. For standard magnets the passive shimming method is usually sufficient (i.e. the superconducting magnet has no superconducting shim or correction coils and no complete resistive shim set). Only a few resistive active shims are necessary to correct small changes caused by the environment or influences from the patient or patient bed. Data from typical MRI magnets are summarized in table G2.2.1. Figure G2.2.20 shows a very compact, actively shielded MRI routine system.

At low temperatures the specific heat of all materials is reduced which means that a very small amount of energy will cause a substantial increase in temperature. Small disturbances of the system may therefore force the magnet from the superconducting to the normal state, a transition for which the expression ‘quench’ is commonly used. Since superconducting magnets for NMR have no connection to the power supply under normal operating conditions, all magnetic field energy is transformed to heat in the magnet structure within the cryostat. This heat load is sufficient to evaporate the liquid helium around the magnet, thus generating a large volume of gas which escapes from the cryostat with considerable pressure and noise. The noise and the cloud of coil gas are the first indications to the user that a quench has occurred. Reasons for such a quench are mechanical instabilities such as wire movements and/or sudden jumps of flux bundles. The quench is, of course, an unwanted phenomenon but since it must always be reckoned with, the design of the magnet must ensure that the system will survive a quench. Special care is taken during the construction of the main coils of the magnet to ensure mechanical stability of the windings, Copyright © 1998 IOP Publishing Ltd

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Figure G2.2.20 MRI magnet. Courtesy of Philips.

since very small movements of wires (due to the Lorentz force) can generate enough heat to initiate a quench. The mechanical stabilization is often done by impregnation of the coil with wax or an epoxy resin. A network of damping resistors ensures that the stored energy of the magnetic field is distributed between the magnet sections and the damping resistors, if the magnet is forced into the normal state. The accumulated experience in the production of successful MRI magnets has resulted in the recognition of many simple but important techniques necessary to obtain a high-quality product which, under normal circumstances, does not quench at the customer site. When energizing a magnet, the flux structure in the wire needs a certain time to find its metastable or stable distribution. Therefore, the magnet has to be energized slowly in order to avoid a quench. The higher the magnetic field the higher the possibility that mechanical relaxation processes may generate a quench. The higher the magnetic field, the smaller the charging voltage to allow the system to find its equilibrium. In order to stabilize the magnet at the nominal field, one normally applies a small overshoot for a brief period before the current is then lowered to the desired value. Despite such a careful energization procedure, the drift will need several hours to reach the stability of approximately 10−8 h−1 which is necessary for MRS. In the early days of superconductivity the magnets quenched at fields much lower than the predicted or calculated values, while repeating the loading procedure demonstrated a progressive improvement in the performance. This effect is known as training. Due to improvements in the wire quality, the winding technique and in the impregnation of the coil, such training effects do not often occur at the customer site but, nevertheless, they are sometimes very frustrating during the development of new magnets. Copyright © 1998 IOP Publishing Ltd

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G2.2.4.3 Superconducting wire Today, thousands of superconducting metals and alloys are known (Rietschel et al 1994). However, only a few of them are useful for superconducting magnets. In this section we restrict ourselves to classical, low-temperature superconductors because high-temperature superconductors are not currently available as a commercial product for MRI magnets. The critical fields of high-temperature superconductors are in general higher than the critical fields of metallic superconductors but due to the high transition temperature and the complex structure of high-temperature superconductors their critical fields and critical currents are very complex. In table G2.2.2 typical data for metallic superconductors of interest are given. Besides Nb-Ti, Nb3 Sn is a very important material which can be used to generate high fields of up to 20 T. However, NbSn belongs to a class of intermetallic compounds which are very brittle and difficult to handle during 3 the manufacturing process of the wire and the winding of the superconducting coil. Table G2.2.2 also presents materials which may be interesting in the future. Nb3(AlGe) together with the Chevrel compound PbMo6S8 demonstrate high critical fields; however, critical currents have not yet been optimized. NbCN superconductors are different in that they are grown on bundles of carbon fibres and thus demonstrate very promising mechanical properties.

Nb-Ti is the most important alloy for fields below 9 T. With only a few exceptions, Nb—Ti is the material for the superconducting wire used in MRI coils. In general, at low fields the limiting factor is not the critical current of these wires but the forces and mechanical stress within the coil. The forces in a superconducting magnet are determined by three major factors: Lorentz forces, forces due to the winding process and the forces due to the thermal contraction during the cooling down of the magnet. In MRI magnets which have rotational symmetry (z axis) the Lorentz forces acting on an isolated loop placed in the background field of the solenoid can be easily calculated. In the mid-plane the corresponding tension in the wire is given by the overall hoop stress

σ = Bz (r ) Jr. The current density J and the stress σ are averaged values over the cross-section of the wire. Typical values for σ are 150 MPa to 180 MPa for MRI wires with a large copper-to-superconductor ratio. For a typical radius r of 0.5 m the hoop-stress limitation provides a relatively small effective current density in the wire. This situation is typical for MRI magnets: the most important factor is the stress load of the magnet whereas the critical current density is normally not a limiting factor. Actually the overall stress analysis, taking account of all major factors and the two-dimensional geometry, is a very complicated problem. The stress in the radial direction normally decreases from a maximum to small values at the outer parts of the coil. At the coil ends, axial forces become very important due to the radial components of the magnetic field. The best analysis of the general problem is obtained if Copyright © 1998 IOP Publishing Ltd

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finite element methods (FEMs) are used (Davies et al 1991). The correct choice of the graduation concept for different cross-sections of wire, a reasonable pre-tension during winding and a careful choice of the material for the coil-former provides the best results for the magnet coil. The superconducting coil is wound of wire with a total length of several thousand metres and could consist of wire pieces with a varying cross-section (the principle of graduated coils; Laukien 1993). The different sub-sections must be joined together so that the joints themselves become a full superconducting connection since even a resistance of the order of 10−9 Ω would cause an unacceptable drift of the magnet. In a very simple coil the ends of the wire must be connected to the main switch. In a 2 T coil up to ten joints may be used depending upon the actual design employed. The joint technique and the test of joints are the critical points in the production of superconducting magnets and therefore are dealt with in a highly confidential manner for a commercial product. Nb—Ti wires are normally cold-welded whereas for multifilament wires this might be a very difficult procedure. Nb3Sn joints and joints between Nb—Ti and Nb3Sn (so-called hybrid joints) are much more difficult to realize at the quality necessary for NMR magnets since in this case the reaction procedure for producing Nb3Sn complicates the situation and the mechanical instability of the reacted wire makes the production difficult. It is a distinct advantage that in MRI magnets normally only Nb—Ti wire can be used because the joints are so much easier to handle. Superconducting wires for MRI magnets are relatively simple when compared with those used in fusion-research magnets. The wire consists mainly of copper with a small amount of superconductor (i.e. the ratio α , defined as the ratio of the cross-section of copper to superconductor, is very high). The number of filaments of superconducting wire used in MRI magnets is in general small compared with the number of filaments in laboratory magnets or magnets used in high-energy physics. The overall weight of the coil is determined mainly by the amount of copper. Figure G2.2.21(a) shows the cross-section of a typical MRI wire consisting of superconducting filaments (1) and the copper matrix (2). An alternative way to avoid complicated wire processing is the use of a standard wire with a small α , inside a separate copper channel (3) which is shown in figure G2.2.21(b).

Figure G2.2.21. Superconducting wire for MRI magnets: (a) standard wire; (b) wire with copper channel.

G2.2.4.4 Cryostat concepts The preferred operational temperature for today’s technical superconductors is 4.2 K and consequently the applied superconductivity is closely related to helium technology. Common to all superconducting NMR magnets is the persistent mode of operation because the NMR technique requires a very stable and time-independent field. The cooling down procedure and the charging of the magnet is normally done only during the installation. After obtaining a stable field and shimming the magnet, the magnetic field should be more or less in the background and available in the same way as the field of a permanent magnet. This operation mode is a big challenge for the cryogenic part of the superconducting magnet. The optimal state of the cryogenic part occurs when it is not obvious to the user of the equipment. Of course this goal has not yet been achieved, but the enormous development of cryostat concepts has led to products which can easily be used in hospitals and by customers who are not cryogenic specialists. The cryostat represents the housing of the superconducting coil and of the container for the cryogens if present. For MRI the bore of the magnet is normally horizontal and the containers and shields are arranged Copyright © 1998 IOP Publishing Ltd

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as concentric cylinders around the horizontal axis. Conventional technology is based upon a bath cryostat (figure G2.2.22) where the main coil (3) and the shielding coil (4) inside the helium can are surrounded by a highly reflective radiation shield (5) and a second shield (6) at a temperature of approximately 80 K. The radiation shield is cooled by the evaporating He gas and the second shield can be a ring container for liquid nitrogen. The alternative cooling concept uses small cryocoolers and will be discussed below.

Figure G2.2.22. A horizontal bath cryostat: 1—vacuum vessel; 2—bore; 3—main coil; 4—shielding coil; 5–20 K shield; 6–80 K shield; 7—turret.

Between the outer shell and the 80 K shield vessel conventional superinsulation is used with up to 100 layers and includes appropriate spacer material. Assuming good assembly, which ensures an absence of free holes thus preventing radiation from the surfaces at 300 K, this part normally is easy to handle. The room-temperature bore of the magnet makes a major contribution to the losses and is the main difference between storage vessels for cryogenic liquids and MRI cryostats. Some 60 to 70% of the losses at 4 K are due to the shield structure of the warm bore. The distances of the different shields in the bore are minimized to values of only 10 mm in order to allow optimal access to the magnetic field. For the shields themselves usually aluminium sheets having a thickness of a few millimetres are used. These sheets are coated with highly reflective aluminium foils as multilayer insulation. Commercial superinsulation is, in general, not optimized for the temperature range between 80 K and 20 K. For the very tight geometry existing between the shields it is very difficult to find an optimum position where there is a minimal contribution from radiation and heat conduction. In addition to the radiation losses, the losses due to heat conduction of the suspension system contribute most the overall cryogenic performance. In vertical NMR magnets the suspension is normally obtained by a two- or three-turret system, where thin, stainless tubes serve simultaneously as a suspension and access to the superconducting coil. This method uses the enthalpy of the evaporating cold He gas very effectively. The cold gas cools the intermediate shield and the suspension tubes (Brechna 1977) via an integrated heat exchanger. The cooling method operates in a feedback mode and reduces the He consumption to a minimum level. However, for horizontal MRI magnets the weight of the superconducting coil is normally so large that additional suspension elements become mandatory. Examples of these elements, which are constructed of glassfibre-reinforced material in the shape of rods or loops, are shown in figure G2.2.23. Figure G2.2.23(a) shows a titanium rod screwed to an eye made of stainless steel; (b) illustrates a wound glassfibre-reinforced loop and (c) a squeezed connection of a glassfibre-reinforced rod. The horizontal cryostat has at least one turret at the top of the outer vacuum vessel. The turret may not exactly be vertical but rather inclined in order to minimize the overall height of the system. The cryogenic liquids are filled through this turret into the corresponding containers. Energizing the magnet requires demountable current leads which must be inserted in the cryostat. After the charging procedure the current leads are removed and within a period of some days the magnet system reaches an extremely Copyright © 1998 IOP Publishing Ltd

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Figure G2.2.23. Suspension elements: (a) titanium rod; (b) glassfibre loop; (c) squeezed connection.

Figure G2.2.24. Pressure increase during a quench.

low evaporation rate. The cryostat, as a container for low-temperature gases, falls under pressure vessel regulations. Large systems require careful inspection and an acceptance certificate from an expert, e.g. British Standard or TÜV. The cryostat is protected by discs that burst under excess pressure which can occur, for example, during a quench. The installation of magnet systems must include an exhaust pipe, of correct dimension, to remove the cold gases from the vicinity of the examination room. The control of the filling height is performed by electrical control units which have an alarm device which is triggered when disturbances occur. Figure G2.2.24 shows the pressure increase as a function of time during a typical quench. During normal operation the He vessel is connected to the exhaust pipe by only a small bypass. If a quench occurs most of the magnetic energy stored will heat up the superconducting coil to a temperature of some 50 to 60 K on a timescale of 2−10 s. The warm coil will evaporate the stored liquid helium within another few seconds which leads to a sudden pressure increase in the vessel. At the very beginning the pressure increase is nearly linear. After 3 s the pressure of the bursting disc is reached and its bursting leads to an intermediate pressure drop. At this time a mixture of liquid and gaseous helium leaves the vessel with considerable noise. If the diameters of the bursting discs and exhaust lines are designed correctly the pressure increase will stay well below the limit of the He vessel and will diminish within a minute. During the quench the temperature of the coil will increase to 50–60 K, the detailed temperature behaviour being dominated by the T 3 dependence of the specific heat of the materials. Copyright © 1998 IOP Publishing Ltd

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During the last decade various improvements have been introduced to the cryostat concept to obtain long refill intervals for cryogenic liquids and low maintenance costs. External or internal refrigerators or liquefiers have been discussed from the very beginning of superconducting MRI magnet development because helium consumption in a bath cryostat is not a very attractive concept for the user. The installation of a normal He-recovering system with a separate liquefier, however, is not very economical because the He consumption of a bath cryostat is already relatively small. An exception would be if several NMR magnets could be supported simultaneously by a central liquefier station. In some cases a simple He-gas recovery system with a well organized transfer of the compressed gas to an external liquefier plant can be helpful. However, in general the most economical way is the external delivery of liquefied gases by a specialized service. Closed-cycle helium liquefiers, with an adapted liquefier power requirement of 0.3 to 0.4 W to cool one magnet, seem to be very attractive. However, these systems are very expensive and reliability represents a serious problem since maintenance intervals of up to 1 year (without any service during this time) are necessary for the concept to be competitive within a hospital environment. An alternative to this approach is a three-stage Gifford-McMahon (GM) refrigerator with a Joule-Thomson expansion as a third stage. However, the restrictions mentioned for a closed-cycle helium liquefier are also valid for this three-stage concept and therefore these methods are used in special cases only. Today relatively simple two-stage GM cryocoolers are accepted worldwide as standard cooling devices. One stage replaces the N2 shield at a temperature of 60–80 K and the other one cools the intermediate shield at a temperature of 20–30 K. By keeping the two shields at these temperatures the He consumption of large MRI magnets can be reduced to a very low level of some 10 ml h−1. The refill interval for liquid He is longer than one year. This type of relatively simple refrigeration system is the key to an economic superconducting MRI magnet, although there is still considerable room for improvement. All active-cooling systems may induce mechanical or electrical disturbances in the magnet and they also have a very inconvenient acoustical noise spectrum which raises problems, especially in a hospital. The reliability of refrigerators could be improved further and a better efficiency would, of course, reduce the overall power consumption. An important step towards an economic and easy-to-use magnet is represented by the concept of the cold-transportable magnet. With improvements in the suspension system and the thermal insulation it now has become possible to ship the magnet cold from the factory to the customer. Only one cooling-down procedure in the factory is necessary for the magnet. Usually no special transport fittings are necessary for the cold shipment. Only appropriate regulations for the transportation of cold gases must be considered if the liquid helium is going to be transported in the helium vessel. The magnet manufacturer guarantees the transportability under certain conditions (e.g. by limiting acceleration in the vertical and horizontal directions). By taking certain restrictions into account, which for example can exclude transportation by train, the cold magnet system can be shipped directly to the customer and installed there within a minimum of time. This procedure saves considerable time and expense. Several additional features have contributed to simplifying the use of superconducting MRI magnets. In addition to the refrigerator and the cold transportable system, a compact design of the magnet should be stressed. The minimal free access within the bore tube and the reduction of all distances within the bore tube, together with extremely short but homogeneous coils, have resulted in a more compact design. Although these systems include an active shield and cold transportability, the weight of superconducting magnets has nevertheless decreased by a factor of two over the last decade without compromises in the homogeneity and central field (figure G2.2.25). Mobile MRI magnets have been used extensively in the USA. The basic idea is that the costly equipment should be shared by several hospitals. These mobile superconducting magnets represent a challenge for magnet and cryogenic engineers because they include cold transportability and charging of the magnet together with stabilization of the magnetic field within a short period of time. Although the

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Figure G2.2.25. A compact actively shielded magnet. Courtesy of Oxford Magnet Technology.

main requirements for the magnet have been fulfilled, the overall concept of the mobile MRI system has lost some of its attraction as stationary systems have become more economical.

G2.2.5 Recent developments and future aspects G2.2.5.1 Medical applications of NMR The fastest imaging technique is echo planar imaging (EPI) as described below. (a) Echo planar imaging (EPI) The basic idea (Mansfield 1977) involves essentially the same sequence as described in the section on ‘MRI myelography’ but refocusing of the spins is not achieved by a 180° RF pulse but instead with a repeated gradient inversion. The significant problem with this method is that the switching of the magnetic field gradients must be very fast (less than 100 µs) and must involve large amplitudes (e.g. for a resolution of approximately 2.5 mm/pixel one needs a gradient strength of about 20 mT m-1 switched in this time frame). Because of these demanding technical requirements this technique has only recently been applied in clinical work and thus still does not constitute part of a routine MRI examination. The (read) gradient, in practice, is normally switched in a sinusoidal manner which is technically easier to realize than a rectangular switching at the prerequisite speed and amplitude. This then necessitates special reconstruction algorithms, not only the standard two-dimensional Fourier transformation. However, with a reduced data matrix (64 × 128 pixel) total scan times of 50 ms are possible. Impressive static and moving EPI images of the heart without ECG triggering, or of gut peristalsis, justify the enormous efforts involved in this technique. Copyright © 1998 IOP Publishing Ltd

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(b) MR spectroscopy Despite many efforts MRS is still in the research stage and thus does not represent a standard method for studying metabolism in a clinical environment. The reason for this is that the interesting chemical compounds are in concentrations (≈ 10 mM) typically 104 times less than those of water protons used for imaging (≈50 M). A prerequisite is therefore a very high magnetic field (B ≥ 1.5 T). The volume selection of a certain pixel in the body is performed with the same localization technique used for imaging. The spectrum itself must be acquired in the absence of any gradient field, otherwise the chemical shift information desired would be destroyed. A principal problem is the inherently poor filling factor. Let us consider the aquisition of a signal from a 1 cm3 voxel in the brain. The receiving coil must cover the entire head. Obviously only a very small part of the coil volume creates the signal but the entire coil volume creates noise. Very interesting results have already been published on this subject but the majority of them are from research centres. (c) Functional imaging It would be very interesting to combine the morphological information acquired from MRI with the study of metabolism. Such studies are possible today only in a hospital environment, using techniques associated with nuclear medicine (e.g. positron-electron tomography, or PET) with the inherent problem of radioactivity introduced into the body. For some years now functional mapping of brain activities using MRI techniques has been a very exciting field of research (Merboldt et al 1995). However, with MR one faces the problems of sensitivity mentioned above. In order to analyse the interesting compounds of metabolism (e.g. in the brain) one has to suppress the very dominant water signal, otherwise the interesting compounds such as NAA or lactate (which exist in concentrations 104 times smaller) cannot be detected. The suppression of the water signal can be achieved by a selective RF pulse which excites only the nuclear spins of the water, followed by a dephasing magnetic field gradient. This requires a very homogeneous magnetic field over the entire volume, otherwise in some areas of the excited volume the spins from the compounds of interest will be dephased. Even more demanding is the detection of lactate with special editing sequences. The lactate signal consists of a doublet (due to spin-spin interaction) possessing a separation of only 7 Hz. In a magnetic field of 2 T, corresponding to an H-resonance frequency of 84 MHz, this requires a relative homogeneity of about 10-7 over the entire volume (i.e. with the head this would involve a sphere with a diameter of 250 mm). This so-called chemical shift imaging, in which one obtains localized spectra over the entire volume, is limited additionally by the acquisition time, since it is very difficult for a patient to lie motionless for approximately 20 min. Therefore these spectra are often acquired from only a small VOI (e.g. 8 ml) which is defined via an MR image acquired previously. G2.2.5.2 Requirements for the magnets The MRI process requires magnetic fields with high homogeneity and very good stability. In addition the necessary gradient fields must be switched very fast without any negative influences from eddy currents and, moreover, the stray field of the magnet should be as small as possible. Other important features are the weight of the magnet and the free access for the patient and the system operators. Of course, the main hardware part of an MRI system should also satisfy the requirements for MR at a reasonable cost and in such a way that the method serves the patient competently. The development of superconducting magnets during the first decade of hospital applications was influenced by the points summarized above and this has led to a product which fulfils at least some of the requirements. The overall question for the magnet manufacturer is the field strength, which more or less determines the cost of the system. Unfortunately there is no optimum field strength for all MRI applications (table G2.2.3). For routine clinical applications medium or even low fields are acceptable whereas research-orientated MR techniques such as spectroscopy and functional imaging usually require

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the highest field possible. The existing technology of superconducting magnets can provide the necessary field strength; however, the overall cost is the crucial point. In figure G2.2.26 the cost for an actively screened magnet is plotted versus the field strength, assuming a normalized price of 1 for a typical magnet at a field strength of 1 T. At a field of up to 0.5 T cost is almost field independent because the housing of the magnet, the coil former and the gradient system determine the basic expenditures. At field strength above 3 T the cost is clearly dominated by the superconducting wire because the amount of wire needed varies quadratically with the field strength, as long as the critical current is not reached.

Figure G2.2.26. Costs versus field strength.

The upper field limit for nonscreened superconducting magnets based on Nb—Ti wire is between 6 and 7 T and for active-screen versions approximately 60% of these values. These magnets are, however, extremely costly because they need a very large amount of superconducting wire and because the number of these magnets to be installed worldwide will be very low. The main reason for higher fields is the signal-to-noise ratio, which increases with increasing field strength. However, there are also disadvantages which make the imaging process more difficult at higher fields. Examples of these disadvantages are the penetration depth and increasing T1 with increasing field strength. In addition, one has to optimize high-frequency equipment such as RF coils, electronics, etc, at the same time, which could become problematic. If, for some reason, nuclei other than protons are used for MR, the arguments provided above are not valid and higher field strength would be necessary, as in normal MRS. Copyright © 1998 IOP Publishing Ltd

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As explained, homogeneity has been drastically improved by the method of passive shimming, but even with standard imaging techniques, the relative homogeneity should increase with increasing field strength since the absolute homogeneity should be constant in order to realize the advantages of the high fields. Moreover, it is precisly the homogeneity which should improve with effective use of new techniques such as EPI and functional imaging. Homogeneity is also influenced by the patient and other actual MRI conditions. Therefore automatic mapping and shimming procedures are very helpful and already in use. An essential part of the superconducting magnet is the cryostat. Its weight has decreased drastically for the same magnet specifications and the cryogenic support required has been minimized, at least for routine systems using a refrigerator as a cooling device. The overall performance of these refrigerators and their reliability, however, could be improved. An interesting approach is the development of pulse-tube refrigerators (Liang et al 1996). By replacing the mechanical pistons with an alternative mechanism one could improve the reliability, reduce the acoustical emission and obtain better efficiency in realizing reasonable cooling power. Classical GM refrigerators have already been used to cool superconducting magnets made of Nb3Sn wire (Laskaris et al 1995) without the use of any cryogenic liquid. Up to now only classical cylindrical superconducting magnets have been discussed. They produce the desired field in an effective way but enclose the patient. This closed cylinder is responsible for claustrophobia which can be a very serious problem for the patient and also does not allow free access to the patient. In figure G2.2.27 two magnets are shown which, at least partially, avoid these disadvantages. A typical design is a transverse coil which consists of two main rings and several sub-coils. This eighth-order coil permits access to the patient in the transverse plane and is one possibility for interventional surgery, where manipulation and imaging are performed at the same time. For certain disc magnets the homogeneity region can even be outside the magnet, to provide very easy access to the VOI. Unfortunately the efficiency for producing a certain field strength in a given VOI decreases, in general, with increasing free access. The transverse magnet, which has already been manufactured (Laskaris et al 1995), does possess relatively good efficiency, whereas the disc magnet provides only a relatively small field strength. With the exception of low-field magnets, all these more or less open magnet structures require superconducting coils to obtain reasonable fields for MRI.

Figure G2.2.27. Magnet designs with good access: (a) transverse magnet; (b) disc magnet.

G2.2.5.3 High-temperature superconductors and MRI Being one of the most important applications of superconductivity, magnets for MR, especially MRI, are regarded as potential candidates for high-temperature superconductors. Of course, to be competitive, new Copyright © 1998 IOP Publishing Ltd

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materials must exhibit properties similar to those of the material already in use. As shown previously, the relatively simple Nb—Ti wire is normally used for MRI magnets because the field strength is not limited by the critical current but by the stress limitation inside a magnet coil. Using Nb3Sn in this case would not be reasonable, as long as the temperature is 4.2 K. New superconducting wires therefore require good mechanical properties which should, at least, be comparable to those of Nb—Ti wires. However, this statement does not necessarily imply that the high-temperature superconductors must be in the form of wires. Persistent mode operation gives superconducting MR magnets an enormous advantage, and it is hard to imagine that NMR magnets which use, at least partly, high-temperature superconductors would be very useful without the persistent mode, since this mode provides the temporal stability necessary for MR. The true persistent mode simultaneously requires use of the joint technique between all parts of the different superconductors. As this joint technique is already a crucial factor for classical superconductors, it could prove even more difficult for new materials (Hase et al 1996). The use of a small power supply to compensate the field decay in a non-persistent coil may be beneficial if the stability of this power supply is excellent and if the thermal losses due to a permanent connection to the coil are acceptable. The field strength of superconducting MRI magnets made of Nb—Ti at 4.2 K is not limited by the critical parameters but by the stresses in and cost of the coil. The use of high-temperature superconductors would therefore only be meaningful if an alternative temperature could be used, which would then present distinct advantages for the whole system. This temperature could be at 78 K or some other reasonable temperature which could be easily established by a refrigerator. The high-temperature superconductor should have no resistance due to thermally activated processes and have suitable stabilization to avoid damage to the coil from a quench. At temperatures well above 4.2 K the quench behaviour, of course, will be very different to quenches at low temperatures. Assuming the same magnetic energy, the averaged temperature increase will be only a few degrees but without reasonable stabilization local burning could nevertheless occur especially if the quench propagation velocity is slow. The cryostat concept for temperatures well above 4.2 K would not be very different to existing technology. A pressure vessel would still be necessary and the number of heat shields probably would be the same unless a working temperature higher that 78 K could be tolerated. It is very likely that optimized refrigerators could be used as a cooling device only for temperatures higher than 20 K. The higher the temperature the better the overall efficiency of the refrigerator system. An integrated refrigerator cryostat for higher temperatures could lead to a compact and easy-to-use superconducting magnet system. As superconducting magnets made of high-temperature superconductors are not yet available, auxiliary equipment such as level indicators or current leads can prove very helpful. RF coils and gradient coils made of high-temperature superconductors have also been discussed. In both cases there are some theoretical advantages; however, the losses are not simple resistive losses but losses connected with the time-dependent RF and gradient fields. In addition one would require cryostats which do not influence the desired time dependence of the fields. A relatively simple application of HT superconducting coils could be their use in electromagnets with iron yokes. In this case the fields at the conductor are low, stability of the field is determined mainly by the temperature dependence of the yoke and a power supply concept has already been accepted. The demands for the new material would be rather moderate with respect to stability, quality of joints, etc, but nevertheless the power consumption would be drastically reduced. Superconducting MRI magnets made of high-temperature superconductors would, of course, replace existing superconducting magnets and therefore initially there would be no new market. Assuming that high-temperature superconducting magnets would be easier to use and/or cheaper to operate, these new products should be quickly accepted by customers and therefore should represent an essential part of an MRI system.

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G2.2.6 Remarks G2.2.6.1 Installation planning For the overall success of MRI as a diagnostic tool, it is necessary that the instrument be easily installed. At first, this aspect would appear to be only marginally dependent on superconductivity; however, it is nevertheless closely related to the success of superconducting magnets and therefore should be explained in more detail. The magnetic field of a dipole is given by the following expression (Thome and Tarrh 1982)

In equation (G2.2.9) md is the dipole moment magnitude, r is the distance from the dipole centre to the point of interest and ir , iq are the unit vectors in spherical coordinates. The stray field influences the environment as shown in table G2.2.4. As already mentioned, a field level of 0.5 mT ( i.e. 500 µ T ) is, in general, accepted as the limit for the stray field. This value must be compared with the earth field of some 20–30 µ T. However, even such a small field of 0.5 mT is not acceptable for the proper use of colour monitors. For an unscreened 2 T magnet the axial distance from the centre to the 0.5 mT contour is typically 12 m and the radial distance is approximately 8 m; the corresponding dipole moment is md = 6 × 106 A m2. Typical stray fields of nonscreened and screened whole-body magnets are shown in figure G2.2.28 ( B0 = 2 T ) and figure G2.2.29 ( B0 = 1.2 T ). In both figures the contours of equal field strengths are plotted at equidistant intervals of 0.5 mT; the inner contours refer to a value of 5.0 mT and the outer contours refer to a value of 0.5 mT.

The installation of an MRI magnet, especially a superconducting one, therefore represents a serious problem if no additional precautions are taken. The simplest solution is to install the magnet in a separate location far away from other facilities. The necessary separation may prove inconvenient, the disturbance on the neighbouring electronics still exists and, of course, the free space required is not always available. The most elegant solution for such magnets is active shielding of the main magnet as explained above. This technique is restricted to fields smaller than, or equal to, 2 T for both physical and economic reasons. For higher fields and other special cases passive shielding with iron is an adequate solution. The shield can be arranged as a body shield around the cryostat or as a room shield situated at a distance of several metres from the magnet. In both cases the amount of iron necessary to obtain effective shielding depends on the dipole moment of the magnet and the desired level of screening. Typical weights for a 2 T magnet are 20–30 t for a body shield and 30–40 t for a room shield. The smaller the distance between the shield and the magnet, the more important the symmetry of the arrangement because large effective forces may result if the iron cage is not symmetrical about the magnet centre. An effective body shield requires, in general, a special design of the superconducting magnet since the iron generates an unacceptable field contribution in the centre. A room shield is less effective than a body shield. However, Copyright © 1998 IOP Publishing Ltd

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Figure G2.2.28. Stray fields of two MRI magnets ( B0 = 2.0 T ): (a) a nonscreened magnet; (b) a room shield.

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Figure G2.2.29. Stray fields of two MRI magnets ( B0 = 1.2 T ): (a) a body shield; (b) an actively shielded magnet.

it does not influence the homogeneity of the magnet so a standard magnet can be used in this instance. The concept of a room shield is suitable to all field strengths. G2.2.6.2 Marketing aspects The medical application of MRI became very important shortly after the introduction of this new diagnostic imaging modality. Today several thousand units have been installed worldwide, the majority of which use superconducting magnets. Table G2.2.5 shows the number of instruments sold between the years 1991 and 1996 in Europe and worldwide sales (Ward 1994); the figures for 1994, 1995 and 1996 represent marketing projections. The figures include all units, from low-field to high-field systems as well as research-oriented systems with very high fields of 3 T and more. In 1992 the greatest number of systems was installed. The decrease experienced since then is dependent on several factors. There is certainly a component due to the market saturation that occurred after the first optimistic phase. Cost-effectiveness in the health services is also another factor which should not be forgotten. In spite of the significant changes experienced since 1992, MRI is the most important market for applied superconductivity alongside the MRS market and that for magnets used in nuclear and particle Copyright © 1998 IOP Publishing Ltd

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physics. The market is more or less continuous and certainly does have a large potential for increase, if one takes into account the situation around the globe. In most of Europe the ratio is one system for every 300 000 people, whereas in the USA, Japan and in Switzerland the above-mentioned figure is around one for every 100000. Low-field systems, however, are becoming more and more attractive and here it is possible to use nonsuperconducting magnets. To supply the non-expert with some figures of the market one might take a unit price of $1 million as a figure for a complete 1 T MRI system. About one third of this price is for the bare superconducting magnet. The value of the superconducting wire which is used inside the superconducting magnet depends strongly on the magnetic field strength, but for a simple approximation, one third of the magnet price is a reasonable figure. Assuming 1000 systems a year, the superconducting magnet portion can be estimated to be some $300 millions while the value of the superconducting wire may be some $100 millions. Worldwide, some ten companies are currently selling MRI systems, among which are smaller companies specialized in the research-orientated market or in producing dedicated systems. There are approximately five companies for superconducting wire and for complete magnet systems in the world. Future development will be determined by the need for routine instruments and innovations which might have an enormous impact on superconducting magnets. New techniques such as EPI and fast imaging require high-field magnets. Low-field and medium-field magnets may be very important for new applications with a large market potential such as that for diagnostic scanning. When the number of systems increases, the price per unit will consequently decrease but special-design low-cost systems may also have their market niche. G2.2.6.3 Magnet alternatives Without question, superconducting magnets are the dominating systems used for MRI today and this will also be the case in the future. The potential of high-temperature superconductors reinforces this statement ( i.e. with the high-current application of the new superconductors, the chances for superconductors are naturally improved ). As pointed out above, the basic requirements for MRI can also be achieved through use of conventional magnet technology ( e.g. resistive electromagnets and permanent magnets ). Both types of magnet were used at the very beginning of MR development. However, during the last few years new products based on these conventional magnets have been introduced. These alternative systems are basically restricted to the low- and mid-field range but are still interesting products with new features. The classical procedure for the generation of homogeneous fields involves air-coil systems (figure G2.2.30(a)) made of copper or aluminium conductors (Müller 1990). In order to achieve free access sufficient for a human body and reasonable homogeneity, power requirements of up to 100 kW are required to produce a magnetic field of about 0.2 T. The power consumption varies with the square of the field strength, for fixed geometry, and together with the technical cooling problem restricts the fields attainable Copyright © 1998 IOP Publishing Ltd

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to moderate values. The advantage of this type of magnet lies in its exceptional adjustment capability and its almost eddy-current-free behaviour, which makes complicated actively shielded gradient units obsolete. The stray field of resistive magnets can be drastically reduced by an iron shield (figure G2.2.30(b)) which fits closely to the coils and causes an increase in the field strength of some 20%. The concept of the iron shield can also be transferred to superconducting magnets (body shield). However, the higher field strength of such magnets results in extremely heavy units and the field increase due to the shield is not significant. Resistive magnets are well suited for low-field and low-cost systems, if the medical information obtained from these systems is sufficient for diagnostic purposes.

Figure G2.2.30. Basic magnet designs: (a) an air coil; (b) a shielded coil; (c) an electromagnet; (d) a permanent magnet.

The standard design for higher magnetic fields is to have magnets with an iron yoke (figures G2.2.30(c) and (d)). The iron yoke (3) guides the magnetic flux in the outer region of the magnet and consequently the magnetic field in the gap is increased and the stray field is more or less screened. In technical systems only iron and iron alloys can be used as yoke material with the saturation magnetization M 1.6 × 106 A m-1 (magnetic induction B 2.0 T). This saturation determines the overall design of the magnet, especially the weight. The magnetic field can be produced by coils (1) or by permanent magnets (4). For a typical whole-body imaging magnet some 20 kW of power consumption is expected for a magnetic field of 0.2 T; a reasonable maximum field is some 0.5 T. The standard design of the yoke is an H-shape or a C-shape as shown in figure G2.2.31. The C-yoke allows a more or less open magnet design, which is a very attractive alternative to the closed structures of air coils and, in particular, conventional superconducting magnets. With these systems the claustrophobic effect can be partially avoided and, moreover, the systems permit interventional applications. Permanent magnets have been used for a long time in compact routine MR spectrometers employed for the examination of small samples possessing a volume of several cubic centimetres. From an overall point of view superconducting magnets which run in a persistent mode are one type of macroscopic permanent magnet. The undoubted advantage of permanent magnets is the fact that no main power supply is necessary for the generation of the static magnetic field. To achieve the necessary field strength and homogeneity, large quantities of permanent magnetic material are required (Zijlstra 1985). These result in large weights and a high cost for these magnets. Even with modern permanent material like NdBFe the magnetic induction is restricted to values less than 0.5 T. An intrinsic problem is the poor temperature stability of the magnetic field and the technical problem of realizing the first magnetization of the entire magnet assembly. In addition, the field of a permanent magnet cannot be switched off like that of other systems, which may prove to be a problem during transportation and in the case of an emergency. Copyright © 1998 IOP Publishing Ltd

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Figure G2.2.31. A C-magnet. Courtesy of Siemens.

Yoke magnets, as well as permanent magnets, have pole faces which are shaped in such a way that the required homogeneity is obtained. For a given field strength an air gap of 0.5–0.6 m determines the amount of permanent material required or the power consumption needed in the case of an electromagnet. For MRI the eddy-current behaviour of these magnets and the stability with respect to time-dependent gradient fields is important because the gradient coils are located between the patient and the pole pieces. Without actively shielded gradients certain imaging techniques can be carried out only with great effort and at great expense. Therefore actively shielded planar gradient systems are necessary to avoid eddy-current effects and unacceptable irreversible effects (from the gradient fields) due to changes in the magnetization.

G2.2.7 Summary Today a wide range of superconducting magnets for MRI is available. They provide magnetic fields of up to several tesla, at low operating costs and with very high stability when compared with other methods of generating high fields. The small premium paid for this exceptional performance is that the user must accept the simple techniques of cryogenics. In some cases this means that one has to handle liquefied gases such as helium and nitrogen. For routine magnets even this handling of liquefied gases is not necessary during a service interval of one year. The recent developments of new MRI techniques such as spectroscopy and functional imaging or EPI have demanded essential improvements in the magnet systems. Very important examples of such improvements are the active shielding of the main magnetic field and the field of the gradient coils. These improvements and other new features have been produced with existing technology. The field strength necessary for sophisticated MRI applications is limited mainly by the cost and not by the existing technology of superconducting magnets. Routine applications and interventional surgery require additional new ideas for magnets. Although the field strength itself could be moderate in these cases, superconducting technology is certainly necessary in order to find a reasonable solution for customers who are essentially radiologists. Indirectly, of course, the patient is also involved with his/her request for information and therapy. Copyright © 1998 IOP Publishing Ltd

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It is obvious that the application of high-temperature superconductors may have an enormous influence on the technology of MRI magnets themselves, as well as on the marketing acceptance of MRI. New superconducting material must be competitive with respect to conventional superconductors. As MRI is a very useful diagnostic tool as well as a very important commercial technique, the application of high-temperature superconductors in the field of MRI requires the utmost effort so that MRI will be at the forefront of the new technologies. In the future it is possible that the different fields of MR, which are historically distinct areas in physics, chemistry and biology, will eventually overlap in the field of MRI as each discipline discovers applications using the others’ techniques. What then will be the demands on the hard-pressed magnet manufacturers? Acknowledgments We are indebted to our colleagues at BRUKER with whom we had the pleasure to cooperate for so long. Many thanks especially to Dr David Philips for reading the manuscript. We appreciate the support and contributions of other industrial partners, who helped with material and information in a very easy-going manner. Thanks are also given to H Bachmann for the help in producing drawings for this contribution. This work has been partially supported by the Arbeitsgruppe Innovativer Projekte (AGIP) and by the science foundation of the Fachhochschule Hildesheim/Holzminden.

References Abramowitz M and Stegun I 1965 Handbook of Mathematical Functions (New York: Dover) Binns K, Lawrenson P and Trowbride C 1992 The Analytical and Numerical Solution of Electrical and Magnetical Fields (Chichester: Wiley) Brechna H A 1973 Superconducting Magnet Systems (Berlin: Springer) Davies F J, Elliot R T and Hawksworth D G 1991 A 2 tesla actively-shielded magnet for whole body imaging and spectroscopy IEEE Trans. Magn. MAG-27 1677–80 Garrett M W 1967 Thick cylindrical coils systems for strong magnetic fields with field or gradient homogeneities of the 6th to 20th order J. Appl. Phys. 38 2563–86 Hase T, Shibtani K, Hayashi S, Shimada M and Ogawa R 1996 Fabrication of superconductively jointed silver-sheathed Bi-2212 tape Cryogenics 36 21–5 Hennig J, Nauerth A and Friedburg H 1986 RARE imaging: a fast imaging method for clinical MR Magn. Reson. Methods 3 823 Hoult D I and Lee D 1985 Shimming a superconducting nuclear-magnetic-resonance imaging magnet with steel Rev. Sci. Instrum. 56 131–5 Jehenson P, Westphal M and Schuff N 1990 Analytical method for the compensation of eddy-current effects induced by pulsed magnetic field gradients in NMR systems J. Magn. Reson. 90 264–78 Laskaris E T, Ackermann R A, Dom B, Gross D, Herd K G and Minas C 1995 Cryogen-free open superconducting magnet for interventional MRI applications IEEE Trans. Magn. MAG-5 163 Laukien D D and Tschopp W H 1993 Superconducting NMR magnet design Concepts Magn. Reson. 6 255–73 Liang J, Ravex A and Rolland P 1996 Study on pulse tube refrigeration Cryogenics 36 87–111 Mansfield P 1977 Multiplanar image formation using NMR spin echoes J. Phys. C: Solid State Phys. 10 55 Mansfield P and Chapman B 1986 Active magnetic screening of coils for static and time-dependent magnetic field generation in NMR imaging J. Phys. E: Sci. Instrum. 19 540–5 Merboldt K-D, Krueger G, Haenicke W, Kleinschmidt A and Frahm J 1995 Functional MRI of human brain activation Magn. Reson. Med. 34 639–44 Müller W H G 1990 Magnetsysteme für Kernspintomographie Radial. Diagn. 31 387–400 Rietschel H, Hoenig H E, Bogner G and Jérome D 1994 Superconductors Ullmann’s Encyclopedia Indust. Chem. 25 705–44 Roméo F and Hoult D I 1984a Magnet field profiling: analysis of a correcting coil design Magn. Reson. Med. 1 44–65

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Sauzade M D and Kan S K 1973 High resolution nuclear magnetic resonance spectroscopy in high magnetic fields Advances in Electronics and Electron Physics vol 34, ed L Marton (New York: Academic) pp 1–92 Thome R and Tarrh J 1982 MHD and Fusion Magnets (New York: Wiley) Turner R 1988 Minimum inductance coils J. Phys. E: Sci. Instrum. 21 948–52 Turner R 1993 Gradient coil design Magn. Reson. Imaging 11 903–20 Vlaardingerbroek M 1996 Magnetic Resonance Imaging (Berlin: Springer) Ward P 1994 Diagnostic Imaging Global, Market Trends in the Diagnostic Imaging Market (San Francisco: Miller Freeman) Zijlstra H 1985 Permanent magnet systems for NMR tomography Philips J. Res. 40 259–88

Further reading Komarek P 1995 Hochstromanwendungen der Supraleitung (Stuttgart: Teubner) Montgomery D B 1980 Solenoid Magnet Design (New York: Hunting) Morneburg H 1995 Bildgebendesysteme für die medizinische Diagnostik (Erlangen: Publicis MCD) Rinck P 1993 Magnetic Resonance in Medicine (Oxford: Blackwell) Slichter C P 1990 Principles of Magnetic Resonance (Berlin: Springer) Stark D D and Bradley W G 1992 Magnetic Resonance Imaging (St Louis: Mosby Year Book) Wilson W 1983 Superconducting Magnets (Oxford: Clarendon)

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G2.3 SQUID sensors for medical applications O Dössel, B David, M Fuchs and H-A Wischmann

G2.3.1 Biomagnetic imaging ‘Biomagnetic imaging’ or ‘magnetic source imaging’ is a new modality for functional diagnosis. The basic idea is that every activity of the brain is connected with neuronal ionic currents and every beat of the heart is generated by ionic depolarization currents. These currents create magnetic fields that can be measured noninvasively outside the body. By sampling these fields at many locations simultaneously, a map of the field is determined from which the location of the sources inside the body can be calculated. The electric potentials at the surface of the body (EEG—electroencephalography and ECG— electrocardiography) are well known quantities for medical diagnosis. Usually only the time course of the signals is interpreted. Since the body consists of different tissues with various electrical conductivities it is very difficult to determine the location of the source from electrical measurements. It can be easily shown, however, that very simple volume conductor models are adequate to interpret the magnetic field data for localization of the sources (MEG—magnetoencephalography and MCG—magnetocardiography). An extensive survey on MEG can be found in the work of Hämäläinen (1993). The magnetic fields to be measured are extremely small: several tens of pT for the heart and some 100 fT for the brain. Only a SQUID-based superconducting measuring system can reach the noise level required for valuable biomagnetic measurements. For that reason, biomagnetic imaging is one of the most important application of superconducting devices today. G2.3.2 SQUID sensors The preparation and optimization of SQUID sensors is described elsewhere in this handbook. For medical applications very low-noise SQUIDs with efficient coupling to external magnetic flux are a prerequisite. In particular, very low 1/f noise is essential, since biomagnetic signals are found mainly in the frequency range between 1 Hz and 100 Hz. For commercial multichannel SQUID magnetometers thin-film Nb/Al2O3/Nb d.c. SQUIDs are used with a noise level below 10 µ φ0 Hz-1/2. Meanwhile, high-Tc SQUIDs based on YBaCuO have now reached a noise level at which good biomagnetic measurements are possible. G2.3.3 Magnetometers and gradiometers As explained in the chapter on SQUIDs (chapter I2), for low-noise performance they must have a low inductance, so they have a small loop and thus a relatively small direct sensitivity to external flux. In order to couple external flux into the SQUID either a ‘multiloop configuration’ is used or a ‘flux-transformer scheme’ (figure G2.3.1). Copyright © 1998 IOP Publishing Ltd

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Figure G2.3.1. Magnetometers with a ‘multiloop configuration’ (upper left) and a ‘flux-transformer scheme’ (upper right); a vertical first-order gradiometer (middle left) and a second-order gradiometer (middle right); a planar gradiometer (bottom).

In the multiloop configuration the SQUID itself consists of many loops that are connected in parallel. Pickup areas of 8 mm diameter with a SQUID inductance of 400 pH have been realized (Drung et al 1991). The flux-transformer scheme uses a pickup loop as an antenna and a coupling coil on top of the SQUID that together form a superconducting closed circuit (Carelli et al 1982). When a magnetic field is coupled to the pickup loop, flux quantization requires that the total flux in the transformer remains constant. As a result a persistent supercurrent is generated, producing a flux via the input coil in the SQUID. The flux transfer coefficient can be defined as

where ΦS is the flux coupled to the SQUID, Φp the flux at the location of the pickup coil, LS , i , P , c a b the inductances of the SQUID, input coil, pickup coil and cable respectively and NP the number of turns in the pickup coil. A simple analysis shows that the optimum sensitivity is obtained when the inductances of the pickup loop Lp and the input coil Li are equal. Also it becomes clear that it is advantageous to have a pickup coil with a large area but a small inductance. The best flux transfer coefficients typically reached are only a few per cent. Nevertheless the sensitivity is increased by orders of magnitude due to the large area of the pickup coil (e.g. 300 mm2 ) compared with the SQUID hole (e.g. 0.01 mm2 ). Copyright © 1998 IOP Publishing Ltd

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Using a gradiometer configuration for the pickup loop, the effective coupling of external signals can be combined with an external noise rejection. Coils wound into opposing directions carry out a subtraction of fields at neighbouring locations. Homogeneous fields originating from distant sources are suppressed. Fields with large gradients from nearby sources are only slightly suppressed (Carelli et al 1982). First-order and second-order gradiometers, measuring δ Bz /δ z or δ 2Bz /δ z 2 , made of superconducting wire are commonly used. They have the advantage that the coil in the distant plane measures only a small amount of biomagnetic signal. However, the precise three-dimensional (3D) construction is not easy. Any sloppiness during the fabrication produces a large imbalance, leading to bad suppression factors of homogeneous fields. The best values achieved with wirewound gradiometers are in the order of 10−3. Planar gradiometers measuring δ Bz/δx and δ Bz /δ y can be fabricated with thin-film technology, so the required precision can be achieved with excellent reproducability (Ahonen et al 1991), but since both loops contain large amounts of biomagnetic signals the required signal-to-noise ratio for the difference signal is not easy to reach. Gradiometers measuring field components that are not roughly normal to the surface of the body ( Bx or By ) are often proposed ( Tesche 1992 ). It can easily be shown that these signals originate mainly from the volume currents and not from the impressed currents ( see section G2.3.6 ). They do not contain additional information compared to the electrical measurements ( EEG/ECG ), so it does not make sense to use a complicated superconducting system to measure them. G2.3.4 SQUID electronics Small signals in the microvolt range with a source impedance of some ohms have to be measured without introducing additional noise. Commonly, an impedance-matching transformer is used, a signal modulation together with a lock-in amplifier and a controller circuit that locks the SQUID in the steepest point of its characteristics (Clarke 1976) (figure G2.3.2). Bias modulation schemes are used in order to suppress 1/f noise coming from fluctuations of the critical Josephson currents in the SQUID (Dössel et al 1991, Foglietti et al 1987, Koch et al 1983). Drung (1992) proposed a very simple new electronics scheme, in which the output signal of the SQUID is directly coupled to a differential amplifier. Using an ‘additional positive feedback’ the U/Φ characteristics can be steepened so that the white-noise contribution of the electronics is competitive with the conventional scheme, but an additional suppression of 1/f noise is not possible. Digital readout schemes (Gotoh et al 1993) may be an advanced alternative for future systems. They will probably not be cheaper or better than the conventional schemes, but they offer the chance of reducing the number of leads into the liquid helium significantly by using multiplexing techniques. Especially for systems with more than 100 channels the leads are not only difficult to fabricate but they are also the largest heat channel into the liquid helium and determine the overall helium evaporation rate. G2.3.5 Multichannel systems G2.3.5.1 Gradiometer arrangements The first important aspect for the general systems design is the geometrical arrangement of the sensors and the outline of the dewar (figure G2.3.3). In planar arrangements all gradiometers are parallel and in one plane. These systems are optimized for MCG measurements. Using a dewar with curved bottom plates, it is possible to come closer to the head which is better for MEG measurements. In planar and curved dewars, typically an area of 160–200 mm diameter is covered with gradiometers. For many events in the brain and heart the region of interest is well known beforehand and the complete field distribution can be sampled with these systems.

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Figure G2.3.2. Typical SQUID electronics.

If larger areas of the field distribution must be measured it is possible to move the system to any region of interest and combine the measured fields for the current reconstruction, but this is only possible for events for which the time course and amplitude can both be reproduced. The first step to time coherent measurements of larger areas is the combination of two dewars. These two dewars can be positioned, to a large extent independently, close to the body for both brain and heart measurements. Thus, signals from the left and right hemisphere or from the front and side of the torso can be recorded simultaneously. The second step for measuring even larger areas is the construction of dedicated brain or heart systems. Large-area brain systems are made of helmet-type dewars that fit many heads with an acceptable margin. Helmets will always be too small for some heads and too large for others, but they make it possible to sample the complete field distribution from many patients in a single shot. An important prerequisite for helmet systems is, of course, that the density of channels is large enough that the spatial sampling theorem is not violated (gradiometer distance about 25 mm). Dedicated heart systems today are planar with a diameter of, for example, 300 mm, but arrangements on a cylinder segment that also covers the left side of the torso are also possible. G2.3.5.2 Shielded environment versus unshielded environment Nearly all multichannel systems installed today are operated in a magnetically shielded environment. Several layers of highly permeable material are combined to achieve a shielding factor of 1000 to 10000, Copyright © 1998 IOP Publishing Ltd

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Figure G2.3.3. Various dewar arrangements for multichannel systems.

depending on the frequency. The optimum geometry for a shielding chamber is of course a sphere, but the most practical solution is a cube. These chambers have a typical weight of 6 t and a typical price of US$500000. The cubic form creates disturbances of external homogeneous fields and leads to undesired gradients. So cubic chambers only make sense with a minimum shielding factor of the order mentioned above—there is no cheap solution in between. There is an ongoing discussion as to whether it is possible to carry out biomagnetic measurements of high quality in an unshielded environment. Active shielding systems have been proposed (Matsumoto et al 1992, ter Brake et al 1993) as well as systems that carry out a software correction using several additional magnetometers and gradiometers that measure the environmental noise (Vrba et al 1993). G2.3.5.3 International system developments At the Technical University of Twente, The Netherlands, a 19-channel system was developed (ter Brake et al 1992). It contains custom-made SQUIDs, wirewound gradiometers and a curved gradiometer arrangement. In Moscow, Russia, a multichannel system was set up that consists of several small single-channel units that can be combined in a very flexible way to form various arrangements (Matlashov et al 1992). The system developed at the PTB Berlin, Germany, is made with integrated magnetometers (Drung et al 1991) and uses simplified SQUID electronics based on the ‘additional positive feedback’ (APF) technique (Drung 1992). The magnetometers are combined to form 37 gradiometer signals. It is a planar Copyright © 1998 IOP Publishing Ltd

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arrangement installed in a heavily shielded environment. A new development contains 55 measuring channels realized with 89 SQUID magnetometers. The biomagnetism group in Rome, Italy, has developed an 11-channel planar (Casciardi et al 1993) and a 28-channel curved (Foglietti et al 1993) system. The development of a 61-channel heart and a 159-channel brain system has now started, funded by the European Community. The Siemens system in Erlangen, Germany (Hoenig et al 1991), consists of 37 channels with Nb/Al2O3/Nb SQUIDs. The gradiometers are in a planar arrangement and made of a foil with a structured superconducting film that is wrapped around a holder. The Philips system in Hamburg, Germany (Dössel et al 1993a), follows the classical approach with NbN/MgO/NbN or Nb/Al2O3/Nb SQUIDs and wirewound gradiometers. It is modular in such a way that every channel can be replaced without warming up the dewar. Two dewars with 31 channels each can be combined to form one system. The TU Helsinki†, Finland, has developed a helmet system that is offered under the label of Neuromag. It consists of 122 planar thin-film gradiometers measuring δ Bz /δ x and δ Bz /δ y at 61 locations. The US company BTL‡ , San Diego, USA, installed many 37-channel systems with curved-bottom, wirewound gradiometers and Nb/Al2O3/Nb SQUIDs. They also developed an upgraded system using a second dewar with 37 channels. The Canadian company CTF, Vancouver, Canada, has developed a 64-channel helmet system ( Vrba et al 1993 ) with wirewound gradiometers. Using additional magnetometers and gradiometers and a software correction scheme measurements in unshielded environments are possible. At the SSL§ in Inzai Chiba, Japan, a planar 64-channel and a 128-channel system were developed. A 256-channel prototype heart system and a helmet-type system are also being developed. G2.3.6 Reconstruction algorithms G2.3.6.1 The inverse problem The general problem in determining the electric current distribution inside a body from the magnetic field and/or electric potential data measured outside this body is called the ‘inverse problem’. Unfortunately, this problem does not have a unique solution (Sarvas 1987). Many different current distributions may create a similar field pattern, but not all of these current distributions are physiologically meaningful. Using simple model assumptions, the problem can be solved and important information of diagnostic value can be extracted. In the following, the most important models will briefly be described. All have in common the assumption that every current distribution is a sum of many elementary physiologial sources that can be modelled as ‘current dipoles’. Current dipoles are small and localized current paths (e.g. passing through an ensemble of cell membranes) together with broad volume currents flowing back through the surrounding tissue closing the circuit. Both the small current path (‘impressed current’) and the volume currents contribute to the magnetic field. A detailed description of the volume currents depends on the geometries and conductivities in the surrounding tissue. Often simplified assumptions are made based on these conductivities, e.g. a homogeneous spherical or half-space model (Cuffin and Cohen 1977). If the exact distribution of conductivities is known for the individual patient, e.g. from 3D magnetic resonance images, this can be considered in any of the presented models, but the computational effort increases drastically (Forsman et al 1992, Hämäläinen and Sarvas 1987).

† Neuromag Ltd, c/o Low Temperature Laboratory, Helsinki University of Technology, 02150 Espoo, Finland. ‡ Biomagnetic Technologies, San Diego, CA 92121-3719, USA. § Superconducting Sensor Laboratory, 2-1200 Muzaigakuendia, Inzai Chiba, 270-13 Japan.

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Figure G2.3.4. Contour plot and reconstructed dipole for a somatosensory evoked field (medianus nerve stimulation, magnetic response after ~20 ms (M20)). Top: measured time course; bottom: contour plot and best-fit dipole.

G2.3.6.2 Dipole localizations The most simple model assumption is that at a given instant of time only one single localized source is active. The location and orientation can be found using a simple least-squares fit procedure (‘moving dipole’, figure G2.3.4). If it can also be assumed that during a specific time range the dipole stays at a fixed location and varies in orientation and/or strength, only an extended least-squares fit with an Copyright © 1998 IOP Publishing Ltd

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improved signal-to-noise ratio need then be performed (‘rotating dipole’ and ‘fixed dipole’). Finally, if there is evidence that two or three separate dipoles were active simultaneously, their locations can be fitted with a multidimensional least-squares fit and the individual time courses can be extracted using a ‘singular-value decomposition’ (SVD). The localizing power of a multichannel SQUID system can be tested using phantom dipoles that are placed at well-known positions below the instrument. The best overall accuracy achieved so far is about 2 mm (Dössel et al 1993b). G2.3.6.3 Dipole probability distribution Different scanning techniques have been developed in which a predefined region of interest is subdivided into small voxels. For every voxel the goodness of fit to a best current dipole inside this voxel is calculated. With this technique, the number of active areas and an estimate of the extension of an active region can be determined. Performing an SVD of the spatio-temporal data matrix and eliminating the noise subspace, temporally independent source distributions can be separated (‘MUSIC’ (Mosher et al 1992)). G2.3.6.4 Current density imaging In this case first the ‘Biot-Savart matrix’ or the ‘lead field matrix’ A is set up. It combines the vector components J of the current dipoles to be reconstructed in every voxel of a predefined region of interest with every sensor signal B in the SQUID array B = AJ where B is the array of measured flux densities, A the Biot-Savart matrix and J the array of x, y and z components of the current density in every voxel of a predefined region. The basic task is to find the inverse matrix A−1. Since in general we have to deal with the underdetermined case, where the number of unknown parameters exceeds the number of measured values by far, the inverse matrix is constructed using linear estimation theory J ’ = AT (AAT + α 2 1 ) −1 B

∆2 = || B − AJ ’ || 2 + α 2 || J ’ ||2

α 2 = || dB || 2/ || J ’ || 2 where α is the regularization parameter, ∆2 the cost function to be minimized and dB the noise of measured flux density. This so-called ‘Moore-Penrose pseudoinverse’ delivers a current distribution that fits well to the measured data and minimizes the total current ( ‘minimum-norm estimate’ ). To avoid noise artefacts a regularization parameter α 2 has to be introduced which is correlated to the noise of the measured data dB ( Illmoniemi et al 1984, Wischmann et al 1992 ). G2.3.6.5 Overlay with morphology Current dipoles and current images as such are only of limited value for medical diagnosis if they are not overlaid with images of the morphology, e.g. from x-ray imaging or magnetic resonance imaging ( MRI ). Coil systems consisting of three orthogonal coils are often used as markers in SQUID imaging. A small current is fed through all individual coils sequentially and the magnetic field pattern is measured with the multichannel system. Since the model of the source is accurately known, the location of every coil system can be reconstructed from the measured data very accurately (≈1 mm ( Fuchs et al 1994 )). Copyright © 1998 IOP Publishing Ltd

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Figure G2.3.5. An overlay of a reconstructed single current dipole on three orthogonal magnetic resonance images (median nerve stimulation, M20). The centre of the plotted dipole arrow is the reconstructed location.

Since one has to deal with a 3D problem, usually a set of 3D magnetic resonance images is taken of every patient. If the location of at least three coil systems is marked, e.g. with oil capsules during the MRI, the same landmarks can be found in the morphological images and the transformation matrix can be set up. Finally the dipoles or current densities can be depicted in the morphological images ( figure G2.3.5 ). Copyright © 1998 IOP Publishing Ltd

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Figure G2.3.6. A segmented cortex. Left: a dipole probability scan; right: a cortical current density image (median nerve stimulation, M20).

G2.3.6.6 Morphological constraints In many cases it is known that the sources must be on the surface of the cortex or the myocardium. In these cases the ‘region of interest’ inside the body is a two-dimensional manifold which reduces the ambiguity of the problem significantly. If the individual cortex or the individual myocardium is segmented from 3D magnetic resonance images and an accurate coordinate system matching between the SQUID multichannel system and the MRI system is carried out, this morphological constraint can be used for dipole localizations, probability scans and current density imaging (Fuchs et al 1994). The resulting images are much sharper and more brilliant than in large-volume reconstructions (figure G2.3.6). G2.3.7 Biomagnetic measurements Many clinical research groups worldwide have started biomagnetic investigations (Hoke et al 1992). The most important applications discussed today are: ( i ) Brain: epilepsy mapping of functional areas prior to neurosurgical treatment Alzheimer’s disease Parkinson’s disease schizophrenia mania, depression, phobia transmission disorders of auditory, visually and somatosensory stimuli ischaemia and stenosis tinnitus functional brain damage after head injury and stroke pain ( ii ) Heart: Wolff-Parkinson-White syndrome

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ventricular tachycardia myocardial infarction classification and risk analysis graft rejection after heart transplantation evaluation of foetal heart beat ( iii )Miscellaneous: peripheral nerve disorders liver susceptometry tracers in gastroenterology. Many of these applications are in an infant stage where clinical evaluation has just begun. Extended investigations have been carried out, e.g. for the diagnosis of focal epilepsy (Hellstrand 1992, Stefan et al 1992), with the result that biomagnetic imaging is an important tool for noninvasive localization of epileptic foci. Also, presurgical neuromagnetic brain mapping has already proven its diagnostic value for optimized intervention planning (Gallen et al 1993). Ischemia and stenosis, especially after transient ischaemic attacks (TSAs), have been investigated intensively (Vieth 1990). The applications in cardiology have concentrated very much on heart arrhythmias (Moshage et al 1991). It has been shown that accessory paths or reentering excitations can be localized noninvasively as a preparation for catheter ablation to interrupt the arrhythmogenic bundle. References Ahonen A I, Hämäläinen M S, Kajola M J, Knuutila J E T, Lounasmaa O V, Simola J T, Tesche C D and Vilkman V A 1991 Mulitchannel SQUID systems for brain research IEEE Trans. Magn. MAG-27 2786–92 Carelli P, Modena I and Romani G L 1982 Detection coils Biomagnetism (NATO ASI Series A) vol 66, ed S W Williamson et al (New York: Plenum) pp 85–99 Casciardi S, Del Gratta C, Di Luzio S, Romani G L, Foglietti V, Pasquarelli A, Pizella V and Torrioli G 1993 11 channel magnetometer for biomagnetic measurements in unshielded environments IEEE Trans. Appl. Supercond. AS-3 1894–7 Clarke J 1976 Superconducting quantum interference devices for low frequency measurements Superconductor Applications: SQUIDs and Machines ed B B Schwartz and S Foner (New York: Plenum) pp 67–124 Cuffin B N and Cohen D 1977 Magnetic fields of a dipole in a special volume conductor shape IEEE Trans. Biomed. Eng. BME-24 372–81 Dössel O, David B, Fuchs M, Krüger J, Lüdeke K-M and Wischmann H-A 1993a A 31-channel SQUID system for biomagnetic imaging Appl. Supercond. 1 1813–25 Dössel O, David B, Fuchs M, Krüger J, Lüdeke K-M and Wischmann H-A 1993b Localization of current dipoles with multichannel SQUID systems Rev. Sci. Instrum. 64 3053–60 Dössel O, David B, Fuchs M, Kullmann W H and Lüdeke K-M 1991 A modular low noise 7-channel SQUID-magnetometer IEEE Trans. Magn. MAG-27 2797 Drung D 1992 Investigation of a double-loop DC-magnetometer with additional positive feedback Superconducting Devices and Their Applications ed H Koch and H Lübbig (Berlin: Springer) pp 351–6 Drung D, Cantor R, Peters M, Ryhänen T and Koch H 1991 Integrated dc SQUID magnetometer with high dV/dB IEEE Trans. Magn. MAG-27 3001–4 Foglietti V, Gallagher W J and Koch R H 1987 A novel modulation technique for 1/f noise reduction in dc SQUIDs IEEE Trans. Magn. MAG-23 1150–3 Foglietti V, Pasquarelli A, Pizzella V, Torrioli G, Romani G L, Casciardi S, Gallagher W J, Kelchen M B, Kleinsasser A W and Sandstrom R L 1993 Operation of a hybrid 28-channel neuromagnetometer IEEE Trans. Appl. Supercond. AS-3 1890–3 Forsman K, Nenonen J, Purcell C and Stroink G 1992 Biomagnetic inverse solutions with a realistic torso model Biomagnetism: Clinical Aspects ed M Hoke et al (Amsterdam: Excerpta Medica) pp 819–23 Fuchs M, Wagner M, Wischmann H-A, Ottenberg K and Dössel O 1994 Oscillatory Event Related Brain Dynamics (NATO Advanced Study Institute) (New York: Plenum)

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Gallen C, Sobel D F, Waltz T, Aung M, Copeland B, Schwartz B J, Hirschkoff E C and Bloom F E 1993 Noninvasive presurgical neuromagnetic mapping of somatosensory cortex Neurosurgery 33 260–8 Gotoh K, Fujimaki N, Imamura T and Hasuo S 1993 8-channel-array of single-chip SQUIDs: connection to Josephson multiplexer IEEE Trans. Appl. Supercond. AS-3 2601–40 Hämäläinen M and Sarvas J 1987 Feasibility of the homogeneous head model in the interpretation of neuromagnetic fields Phys. Med. Biol. 32 91–7 Hämäläinen M, Hari R, Ilmoniemi R J, Knuutila J and Lounasmaa O V 1993 Magnetoencephalography— theory, instrumentation, and applications to noninvasive studies of the working human brain Rev. Mod. Phys. 65 413–92 Hellstrand E 1992 Magnetoencephalography and epilepsy Electromedica 60 67–73 Hoenig H E, Daalmans G M, Bär L, Bommel F, Paulus A, Uhl D, Weisse H J, Schneider S, Seifert H, Reichenberger H and Abraham-Fuchs K 1991 Multichannel DC SQUID sensor array for biomagnetic applications IEEE Trans. Magn. MAG-27 2777–85 Hoke M, Erne S N, Okada Y C and Romani G L (ed) 1992 Biomagnetism: Clinical Aspects (Amsterdam: Excerpta Medica) Ilmoniemi R J, Hämäläinen M S and Knuutila J 1984 The forward and inverse problems in the spherical model Biomagnetism: Applications and Theory ed H Weinberg, G Stroink and T Katila (New York: Pergamon) pp 278–82 Matlashov A, Bakharev A, Zhuralev Y and Slobodchikov V 1992 Biomagnetic multi-channel system consisting of several self-contained autonomous small-size units Superconducting Devices and Their Applications ed H Koch and H Lübbig (Berlin: Springer) pp 511–6 Matsumoto K, Yamagishi Y, Wakusawa A, Noda T, Fujioka K and Kuraoka Y 1992 SQUID-based active shield for biomagnetic measurements Biomagnetism: Clinical Aspects ed M Hoke et al (Amsterdam: Excerpta Medica) pp 857–61 Moshage W, Achenbach S, Weikl A, Göbl K, Abraham-Fuchs K, Schneider S and Bachmann K 1991 Progress in biomagetic imaging of heart arrhythmias Frontiers in European Radiology vol 8, ed Baert and Heuck (Berlin: Springer) pp 1–19 Mosher J C, Lewis P S and Leahy R M 1992 Multiple dipole modeling and localization from spatio-temporal MEG data IEEE Trans. Biomed. Eng. BME-39 541–57 Sarvas J 1987 Basic concepts of the biomagnetic inverse problem Phys. Med. Biol. 32 11 Stefan H, Schneider S, Feistel H, Pawlik G, Schüler P, Abraham-Fuchs K, Schlegel T, Neubauer U and Huk W J 1992 Ietal and interictal activity in partial epilepsy recorded with multichannel magnetoencephalography Epilepsia 33 874–87 Tesche C D 1992 Exploiting lead field analysis to obtain current source reconstructions and a figure of merit Biomagnetism: Clinical Aspects ed M Hoke et al (Amsterdam: Excerpta Medica) pp 735–9 ter Brake H J M, Flokstra J, Houwman E P, Veldhuis D, Stammis R, van Ancum G K, Martinez A and Rogalla H 1992 A 19-channel DC SQUID based neuromagnetometer Biomagnetism: Clinical Aspects ed M Hoke et al (Amsterdam: Excerpta Medica) pp 847–51 ter Brake H J M, Huonker R and Rogalla H 1993 New results in active noise compensation for magnetically shielded rooms Meas. Sci. Technol. 4 1370–5 Vieth J 1990 Magnetoencephalography in the study of stroke Advances in Neurology vol 54, ed S Sato (New York: Raven) pp 261–9 Vrba J et al 1993 Whole cortex 64 channel SQUID biomagnetometer system IEEE Trans. Appl. Supercond. AS-3 1878–82 Wischmann H A, Fuchs M and Dössel O 1992 Effect of the signal to noise ratio on the quality of linear estimation reconstructions of distributed current sources Brain Topogr. 5 189–94

Copyright © 1998 IOP Publishing Ltd

G3 Superconducting magnets for thermonuclear fusion

J-L Duchateau

G3.0.1 Introduction Although it was originally applied to magnets for fundamental research and high-energy physics, superconductivity has now invaded the sphere of fusion magnets. In fact, confining very hot deuterium-tritium plasmas, thanks to high fields, now appears to be the most promising way to control thermonuclear fusion and produce a new kind of energy. Because of the huge size of these magnets, superconductivity stands out as the obligatory path for sparing the power inputs of future plants. Superconducting magnets are considered for the main confinement field of these machines for both torsotrons/heliotrons and Tokamaks. Three existing Tokamaks ( T15, Tore Supra and Triam) operate with superconducting magnets at the toroidal field system. In addition, superconducting magnets are also now considered for poloidal field systems, control magnets and the ohmic heating coils of Tokamaks. Today no project of magnetic confinement fusion of significant size exists without involving superconducting magnets. The most outstanding project is of course the ITER project (International Thermonuclear Experimental Reactor), in which Europe, the United States, Japan and the Russian Federation work together to perform ignition on a Tokamak. The engineering design phase of this project has started and will take six years. The design of these magnets is particularly difficult to achieve. Simultaneously a very high current (50 kA), a very high voltage (5 kV) and a high field (13 T ) are required, pushing engineers to develop new kinds of conductor. In fact 1500 t of Nb3Sn wires, cabled in a so-called cable-in-conduit cooled by a forced flow of helium, will be necessary in projects such as ITER. This is really a challenge and will, if successful, definitively ensure the industrialization of Nb3Sn. Until now, in spite of its high critical fields and temperatures, this superconductor has not been as widely accepted as Nb—Ti in important projects. G3.0.2 Superconductivity for fusion G3.0.2.1 Why fusion? As a consequence of economic growth and population explosion, world energy demand continues to rise. Copyright © 1998 IOP Publishing Ltd

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Fusion, in contrast to fission, cannot at present be considered as an energy source. It might become commercially available around the middle of the 21st century. In the future it appears that the capability of supplying large energy demands will be limited to three major sources: coal, fission and fusion. Coal reserves seem sufficient for 200–400 years depending on consumption, but considerations of economic independence and also of CO2 and SO2 emissions may limit its massive use in Europe. As regards fission, commercially exploitable uranium reserves are estimated to be sufficient for about 60 years at the current rate of consumption. This capability could be multiplied by about 50 by the use of fast breeder reactors (Colombo 1990). In competition with fission, as a possible major source of energy, fusion offers environmental advantages. Neither long-life radioactive wastes, nor actinides, are produced by fusion. Operating with a very limited quantity of tritium in the hot region (a few grams), the fusion reactor is basically unable to generate catastrophic uncontrolled chain reactions. The actual mineral reserves of lithium may be equated with about 1500 years of available energy for humanity, but the possibility of a deuterium—deuterium reaction being exploited in the distant future or the extraction of lithium from seawater makes fusion appear to be a virtually inexhaustible source of energy. However, fusion has still to demonstrate this virtual capacity. No doubt the technical results of fusion in the next 20 years will help to place this source of energy among the other, including the renewable, sources. G3.0.2.2 The way to fusion The binding energy per nucleon is a function of the nuclear mass. Nuclear energy can be produced in two ways: by breaking heavy nuclei ( fission ) or sticking light nuclei together ( fusion ) ( figure G3.0.1). The fusion deuterium-tritium ( DT ) reaction is the easiest to achieve D + T → He 4 (3.56 MeV ) + n( 14.03 MeV ).

Figure G3.0.1. Binding energy per nucleon as a function of the atomic weight. Copyright © 1998 IOP Publishing Ltd

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The main objective is to reach the ignition regime in which a thermonuclear industrial plant can operate. In this regime, the energy of the α particles ( He4 ) is sufficient to compensate the losses; for example it keeps the plasma at its operating temperature without an external power source. The neutron energy is thus available to produce the net energy of the plant. Consideration of the energy balance leads to the well known Lawson criterion (Adam et al 1987). If the temperature T is in the range 10–20 keV and 108 − 2 × 108 K

n τ > 10 2 0 (m−3 s keV )

where n is the plasma density (particles per cubic metre), τ the confinement time (seconds) and T the plasma temperature ( keV ). Confining very hot low-density plasmas by high magnetic fields is one of the ways to control thermonuclear fusion. The temperature is increased by the Joule effect associated with a current induced inside the plasma and by additional means such as neutron beams and high-frequency electromagnetic waves. Fundamental considerations of plasma stability lead to increases in the size of the machines, the magnetic field and the plasma current. Deuterium can be produced in significant amounts from water but tritium does not exist in nature and has to be produced in reactors from the lithium of the blankets surrounding the hot region by capturing the neutrons of the DT reaction. G3.0.2.3 Magnetic configuration As has been seen, thermonuclear plasmas require very high (10–20 keV) temperatures for large plasma volumes. Only magnetic fields may be used to balance the pressure associated with these temperatures. The basic idea of magnetic confinement is that, to a first approximation, a plasma particle travels along a field line. Classically it can be considered in fusion machines that these field lines are established on magnetic surfaces. If these surfaces cross the in-vessel wall, particles must be prevented from hitting this wall by means of magnetic mirrors. Historically, investigations on magnetic confinement have produced, with more or less success, many different systems. It has to be pointed out that magnetic confinement fusion research is an outstanding example of international cooperation. In practice, worldwide the main emphasis has been put on toroidal machines with major funding given to Tokamaks. (a) Tokamaks This machine was discovered in the mid-1950s at the Kurchatov Institute in the former USSR. In the 1960s, this kind of machine had provided excellent results in comparison with alternative magnetic configurations. Nowadays thermonuclear reactions are nearly reached in large Tokamaks such as JT60 in Japan, TFTR in the USA and especially JET in Europe—JET as the world’s largest Tokamak has provided a significant contribution to progress in Tokamak physics. In toroidal machines, the basic confinement line fields, curved and closed along a torus, are produced by coils external to the plasma (toroidal field coil). In addition to that, to prevent the vertical thermal drift of the particles, a rotational transform of the field is necessary. Field lines are helices. In Tokamaks this rotational transform is produced by a huge current circulating in the plasma itself (figure G3.0.2). This current mainly arises thanks to a central solenoid acting like the primary of a transformer, the plasma being the secondary. The central solenoid is only one part of the poloidal field coil, the other part is made of large diameter so-called ring coils which play a role in the plasma equilibrium. Copyright © 1998 IOP Publishing Ltd

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Figure G3.0.2. Presentation of a Tokamak.

(b) Stellarators In this second kind of toroidal machine, the rotational transform is produced not with plasma current but only by helicoidal external windings. The state of the art in this case is less advanced than for Tokamaks. The main interest in these machines lies in their intrinsic capability of steady operation in comparison with the pulsed mode of Tokamaks and the absence of disruptions classically associated with Tokamak plasma currents. A sketch of Wendelstein VII-X, an advanced stellarator design of the IPP-Euratom Association approved by Euratom, is presented in figure G3.0.3. The magnet system is superconducting using a

Figure G3.0.3. Sketch of the basic configuration of Wendelstein VII-X. The lines on the plasma surface indicate the magnetic field lines and meridional sections respectively. Courtesy of Max-Planck-Institut für Plasmaphysik. Copyright © 1998 IOP Publishing Ltd

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forced flow Nb-Ti cable-in-conduit with a maximum field on the conductor of 6.2 T (Sapper et al 1993). The jacket material is made of a special aluminium alloy which will be hardened after winding at a moderate temperature. It will be built at Greifwald in Germany. A similar concept is the torsatron concept illustrated by the construction at Nagoya in Japan of the Large Helical Device (LHD). The on-site winding of the two superconducting helical coils composing the main magnetic system started at the beginning of 1995 and will take 18 months (Imagawa et al 1995). In phase I of the project the conductor will be cooled by a helium bath at a temperature of 4.4 K. The maximum field in the coil is 6.9 T and the total stored energy is 920 MJ. (c) Mirror machines A word has to be said about mirror machines which were one of the important approaches to magnetic confinement before 1980 but seem to have been left aside. The simplest magnetic mirror is produced by two parallel solenoids with current circulating in the same direction. This basic configuration has to be complicated for plasma stability, to keep the field minimum on the plasma. One of the possible configurations is the famous baseball configuration (see figure G3.0.4(a)) allowing a field increase in every direction with only one coil having a tennis-ball-seam shape. One of the most important superconducting systems in the world has been assembled in the USA to build the mirror machine MFTF-B. However, this machine has never been put into operation because of cuts in the fusion programme. For this machine the Ying-Yang configuration (figure G3.0.4(b)) has been preferred (Wilson 1983).

Figure G3.0.4. (a) A baseball coil; (b) Ying-Yang coils. Reproduced by permission of Oxford University Press.

G3.0.3 Superconducting Tokamaks It has taken a long time for plasma physicists to accept superconducting coils in their Tokamaks. While admitting the long-term necessity of superconductivity for fusion the tendency was to delay this technological mutation as far as possible. Difficulties in achieving high current densities in the presence of cryostats, instability and losses of superconductors under fast changing fields, uncertain reliability and industrial availability of high-field superconductors were severe drawbacks which explain this cautious attitude. At the beginning of the 1980s, the increasing size of the Tokamaks, and their associated longer plasma burn time pointed out, in the first stage, the competitivity of superconducting toroidal field systems in comparison with the complicated previous machines with copper coils and flywheel generators. Thereby and logically two large devices have been built in the last few years: Tore Supra in Cadarache and T15 at the Kurchatov Institute in Russia. Copyright © 1998 IOP Publishing Ltd

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Besides these two machines a smaller machine Triam has been built at the Kyushu University in Japan where it has been operating successfully since 1986.

G3.0.3.1 Opening the way: the Large Coil Task experiment In the framework of an important international task to prepare for the introduction of superconducting magnets into fusion experiments, six superconducting coils forming a torus were assembled and tested in 1986 at Oak Ridge National Laboratory (Beard et al 1988). Even if, since then, the fusion requirements for superconducting coils have been drastically increased, imposing larger sizes, higher current and higher fields making Nb3Sn quite compulsory, the Large Coil Task (LCT) stands out as a major milestone for fusion technology. Each of the six coils had a 2.5 m × 3.5 m D-shaped bore, a current between 10 and 18 kA, and was designed for stable operation at 8 T. This level of field was reached on all the coils corresponding to a maximum total stored energy reached of 600 MJ (see table G3.0.1). As an ultimate test, the six coils reached the maximum field of 9 T. A total stored energy of 944 MJ was achieved, but the test was terminated at this level by the dump of the torus caused by the resistive heating of the Westinghouse (WH) coil and subsequent quench. The six conductors, which can be considered to a certain extent as the predecessors of the modern conductors for fusion, are presented in figure G3.0.5.

Extensive experience has been accumulated on cooling down, hydraulics, quench detection, stability and mechanics.

G3.0.3.2 Triam The construction of Triam-1M (Nakamura et al 1989) started in 1982 and this Tokamak was put into operation in 1986 at the Kyushu University in Japan. One of its major features is to operate with a peak field of 11 T on a superconductor, which is not so far from the ITER requirements. The whole machine is situated in a vacuum tank (figure G3.0.6) of 4 m diameter and 3.6 m height. The poloidal coils except the ohmic heating coil are situated inside the toroidal field (TF) coils. The toroidal field system is made of 16 D-shaped coils. Pool boiling conditions associated with monolithic Nb3Sn surrounded by copper and aluminium sections (figure G3.0.7) ensure steady-state cryostability. The main toroidal field system features are presented in table G3.0.2. Copyright © 1998 IOP Publishing Ltd

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Figure G3.0.5. The six conductors of the LCT experiment. Reproduced by permission of Elsevier Science BV. Copyright © 1998 IOP Publishing Ltd

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Figure G3.0.6. The Tokamak Triam-IM. Courtesy of Kyushu University.

Figure G3.0.7. Cross-sectional view of the Triam-IM conductor. Reproduced by permission of Elsevier Science Publishers Ltd.

G3.0.3.3 T15 The Tokamak T15 was completed in October 1988 at the Kurchatov Institute in Moscow and the first shots with plasma were produced in December 1988. T15 is the largest Tokamak using Nb3Sn (react and wind process) as a superconductor for the toroidal field system. The toroidal field system is made of 48 circular coils located in pairs in stainless steel cases and shared through 12 assembly modules (Alkhimovich 1991). The main parameters of T15 are presented in table G3.0.3. Copyright © 1998 IOP Publishing Ltd

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(a)Conductor The cable is made of 11 twisted superconducting bronze wires. These wires are jacketed by a copper structure in which two helium channels are provided (figure G3.0.8). Normal operation at 4.5 K has not yet been possible due to cryogenic problems with the helium

Figure G3.0.8. T15 conductor: 1-copper; 2-Nb3Sn in a bronze matrix; 3-cooling channels. Reproduced by permission of Elsevier Science Publisher. Copyright © 1998 IOP Publishing Ltd

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liquefier. Moreover, it seems that a resistive voltage distributed along the turns of some coils exists in the toroidal field system. The origin of this voltage, already observed in the Nb3Sn LCT coil of Westinghouse, might be related to the strain degradation during winding after the reaction. Resistive losses of around 50 W are produced in this process in each coil. In spite of this, the nominal current 3.9 kA has been reached in transients and quite normal operation is possible at 3 kA at a temperature of 9 K (Alkhimovich 1991). G3.0.3.4 Tore Supra This Tokamak (figure G3.0.9) is practically the same size as T15. Tore Supra’s first operation took place in 1988 at the Cadarache Nuclear Centre (France). At that time a conservative approach was taken with the conductor, based on the very industrial Nb-Ti coupled with quite new and less established one atmosphere Hell cooling (Turck 1995). (a) Presentation of the Tokamak The main characteristics of Tore Supra are presented in table G3.0.4.

Figure G3.0.9. The Tokamak: Tore Supra. Copyright © 1998 IOP Publishing Ltd

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In principle devoted to long-pulse operation in ‘next step’ relevant plasma conditions thanks to its superconducting coils, Tore Supra takes a specific part in the Euratom fusion programme. 30-200 s cycles currently allows the study of significant long times of confinement, density profiles, plasma heating of different kinds, impurity control and plasma instabilities. Moreover, Tore Supra brings precious information and significant experience on the operation of superconducting coils in the fusion environment. (b)The conductor The conductor and its characteristics are presented in figure G3.0.10 and table G3.0.5. The manufacture of this conductor (40 t) has been equally shared between two companies: Alsthom Atlantique and Vacuumschmelze.

Figure G3.0.10. Tore Supra conductor (AISA version).

The basic choice has been to take better part of Nb-Ti by operating the coils in a superfluid helium bath at 1.8 K and atmospheric pressure (Claudet bath). The other choice could have been Nb3Sn in a helium bath at 4.2 K and atmospheric pressure but the industrial development of this superconductor, though far better as concerns the critical properties, was judged not sufficiently advanced at the time of the design (1979) and of the go-ahead (1981). In fact at 1.8 K, the critical current density of Nb-Ti is in the same range as the critical current density of Nb3Sn (4.2 K). In addition, superfluid helium offers outstanding properties for heat conduction which Copyright © 1998 IOP Publishing Ltd

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allows the enthalpy between 1.8 K and Tλ ( 3 × 105 J m−3 ) to be fully available to stabilize the conductor. For the nominal field and current values (9 T and 1400 A) the conductor satisfies stability requirements of two kinds. ( i ) In the case of a sudden temperature increase caused by an accidental perturbation the conductor can recover after a heat deposition increasing the temperature of a whole pancake up to 15 K. ( ii ) In the case of a plasma current disruption the conductor is subjected to variable fields around 0.3-0.7 T (parallel and transverse) with an associated time constant limited by the steel casing to 10-20 ms The conductor does not at any point reach a temperature higher than 3 K. (c) The toroidal field system The main parameters of the Tore Supra toroidal field system are presented in table G3.0.6. The 18 coils composing the toroidal field magnet have been manufactured by an Italian company: Ansaldo. They are designed to form a complete and rigid torus after assembly, only the weight is transmitted to room temperature through six legs. Each cold winding is encased in a strong stainless steel casing which provides most of the mechanical strength.

Figure G3.0.11 shows one module of 1/6 th of the magnet. Besides the strength, the thick casing acts as a separate vacuum envelope around each coil, as a barrier against a possible helium leak and as a magnetic shield against fast field variations (20 ms time constant). A cutaway drawing of the coil in figure G3.0.12 shows the 4.5 K cooling channels in the casing, along with the other elements of the coil. To ensure the thermal insulation of the 1.8 K vessel from the 4.5 K case, a new insulating material has been developed, using alumina powder as a filler in a polyimide matrix. Copyright © 1998 IOP Publishing Ltd

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Figure G3.0.11 One module of the Tore Supra toroidal field system.

(d) Normal operation and protection The superconducting toroidal field system stands out as one of the major utilities of Tore Supra. It is operated fully automatically daily thanks to a control computer. The toroidal field is routinely increased within half an hour in the morning for the whole day. A great deal of attention has been paid to the design of the safety system in order to ensure reliable operation (Duchateau et al 1991). This aim has been achieved mainly by doubling critical components and ensuring quench detection through five independent detectors. In addition the daily automatic check of the system before starting it every morning makes major electronic failures visible when the system is not in operation. In the case of quench a fast discharge of the energy is triggered by the safety detection system. During this discharge the voltage across any coil does not exceed 500 V. The time constant of the current in any coil except the quenched coil is 120 s. As concerns the quenched coil, a particular circuit has to be considered because of the increase of the resistance of this coil. In a kind of autoprotection mode the time constant of this coil is thus around 20 s. Copyright © 1998 IOP Publishing Ltd

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Figure G3.0.12. Sketch of a superconducting coil of the Tore Supra toroidal field system.

(e) Status after eight years of operating experience The first cooling down of the machine took place in April 1988. In the first phase it was decided to operate the whole machine with a reduced toroidal field (2 T) in order to get enough information on the behaviour of all systems and on synergistic effects before increasing the field to the nominal value. In November 1989 the current was increased step by step up to 1455 A corresponding to 4.5 T at the plasma axis and to 9.3 T on the conductor with no accommodation. The superconducting magnet of Tore Supra is one the largest superconducting magnets in operation in the world. By its long and successful operation it has demonstrated that superconductivity is a good solution, and even perhaps the only solution, for magnetic fusion. The main exploitation problems are related to the electrical pertubations brought to the quench detection system during heating by means such as neutral beams or the radiofrequency ion cyclotronic wave. These perturbations can trigger a dump of the system which means a typical shut down of 2 h. As an example, five dumps of the system occurred in 1995, none of them being related to a quench of the magnet. The total number of cycles for the current is presented in table G3.0.7.

11471 poloidal shots have been performed since 1988 with a plasma current established for more than 1 s. The achievement in 1996 of several shots with a duration longer than 1 min must be emphasized. Copyright © 1998 IOP Publishing Ltd

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Two of them have even lasted 2 min with a plasma current of 0.8 MA. Thanks to its superconducting toroidal field system, it is envisaged that in the near future Tore Supra will be able to perform very long 1000 s shots comparable to ITER shots. The conceptual design associated with this next operation phase is ready and the associated necessary transformations are under discussion. G3.0.4 Future fusion machines: the increasing importance of superconductivity G3.0.4.1 Superconductivity, an obligatory path for future Tokamaks The anticipated increase in size of future Tokamaks, in projects beyond JET, TFTR and JT60, is such hat the use of copper coils will no longer be possible. This is evident from table G3.0.8.

The advantages of the superconductivity approach have become more apparent with time. Certainly the difficulties with the industrial production of Nb3Sn, for instance, are present in the minds of everybody but what will be gained in return then is: ( i ) a considerable efficiency increase thanks to the decrease in the energy consumption; ( ii ) a natural and easy steady-state regime; ( iii )an increase in current density which means size and cost reductions. A word has to be said about possible use of high-Tc superconductors in the future for fusion applications. All the constraints of superconductors for fusion have to be fulfilled and in particular the following: high Jc at high field, addition of a normal metal for stability and protection, maintaining quality under stresses in tension and in compression and under irradiation. None of the high-Tc superconductors seems to fulfil these conditions at present. Besides, it is to be noted that operating at high temperature, 80 K for instance, will not result in such an important energy gain (0.5 MW is in balance for instance for Tore Supra). Simplification of the internal cryogenics, suppression of shields and decrease of the refrigerator cost will certainly be associated with the use of such materials and that has to be taken into consideration. Nevertheless high-Tc superconductors might see their first application in fusion systems in more restricted places such as the numerous current leads of the toroidal field system. A recent acceleration in the use Copyright © 1998 IOP Publishing Ltd

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of a certain kind of high-Tc material can be pointed out. Multifilaments of bismuth oxides have been developed by several companies (Sheahan 1994) by the so-called powder-in-tube method. No application of these materials at 77 K can be envisaged, but their behaviour in the range 20-30 K at very high fields makes them very attractive potentially for fusion applications. G3.0.4.2 ITER 11 September 1992 stands out as a major milestone along the path of international co-operation on fusion. Delegations from the four ITER parties, the European Community, Japan, the Russian Federation and the USA, decided to initiate the ITER Engineering Design Activities ( EDA ), a phase which was to have a duration of six years. The information developed during this ITER EDA phase will provide the basis for future decisions on the construction of ITER, a reactor aimed at demonstrating the scientific and technological feasibility of fusion energy for peaceful purposes. The budget for this phase is one billion dollars. Three EDA co-centres of equivalent importance have been chosen. The centre of Naka in Japan is of major interest for superconductivity because all the activities concerning superconducting coils are managed from there. The European Euratom Associations have already taken an important part in the ITER design, taking advantage of the work developed in the framework of the Next European Torus ( NET ) project ( NET 1993 ). All these European laboratories continue to participate actively in the EDA phase. One of their major contributions during this EDA phase, as concerns the superconducting magnets, is the design and testing (in relation with the industry) of the toroidal field model coil as a representation of the toroidal field system of ITER. Another model coil, the central solenoid ( CS ) model coil, is in the charge of Japan and the USA. This project is a real challenge. For the first time superconducting conductors will have to carry very large 40-60 kA currents at high fields ( 12-13 T ) in both the toroidal field system and the central solenoid. For the first time superconducting magnets will have to sustain a 5 kV voltage to ground routinely. This current and this voltage are of the same order of magnitude as those of the stator of the large electrical generators associated with nuclear plants. ( a ) N b 3S n o r N b - T i In recent years, the general trend towards increasing fields up to 13 T has made compulsory the choice of Nb3Sn as the basic superconductor material both for poloidal field coils and for toroidal coils (table G3.0.9).

As can be seen from figure G3.0.13 (Turck 1991), the critical current densities available around 13 T with Nb3Sn (typical modified jelly roll (MJR) strand) are well above those available with Nb-Ti. Moreover the critical temperature and fields presented in table G3.0.9 emphasize the advantage of Nb3Sn as concerns stability. Nb-Ti is quite competitive up to fields of about 11 T, taking into account that this material has advantages as regards losses (very low effective diameters), residual resistivity ratio (RRR) and of course Copyright © 1998 IOP Publishing Ltd

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Figure G3.0.13. Nb-Ti and Nb3Sn critical current densities at 4.2 K and 1.8 K.

Figure G3.0.14. Sensitivity to strain for a typical Nb3Sn strand.

costs. But beyond 11 T the choice is Nb3Sn or no fusion project at all. One of the difficulties related to the use of Nb3Sn is the necessity of using huge furnaces to react coils for typically three weeks, at temperatures of around 650°C to form the Nb3Sn phase. The question of insulation under these conditions has not been completely solved. Another difficulty is that Nb3Sn, unlike Nb-Ti is very sensitive to strain. This sensitivity to strain is pointed out in figure G3.0.14 for a typical MJR strand. This strain appears during the cooling phase after the reaction due to the differential thermal contraction of the conductor components (Cu, bronze, Nb3Sn, stainless steel). This strain is typically -0.2 to -0.3% for a basic strand but can be up to -0.7% for a strand embedded in a conductor because of the large section of stainless steel tightly bonded to the strands. About 1500 t of Nb3Sn will be present in the magnetic system which is certainly a very important new step for superconductivity, but a large place still exists for Nb-Ti (about 500 t) in the machine for Copyright © 1998 IOP Publishing Ltd

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the external poloidal field coils due to the very low price of the material (a factor of five cheaper than Nb3Sn) and of the magnet manufacture. Moreover, the forced-flow cable-in-conduit is the only type of conductor possible for ITER and experience with such conductors is, so far, not so important. No doubt, in the case of success, the impact of ITER will be very great on other topics such as energy storage and, more generally, the production of large fields in large volume. A sketch of the machine is presented in figure G3.0.15. All the data given hereafter have to be considered as very preliminary (ITER 1995).

Figure G3.0.15. A 3D view of the ITER magnet systems. Courtesy of ITER.

(b) The toroidal field system The ITER toroidal field system is made of 20 coils forming a complete torus. Considering one turn of a magnet, a very particular problem of this system is to find the convenient shape leading to no bending stresses on the turn. As the toroidal field and thus the perpendicular force vary all along the turn, the radius on each point has to be adjusted to find the so-called bending-free D-shape. General data concerning the toroidal field magnet are presented in table G3.0.10. The conductor has a circular cross-section and is pancake wound embedded in stainlesssteel-insulated reinforcing radial-grooved plates. Each of the 20 coils is taken inside a stainless steel case which in addition to the plates resists the hoop stress. The out-of-plane forces created by the poloidal field system are resisted by an outer intercoil structure placed between the coils and the outside of the machine and by a crown torsion cylinder. In contrast to the case of Tore Supra, the toroidal field system is mechanically linked to the central solenoid which supports the toroidal field centring forces. As concerns the coil manufacture, Copyright © 1998 IOP Publishing Ltd

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the wind and react process is used. Each layer of conductor can be reacted independently. The electrical dry insulation is added during the assembly phase after the reaction. (c) The central solenoid The central solenoid is made of one simple block. It is layer wound which makes it possible to grade and install the joints between layers outside the coil at the top and bottom. This allows some increase in the overall current density and some gain in the available flux swing. This also makes it possible to save on the use of superconductor. General data concerning the central solenoid magnet are presented in table G3.0.11. The conductor is a circular Nb3Sn cable in an externally square-shaped jacket. The heat load of the inner layer is particularly high as this layer sees all along its length the maximum field variation. The situation is diffrent in a pancake organization where, due to the decrease in field along the pancake, the heat load associated with the field variation is not so high. This layer will be wound four or six conductors in hand to reduce the cooling length and the pressure drop. The central solenoid operates in pulse mode. Its capacity to accept several consecutive shots has to be carefully studied in relation to the heat deposition as a function of time and the residence time of helium

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inside the considered layer which depends on the mass flow rate. For the coil fabrication, the huge size of this magnet means careful preparation is required. The wind and react process is used, in separate layers or in groups of layers. The best solution is to insulate the winding after reaction. (d) Conductor (i) General outlines. Very high current (about 40 to 60 kA), associated with high voltages to ground (about 5 kV) and high fields (12 to 13 T), is required for fusion applications. Only forced-flow cables-inconduit seem able to sustain with reliability this high level of specification through high wetted perimeters, local mechanical reinforcement and safe insulation. It is to be noted that such a concept does not have a very long history in comparison with the classical bath-cooled conductor, but in the framework of the fusion programme significant experience has now been accumulated both in conductor design and tests and in coil fabrication and tests. As concerns the fabrication of the conductor, one possible way of making it is to draw a circular bundle of conductor through a stainless steel jacket, hundreds of metres in length. The conductor is manufactured in two steps: a cabling process followed by a jacketing process. One of the most difficult problems to solve is the additional compressive strain induced on the superconductor during heat treatment by the differential thermal contraction between strands and jacket (Steeves et al 1984). This compressive strain might be kept at a limited value (>-0.5%) in the case of stainless steel if the void fraction is sufficiently high (around 40%), i.e. with a mechanical bond of bad quality between the jacket and the strands which is not compatible with the required mechanical quality of the cable. Other ways to keep this compressive strain at a low value are under investigation. One way of solving the problem is to use as the reference material for the jacket Incoloy 908 whose thermal contraction matches the thermal contraction of Nb3Sn better than stainless steel. It is to be noted, however, that no real industrial experience exists for this material. Two prototypes of conductors manufactured by ABB and LMI have been successfully tested in the Fenix test facility of Lawrence Livermore Laboratory (USA) (Bruzzone et al 1993) (figure G3.0.16).

Figure G3.0.16. A 40 kA conductor developed in the framework of the NET programme: (a) ABB conductor; (b) LMI conductor. Reproduced by permission of Elsevier Science Publisher.

In 1993, a new kind of conductor developed by CEA and manufactured by Dour Metal Industry was tested in the Sultan test facility of the Paul Scherrer Institute (Switzerland) (Bessette et al 1992), which is now the devoted facility in Europe for such kinds of test (figure G3.0.17). One can notice: •

the circular arrangement of the bundle of conductor protected by a stainless steel wrapping which prevents any deformation after cabling and acts as a protection during transport; this shape allows it

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Figure G3.0.17. A 40 kA conductor developed in the frame of the NET programme CEA-Dour Metal Industry conductor. The sides of the white square are 40 mm long.

to be drawn through the jacket during the fabrication phase; the jacket is fabricated and leak tested in a preliminary phase; • the central hole, which may prevent any severe blockage of helium due to metallic particle

accumulation and ensure a low hydraulic pressure drop;

• the six-petal arrangement forming a vault without any extra support, each petal being surrounded

by a stainless steel wrapping to cut out large coupling currents and keep the overall cable time constant at a low level.

The conductor design chosen by ITER is in fact very close to this concept. Different grades of conductors have now been designed for use in different parts of the machine. The conductors of the model coils of ITER, very representative of these conductors, are presented in figure G3.0.18 (Okuno et al 1994). The fabrication lines for these conductors are now in operation. In Europe two companies are involved in this activity: EM-LMI for the cabling and Ansaldo for the jacketing. About 6 km of conductor will be fabricated for the central solenoid model coil and 1 km for the toroidal field model coil.

Figure G3.0.18. Conductors of the ITER model coils. Reproduced by permission of Elsevier Science BV. Copyright © 1998 IOP Publishing Ltd

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(ii) Available cable space current density. The composition of the conductor inside the jacket, i.e. the helium section, copper section and noncopper section, has to be adjusted such as to maximize the available cable space current in relation to the design criteria which have to be fulfilled. Hot spot criterion. In the case of a transition to the normal state, the increase of temperature until the complete extinction of current must be limited to about 150 K

where J is the overall current density in the cable (A mm−2 ), ρ the equivalent resistivity of the cable (Ω m), C the equivalent heat capacity of the conductor ( J m−3 K−1 ), T the conductor temperature (K) and B the magnetic field (T). Well cooled criterion. The critical energy, that is to say the maximum energy input acceptable by the conductor without losing superconductivity, can be very high, nearly the energy sink of the helium section of the conductor if this criterion is fulfilled, which is typically more than 500 times the enthalpy of the materials. That is the main interest of the cable-in-conduit concept

ρJ 2 < 4hβα (TC − Tb )/d where d is the composite diameter (m), α the wetted perimeter coefficient, Tc the critical temperature ( K ), Tb , the bath temperature (at the end of a shot) ( K ), β the safety coefficient necessary to take advantage of large critical energies and h the heat transfer coefficient ( W m−2 K−1 ). This critical energy has to be kept at least greater than the energy which appears in the case of a plasma disruption (time range: 100 ms) and greater than the typical mechanical energy release (time range: a few milliseconds). The question of the helium energy sink to be taken into account is always a source of discussion and controversy. It is of course very dependent on the thermodynamic process being considered. In view of the very long length of the cooling channel which is submitted to variation in the disruption field (more than 100 m), our opinion is that the internal energy (isochore model) of helium must be taken into account for this evaluation which leads to far smaller values than the enthalpy at constant pressure which is sometimes considered. Evaluation of Tb . The evaluation of Tb is not so easy to perform; it has to be done at any place and time. A typical current variation in the central solenoid coil is presented in figure G3.0.19 together with the field at the inner layer of the central solenoid coil. The greater part of the losses appears at the beginning of the shot when the field decreases from 13 T to −6 T corresponding to a temperature increase greater than 2K. The calculation of the helium temperature at a given time and location must take into account the heat deposition and its repartition along the layer from the beginning of the run. This calculation is complicated by the two helium channels whose hydraulic properties are very different. Several hydraulic codes have been developed which help to solve this problem. Temperature margin . TC S − Tb = ∆T m a r g where Tc s is the current sharing temperature. Both Tc s and Tb are functions of the location along the channel and of the time. The difference between Tc s and Tb has to be kept higher than, typically, 2 K; this is the safety margin. In particular, for the central solenoid, which operates in pulsed mode, the temperature increase accumulation must be avoided. The transit time of the helium across the channel is a very important characteristic: it will take about 2000 s for the helium flow to travel across the central solenoid inner layer which is also the time duration of an ITER pulse. Copyright © 1998 IOP Publishing Ltd

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Figure G3.0.19. Description of a central solenoid cycle.

(e) Strand (i) Nb3 Sn strand. As a result of the high field specification, the only possibility for fusion at high field is to use Nb3Sn wires. A lot of work has been done recently (ABB 1988, Bruzzone et al 1995, LMI 1988) to estimate whether the industry would be able to supply in large quantities Nb3Sn strands fitting fusion specifications regarding hysteresis losses (between 200 and 600 mJ cm-3 nonCu for a ±3 T cycle), RRR (>100), Jn o n C u (between 600 and 700 A mm−2 at 12 T and 4.2 K) and unit lengths (>3 km). So far the situation is not very clear, even if major improvements have appeared. The fabrication of the model coils will help clarify on large quantities what the real available strands are. The possible processes. Three kinds of process are in competition for producing the Nb3Sn strand. •

The modified jelly roll (MJR) strand. Much experience has been accumulated on this strand both in Europe and in the USA. Significant lengths of cabled 40 kA conductors have been manufactured and tested, using this material. A typical strand produced by the Teledyne Wah Chang Company is presented in figure G3.0.20. There is no bronze in the strand during its fabrication. During the three stages of heat treatment the tin diffuses from the 19 central cores to form the bronze (first two stages) then the Nb3Sn from the niobium mesh (last stage). The copper shell is protected from tin pollution by a vanadium barrier. Large unit lengths and high current density (650 A mm− 2 at 12 T ) are currently associated with this material in relation to high hysteresis loss and high effective filament diameter (∼35 µ m) due to bridging. Vanadium has now been abandoned for a double tantalum and niobium barrier. • The bronze route strand. This is the oldest process. A typical strand produced by Vacuumschmelze is presented in figure G3.0.21. The tin for the filaments is provided from a bronze matrix existing before the reaction. Pollution of the pure copper shell is prevented by a tantalum barrier. The difficulty of drawing large unit lengths, including important hard bronze parts, now seems to have been overcome. Critical current densities are smaller than for the MJR process ( 550 A m2 at 12 T ) but very small effective diameters can be achieved (5 µ m < d e f f < 10 µ m). The quality of this product is certainly very homogeneous and perhaps more con trolled than the other processes. Copyright © 1998 IOP Publishing Ltd

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Figure G3.0.20. A 0.73 mm MJR strand produced by Teledyne Wah Chang, Albany as part of the NET development (used for the CEA 40 kA conductor).

Figure G3.0.21. A 0.81 mm bronze strand produced by Vacuumschmelze for the central solenoid model coil.

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The internal tin Nb3Sn strand. A typical strand produced by EM-LMI is presented in figure G3.0.22. The arrangement is very similar to the MJR strand. The anti-diffusion barrier is made of tantalum. In principle, the MJR strand and the internal tin strand are supposed to present the same kind of result but less practical experience exists with this wire.

Figure G3.0.22. A 0.73 mm internal tin strand produced by EM-LMI for the toroidal field model coil.

The critical current. The critical noncopper current density can be very well represented by the following law (Summers et al 1991)

for ternary Nb3Sn

Tc can be numerically calculated as the value giving 0 for Jc . The n value. During the current-sharing regime the current-voltage ( I-V ) characteristic can be modelled in the form: V = α Ι n . Low n values have sometimes been seen in the past and are not acceptable, due to the resulting high power dissipation at low current in coils made of long lengths of conductor. Specifications of the n value of around 20–30 are assumed and seem to be met without difficulties. Copyright © 1998 IOP Publishing Ltd

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Residual resistivity ratio ( RRR). Low RRR (20–40) values are periodically pointed out, in particular in chrome plated strands. These poor RRR values are not very well understood. They appear after long duration heat treatments and may be related to pollution coming from the Cr bath or, which is worse, even from leakage through the anti-diffusion barrier. This has important consequences as concerns stability and protection. Chrome plating. The strands will be chromium plated with a 2 µm thick layer to prevent sintering of the strands during the heat treatment. As a result of its well known mechanical hardness, the chromium is also assumed to limit the area of the contact surface between two neighbouring strands in the cable, thus reducing the coupling currents between these two strands. This role is not completely understood and may depend on the quality and the nature of the chrome plating which can greatly vary from one vendor to another. The strand diameter. Much experience has been accumulated in cabling strands with diameters in the range 0.7–0.8 mm. Lower diameters lead to an increasing number of stages in the cable and thereby increasing costs. Larger diameters make stability considerations more difficult and may be connected to breakage during the cable manufacture. (ii) Nb-Ti strand. The Nb-Ti strands which could be used for the external poloidal field coils of ITER are very similar to those used for accelerator magnets. No new developments are in principle needed. The only specific point which is requested is a thin external cupro-nickel layer which helps to limit and control the time constant of these large cables. This strand is available in standard production. The very low achievable filament diameter (5 µm) is an advantage in pulsed operation. Such a wire produced by GEC Alsthom is presented in figure G3.0.23.

Figure G3.0.23. A 0.8 mm Nb-Ti strand produced by GEC Alsthom for fusion applications.

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References ABB 1988 Contract Report NET 88/745 Adam et al 1987 La fusion thermonucléaire contrôlée par confinement magnétique Collection du CEA Alkhimovich V A 1989 First tests of T15 toroidal field superconducting system Proc. Int. Conf. on Magnet Technology, MT 11 (Tsukuba, 1989) (Amsterdam: Elsevier) Alkhimovich V A 1991 The current capacity tests of the tokamak T15 Nb3Sn. Toroidal coil assembly IEEE Trans. Magn. MAG-27 2057 Aymar et al 1995 ITER Interim Design Report June 1995 Beard D S et al 1988 The IEA Large Coil Task Fusion Eng. Design 7 No 1 and 2 Bessette D, Ciazynski D, Decool P, Duchateau J L and Kazimierzak B 1992 Fabrication and test results of the 40 kA CEA conductor for NET/ITER Proc. 17th SOFT (Rome, 1992) p 788 Bruzzone P et al 1993 Test results of full size 40 kA NET/ITER conductor in the Fenix test facility IEEE Trans. Appl. Supercond. AS-3 357 Bruzzone P, Mitchell N, Steeves M, Spadoni M, Takahashi Y and Sytnikov V 1996 Conductor fabrication for the ITER model coils IEEE Trans. Magn. MAG-32 2300 Colombo V 1990 Fusion Programme Evaluation Board Report July 1990 Duchateau J L, Bessette D, Ciazynski D, Pierre J, Rouanet E, Riband P and Turck B 1991 Monitoring and controlling TORE SUPRA toroidal field system. Status after a year of operating experience at nominal current IEEE Trans. Magn. MAG-27 2053 EM-LMI 1987 Contract Report NET/87.745 B/S.O/ Imagawa S et al 1995 Construction of helical coil winding machine for LHD and on-site winding IEEE Trans. Magn. MAG-32 2248 Nakamura Y, Nagao A, Hiraki N and Itoh S 1989 Reliable and stable operation on the high field superconducting tokamak, TRIAM-IM Proc. Int. Conf. on Magnet Technology, MT 11 (Tsukuba, 1989) (Amsterdam: Elsevier) Okuno et al 1994 Status of the ITER magnet R&D program Proc. 18th Symp. on Fusion Technology (Karlsruhe, 1994) (Amsterdam: Elsevier) p 845 Sapper J and Sapper W 7-X Technical Group 1993 Superconducting coil for the Wendelstein 7-X stellarator Fusion Eng. Design 20 23–32 Sheahen T P 1994 Introduction to High Temperature Superconductivity (New York: Plenum) Steeves M M, Hoenig M O and Cyders C J 1984 Effects of Incoloy 903 and tantalum conduits on critical current in Nb3Sn. Cable in conduit conductors Adv. Cryogen. Eng. 30 883 Summers L T, Duenas A R, Karlsen C E, Ozeryanski G M and Gregory E 1991 A characterization of internal Sn Nb3Sn superconductor for use in the proof of principle coil IEEE Trans. Magn. MAG-27 1763–6 Toschi R et al 1993 NET predesign report Fusion Eng. Design 21 Turck B 1991 Recent developments in superconducting conductors Fusion Eng. Design 14 135 Turck B 1995 Six years of operating experience with TORE SUPRA, the largest Tokamak with superconducting coils IEEE Trans. Magn. MAG-32 2265 Wilson M 1983 Superconducting Magnets (Oxford: Oxford University Press)

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G4 Superconducting magnets for particle accelerators (dipoles, multipoles)

R Perin and D Leroy

G4.0.1 Introduction High-energy physics, with its continuous quest for higher particle beam energies to probe matter at smaller and smaller sizes, has been for almost three decades the prime mover in the development of superconducting magnets for guiding and focusing the beams in circular particle accelerators. Particle accelerators are complex machines in which electrically charged particles are accelerated to higher energy by means of electric fields, while they are maintained and focused in their trajectories either by means of electric fields or, more powerfully, by magnetic fields. General descriptions of these machines can be found in many books (e.g. Livingood 1961). Accelerators are fundamentally of two different types: circular and linear. In linear accelerators particles travel on a straight path and the use of magnets is limited to periodic focusing of the beam, for which classical resistive magnets (quadrupoles and higher multipoles) are in general employed. An application of superconducting magnets is sometimes found at the end of linear accelerators to deflect the beam from its original direction to different areas of experiments, but this is in general limited only to a single magnet or a few dipole and quadrupole magnets. In circular particle accelerators (and storage rings), the particles are maintained in their circular (or quasi-circular) orbit inside a vacuum chamber for millions of revolutions by dipole bending magnets which cover most of the beam paths (see figure G4.0.1). In addition, many other magnets are needed to keep the beam transverse size small (focusing quadrupoles) and stable (multipole corrector magnets). Superconducting magnets have become essential components of hadron (protons or heavier particles) accelerators/colliders, which are so far the most massive applications of superconductivity, and also of compact accelerators, e.g. ‘portable’ synchrotron light sources and small cyclotrons for medical applications (see chapter G5). This chapter presents only the main aspects of the magnets of the high-energy accelerators/ colliders currently in operation or under construction. Part of what follows was presented at the CERN Accelerator School on Applications of Superconductivity in 1995 (Perin 1995). Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.1. A schematic diagram of a circular particle accelerator/collider.

G4.0.2 Motivation for the use of superconducting magnets. Advantages in terms of cost, space and energy consumption For proton (hadron) accelerators/colliders, synchrotrons are at present the most efficient and economical machines to accelerate, store and collide high-energy beams with the intensity necessary to produce in sufficient number the rare events that high-energy physicists want to study. As the attainable beam energy in circular machines is proportional to the radius and bending magnetic field, any forward step in energy corresponds to an increase of size or magnetic field or both; hence the push towards higher field, the main motivation being economy in terms of space and capital investment, and social acceptability from the points of view of ground occupancy and energy consumption. Figure G4.0.2 shows schematically how the cost of the magnets per tesla metre varies with field level (Perin 1990). The plots include the cost of magnet cryostats, but not the cost of the cryogenic plants which varies little with field level for the same operating temperature, the size of the plants usually being determined by a desired cool-down time. The cost increase for a 1.8 K superfluid helium system with respect to a standard 4.2 K installation is relatively small and can be estimated to be 8–10% of the cost of the cryomagnet system. The savings in global cost of the accelerator facility are in fact much higher than indicated in figure G4.0.2, as civil engineering, infrastructure and installation costs decrease in inverse proportion to field in the first approximation. A plot of global cost versus field— which would depend, of course, on other factors related to the geographic location—would, therefore, show a much greater advantage in going to high magnetic fields. When considering running costs, this trend is dramatically reinforced. Electric power consumption, which is mainly determined by cryogenics, is in fact almost independent of field level. When comparing with the power consumption of classical magnets the gain is striking, e.g. savings of a factor of 60 are estimated for the CERN Large Hadron Collider (LHC) (LHC Study Group 1995). The way to high fields is, however, a very difficult uphill one, as illustrated in figures G4.0.3 and G4.0.4, because of intrinsic characteristics of the superconductors, rapidly rising electromagnetic forces and stored energy which severely complicate the problems of the force-containment structure and magnet protection at resistive transitions. Moreover, field errors at low field due to persistent currents in the superconducting filaments tend to be greater in high-field magnets as they require more superconductor. Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.2. Cost of magnets and cryostats in terms of bending power (tesla × metre) for 50 mm diameter coil-aperture dipoles.

Figure G4.0.3. Current density ( Jc ) in commercial superconductors (wires, non-Cu part), and coil thickness (w) in dipole magnets with graded current density and average ratio Cu/Sc = 1.7:1.

G4.0.3 Special features of superconducting magnets for particle accelerators There are some fundamental differences between superconducting magnets for accelerators and other superconducting magnets (e.g. large solenoids for particle detectors, toroidal coils for nuclear fusion machines) that make their design and construction a very special branch of the technology: ( i ) the need to use very high current densities to economically produce the required high bending/ focusing fields; ( ii ) the complex and nonuniform repartition of electromagnetic forces; ( iii )the extreme precision of the magnetic field distribution in small apertures; ( iv )the high degree of reproducibility and reliability. Some of these characteristics make the task of building superconducting accelerator magnets singularly complex and difficult. In particular: Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.4. Electromagnetic force components Fx , Fy and stored electrical energy E per unit length versus magnetic field in 50 mm diameter coil-aperture dipole magnets.

• • • •

point (i) precludes cryogenic stabilization; point (ii) leads to quite elaborate mechanical structures; point (iii) demands an unprecedented dimensional precision in components and manufacture; point (iv) requires manufacturing methods adapted to large-scale industrial production.

G4.0.4 Dipoles and quadrupoles: their importance and development As mentioned above, in circular machines the attainable beam energy is proportional to the radius and bending magnetic field; therefore, there is a great incentive to develop dipoles for higher and higher fields. As a consequence the dipole magnets, which are the most important components of the accelerators from the point of view of cost, are bound to become technologically the most advanced and critical. In parallel, the focusing elements (quadrupoles) have to become stronger while being kept as short as possible. In the 1970s superconductors were developed to an adequate level and a sufficient understanding of superconducting accelerator magnets was accumulated through the effort of several laboratories for the construction of large numbers of superconducting magnets to be envisaged. Historically the first superconducting magnets which operated reliably in an accelerator/collider were eight high-gradient, large-aperture, quadrupoles for the low-beta insertion of the Intersecting Storage Rings (ISRs) at CERN (Billan et al 1979), but the first superconducting accelerator was the Fermilab Tevatron (Cole et al 1979, Cooper et al 1983, Hanft and Brown 1989, Koepke et al 1979, Tollestrup 1979). The importance of this machine is universally recognized: it has been for many years the largest application of superconductivity and a true prototype of a new generation of particle accelerators. The next superconducting accelerator machine was the proton ring of the HERA collider built at DESY, Hamburg in the 1980s and currently in operation (Auzolle et al 1984, Kaiser 1986, Meinke 1991, Perot and Rifflet 1991, Schmüser 1992, Wolff 1988). The Relativistic Heavy Ion Collider (RHIC) at Copyright © 1998 IOP Publishing Ltd

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Brookhaven National Laboratory (BNL) is almost complete (Brookhaven National Accelerator Laboratory, updated yearly; Thompson et al 1991, 1993, Wanderer et al 1990). The manufacture of the main dipole and quadrupole magnets was successfully completed in industry at the end of 1995. Two large projects, the SSC (Dahl et al 1985, Ducos et al 1994, Goodzeit et al 1990, Jackson 1988, Kuzminski et al 1993, Lietzke et al 1994, Ogitsu et al 1993, Peoples 1989, Sanford and Matthews 1990, Schermer 1994, Spigo et al 1994, Taylor et al 1991, Wanderer et al 1993) in the USA and the UNK (Alexandrov et al 1994) in Russia, have unfortunately been discontinued. The LHC at CERN (Acerbi et al 1994, Ahlbäck et al 1994, Baze et al 1992, Bona et al 1997, Leroy et al 1989, Perin 1991, 1994, 1996, Rifflet et al 1994, LHC Study Group 1995, Shintomi et al 1996, Yamamoto et al 1993) is at present the most important and advanced project. Its construction was approved in December 1994. The evolution of field and field gradient of superconducting main magnets for various accelerators are given in table G4.0.1. Improvements in the performance of superconductors, better insulation systems and force-containment structures and refinements in manufacturing have permitted the field and gradient to be increased. A bold step is being taken with the LHC by using the superfluid-helium technique first used at Tore Supra (Claudet and Aymar 1990), thus enhancing the performance of the traditional Nb-Ti superconductor, and by adopting the two-in-one configuration first proposed at BNL (Blewett 1971) which reduces costs and space occupancy considerably.

In modern proton or heavier-particle high-energy accelerators the share of the magnet system in the cost of the facility is considerable and predominant in some cases, e.g. the LHC, which will be installed in the existing LEP tunnel and will use the existing infrastructure (figure G4.0.5). The main dipoles and quadrupoles account for the largest share of the cost of the magnet system (figure G4.0.6). This chapter mainly deals with these two types of magnet. The design and construction of the corrector magnets, which are in general less demanding and can, therefore, be dimensioned with larger margins, follow the same pattern. G4.0.5 Categories of superconducting magnets for accelerators Three different classes of magnets can be defined with respect to the way in which the desired field quality is achieved. Type (a). Magnets in which the field distribution is dominated by the coil configuration. The ISR and LEP quadrupoles, the Tevatron, HERA, RHIC, (SSC) and LHC main dipoles and quadrupoles pertain to this class (figure G4.0.7). Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.5. Cost breakdown of the LHC machine components.

Figure G4.0.6. Cost breakdown of the LHC magnet system.

Type (b). Iron-dominated magnets, also called superferric, in which the iron (low-carbon steel) pole shape determines the field pattern (examples of these are the HERA correctors, and the RHIC sextupoles (Linder et al 1994)) (figure G4.0.8). Type (c). Magnets in which the field distribution results from the interplay of coils and yoke, both strongly contributing to produce the desired field. Examples are some of the model magnets built at TAG and the configurations of the ‘Block-Coil Dual Dipole’ and the ‘Pipe’ magnet (McIntyre et al 1994) (figure G4.0.9) recently proposed by Texas A&M University and LBL.

G4.0.6 Superconductors Figure G4.0.10 shows an accelerator-type superconducting cable and its electrical insulation. In recent years, industrial firms in many countries, stimulated by accelerator projects—in particular HERA, RHIC, Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.7. Example of a coil-dominated magnet: the LHC dipole.

Figure G4.0.8. Example of an iron-dominated (superferric) magnet: the RHIC arc sextupole.

SSC, UNK and LHC—and supported by universities and laboratories, have worked hard to produce conductors which meet the requirements of particle accelerators, i.e. which have • high current density in the coil for efficient magnet design and a small spread of current densities throughout production for field uniformity; • low magnetization at low field to limit the magnetic field errors at injection in the accelerator. Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.9. Example of a ‘mixed’ magnet in which the field configuration is determined by the conductor and iron: the Block-Coil Dual Dipole.

Figure G4.0.10. Typical superconducting cable and its insulation.

Superconducting accelerator magnets use high-current superconducting cables having an overall high current density. High currents permit the design of superconducting magnets made with a small number of turns and a low self-inductance. These magnets have an efficient mechanical design and produce lower voltages when quenching. Above about 3 kA the conductors consist of cables made of superconducting strands transposed and flattened in a compacted, quasi-rectangular, shape. An overall high current density allows the high fields needed by modern accelerators to be achieved in magnets having a more compact and consequently more cost-efficient design. Besides the current density in the superconducting material itself, the other parameters related to the cable overall current density are: • the amount of copper around the filaments, • the cable compaction, • the electrical insulation. Copyright © 1998 IOP Publishing Ltd

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As an example, a typical LHC insulated superconducting cable consists of 30% superconductor, 48% copper, 17% electrical insulation and 5% void to be filled with helium. Chapter B8 reviews the commercially available superconducting wires and cables. The practical superconducting materials for accelerator magnets are still Nb-Ti and Nb3Sn. Nb-Ti, which is produced industrially in large quantities at a competitive price, is used in today’s accelerator magnets. Nb3Sn, which presents more technological constraints with respect to magnet manufacture, could be used in a small series of beam handling magnets, built according to the wind-and-react techniques. High-field accelerator magnets have been built with materials of the two types. Figure G4.0.11 shows the current densities of conductors produced in the USA and Japan for the SSC magnet development programme. A current density Jc of 2700 A mm-2 at 5 T, 4.2 K can be at present specified for large quantities of Nb-Ti superconductor, with an expected rejection rate not exceeding 5%. Progress has been made also in reducing Jc spread in production: a histogram of measurements on the cables produced by one manufacturer for the HERA dipoles is presented in figure G4.0.12. All cables had an Ic in excess of the specified value ( Ic = 8000 A at 5.5 T, for 4.6 K ) and the Ic standard deviation was 2%. By the use of superfluid helium, the operating temperature of the superconducting magnets can be reduced from 4.2 K to 1.9 K, leading to an increase of the operating magnetic field. At the same critical

Figure G4.0.11. Current density in the superconductor part of cables produced from R&D billets for the SSC magnet development programme. Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.12. Critical current in kA at 5.5 T and 4.6 K in HERA cables produced by ABB.

current density, the shift in field is 3 T for Nb-Ti and at least 1.5 T for (Nb-Ti)3Sn. Figure G4.0.13 shows the shift in field in the same type of strand produced by different suppliers. It appears that the manufacturing process is an important parameter in the field shift since the data are obtained with the same composition of the raw material produced by the same supplier. A higher field shift is correlated to a higher B*c 2 , which is known to be related to the composition of the Nb-Ti alloy but is also related to the manufacturing process as suggested by figure G4.0.14. The state of the art in the construction of superconducting magnets seems to indicate that when a superconducting material can carry a critical current density of about 1000 A mm−2 at a given field B, a magnet can be studied and built for this given field. Accelerator dipoles of 10 T can be built using Nb-Ti and dipoles of 14-16 T can be envisaged for the future using Nb3Sn material with additions of Ta or Ti. Figure G4.0.15 shows the critical current obtained in the framework of the LHC development programme. A current density of 1000 A mm−2 at 10 T and 1.9 K may be specified for large quantities of Nb-Ti superconductor. This is the value at the 3σ limit of the average value of the critical current distribution which has a standard deviation of 3%. The magnetization has a strong influence on the field errors in magnets for accelerators. During the injection of the particles in the accelerator, which takes place at relatively low fields, the field errors are dominated by the magnetization due to the superconducting filaments Mf = kdJc where d is the filament diameter and Jc is the current density at low field and k = ¾πµ0 . The accelerator magnets require small filaments in the range of 5-10 µm diameter to limit the field errors. Strands of 1.3 mm diameter having 23 000 filaments 5 µm in diameter have been produced for the LHC development programme. The magnetization in the strands depends on the proximity of the filaments in the final stage. It is safe to have at least 1 µm spacing between filaments to avoid coupling between them. For reasons of uniformity the magnetization must be kept constant within 2% throughout the production. The magnetization varies between different suppliers, as shown in figure G4.0.16, but it may be possible to accept a variation from supplier to supplier of up to 20% by means of a dedicated repartition of the magnets in the ring. Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.13. Shift in field depending on strand manufacturing process for two types of strand. Dark points: 1.065 mm diameter. Open points: 0.865 mm diameter.

Figure G4.0.14. Correlation between B c*2 and the ‘shift’ in field between 4.2 K and 1.9 K for Nb-Ti.

The copper-to-superconductor ratio λ plays an important role in critical current Ic and in the strand magnetization, Ms , the two major values for an accelerator-type magnet, according to the following relations:

where D is the strand diameter, N is the number of filaments, Jc the critical current density at a given field and temperature. The absolute value of the copper-to-superconductor ratio is determined by criteria related to the stability of superconductors and to the protection of the magnet in case of a quench. It varies from 1.3 to 1.9 in conductors for accelerators. The variation of the copper-to-superconductor ratio has to be tightly controlled during the wire production. Because a billet is made of machined subelements like hexagonal sets (Nb-Ti + Cu), tolerances of fabrication in the billet assembly lead to variations in the Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.15. Critical current density of LHC strands (noncopper part) at various temperatures.

Figure G4.0.16. Magnetization measurements of the strands at 2 K for two types of wire produced by different suppliers.

copper-to-superconductor ratio. In addition, during the drawing of the billet, the copper-tosuperconductor ratio can vary between the beginning and the end of the billet. This parameter has to be controlled continuously. Figure G4.0.17 shows the variation along the length. A variation of 0.1 for a conductor having a copper-to-superconductor ratio of 1.6 is equivalent to a variation of 3% in the critical current density. For very high-field magnets, large-aspect-ratio (width/thickness) keystoned Rutherford-type cables are required. The largest ones from 17 mm to 23 mm have been developed by industry for the LHC magnet development programme. Improvements in cabling methods, together with a better understanding of wire quality for cables, led to a reduction of Jc due to cabling not exceeding 1 to 3%. The volume compaction coefficient is given by P = n πD 2/4wt cos α Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.17. The variation of the copper-to-superconductor ratio λ along the strand length during manufacture.

where n is the number of strands, D the strand diameter, w the cable width, t the cable mean thickness and α the cabling angle. The compaction is a compromise between the necessary mechanical coherence of the cable required for the coil fabrication and the degradation in current density. It has been observed in the LHC cables that the degradation increases rapidly above ∼92.5% compaction. Lower compaction allows helium penetration into the cable and better thermal stability, but too low a compaction makes the winding difficult and could easily allow wire movements. Compactions in the range 89 to 91% are generally used in the superconducting cables for accelerators. The field errors when the field is ramped are dominated by the interstrand resistance between strands of the two layers of the cable (crossover resistance). The resistance between adjacent strands of the same layer can be a factor of 50 smaller than the crossover resistance. Depending on the type of magnet and the type of accelerator, the interstrand resistance has to be kept to a minimum value; for LHC it is 10 µ Ω per contact. This value must be guaranteed at the end of the magnet manufacture, i.e. not only after the cable manufacture, but also after the winding, the temperature cycle imposed by the coil fabrication and the magnet assembly in the mechanical structure. As an example, the use of all-polyimide electrical insulation imposes a heat treatment at 185° for 2–3 minutes during the coil fabrication. The strand coating is so thin that the contact resistance is governed by the presence of oxides. In the RHIC magnets the oxide layers are created on the bare copper strands during the coil fabrication. In the Tevatron and in the HERA machines the strands are covered by a layer of SnAg 5 % . If a low value of the interstrand resistance leads to high field errors during ramping, high interstrand resistances are to be avoided to prevent instabilities in the magnet due to a nonuniform distribution of currents between the strands of the cable. Certain phenomena in the interplay between contact resistance and stability are not completely understood. For instance, in a magnet, the nonuniform field distribution all along the cable combined with the nonuniform junction resistance between the strands induces a nonuniform distribution of currents among the strands of the same cable. When an energy perturbation occurs, a strand may start to quench locally, and be cooled down through the heat-exchange surface while the increased local voltage forces a redistribution of currents among the strands. All these complex phenomena, which are of primary importance in accelerator magnets, are under thorough study and would need a complete dedicated review. Copyright © 1998 IOP Publishing Ltd

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G4.0.7 Magnetic design In beam-guiding magnets, the problem is essentially bidimensional because, apart from the localized regions of the ends, the field distribution has to be the same in all planes perpendicular to the beam axis. Type (b) magnets (from section G4.0.5) are designed and built in much the same way as classical resistive magnets, the main difference being that they allow much higher current densities in the windings. So, for the magnetic design, use is made of the well-developed finite-difference and finite- element computer codes. For type (c) magnets the same methods apply, since the iron yoke plays an important role. The design of type (a) magnets is more typical of superconducting accelerator magnets. The resulting coil configuration is in general an approximation to the cosnθ distribution (see also chapter C1). Generally the design starts with analytical methods, integrating the Biot-Savart law describing the field (induction) induced by a current flowing through a (infinitely) thin wire

The windings are subdivided into small regular parts composed of rectilinear current lines whose contributions to the field can be computed analytically and added up by means of relatively simple computer programs. In this first phase of the design, iron is in general assumed to have infinite magnetic permeability ( µ = ∞). Only in a second phase are the effects of the real iron characteristics (i.e. remanence, variable permeability and saturation) analysed. This is done by means of computer programs solving the partial differential equations by finite-difference or finite-element numerical methods. More details on the methods of magnetic design can be found in the literature (e.g. Mess and Schmüser 1989, Perin and van der Meer 1967, Perin et al 1979, Russenschuck 1995). G4.0.8 Field quality Field quality is, of course, of great importance to the accelerator designer, as it directly affects beam optics and stability of the beam(s). G4.0.8.1 Definition of multipole field components The field in accelerator magnets can usually be treated as two dimensional since the beam integrates through the field. It can then be expressed as a power series:

where B1 = magnitude of dipole field in the y (vertical) direction; bn = normal multipole coefficient an = skew multipole coefficient Z=x+y Rr = reference radius n = 1: dipole n = 2: quadrupole n = 3: sextupole, etc. The coefficients an and bn are dimensionally dependent, so they must be defined at a given reference radius. Copyright © 1998 IOP Publishing Ltd

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G4.0.8.2 Sources of field errors One may distinguish two types of field error: • Those respecting the magnet basic symmetry (n = 3, 5, 7, 9,… for dipoles; n = 6, 10, 14,… for quadrupoles; etc). These are the so-called ‘allowed’ components which can be limited to acceptable values by tuning the design, but may re-appear at the manufacturing stage. • Those which violate the basic symmetry of the magnet, producing the so-called ‘un-allowed’ multipoles, and which result mainly from manufacturing errors.

Field errors have different origins: conductor placement errors, iron saturation, coil deformation under electromagnetic forces, persistent and coupling (eddy) currents. (a) Conductor placement errors The field errors originating from mis-positioning of conductors or complete coils can be computed analytically at the design stage from the known possible or imposed manufacturing tolerances. Some examples are given for the LHC dipole, which has a coil aperture diameter of 56 mm, in terms of the most sensitive multipole components an , bn defined at the reference radius Rr = 10 mm. • A 0.1 mm increase of the coil azimuthal size (which incidentally would correspond to a decrease in compression of about 30 MPa) while current and coil thickness remain unchanged, produces a sextupole and a decapole

b3 = 1.2 × 10−4

b5 = 0.03 × 10−4.

• A 0.1 mm vertical off-centring of the coils with respect to the yoke produces a skew quadrupole

a2 = 0.15 × 10−4 at 0.58 T (injection field) and 0.09 × 10−4 at 8.4 T (nominal field). • A 1 mm difference in length between lower and upper coil of the same aperture in a 14m long LHC dipole ( ∆L/L = 7 × 10− 5 ) is equivalent to an ‘integrated two-dimensional’ skew quadrupole

a2 = 0.06 × 10−4. • A 0.1 mm top/bottom asymmetry in coil azimuthal size produces a skew quadrupole

a2 = 2.7 × 10−4. • A 0.1 mm left/right asymmetry of the coil azimuthal size (e.g. originated by a corresponding asymmetry in the curing mould) produces a skew sextupole

a3 = 0.7 × 10−4. Fluctuations of bending power from magnet to magnet may also be originated: • A 0.1 mm increase of coil inner radius, with unchanged current and coil thickness, produces

• A 0.1 mm increase of yoke aperture (coil-yoke distance) produces

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From these examples it can be understood how difficult it is to obtain in superconducting magnets a field quality comparable to that of classical lower-field magnets, where the field distribution is determined by the iron-pole profiles which can easily be produced with 0.01 mm precision. In superconducting magnets the coil geometry is the result of assembling stacks of conductors, typically 15 to 20, which are produced with stringent, but not infinitely small, tolerances (in cables the best that can be achieved nowadays is of the order of ±0.0025 mm on the thickness) and insulated by wrapping them with tapes which can be industrially produced with a tolerance of a few micrometres on their thickness. As in general the coils include longitudinal wedges, these can by adjusted during the production to tune down the so-called ‘allowed’ multipoles. (b) Iron saturation The field errors originating from the remanence and the variable permeability vary with excitation and depend strongly on the coil-yoke distance. For cold iron magnets they have to be carefully evaluated, are in general systematic and affect mainly the first higher multipoles (six-pole and ten-pole in dipoles) and can be compensated for either by correction windings in the aperture (e.g. HERA) or by small corrector magnets placed at the magnet ends (e.g. LHC). (c) Coil deformation under the electromagnetic forces These errors vary with excitation and can be computed after the mechanical analysis of the structure and the determination of the deformed coil configurations, using analytical or other computer programs. (d) Persistent currents in the superconductors The persistent-current errors are a particularity of superconducting magnets. They are due to currents induced in the superconducting filaments by field variations and, contrary to normal conductors where resistance rapidly reduces and after a while eliminates eddy currents, they circulate indefinitely as long as the superconductor is kept below its critical temperature. Persistent-current errors affect all field multipole components allowed by the symmetry configuration of the magnet, including the fundamental one. Their relative importance decreases with excitation, but they are particularly disturbing at low field levels and especially at injection. They depend on the previous powering of the magnet and vary with time and, therefore, require a careful study of the magnet excitation cycle. Persistent currents are proportional to the effective diameter of the superconducting filaments, so accelerator magnets use filaments as thin as possible, compatible with economy and quality of production. This subject is extensively treated in the literature (e.g. Holzer 1996). (e) Coupling currents Currents occur during field sweep in multistrand conductors both inside the strands, mainly due to coupling between filaments, and between the strands. They distort the magnetic field, and their effects depend on the geometrical and electrical characteristics of strands and cables (matrix and inter-strand resistance, cable aspect ratio, distribution of superconducting filaments, etc) and, of course, on the field ramp rate. This subject is also treated in the literature (e.g. Devred and Ogitsu 1996). (f) Twist In long and slim objects such as superconducting magnets, twist around the longitudinal axis is difficult to prevent and can produce nonnegligible field orientation errors. In general a tolerance of about 1 mrad is required in dipole magnets, and, although the beam integrates along its path, local twists should not exceed this value by more than a factor of two or three. Excessive twist can be prevented by locking the magnet active part to a structural component of high torsional rigidity, generally a closed cylinder of large diameter, such as a helium vessel, which in many cases is also part of the force-containment structure. (g) Curvature Long dipole magnets are normally bent to follow the particle trajectory. The curvature must be carefully controlled to avoid errors and loss of the very costly useful aperture.

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G4.0.8.3 Estimation of field quality At the design stage of accelerators, forecasts of field quality can be obtained by applying the considerations of section G4.0.8.2. These can be corroborated by existing statistical results, at present from the Tevatron (744 dipoles), from HERA (∼500 dipoles) and recently from RHIC, as well as from LHC models and prototypes. As an example, table G4.0.2 presents the expected error field components in the LHC dipole at injection and for the nominal operation field. Table G4.0.3 shows an estimate of the coupling-current errors for given interstrand contact resistance and field ramp rate: note that, at a given ramp rate, coupling currents produce the same errors at any field level, so that the relative field errors are highest at injection. If the interstrand resistance is uniform throughout the coils, coupling currents only produce allowed harmonics (in this case n = 3, 5, 7, etc). The other components (n = 2, 4) have been estimated assuming an interstrand contact resistance difference of 5 µΩ between top and bottom poles.

An example of twist in 10 m long magnets is shown in figure G4.0.18, where the field orientation in the two apertures of three LHC dipole prototypes of the first generation is plotted. The remarkably good parallelism of the field in the two apertures at all longitudinal positions is noticeable. G4.0.9 Forces and mechanics G4.0.9.1 Some facts • Electromagnetic forces are to a first approximation proportional to B 2

F=I×B

B2

and are extremely high in high-field magnets (see figure G4.0.19). Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.18. Field orientation measured in the two apertures of three 10 m long LHC dipole models (A1, A2, A3).

Figure G4.0.19. Transverse electromagnetic forces in the LHC dipole coils in a 9 T field. Copyright © 1998 IOP Publishing Ltd

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• The specific heat of materials (NbTi, Cu, etc) is very low † at 4.2 K (for Nb-Ti, C 6.5 × 10−4 J g−1 −1 −4 −1 −1 K ; for Cu, C 1 × 10 J g K ) and is about a factor of ten lower at 2 K. • A superconductor stays in the superconducting state when temperature, magnetic field and current density are below their critical values. • The temperature margin, the difference between the conductor working temperature and the critical temperature at the working field and current, is usually very small (1–2 K); in LHC dipoles it is 1.2–1.4 K. • Even extremely small sudden movements of the cable (or even of a strand) or cracking of the insulation generate enough heat to raise local parts of the superconductor above the critical temperature, provoking premature quenching (training). • It is therefore necessary to limit these sudden movements by careful design and construction of the force-containment structure. Gradual elastic deformations are permitted, to a certain extent.

G4.0.9.2

‘Roman arch’ concept for mechanical stability

Coils are built from different materials: conductors, metallic or insulating spacers, insulation and glue (epoxy or polyimide resin) to keep everything together (at least during manufacture). Sometimes the construction is even looser. It is desirable to avoid the insulation sticking to the conductor in order to: • •

let the conductor slide with little friction inside the insulation, to prevent the sudden release of heat in case of small movements; let the liquid helium enter into direct contact with the conductor to carry away heat. This is beneficial to stability and essential in the case of pulsed magnets or if the magnets have to be operated in helium II (≈2 K), and to remove heat deposited by beam losses (e.g. LHC).

Insulation and glues cannot be relied upon to withstand the high tensile stresses that result from the electromagnetic forces. If nothing is done cracks and micro-fissures will develop and the magnet will repeatedly quench (training). As the supporting structure and the coils cannot be infinitely stiff, the solution lies in the application of an adequate pre-compression which will prevent any tensile stress in the coil when electromagnetic forces are applied. One successful idea was to make the coil behave as a Roman arch. For cosnθ winding configurations the external structure applies a radial inward compression which, as in Roman arches, is transformed into azimuthal compression inside the coil which counteracts the formation of tensile stresses that would otherwise appear under the action of the electromagnetic forces (figure G4.0.20). For this concept to work, the coils should not be supported inside.

Figure G4.0.20. The Roman arch analogy.

† ∼10−4 of the room-temperature specific heat.

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A Roman arch can be built from materials of poor or even zero tensile properties (e.g. from dry stones), provided they have a good resistance to compression. The concept was first thought of and applied to the design and construction of the ISR low-beta quadrupoles (Perin 1975). It was then adopted in practically all superconducting main magnets for accelerators, with considerable elaboration and refinement, particularly at the Fermi National Accelerator Laboratory for the Tevatron (Tollestrup 1979) where, among many other new technologies, the laminated collar structure was introduced. In the example of the ISR quadrupoles (figure G4.0.21), where the coils were vacuum impregnated with epoxy resin, the pre-compression was applied by means of aluminium-alloy shrinking rings through the yoke quadrants and a set of stainless-steel spacers.

Figure G4.0.21. The ISR superconducting quadrupole.

A more recent example of strongly pre-stressed magnet coils is the very successful dipole wound with Nb3Sn superconductor which was built at the University of Twente in the framework of a collaboration with CERN and attained a central field of 11 T at the first quench (den Ouden 1994, 1997). In this case the fully epoxy-impregnated coils were strongly compressed before being collared by pre-heated aluminium alloy rings which further compressed them during cooling. Too high pre-compression is, however, not recommended, because, besides making construction and assembly more difficult and the risk of damaging the insulation, it may induce premature quenching. There are examples of magnets which behaved better when the pre-stress was decreased (Wanderer et al 1991). What is essential is either to avoid movements of the conductors or to allow little movements to occur with no or very little heat generation. At too high pre-stress, if conductor movements occur, they may generate enough energy to take the conductor over the stability limit, even at relatively low fields. At lower pre-stress, the same movement may occur but with less generation of energy, insufficient to produce a quench. On the other hand, when the pre-stress is too low, magnets may exhibit an erratic behaviour beyond the field level at which the compression vanishes in some parts of the coils, as shown in a number of LHC dipole models (Siemko 1997). Copyright © 1998 IOP Publishing Ltd

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G4.0.9.3 Force-containment structures The purposes of the force-containment structures are as follows. •

• •

To prevent any sudden movement of the conductors leading to premature quenching and training. To achieve this, the structure must not only be sufficiently strong to withstand the electromagnetic forces without damage, but must also be able to pre-stress the coil in such a way that any tensile stress is pre vented at all excitation conditions. To limit the elastic deformations of the coils, so that the field quality is maintained at all field levels. To guarantee the required structural and dimensional integrity of the magnet.

For magnets working at moderate fields, up to about 5.5 T, coil clamping is in general achieved simply by collars, which are adequately compressed when mounted around the coils and locked in position by means of dowels or keys. Different solutions are, however, adopted for the collar material (aluminium alloys, nonmagnetic steels, steel-yoke laminations) and locking system (dowel rods through quasi-circular holes, keys inside grooves). Aluminium-alloy collars produce an increase (or at least not a decrease) of the pre-stress on the coil at cool-down, thanks to their high thermal contraction. In this way the-high pre-stress is applied only when needed and not at room temperature, where creep of insulation or even copper may occur in the long run. Another advantage is lower cost of materials. A drawback compared to stainless or other nonmagnetic steels is that more space (wider collars) is required because of the lower elastic modulus and tensile/compressive strength. In some cases, however, the coil-to-yoke distance is dictated by the required field quality. In addition, aluminium alloys are more sensitive to stress concentration under fatigue conditions, so that it is not advisable to use grooves and keys to lock them. A problem with the present collaring systems using rods is that the coils have to be compressed more than is strictly necessary in order to produce the clearances (typically 0.1 mm) which permit the insertion of rods and to compensate for the elongation of the collar ‘legs’ under the reaction of the coils when releasing the press. With regard to the latter, austenitic steels behave better than aluminium alloys owing to their larger elastic modulus (10–20% higher pre-stress remaining after releasing the pressure). On the other hand, with austenitic-steel collars one is bound to lose a percentage (30–40%) of the pre-stress during cool-down. It is recognized that to achieve efficient designs for fields above 6 T, it is necessary that the rest of the structure (yoke + other components) contributes to the containment of the forces. So, in the HERA dipoles, which were designed for a lower operation field, above 6 T the collars come into contact in the horizontal plane with the yoke, which acts as a stopper to limit the deformation of the coil-collar assembly (Kaiser 1986). This, however, happens after a nonnegligible radial outward elastic expansion (∼0.1 mm) of the assembly in the median plane. For the LHC 8–10 T prototype dipoles (figure G4.0.7) a new type of mechanical structure was designed in which aluminium-alloy collars are surrounded by a vertically split yoke (with an open gap at room temperature) clamped by a stainless steel shrinking cylinder. The collars are clamped around the coils at room temperature with a moderate pressure to avoid risk of room-temperature creep. During the cooling process the shrinkage of the collars increases the pre-compression in the coils, while the yoke halves, which are actuated by the outer shrinking cylinder, move horizontally inwards, applying additional compressive forces to the collar-coil assembly in the direction exactly opposite to the main action of the electromagnetic forces. The gap between the two parts of the yoke closes at a predetermined lower temperature and when cool-down is completed a compressive force is produced at the mating face between the yoke halves. If this force is equal to or larger than the horizontal resultant of the electromagnetic forces, the gap remains closed in all operating conditions and the split iron behaves as a single stiff solid body. Coil displacement and deformation under the electromagnetic forces are, therefore, greatly reduced compared with those in a simple collar-clamping system, with beneficial effects on magnet stability and field quality.

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In the example of the LHC dipoles at 9 T field, the horizontal radial elastic expansion of the coil in the median plane is only about 0.05 mm. In the same conditions the maximum compressive stress in the coil inner layer is 81 N mm– 2, to be compared with 126 N mm– 2 computed for a structure of the same dimensions, but without the compressive pre-stress on the split-yoke mating faces. The price to be paid for this more efficient but more complex structure is that its components have to be produced with tighter dimensional accuracy to ensure that they fit together perfectly and the mechanical assembly of yoke and shell has to be tightly controlled, which is not easy to obtain in industrial production. Axial electromagnetic forces in high-field magnets produce large stresses and strains in the coil. A quick way to determine globally the axial force is to simply take the derivative of the magnetic stored energy W with respect to the axial direction z : dW/dz. In the LHC 15 m long dipole, at 9.6 T, close to the conductor short-sample limit, the stored energy per metre length is 670 kJ and the axial force is then 0.67 MN; the axial tensile stress in the coils would be about 100 MPa and the elastic elongation 12 mm. Magnet ends, therefore, have to be adequately supported by the external structure and detailed stress analysis has to be done for any end design. Part of the axial force is transferred by friction from the coils to the outer structure, usually the shrinking cylinder, via collars and yoke. The rest is taken by end-plates and from them transferred to the strongest longitudinal elements, normally the shrinking cylinder. The proportion of the axial force transferred by each of these two routes depends on the type of construction. Measurements have shown that with the LHC dipole construction type only 15–20% of the axial electromagnetic force is taken by the end-plates. Figure G4.0.22 is a longitudinal section of the magnet showing the special bolts which carry the force from the coil end-pieces to the end-plates.

Figure G4.0.22. A longitudinal section of the LHC dipole.

In lower-field magnets and generally in quadrupoles one may choose to let the coils be free to expand axially (e.g. in the LHC main quadrupoles). Design and stress analysis of the force-supporting structures are normally first carried out using the classical mechanical engineering methods and then refined with the aid of modern computer packages. This software can calculate the electromagnetic force distribution, the behaviour (stresses and strains) of the whole magnet and of each of its components under all operational conditions and during assembly. Many examples of this procedure can be found in the literature (e.g. Perin etal 1995). Copyright © 1998 IOP Publishing Ltd

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These design methods, however, rely on a thorough knowledge of the material properties, especially those of the coil package, which requires careful measurements of Young’s moduli and thermal expansion.

G4.0.10 Collider

The world’s largest application of superconductivity: the CERN Large Hadron

G4.0.10.1 General The Large Hadron Collider (LHC) is a superconducting accelerator/collider for protons, heavy ions and electron—proton collisions in the multi-teraelectronvolt energy region, which will be installed at CERN in the 27 km tunnel of LEP. It mainly consists of a double ring of high-field superconducting magnets operating in superfluid helium at a temperature of 1.9 K. To reach the desired beam energy ( 7 TeV for protons) the main bending dipole magnets will operate at 8.3 T and the focusing quadrupoles at 220 T m−1 gradient. These main magnets have a two-in-one configuration (see figure G4.0.7) with the magnetic channels for the two counter-rotating beams placed in common yokes and cryostats. The LHC will have nearly 10 000 superconducting magnetic units, including about 1300 14 m long main dipoles, about 400 3 m long main quadrupoles and a large number of corrector and special magnets. The conceptual design of the machine is reported in a book (LHC Study Group 1995) and a schematic layout is shown in figure G4.0.23. The main parameters for proton—proton operation are listed in table G4.0.4. Heavier ions can also be accelerated and collided, e.g. for lead (Pb) ions the centre-of-mass collision energy will be up to 1150 TeV with a luminosity of 1027 cm−2 s−1.

Figure G4.0.23. Schematic layout of the LHC. The beams of its two rings cross at four points around the circumference, where the planned experiments (ATLAS, CMS, ALICE and LHC-B) take place. Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.24. The LHC with an electron machine on top.

In addition there will be a unique possibility of colliding the high-energy protons with the electrons of a machine which could be installed using LEP components in the space kept free above the LHC (figure G4.0.24). G4.0.10.2 The magnet system In the long arcs (figure G4.0.23), the LHC is made up of regular cells of a bending/focusing configuration which is repeated periodically around the rings. There will be 23 cells, each 106.90 m long, in each octant. Each half-cell (figures G4.0.25 and G4.0.28) will be mainly composed of three dipoles about 14.2 m long and one 3 m long quadrupole. In addition to the regular arcs, there are other magnets in the so-called dispersion suppressor sections and on either side of each crossing point. In total, about 24 km of the LHC circumference will be occupied by superconducting magnets of different types. (a) Main dipoles The main parameters of the dipole magnet are listed in table G4.0.5 and its cross-section is shown in figure G4.0.7. The basic design features, confirmed by the results of the research and development programme, are cooling with superfluid helium at 1.9 K, a two-in-one configuration, two-layer coils, Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.25. Schematic layout of the LHC half-cell.

aluminium-alloy collars common to both apertures, a vertically split yoke and a stainless steel shrinking cylinder. (b) Main quadrupoles The main parameters of the lattice quadrupoles are listed in table G4.0.6 and a cross-section is shown in figure G4.0.26. G4.0.10.3 Cryogenics The magnets are immersed in a static superfluid-helium bath at atmospheric pressure and cooled by heat exchange with saturated superfluid helium flowing through a tube running through the magnet chain of each half-cell (see figure G4.0.27). The sub-cooled helium at 2.2 K is expanded to saturation through a Joule-Thomson valve and sent to the end of the half-cell, from where it returns, gradually vaporizing from heat exchange with the static bath in the heat exchanger tube. The loop is maintained at the saturation pressure of about 10 mbar. By exploiting the latent heat of vaporization, the temperature of each magnet is well controlled (<1.95 K) and independent of its distance from the cryoplant. The helium refrigeration Copyright © 1998 IOP Publishing Ltd

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Figure G4.0.26. Schematic cross-section of the lattice quadrupole.

Figure G4.0.27. The magnet cooling system.

capacity will be provided by eight 18 kW cryoplants located near the even points (figure G4.0.23) where most of the infrastructure already exists. Half of them are the four existing LEP cryoplants which will be boosted from 12 to 18 kW. Copyright © 1998 IOP Publishing Ltd

References

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Figure G4.0.28. A prototype half-cell of the LHC under test at CERN.

G4.0.10.4 Protection at quench The quench protection system is based on the so-called ‘cold diode’ concept. In a group of seriesconnected magnets, if one magnet quenches then the magnetic energy of all the magnets will be dissipated in the quenched magnet, thus destroying it. This is avoided by by-passing the quenched magnet and then rapidly de-exciting the unquenched magnets. The diodes are installed in the He-II cryostat of each dipole and quadrupole unit. In fact in each twin dipole or quadrupole, a set of two series-connected diodes is placed across the terminals. This solution provides safe blocking voltage at current ramp and a welcome redundancy: in case of failure of one diode, the LHC can still run, albeit at reduced ramp rate ( LHC Study Group 1995). The large stored energy ( 550 kJ m–1 ) of the magnets coupled with the relatively low natural quench propagation speed ( 10 to 20 m s–1 ) makes it necessary to detect quickly a developing quench and fire strip heaters which spread the quench over the full volume of the magnet. References Acerbi E, Bona M, Leroy D, Penn R and Rossi L 1994 Development and fabrication of the first 10 m long superconducting dipole magnet prototype for the LHC IEEE Trans. Magn. MAG-30 1793–6 Ahlbäck J, Ikäheimo J et al 1994 Electromagnetic and mechanical design of a 56 mm aperture model dipole for the LHC IEEE Trans. Magn. MAG-30 1746–9 Alexandrov A et al 1994 Investigation of the string of four UNK superconducting magnets Proc. 4th Eur. Particle Accelerator Conf. (London, 1994) vol 3, ed V P Suller and C Petit-Jean-Genaz (Singapore: World Scientific) Auzolle R, Patoux A et al 1984 Construction and test of superconducting quadrupole prototypes for HERA J. Physique Coll. 45 Cl 263–70 Baze J M, Cacaut D et al 1992 Design and fabrication of the prototype superconducting quadrupole for the CERN LHC Project IEEE Trans. Magn. MAG-28 335–7 Billan J et al 1979 A superconducting high-luminosity insertion in the intersecting storage rings (ISR) IEEE Trans. Nucl. Sci. NS-26 Blewett J P 1971 200-GeV intersecting storage accelerator Proc. 8th Int. Conf. on High Energy Accelerators (Switzerland, 1971) ed M H Blewett and N Vogt-Nilsen (Geneva: CERN) pp 501–4 Copyright © 1998 IOP Publishing Ltd

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Bona M, Penn R and Rossi L 1997 Status of the construction of the first 15 m long superconducting dipole for the LHC Proc. MT15 (Beijing, 1997) Claudet G and Aymar R 1990 Tore Supra and He II cooling of large high field magnets Adv. Cryogen. Eng. 35 55–67 Cole F T, Donaldson M R et al (eds) 1979 A Report on the Design of the Fermi National Accelerator Laboratory Superconducting Accelerator (Batavia: FNAL) Cooper W E, Fisk HE et al 1983 Fermilab Tevatron quadrupoles IEEE Trans. Magn. MAG-19 1372–7 Dahl P, Cottingham J et al 1985 Performance of four 4.5 m two-in-one superconducting R&D dipoles for the SSC IEEE Trans. Nucl. Sci. NS-32 3675–7 den Ouden A, Wessel S, Krooshoop E, Dubbeldam R and ten Kate H H J 1994 An experimental 11.5 T Nb3Sn LHC type of dipole magnet IEEE Trans. Magn. MAG-30 2320–3 den Ouden A and ten Kate H H J 1997 Quench characteristics of the 11 T Nb3Sn dipole magnet MSUT Proc. MT-15 (Beijing, 1997) Devred A and Ogitsu T 1996 Ramp rate sensitivity of SSC dipole magnet prototypes Frontiers of Accelerator Technology (Singapore: World Scientific) Ducos G, Giacometti J et al 1994 High energy booster quadrupole cold mass development and industrialization program Supercollider 5: Proc. 5th Int. Symp. on the Super Collider (San Francisco, CA, 1993) ed P Hale (New York: Plenum) pp 105–8 Goodzeit C, Wanderer P et al 1990 Status report on SSC R&D dipole magnet tests results New Technologies for Future Accelerators vol III, ed G Torelli pp 147–60 Hanft R W and Brown B C 1989 Studies of time dependence of fields in Tevatron superconducting dipole magnets IEEE Trans. Magn. MAG-25 1647–51 Holzer B J 1996 Impact of persistent currents on accelerator performance Proc. CERN Accelerator School on Superconductivity in Particle Accelerators (Hamburg, 1996) CERN 96–03 Jackson J D (ed) 1988 Conceptual Design of the Superconducting Super Collider SSC-SR-1020, March 1986, revised September 1988 Kaiser H 1986 Design of superconducting dipole for HERA 13th Int. Conf. on High Energy Accelerators (Novosibirsk, 1986) and DESY-HERA 1986–14 Koepke K, Kalbfleisch G et al 1979 Fermilab doubler magnet design and fabrication techniques IEEE Trans. Magn. MAG-15 658–61 Kuzminski J, Bush T et al 1993 Quench performance of 50-mm aperture, 15-m long SSC dipole magnets built at Fermilab Int. J. Mod. Phys. A (Proc. Suppl.) 2B 588–91 Leroy D, Perm R, Perini D and Yamamoto A 1989 Structural analysis of the LHC twin-aperture dipole Proc. 11th Int. Conf. on Magnet Technology (Tsukuba, 1989) LHC Study Group 1995 LHC, the Large Hadron Collider CERN/AC/95–05 (LHC) Lietzke A F, Barale P et al 1994 SSC quadrupole magnet performance at LBL Supercollider 5: Proc. 5th Int. Symp. on the Super Collider (San Fransisco, CA, 1993) ed P Hale (New York: Plenum) pp 109–12 Lindner M et al 1994 Construction details and test results from RHIC sextupoles IEEE Trans. Magn. MAG-30 1730–3 Livingood J J 1961 Principles of Cyclic Particle Accelerators (New York: Van Nostrand) Mclntyre P M, Scanlan R M and Shen W 1994 Ultra-high-field magnets for future hadron collider Proc. ASC-94 (Boston, 1994) Meinke R 1991 Superconducting magnet system for HERA IEEE Trans. Magn. MAG-27 1728–94 Mess K H and Schmiiser P 1989 Superconducting accelerator magnets CERN Accelerator School on Superconductivity in Particle Accelerators CERN 89–04 Mess K H, Schmuser P and Wolff S 1996 Superconducting Accelerator Magnets (Singapore: World Scientific) Ogitsu T, Akhmetov A et al 1993 Mechanical performance of 5-cm aperture, 15-m long SSC dipole magnet prototypes IEEE Trans. Appl. Supercond. AS-3 686–91 Peoples J for the Team at BNL, Fermilab, LBL and the SSC Central Design Group 1989 Status of the SSC superconducting magnet program IEEE Trans. Magn. MAG-25 1444–50 Perin R 1975 Mechanical stability of superconducting quadrupole coils Proc. 5th Int. Conf. on Magnet Technology (Rome, 1975) pp 551–6 —1990 High-field superconducting magnets for particle accelerators New Techniques for Future Accelerators III ed G Torelli (New York: Plenum) pp 87–106

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—1991 The superconducting magnet system for the LHC IEEE Trans. Magn. MAG-27 1735–42 —for the LHC Magnet Team 1994 Status of the Large Hadron Collider magnet development IEEE Trans. Magn. MAG-30 1579–86 —1995 Field, forces and mechanics of superconducting magnets Proc. CERN Accelerator School on Superconductivity in Particle Accelerators (Hamburg, 1995) CERN 96–03 —1996 Superconducting magnets for the Large Hadron Collider Project Report CERN, Geneva Perin R, Perini D, Salminen J and Soini J 1995 Finite element structural analysis of LHC bending magnet Proc. MT14 (Tampere, 1995) Perin R and van der Meer S 1967 The Program ‘MARE’ for the Computation of Two-Dimensional Static Magnetic Fields CERN 67–7, Intersecting Storage Rings Division (Geneva: CERN) Perin R, Tortschanoff T and Wolf R 1979 Magnetic Design of the Superconducting Quadrupole Magnets for the ISR High-Luminosity Insertion CERN-ISR-BOM/79–2 (Geneva: CERN) Perot J and Rifflet J M 1991 Measurement data taken during the industrial fabrication of the HERA superconducting quadrupoles Supercollider 3: Proc. IISSC (Atlanta, GA, 1991) ed J Nonte (New York: Plenum) pp 313–24 Rifflet J M, Cortella J, Deregel J, Genevey P, Perot J, Vedrine P, Henrichsen K H, Rodriguez-Mateos F, Siegel N and Tortschanoff T 1994 Cryogenic and mechanical measurements of the first two LHC lattice quadrupole prototypes Proc. 4th Eur. Particle Accelerator Conf. (London, 1994) ed V P Suller and C Petit-Jean-Genaz (Singapore:World Scientific) pp 2265–7 Russenschuck S 1995 A computer program for the design of superconducting accelerator magnets CERN AT/95–39 (MA), LHC Note 354 Sanford J R and Matthews D M (eds) 1990 Site-Specific Conceptual Design of the Superconducting Super Collider SSCL-SR-1056 Schermer R I 1994 Status of superconducting magnets for the Superconducting Super Collider IEEE Trans. Magn. MAG-30 1587–94 Schmuser P 1992 Properties and practical performance of SC magnets in accelerators Proc. 3rd Eur. Particle Accelerator Conf. (Berlin, 1992) (Gif-sur-Yvette: Edition Frontieres) pp 284–88 Shintomi T et al 1996 Development of a 56 mm aperture superconducting dipole magnet for LHC Proc. Applied Superconductivity Conf. (Pittsburgh, PA, 1996) Siemko A 1997 private communication Spigo G, Cunningham G, Goodzeit C, Orrell D, Turner J and Jayakumar R 1994 Design and performance of a new 50 mm quadrupole magnet for the SSC Supercollider 5: Proc. 5th Int. Symp. on the Super Collider (San Fransisco, CA, 1993) ed P Hale (New York: Plenum) pp 647–51 Taylor C E, Barale P et al 1991 A 40 mm bore quadrupole magnet for the SSC IEEE Trans. Magn. MAG-27 1888–91 Thompson P, Anerella M et al 1993 B series RHIC arc quadrupoles Proc. 1993 IEEE Particle Accelerator Conf. (Washington, DC, 1993) (Piscataway, NJ: IEEE) pp 2766–8 Thompson P A, Gupta R C et al 1991 Revised cross section for RHIC arc dipole magnets Proc. 1991 IEEE Particle Accelerator Conf. (San Francisco, CA, 1991) (New York: IEEE) pp 2245–7 Tollestrup A V 1979 Progress report—Fermilab energy doubler IEEE Trans. Magn. MAG-15 Wanderer P, Anerella M et al 1993 Magnetic design and field quality measurements of full length 50-mm aperture SSC model dipoles built at BNL Int. J. Mod. Phys. A (Proc. Suppl.) 2B 641–3 Wanderer P, Cottingham J et al 1990 Dipole magnet development for the RHIC accelerator New Technologies for Future Accelerators vol III, ed G Torelli pp 175–88 Wanderer P et al 1991 Effect of prestress on performance of a 1.8 m SSC R&D dipole Supercollider 3: Proc. IISSC (Atlanta, GA, 1991) ed J Nonte (New York: Plenum) Wolff S 1988 Superconducting HERA magnets IEEE Trans. Magn. MAG-24 719–22 Yamamoto et al Development of IOT dipole magnets for the Large Hadron Collider IEEE Trans. Appl. Supercond. AS-3 769–72

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G5 Superconducting synchrotron x-ray sources

Martin N Wilson

G5.0.1 Introduction Synchrotron radiation is produced whenever beams of charged particles are constrained to follow a circular orbit by the application of a magnetic field. If the magnetic field strength is increased, the orbit is curved more tightly and, in consequence, the individual photon energies and total radiated power are increased. For this reason, it is often advantageous to use high-field superconducting magnets in synchrotron radiation sources. Synchrotron radiation is the most intense known source of continuous x-ray power. The emitted photon beam is tightly collimated in one direction, and the spectrum of the radiation may be adjusted by changing the magnetic field. For all these reasons, synchrotron radiation has become an important experimental tool in a wide variety of research activities: biological sciences, crystallography, surface science, solid-state physics, microstructural analysis, photoelectron spectroscopy, fluorescence and absorption spectroscopy, photo-chemistry and many others. There is an increasing industrial interest in using the intense and highly collimated x-ray beam for replicating the fine circuit patterns on semiconducting microchips, and also for producing mechanical devices of very small size and high precision. Like all other kinds of electromagnetic radiation, synchrotron radiation is produced by the acceleration of a point charge-in this case, by the centripetal acceleration of motion around a circular orbit. Figure G5.0.1 sketches the electromagnetic radiation emitted by a charged particle moving around a circular orbit under the influence of a magnetic field. At low energies, the emitted radiation follows a classical dipole pattern but, as the energy is increased (figure G5.0.1(b)), relativistic effects cause the radiation to become progressively more sharply peaked in the forward direction. G5.0.2 Some basic expressions A single charged particle travelling around a circle emits energy U per turn, where

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Figure G5.0.1. Synchrotron radiation emission from: (a) a single electron at low velocity, (b) a single electron at high velocity and (c) a continuous circulating beam of electrons, producing a radiation fan of vertical opening angle δ φ.

where γ is the relativistic ratio of total energy to rest mass energy

where e is the charge and ρ is the radius of the circle, all in electrostatic units (esu), see for example Koch et al (1987) or Margaritondo (1988). It is easy to see from equation (G5.0.1) why electrons (or positrons) are always used in synchrotron radiation sources. Because of its small rest mass, the relativistic γ of an electron at a given energy is much higher than that of other charged particles. In comparison with the proton for example, γ at the same total energy is 1836 times higher and the energy emitted per turn is 18364 = 1013 times higher. Thus the world’s largest planned accelerator, the giant large hadron collider ( LHC ) at CERN with a proton energy of 7 TeV, will emit no more synchrotron power than a small electron ring. Converting equations (G5.0.1) and (G5.0.2) to practical units and substituting the rest mass of the electron (0.511 MeV) we find

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where E is measured in MeV and U in eV. The radius of bending is related to magnetic field B by

in esu. Converting to practical units and approximating for electrons at relativistic energies

where B is in tesla and ρ is in metres. When many particles travel in orbit, the total radiation power is simply equal to the sum of their individual contributions. It follows that, to obtain a high radiation power, one needs to have many particles in orbit. Practical x-ray sources therefore seek to maximize the orbiting current by recirculating a beam of electrons in a closed loop-a storage ring. In such an arrangement, beams of electrons from an injection accelerator are successively ‘stacked’ in the ring until circulating beams of a fraction of an ampere are obtained. From (G5.0.3) and (G5.0.5), an electron current I (A) travelling around a circle of radius ρ (m) in a field B ( T ) will radiate power P ( W )

For example, a typical storage ring x-ray source with conventional bending magnets of field ∼1.3 T might store an electron current of 200 mA at an energy of ∼1500 MeV. From (G5.0.6) the total x-ray power emitted is ∼23 kW. Note from (G5.0.6) and (G5.0.5) that increasing the bending field for a given energy will increase the x-ray power and reduce the bend radius. In practice, however, the higher fields of superconducting magnets have been used to enable similar powers to be obtained from electron beams at lower energy, with even greater reductions in radius. As shown in figure G5.0.1(c), when many particles circulate continuously in orbit, the emitted radiation becomes a continuous fan. The opening angle of this fan δφ (radians) is given approximately by

For our typical energy of 1000 MeV, the opening angle δ φ ∼ 1 mrad. At a distance of 10 m, the fan is thus only 10 mm high-a very tightly collimated beam. G5.0.3 Storage rings Figure G5.0.2 is a simplified schematic showing basic features of an electron storage ring complete with its injector. Energetic beams of electrons are produced in the injector, which is usually a linear accelerator (linac), or alternatively a microtron or a small booster synchrotron. These electrons are transported from the injector along the injection beamline to the septum magnet. This is a special design of (resistive) magnet, which uses a thin current sheet or septum to produce magnetic field on one side and zero field on the other side. In this case, the field side is used to bend the injected beam so that it is parallel to the orbit in the ring, while the zero-field side produces no deflection of the beam which is already circulating in the ring. Once inside the storage ring, the injected beam is constrained to follow a closed orbit by the vertical magnetic field of the bending magnets, in this example four in number. An essential player in the injection process is the kicker, which is a pulsed magnet located some distance upstream of the septum. Just before Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.2. A schematic diagram of a simple storage ring, showing how electrons are injected from the linac via the septum magnet, are bent into a closed loop by the four dipole magnets, focused by the quadrupole magnets and accelerated by the radiofrequency cavity. Synchrotron radiation is emitted wherever the electrons are bent by the dipole magnets.

each pulse of electrons is produced by the injector, the kicker magnet is energized to deflect the already circulating beam very close to the septum. In this way, the newly injected electrons are captured into the same closed orbit as the already circulating electrons. When the injection pulse is finished, the kicker magnet is switched off and the orbiting electrons return to their original orbits together with the newly injected electrons. To accumulate sufficient current in the ring, several hundred pulses of electrons must be injected. This process is known as multi-shot injection. To confine the electrons in stable orbits, it is also necessary to provide some transverse focusing effect, otherwise electrons with a small divergence from the correct orbit would be lost after only a few turns. Remember that, in a storage ring, the electrons complete many millions of orbits per second. Focusing is usually provided by magnetic field gradients produced in quadrupole magnets. A single quadrupole magnet produces a focusing effect in one transverse plane, and a defocusing effect in the other. However, if two quadrupole magnets of oppositely directed gradients are used together, the net effect will be one of focusing. This is the principle of alternating gradient focusing (Livingood 1961) which is used in all modern particle accelerators. Although the electrons are completely confined by the transverse focusing fields, they do nevertheless oscillate from side to side about the equilibrium orbit as they progress around the ring, an effect known as betatron oscillation. As the strength of the focusing force is increased, the amplitude of betatron oscillations decreases and their frequency increases. The ratio of the betatron oscillation frequency to the revolution frequency is known as the tune of the ring Q , it is in fact the number of betatron oscillations per revolution. Tunes can be, and usually are, different in the two transverse directions, i.e. Qx and Qy . Exact integer or fractional tunes are to be avoided because they mean that an oscillation will keep returning to the same point on successive orbits. In this way, the effects of small errors can be amplified in a resonance and cause the beam to ‘blow up’ and be lost. As soon as the injected beam is accumulated into an orbit, it starts to lose energy by synchrotron Copyright © 1998 IOP Publishing Ltd

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radiation each time it traverses a bending magnet. This energy loss is replenished by the radiofrequency (RF) cavity, which applies an alternating electric field parallel to the orbit. During the positive parts of the RF cycle, this field accelerates the electrons, but during the negative parts it decelerates them. Responding to this input, the electron beam divides itself into a set of bunches with spaces in between, such that electrons in each bunch gain exactly the right energy to replenish their synchrotron radiation losses. By the principle of phase stability (Livingood 1961), electrons which gain too much energy on one pass through the cavity adjust their phase such that at the next pass they gain less energy and thereby on balance receive an exact replenishment of their synchrotron radiation losses. The effect is one of focusing in the longitudinal direction, very similar to the transverse focusing. The number of bunches around the orbit, known as the harmonic number, is equal to the ratio (which must be an integer) between the RF frequency and the orbital revolution frequency. Cavities typically operate at frequencies between 50 and 500 MHz and produce accelerating voltages of several hundred kilovolts. Some storage rings are injected at their full operating energy, in which case the cavity only serves to replenish energy lost by synchrotron radiation. However, the availability of a cavity also gives the option of accelerating the stored beam. The advantage of so doing is that the injection accelerator can then be of lower energy and therefore smaller and cheaper. When sufficient current has been accumulated at injection energy, acceleration is started by simultaneously ramping the fields in all magnets around the ring. Following the principle of phase stability, the electrons adjust their phase at the RF cavity to gain sufficient energy such that they remain in the same geometrical orbit, with their energy increasing in synchronism with the rising magnetic field. This effect is known as synchrotron acceleration. Throughout the process, the RF cavity provides the energy input for acceleration plus replenishment of the loss via synchrotron radiation. It may be seen from equation (G5.0.6) that the latter increases very rapidly with the increasing energy of the beam and the cavity voltage must accordingly be increased to keep up. When the magnets reach their maximum field, ramping stops, acceleration stops and the beam continues to circulate at its full energy. Stored electron beams can circulate for many hours, but their lifetime is finite because of a number of disruptive effects. The most powerful of these effects is scattering by residual gas molecules in the vacuum vessel of the ring. It follows that, to achieve the best lifetime, one must pay careful attention to maintaining good ultra-high-vacuum (UHV) conditions in the ring. Unfortunately, the beam itself tends to spoil the vacuum because the emitted x-rays, where they hit the interior of the vacuum vessel, cause photo-desorption of any adsorbed gases. Fortunately, this effect falls off with time as the x-ray beam ‘scrubs’ the interior surfaces. The existence of cryogenic surfaces within the vacuum space can be extremely helpful in achieving good vacuum. With a clean vacuum system the circulating beam can survive for a day or longer. In this almost steady state, the circulating loop of electrons is effectively serving the purpose of a transducer, which converts the electromagnetic energy of the cavity into the (much shorter-wavelength) electromagnetic energy of emitted x-rays. G5.0.4 Scaling laws In this section, we discuss the spectrum of synchrotron radiation and show how, by a suitable choice of storage ring parameters, we can match that spectrum to a particular application. We look at the scaling laws with respect to magnetic field and show that the higher bending fields available from superconducting magnets can bring substantial benefits for some applications. As an example, we consider the application which has so far been the largest user of superconducting synchrotrons: the production of microchips by x-ray lithography (IBM 1993). This is a pattern replication process, in which the x-rays are shone through a mask bearing the pattern of the circuit to be replicated. On the other side of the mask is a silicon wafer coated with photoresist, a suitable polymer which is sensitive to x-radiation. By exposing the mask-wafer combination to x-rays, an image of the mask is

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imprinted on the photo-resist, exactly like contact printing in photography. The highly collimated nature of synchrotron radiation means that the shadow cast is extremely sharp, and the high intensity means that exposure times are only ∼1 s. Because x-ray wavelengths are much shorter than those of the visible light used in present day photolithography processes, much finer patterns can be printed onto the silicon wafer. Although not yet in production, it is likely that microchips with circuit patterns <0.2 µm will be made by x-rays. Processes developed to date for x-ray lithography require photons with energies in the range 1000-1800 eV. The emission of x-ray photons by synchrotron radiation follows a universal law (Margaritondo 1988) whereby the number of photons emitted per second per milliradian of bend within a frequency bandwidth of 0.1 % is given by

where hv is the energy in eV of the emitted photons, I is the circulating current in milliamps and y is the ratio

The critical photon energy hvc is defined as

It is that point of the spectrum which has an equal power above and below it. Substituting for the emitted photon energy, we find the power per milliradian in a 0.1% bandwidth

The function yG1( y ), shown in figure G5.0.3, is a universal function for synchrotron radiation, independent of the electron energy, ring size, etc. It may be seen that peak power occurs at y = 1.3. Let us now define a desired photon energy hvw as that which is in the middle of our applications window, i.e. at 1400 eV for x-ray lithography. To put hvw on the peak of the spectral power curve, we must choose hvc such that yw = hvw /hvc is equal to 1.3. However, this choice does not produce the maximum power in the applications window because hvc also occurs as a multiplier in equation (G5.0.11). Optimizing equation (G5.0.11) for maximum power at hvw we find another criterion yw = 0.3. In practice, however, this criterion produces an inconveniently large power at photon energies higher than hvw and yw is therefore chosen to be somewhere between these extreme values (Wilson et al 1993), usually in the range 0.6 to 1. As an example of a ring designed for lithography, figure G5.0.4 shows the power spectrum of the Oxford ring Helios, which was designed with hvc = 1467 eV, i.e. yw = 0.95. Also shown on the curve is the process window for x-ray lithography, extending between 1000 and 1800 eV. From equation (G5.0.10), it may be seen that choosing yw and thereby defining hvc creates a link between the bending field and bending radius, that is ρ ∼ B – 3/2. Rings designed for constant hvc can thus be much more compact if high-field magnets are used and the factor of advantage is stronger than the more familiar linear relationship implied by equation (G5.0.5) for rings at constant electron energy. Another important scaling law concerns the multi-shot injection process described in section G5.0.2. Because electrons cannot be injected exactly into the closed orbit of the already circulating beam, they oscillate about that orbit at the betatron frequency. The next shot cannot be injected until these oscillations Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.3. The function y G1( y ) versus y = hv/hvc , showing a maximum at y = 1.3.

have died away, which they do in a characteristic time called the damping time. Prompt damping of the oscillations is clearly desirable in enabling one to inject at a faster rate. Strong damping can also be very useful in suppressing certain potentially disruptive instabilities which can occur during the injection process. From Sands (1979) the damping time for these oscillations of the electron beam is given by

where Ei is the injection energy, t is the orbit revolution time, U is the energy loss per turn from equation (G5.0.3) and the factor 10– 6 is added to the expression in (G5.0.5) to put U into the same units as Ei . The damping partition coefficient is a number of order unity and is a function of the focusing fields around the orbit. To eliminate t from equation (G5.0.12), we substitute a factor m for the fraction of orbit perimeter occupied by bending magnets. The orbit time t is therefore 2πρ/mc, where c is the velocity of light. Eliminating ρ between equations (G5.0.3), (G5.0.10) and (G5.0.12) we find

Thus, for a fixed critical photon energy (prescribed by the needs of the application), a fixed damping Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.4. The power spectrum of Helios, showing the preferred window for lithography and the critical photon energy ( 700 MeV electrons, 250 mA stored beam in 4.5 T ).

time (prescribed by the injection process) and a fixed and m (prescribed by the layout of the ring), the injection energy is inversely proportional to the maximum field in the bending magnets. We now use these scaling laws to make some general statements about the advantages of superconducting magnets in synchrotron radiation rings. The field of conventional magnets is limited by saturation of the iron to ∼1.8 T. However, synchrotron sources impose additional constraints because it is necessary to leave a clear gap at the outboard side of the magnet, so that the x-rays can emerge. This creates an asymmetry in the magnetic circuit, which means that distortions of the field shape occur at lower fields than in conventional magnets and limits the maximum field to typically ∼1.3 T. Superconducting magnets, of course, are not limited by iron saturation and can achieve much higher fields. However, it is a matter of common experience that the field limitation of transverse field bending magnets is significantly lower than for simple solenoids. In addition, the need to leave an open slot for the emerging x-rays raises serious mechanical stress problems and has so far limited the maximum field to ∼4.5 T. The factor of advantage of superconducting versus conventional magnets is thus (4.5/1.3) = 3.5. From equation (G5.0.10) we see that the bending radius for a given hvc is accordingly reduced by a factor 3.53/2 = 6.5. In the case of x-ray lithography with hvw = hvc = 1400 eV, we find bending radii of 0.5 m for superconducting and 3.2 m for conventional magnets. It is the compactness of superconducting rings which has made them the preferred choice for x-ray Copyright © 1998 IOP Publishing Ltd

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lithography. The extremely clean conditions of a modern wafer fabrication facility mean that the price of floor space is at a premium-thus the compact ring is cheaper to install. Not so obvious, but equally important, is the fact that the compact superconducting ring can be transported intact, as shown in figure G5.0.5. This means that the ring can be commissioned and proven at the vendor’s factory, shipped to the user site as a fully tested unit, and subsequently be up and running quickly. Given the interest cost on the $ 1000m price tag of a modern wafer fabrication facility, the need to minimize delays is obvious.

Figure G5.0.5. Helios arrives at IBM East Fishkill after transportation by road and the roll-on roll-off transatlantic ferry.

A typical injection energy for compact superconducting rings is ∼100-150 MeV. From equation (G5.0.13) we see that a conventional ring to produce the same hvc and having the same damping time, focusing parameters, etc would need to be injected at 3.5 times the energy, that is 350-525 MeV. The cost and space requirements of such a large injector are substantial. G5.0.5 Superconducting magnets: some basic design choices Assuming that the arguments of section G5.0.4 have been sufficiently satisfactory to decide in favour of a superconducting ring, we now consider some of the basic design choices for the magnet. The first and most basic choice, as illustrated in figure G5.0.6, is how many bending magnets there should be in the ring. The usual arguments about ratios of surface to volume say that the cryogenic loss of the system will be better served by having one big magnet than many small ones. In addition, the most significant field errors occur at the ends of magnets. If the number of magnets, and hence the number of ends, is reduced, then the error field as a proportion of the total, will be reduced. Both of the above considerations would seem to point strongly in the direction of just one magnet which, as shown in figure G5.0.6, would be circular and would occupy the whole orbit. Indeed such a layout is adopted for the ring Aurora, which will be described in section G5.0.6. However, although it brings the simplest magnet and most compact ring design, the single magnet does entail some problems. As Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.6. Numbers of magnets in a superconducting ring.

shown in figure G5.0.2, the storage ring orbit must accommodate many functions besides straightforward bending. There must be space for focusing, RF cavity and pulsed magnets for injection, as well as various diagnostic monitoring devices. For this reason, most designs to date have opted for the inclusion of two or more straight sections in which these components may be located. In fact the majority of rings have taken just two straight sections in a layout commonly known as the racetrack. The two 180° superconducting bending magnets are contained in separate cryostats, with all other components being located in the straight sections at room temperature for ease of access. Using more than two bending magnets can be advantageous in rings where it is desired to achieve a very small electron beam size. A large number of straight sections enables one to include many quadrupoles, whose strong focusing effect squeezes the beam down. Small beam sizes are important for certain research applications which require x-ray beams of high brilliance, but they are not required for x-ray lithography. In these rings, however, it has not so far proved attractive to use superconducting bending magnets because the overall ring size is determined more by the length of the straight sections than by the bending magnets. For the superconducting magnets themselves, the first design choice is whether to use iron or not. Iron brings the advantage that it shields the fringing field of the magnet and also makes some contribution to the field within the electron beam aperture, but iron has the big disadvantage of field distortion. At the high fields used in superconducting magnets, the iron is driven heavily into saturation and this causes the field to distort. It is particularly a problem during the ramping process, where the degree of saturation varies continually as the maximum field is approached. As noted in the previous section, the need to leave a slot for the emerging x-ray beam makes this distortion worse. With air-cored magnets, there is no field distortion from saturation and the field shape is linear from injection right up to full energy. In addition, the magnets are very much lighter-typical weight of an iron yoke is 25-50 t (25-50 × 103 kg). As noted earlier, lightness of weight, with the consequent transportability, are important features in an industrial machine. Against these advantages must be set the fringing field and the somewhat larger number of ampere turns needed to produce a field without iron. Fringing fields can create some practical problems, for example they can affect relays, some electronic Copyright © 1998 IOP Publishing Ltd

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circuits and vacuum pumps with moving parts. Electron trajectories in the injected beam can be somewhat distorted. However, with foresight and careful design, all these problems can easily be overcome. The effect of fringe fields on people is not a strong consideration here because the storage ring must be enclosed within a radiation shielded enclosure and the fringe field outside this enclosure is negligible. Although the author’s preference is for coils without iron, this question is certainly not settled, as may be seen in the next section where more than half the rings described use iron. Following on from any decision about iron comes the question of coil shape. As shown in figure G5.0.10, coils designed for compact rings must be bent in the direction of particle motion, and this makes their manufacture complicated. There is a problem of negative curvature at the inboard side of the winding, which has been solved in a variety of ingenious ways, most of which are regarded by their inventors as proprietary. In solving this problem, designers have generally found it more convenient to use coils of a simple rectangular cross-section, rather than the ‘cos Θ’ style of winding used in magnets for high-energy proton rings. Careful design of the end turns is important, for two reasons. Firstly, the peak fields and stresses are invariably to be found in the end windings, and naturally this makes the ends the most likely area for quenching to be initiated. Secondly, the errors due to the end fields become proportionately much more important when the length of the magnet is as small as it is in a compact ring. Higher-order harmonics from the ends account for most of the nonlinear field effects encountered by the electron around its orbit. If the ring is to be injected at low energy, careful attention must be paid to the field quality at injection. Because the injection field is small, in some cases as low as 1/40 of the full energy field, the effect of error components is proportionately much larger than at full field. The two principal sources of error are magnetization of structural materials in the magnet or cryostat and magnetization of the superconducting filaments. In designing the magnet structure and cryostat, care must be taken to avoid the use of magnetic materials. Cryogenic grade austenitic stainless steels of the type 304L and 316L are to be preferred; nitrogen doping in types 304LN and 316LN provides increased strength and reduces the tendency of work-hardened material to become slightly magnetic at low temperature. Particular attention must be paid to welding, which always brings the possibility of ferritic inclusions. When in doubt, always measure the weld permeability, both before and after cooling to low temperature. To reduce the superconductor magnetization, filament diameter should be small; < 10 µ m is sufficient in most instances. For convenience in winding coils of complex shapes, it is generally preferable to have a conductor of reasonably large cross-section-say several square millimetres. The classical way to achieve large conductor size with fine filament size is to use a cable consisting of many strands, each of which contain ∼1000 filaments. The fully transposed ‘Rutherford’ style of cable is usually preferred because it provides good dimensional stability together with full electrical transposition of the filaments. During ramping from injection field to full field, the changing field tends to couple filaments together within the individual wires and to couple wires together within the cable. To reduce this effect, which increases the field error and a.c. loss, both the wires and the cables should be tightly twisted. Some a.c. losses will occur during the ramping, and it is desirable to provide good cooling inside the winding by retaining a certain amount of porosity in the liquid helium. This is usually done by leaving the cable and insulating wrap slightly open. Porosity also provides a valuable contribution to stability against slight movements of the conductor under the influence of electromagnetic stresses. For those rings which are injected at full energy, the field may be left permanently at its full value. In this case, it may be advantageous to operate in the persistent mode, with a persistent current switch across the terminals of the magnet. When the magnet has been charged to its full current, the current leads may be removed if desired, and in this way the cryogenic heat load may be reduced. Persistent operation is not recommended for ramped rings, however, even if the lifetime is such that the ring remains at full field for as long as 24 h. The problem is that persistent current switches have a finite resistance and, during ramping, a fraction of current will short circuit the coil and thus produce a nonlinear ramp, which will Copyright © 1998 IOP Publishing Ltd

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change the tunes Qx and Q y . It is desirable to provide ‘trim’ coils which introduce controllable amounts of higher-order field component, usually quadrupole and sextupole. Because these field contributions are relatively modest, the trim coils have cross-sections of only a few square millimetres and are wound from a single-wire conductor, operating at currents of less than ∼50 A. Trim coils serve two purposes: to correct for errors in the main coils, and to adjust the electron beam parameters. The usual strategy is to measure electron beam parameters at discrete energy levels during the ramp, and adjust the trim coils to produce desired values of tune, chromaticity, etc. The ring’s computer control system is then instructed to learn these discrete values of trim coil energization, and interpolate between them to produce a continuous variation of trim current during the ramp. Cryogenic cooling for all magnets constructed so far has consisted of simple bath cooling with helium boiling under ambient pressure. Typical heat leaks per magnet are in the range 5-15 W and, for this reason, it is almost essential to use a closed-cycle liquid helium refrigerator. Because the suction side of the compressor runs at a pressure of ~100 mbar, the boiling temperature of liquid in the magnet is typically 4.3–4.4 K. The presence of a liquid helium vessel with surface area of several square metres at a temperature of ∼4.4 K raises the obvious possibility of cryogenic pumping. This can be very valuable in the region of the bending magnet, because it provides pumping exactly where it is most needed. As mentioned in section G5.0.3, photo-desorption is the biggest vacuum load in storage rings and of course the x-rays which cause photo-desorption are only emitted in the region of the bending magnet. A surface at 4.4 K in this region can therefore be an extremely effective pump, particularly when the electron beam is almost completely enclosed by the cryogenic surface. To take advantage of the cryogenic pumping effect, it is only necessary to make a common vacuum space for the magnet and electron beam. In fact, this can ease engineering problems by eliminating the need for a vacuum vessel and shield within the magnet aperture. However, the commonality of vacuum does raise one operational problem. If it is necessary to vent the vacuum for any reason, the magnets must first be wanned up, which may take 2-3 days. Because most of the failure-prone components are located in the warm straight sections, the problem can largely be avoided by putting vacuum valves between the magnets and the rest of the ring. Of course, if the problem is in a magnet it will still be necessary to warm it up, but this would be the case with a warm or cold vacuum. The author’s opinion is therefore that the benefits of ‘cold vacuum’ greatly outweigh the disadvantages. However, this is by no means a consensus view, and many rings described in the next section have a warm vacuum, with their cryogenic and electron beam vacuum systems being quite separate. G5.0.6 Operational rings In this section, we illustrate some of the considerations discussed so far by describing the five superconducting compact rings which are currently in operation. Table G5.0.1 summarizes the key properties of these rings. We start with Helios, which is described in rather more detail than the other rings, not because it is any more important, but merely because it is more familiar to the author. (a) Helios Helios (Wilson et al 1993) was ordered by IBM from Oxford Instruments as the world’s first commercial compact ring for x-ray lithography. In designing Helios, Oxford Instruments benefited greatly from a collaboration with the Daresbury Laboratory. Figure G5.0.7 shows a simplified schematic diagram of Helios and figure G5.0.8 shows a picture of it in the clean room at Oxford Instruments after assembly and before commissioning. Helios was delivered to the IBM East Fishkill site in spring 1991; the first electron beams were stored within two months of arrival and full specification was achieved some two months

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Figure G5.0.7. A simplified schematic diagram of Helios.

Figure G5.0.8. Helios after completion of assembly in the clean room at Oxford Instruments.

later. Helios now runs as a routine facility for the development of x-ray lithography techniques and has achieved a remarkably high level of reliability, with x-rays being provided for 99.5% of the scheduled Copyright © 1998 IOP Publishing Ltd

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FIGURE G5.0.9. Schematic cross-section of the Helios bending magnet inside its liquid helium vessel, surrounded by the liquid-nitrogen-cooled radiation shield and outer vacuum tank.

time in 1993. As shown in figure G5.0.9, the cold vacuum bore configuration was chosen for the superconducting bending magnets; there is a common vacuum between both cryostats and straight sections. Around the magnets, there is a clear slot around 180° of the median plane to allow unimpeded exit of the x-rays. Superinsulation cannot be used because the cryogenic space must achieve UHV pressures and it is therefore necessary to surround the liquid helium vessel by a radiation shield. To minimize eddy current effects, the shield is a stainless steel fabrication, with liquid nitrogen cooling pipes welded to it. Of course it must also have an x-ray slot on the median plane. To minimize the amount of room-temperature radiation shining back onto the liquid helium vessel, the x-ray slot has a cooled collimator which reduces the solid angle of room-temperature radiation seen by the liquid helium vessel. Outboard of the nitrogen shield, the x-rays are either transmitted out along beam pipes or are absorbed on water-cooled copper absorbers, which are inclined to minimize reflection back into the cryogenic space. The main bending fields are produced by two pairs of rectangular section coils situated symmetrically about the median plane. Despite the apparent simplicity of this configuration, by careful optimization it has been possible to achieve field qualities of ∼10– 4. In addition to the uniform or dipole field, the main coils are also designed to produce a fixed field gradient of 1.8 T m–1 which, together with the normal quadrupoles in the straights, provides the focusing. Between the main coils and the aperture are located the superconducting multipole trim coils. The quadrupole trim coils are located in the central arc of the dipole and are used to change the fixed gradient so that the tune of the ring may be adjusted. Sextupole trim windings are located at each end of the dipole in the same position as the quadrupole trims. Used in conjunction with the normal sextupole, they allow Copyright © 1998 IOP Publishing Ltd

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the chromaticity of the ring to be reduced to zero. In addition, as shown above and below the magnet structure in figure G5.0.9, radial field trim windings are provided to compensate for any slight departure from vertical of the dipole field or any slight misalignment of the magnets when installed in the ring. The main coils are wound from 11-strand Rutherford cable of cross-section 1.2 mmx 3.6 mm, insulated by a triple wrap of 25 µm Kapton, with additional Kevlar between layers. Each wire, of diameter 0.64 mm and matrix to superconductor ratio of 1.8:1, contains 2000 NbTi filaments of diameter 8.5 µm. The winding is deliberately left porous to liquid helium for improved superconductor stability during ramping of the field. At an operating field of 4.5 T, the cable carries 1039 A and the peak field on the winding is ∼5.4 T. The trim coils are wound from a single strand of the main cable wire and typically operate at currents up to 30 A. The force between the upper and lower halves of the overhanging coil segments in figure G5.0.9 is ∼140 t. It is therefore necessary to support the coils by a massive 316LN stainless steel support structure, rather like a ‘G’ clamp. Deflection of the structure under load is ∼1 mm and this constitutes one of the major field errors which, fortunately, may easily be compensated by the trim coils. Figure G5.0.10 shows the coil end windings, which are turned ‘up and over’ the beam aperture. Advantages of this configuration are a good field profile for the beam, low peak field on the superconductor,

Figure G5.0.10. Cut-away view of the complete dipole magnet inside its liquid helium cryostat, showing the end turns of the superconducting coils. Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.11. Measuring the dipole magnet field shape using a large open test cryostat. The mouse within the dipole aperture is driven along the electron orbit by the large rotary arm.

a large clear vertical aperture and a short overall length-but coils of this shape are difficult to wind. After completion of winding, and before assembling into its operational cryostat, each dipole magnet is tested in a large tub-shaped cryostat of a type normally used at Oxford Instruments for testing medical imaging magnets; figure G5.0.11 sketches the general arrangement. In the course of this test, the magnetic field at all points around the 180° arc is measured using the ‘mouse’ shown in figure G5.0.12. The mouse moves along the electron orbit carrying a fixed array and a rotating array of Hall probes, which measure the field in the usual form of a Fourier expansion

Results from this measurement are fed into the Daresbury computer code ORBIT, which tracks many orbits of electrons around a ring having the measured field characteristics and provides a check on whether Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.12. The field-measuring mouse follows the electron trajectory inside the dipole magnet aperture, measuring field quality as it goes.

the field quality is adequate. After satisfactory completion of their field quality tests, magnets are welded into their helium vessels, carefully leak checked and given a thorough UHV cleaning process, before being assembled, with the radiation shields, into their vacuum vessel as shown in figure G5.0.13. After integration with the ring, the magnet and straight-section vacuum vessels are baked under vacuum to remove adsorbed gases. The ion pumps are switched on and the magnets are cooled to ∼4.4 K, after which the pressure falls to ∼1O–10 mbar. Cooling for the magnets is provided by a Linde TCF20 closed-cycle liquid helium refrigerator of ≈100 W capacity, providing a good reserve of cooling power. As an insurance against refrigerator failure, a buffer storage Dewar of 35001 capacity is provided, sufficient for several days operation if the refrigerator fails. Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.13. Assembling the dipole magnet, with thermal radiation shield, into its outer vacuum vessel.

(b) Super ALIS As shown in figure G5.0.14, Super ALIS and the Normal Accelerator Ring (NAR) provide the heart of a synchrotron radiation complex at the LSI Laboratories of Nippon Telegraph and Telephone Corporation (NTT) (Hosokawa 1993). NAR may either be used as a storage ring in its own right or as a booster synchrotron for Super ALIS which, as shown in figure G5.0.15, is a compact racetrack ring with two 180° superconducting bending magnets. Super ALIS has two possible modes of injection, low and full energy. The low-energy injection takes 15 MeV electrons straight from the linac through the inflector on the right-hand side of figure G5.0.15. Although its high magnetic field and small perimeter give Super ALIS a shorter damping time than a conventional ring, it is still much too long for multi-shot injection at this very low energy. All the beam must therefore be injected in a single high-current pulse from the linac, immediately after which the magnets must be ramped to full energy. Despite the difficulties of this technique, an impressive 200 mA Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.14. Layout of the NTT synchrotron radiation facility, showing the normal ring NAR and superconducting ring Super ALIS.

Figure G5.0.15. Plan of Super ALIS showing the superconducting bending magnets and two kickers for injection at 15 MeV and 600 MeV.

has been stored-a considerable achievement in itself, but still falling short of NTT’s requirement. To achieve higher currents, full energy injection has been used, with NAR running as a booster synchrotron and injecting through the left-hand inflector in figure G5.0.15. In this way much greater currents of up to 740 mA have been stored at the full energy of 600 MeV, and this value is limited only by the available RF power. At the lower energy of 520 MeV, where the RF power requirement is less, a spectacular 1215 mA has been stored. An interesting feature of Super ALIS is the use of ‘wobbler magnets’. These are conventional a.c. magnets arranged in the straight sections such that the plane of the beam can be moved vertically ±15 Copyright © 1998 IOP Publishing Ltd

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mm for the purpose of scanning a lithography mask. The ring was built by Hitachi. It has ten beam ports, of which three are currently connected to lithography steppers. The superconducting bending magnets have warm iron yokes and a room-temperature vacuum space for the beam, which is quite separate from the cryogenic vacuum. Following the usual practice, both bending magnets are connected electrically in series so that they will carry the same current. During the early stages of commissioning, problems were encountered because of differences between the bending magnet fields when ramping. On investigation, it was found that the problem was caused by eddy currents in the copper radiation shields, one of which had a lower impedance than the other. The resulting asymmetry between the two magnets caused substantial beam losses. Fortunately it proved possible to correct the difference by means of a passive compensation circuit ( Yamada et al 1990). (c) Aurora Of all the compact superconducting rings, Aurora ( Yamada 1990 ) is unique in using the constant gradient or ‘weak’ focusing principle. As discussed in section G5.0.3, most accelerators use quadrupole magnets to confine the electron beam by alternating gradient ‘strong’ focusing. A different approach, dating from the early days of accelerators, is to use a gradient which is constant along the entire orbit. Provided the field index n = (−R/B )dB/dR lies between 0 and 1, this constant gradient provides a weak focusing effect in both planes. Accordingly, Aurora has a magnetic field profile which decreases with radius and is constant around the circular orbit which, as shown in figure G5.0.16, is contained entirely within the magnet aperture. Clearly this arrangement gives a very compact design, but the absence of straight sections means that all the functions of injection, RF acceleration, diagnostics, etc must be accommodated within the magnet aperture. These restrictions have led the Sumitomo design team to introduce some very innovative ideas for injection and the RF cavity. As shown in figure G5.0.16, the injected beam, coming from a microtron at 150 MeV, is guided through the main bending field by a magnetic channel and is then brought alongside the closed orbit by an electrostatic infiector. The resonance injection scheme used (Takayama 1987) is essentially a time reversal of a resonance extraction process used previously in cyclotrons. The horizontal tune of the ring is set close to the ‘ half-integer resonance ’ Qx = 0.5 and, during the injection pulse, a ‘ perturbator ’ coil is excited to bring regions outside the closed circular orbit onto the resonance. Electrons injected into this region tend to spill inwards towards the closed orbit and, if the perturbator is progressively reduced to zero during the spill process, a fraction of these electrons is captured into stable closed orbits. As with the conventional multishot scheme, this process may be repeated as many times as necessary to accumulate the required stored current. The RF is chosen to give a harmonic number of two, i.e. just two bunches in the ring. A complicating factor in designing the RF cavity is the need to leave a clear slot on the median plane to let out the emitted x-rays. The cavity configuration chosen is that of two re-entrant quarter-wave resonators. Vacuum pumping is via a NEG (non-evaporable getter) and large-area cryopanels located in an annular chamber outboard of the orbit. As shown in figure G5.0.16, the magnet consists of a massive iron yoke and pole pieces, energized by two superconducting coils. The pole faces are shaped to produce a field index n = 0.7 at the injection field level, which produces Qx ∼ 0.5 as required for injection. Saturation of the iron causes this index to reduce progressively to n = 0.37 as the field is increased to its maximum value. In fact, this change is desirable because it increases the radial tune and decreases the vertical tune in such a way as to increase the Touschek lifetime at full energy. However, it does mean that the stored beam must cross several resonance lines during the ramp. Originally it had been feared that these resonances would cause beam losses and Aurora is therefore equipped with pulsed ‘resonance jumper’ coils to promote a very quick transition through each resonance. In the event, however, these coils were not needed and it has been possible to ramp substantial currents with small losses. For fine tuning of the field shape, six sets of

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Figure G5.0.16. Plan and vertical cross-section of Aurora showing the superconducting coils inside the warm iron yoke.

conventional trim coils are mounted directly on the pole faces. The main superconducting coils are short solenoids of 1984 turns each, with inner diameter 1.48 m, outer diameter 1.8 m and height 162 mm. They are independently supported from room temperature above and below the mid-plane by carbon-fibre-reinforced plastic rods. Although the force of attraction between the coils is reduced by image forces in the iron, it is still ∼ 140 tonnes, which must be taken in tension by the support rods. Separate cryostats are provided for each of the coils, which confers the advantage of allowing the system to be split apart at the mid-plane for maintenance access. As in many cyclotrons, the mid-plane split is brought about by built-in hydraulic jacks which lift the top pole. Field uniformity in the azimuthal direction has been measured to be ±0.15 mT at 1 T and ±2.0 mT at 3.83 T. The radial width Copyright © 1998 IOP Publishing Ltd

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of the ‘good-field’ region, with a uniformity of better than 10− 4, is ∼20 mm. At its usual peak energy of 575 MeV, Aurora routinely stores 400 mA and has achieved beams of 600 mA. Of the nine available beam ports, one is at present connected to a lithography stepper and seven are used for research. At the end of 1995, the whole facility and research team will transfer from Sumitomo Heavy Industries, where it was designed and built, to Ritsumeikan University. (d) Mitsubishi The Mitsubishi ring (Yamamoto et al 1993) is a two-sector racetrack which is injected at full energy from a conventional booster synchrotron. Because the magnets remain more or less permanently at their full energy, they are equipped with persistent current switches and retractable current leads, with the objective of minimizing the cryogenic losses. However, the persistent current switches have been made with a high normal-state resistance so that magnet ramping trials can be carried out later. The cryostats are designed for a target loss rate of only 0.5 1 liquid helium per hour. Compound suspension rods of glass-fibre-reinforced plastic and stainless steel are used to support the weight and electromagnetic forces between the coils and iron shield. Figure G5.0.17 shows the general arrangement and figure G5.0.18 shows a cross-section of the coils in their helium vessel. The main dipole field is produced by two large coils of square cross-section, with turned up ends. Each coil consists of 20 double pancakes, with liquid helium channels between the pancakes. The conductor is a rectangular section monolith of NbTi and copper, insulated with PVF (polyvinyl fluoride) and Kevlar. Although the field quality from the two large coils alone is not sufficiently good for a storage ring, when correction fields from the quadrupole and sextupole coils are applied, good-quality fields are produced. As shown in both figures, the multipole correction coils are of smaller cross-section and have straight rather than turned-up ends. Inevitably, there are large forces of attraction between coils above and below the median plane and these forces are taken by stainless steel members at liquid helium temperature. The electron beam pipe is at room temperature, with radiation screens at 20 K and 80 K between it and the liquid helium vessel. Synchrotron radiation ports at room temperature convey the x-rays through the outer parts of the cryostat, passing between the cold stainless steel members which support the inter-coil forces. Outside the cryostat at room temperature is located the iron shield. Because it is located at some

Figure G5.0.17. General arrangement of the dipole, quadrupole and sextupole coils in the Mitsubishi magnets. Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.18. A cross-section of the Mitsubishi cryostat, showing the warm vacuum pipe surrounded by the liquid helium vessel.

distance from the coils, the iron does not make much contribution to the field, but serves as a shield to carry the return flux from the magnet. By careful location of the iron relative to the coils, it has been possible to minimize the net forces between them and thereby avoid excessive force on the cryogenic supports. (e) NIJI-III Built by Sumitomo Electrical Industries (SEI) with design input from the Electrotechnical Laboratory (ETL) (Takada et al 1991), NIJI-III (which means ‘rainbow’ 3) is different in having four straight sections, with a superconducting dipole at each corner. Initially the ring was operated at the Electrotechnical Laboratory where it was injected at 280 MeV by the linac TELL. In 1993 the ring was moved to the Harima Research Laboratories of SEI, where it is now installed in a new facility with a dedicated linac, initially producing 50 MeV, but later to be upgraded to 100 MeV. As shown in figure G5.0.19, NIJI-III has three quadrupoles in the two long straights and one in the short straights. This arrangement makes it possible to attain a smaller source size than the racetrack rings, but at the price of a somewhat larger footprint. Like Super ALIS, it has a beam-wobbling magnet for lithography scanning. Figure G5.0.20 shows a cross-section of one of the dipole magnets, which have no iron and coils of a cos Θ cross-section like the high-energy proton magnets. Because there is no slot for the emitted x-rays, most of them must be absorbed within the dipole aperture and the only beams to be used are those emerging from the end of each magnet. The beam pipe has its walls at ∼4.2 K and the x-ray absorber is cooled by liquid nitrogen. The quadrupole trim coils, which can change the field index n from 0 to 0.5, are moulded directly onto the beam pipe. Experimentally, it has been found that these trim coils are quenched if the absorber temperature is allowed to rise above 300 K. Accordingly the liquid nitrogen flow to the absorbers is adjusted to keep their temperature below 240 K. The main dipole coils are wound as a double-layer pancake, with each layer consisting of three turns of Rutherford cable wound together, i.e. the coil is six turns thick. The cable is insulated with a double wrap of Kapton. In common with all ‘short’ air-cored magnets, the major source of field error comes at the ends. For NIJI-III the end turns have been designed with specially shaped spacers such that the sextupole error field has equal positive and negative components and the net integrated sextupole error Copyright © 1998 IOP Publishing Ltd

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Figure G5.0.19. Plan view of NIJI-III.

Figure G5.0.20. Cross-section of a NIJI-III dipole.

along the beam trajectory is zero. At the conclusion of its trials at ETL, NIJI-III had achieved a stored beam of 200 mA and beam wobbling had been successfully implemented to scan an area of 50 mm × 50 mm. The ring was then moved to a new purpose-built facility at the Research Laboratories of SEI in Harima, where it is injected by a new linac of 50 MeV, shortly to be upgraded to 100 MeV. ( f ) Diamond and SLS Finally, we mention two new ring proposals in Europe where superconducting dipoles will used as a part of large conventional rings. Diamond is a 3 GeV ring proposed for the UK by the Daresbury Laboratory Copyright © 1998 IOP Publishing Ltd

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(Poole et al 1993) and the Swiss Light Source (SLS) is a 2.1 GeV ring proposed by the Paul Scherer Institute (Joho et al 1994). In both rings, a number of the conventional bending magnets will be replaced by superconducting magnets producing fields of ∼4.5 T. The advantages of so doing are twofold. Firstly the critical photon energy is increased by a factor of ∼3.5 in comparison with the ∼1.3 T conventional magnets. Secondly the tighter curvature makes it possible to get in closer with the first optical element of the x-ray beamline and thereby gather more photon flux. Both rings propose to use a novel design of iron yoke superconducting dipole, invented by Mezentsev and Vobly (Kulipanov et al 1992) which uses the sloping geometry of an iron polepiece to minimize the distorting effects of iron saturation. G5.0.7 Concluding remarks The principal use of superconductivity so far has been in compact rings for x-ray lithography, where their small size and light weight offer real advantages in space saving and transportability. For the future, it seems likely that superconducting dipole insertions will also be used in large conventional rings to provide a harder photon spectrum and closer access to the x-ray source point. Acknowledgments For their contributions to Helios, I am grateful to colleagues at Oxford Instruments and Daresbury Laboratory. For their helpful supply of information used in section G5.0.6, I am grateful to: T Hosokawa, H Yamada, S Okuda, S Nakamura and H Takada. References Hosokawa T 1993 Super ALIS: NTT’s exploring tool for ULSIs Synchrotron Radial. News 6 16 IBM 1993 For a full description of x-ray lithography see the special edition, devoted exclusively to this topic, of IBM J. Res. Dev. 37 May 287–474 Joho W et al 1994 Design of a Swiss light source Proc. 4th Eur. Particle Accelerator Conf. (London, 1994) p 627 Koch E E, Eastman D E and Farge Y 1987 Handbook on Synchrotron Radiation ed G V Marr (Amsterdam: Elsevier) Kulipanov G N, Mezentsev N A, Morgunov L G, Sadjaer V V, Shkaruba V A, Sukhanov S V and Vobli P D 1992 Development of superconducting compact storage rings for technical purposes in the USSR Rev. Sci. Instrum. 63 731–6 Livingood J J 1961 Principles of Cyclic Particle Accelerators (New York: Van Nostrand) Margaritondo G 1988 Introduction to Synchrotron Radiation (Oxford: Oxford University Press) Poole M W, Clarke J A, Smith S L, Suller V P, Welbourne L A and Mezentsev N A 1993 A design concept for the inclusion of superconducting dipoles within a synchrotron light source lattice Proc. 1993 IEEE Particle Accelerator Conf. (Washington, 1993) pp 1494–6 Sands M 1979 The physics of electron storage rings Stanford Linear Accelerator Center Report SLAC-121 Takada H, Tsutsui Y, Emura K, Miura F, Suzawa C, Masuda T, Okazaki T, Keishi T, Hosada Y and Tomimasu T 1991 NIJI-III compact superconducting electron storage ring Japan. J. Appl. Phys. 30 1893 Takayama T 1987 Resonance injection method for the compact superconducting ring Nucl. Instrum. Methods B 24/25 420 Wilson M N, Smith A I C S, Kempson V C, Townsend M C, Schouten J C, Anderson R J, Jorden A R, Suller V P and Poole M W 1993 The Helios compact superconducting storage ring x-ray source IBM J. Res. Dev. 37 May 351–71 Yamada H 1990 Commissioning of Aurora: the world’s smallest synchrotron light source J. Vac. Sci. Technol. B 8 1628 Yamada K, Nakajima M and Hosokawa T 1990 Reduction of the dynamic field deviation in series connected magnets for compact storage ring Proc. 2nd Eur. Accelerator Conf. (Nice, 1990) pp 1154 Yamamoto S et al 1993 Superconducting magnets for compact synchrotron radiation source IEEE Trans. Appl. Supercond. AS-3 821

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G6 Superconducting magnetic separation

JHP Watson

G6.0.1 Introduction This chapter discusses the penetration of superconductivity into magnetic separation and considers the likely impact of the high-transition-temperature superconductors and the consequences of being able to generate high magnetic fields cheaply. For many years magnetic separation has been extensively used by the minerals industry. The market worldwide is approximately $155 million per year. Most of the magnetic separators use iron magnetic circuits and low-power coils or even permanent magnets and so the power savings that the use of superconductors can offer offset by a higher capital cost has not been a commercially attractive alternative. However, in recent years, the use of superconductivity has effectively penetrated into specific sectors of this magnetic separator market. The penetration of superconducting technology into other sectors appears to be imminent. Magnetic separation is achieved by a combination of a magnetic field and a field gradient which generates a force on magnetizable particles such that paramagnetic and ferromagnetic particles move towards the higher-magnetic-field regions and the diamagnetic field particles move towards the lower-field regions. The force Fm on a particle is given as

where χ is the magnetic susceptibility of the particle with volume Vp , B0 is the applied magnetic field, ∆B0 is the field gradient and µ0 is the magnetic constant 4π × 10−7 H m−1. Electromagnets in conjunction with an iron circuit have been used to generate a magnetic field in an air-gap. Field gradients are produced by shaping the poles or by using secondary poles. Secondary poles consist of pieces of shaped ferromagnetic material introduced into the air-gap. The magnetic induction produced in the air-gap in an iron circuit is limited to about 2 T if the separation zone is reasonably large compared with the volume of the iron in the magnetic circuit. The magnetizable particles processed by these machines are separated by being deflected by the magnetic field configuration or they are captured and held by the secondary poles. The particles are released from the secondary poles by either switching off the magnetic field or by removing the secondary poles from the field mechanically. With particles which are large or strongly magnetic, separation can be accomplished with electromagnets which consume modest amounts of electric power. There are a number of ways in which magnetic separation can be achieved, namely, Copyright © 1998 IOP Publishing Ltd

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( i ) where the difference in magnetic properties between the particles to be separated is sufficiently large; ( ii ) where the material, although not sufficiently magnetic, can be attached to something which is sufficiently magnetic for separation to be achieved; ( iii )when magnetic ions to be separated are in solution, a chemical or, as in a case described below, a biochemical treatment is required to produce a magnetic precipitate which can either be extracted itself or attached to a magnetic particle. Separators which use a finely divided system of secondary poles are known as high-gradient magnetic separators. High-gradient magnetic separation is a process in which magnetizable particles are extracted onto the surface of a fine ferromagnetic wire matrix which is magnetized by an externally applied magnetic field. This process allows weakly magnetic particles of colloidal size to be manipulated on a large scale at high processing rates so there are, in addition to the kaolin industry, a large number of potential applications in fields as diverse as the cleaning of human bone marrow, nuclear fuel reprocessing, sewage and waste water treatment, industrial effluent treatment, industrial and mineral processing and extractive metallurgy.

G6.0.2 High-gradient magnetic separation In the late 1960s and early 1970s large-sized, so-called high-gradient magnetic separators (HGMSs) were developed (Kolm 1971, Marston et al 1971) which have made it possible to extract weakly magnetic colloidal particles from a liquid or gas, which carries them through the separator. It is through this industry that magnetic separation using superconducting magnets has stimulated most interest. Kaolin is a white alumino-silicate mineral which occurs in nature as a finely divided particulate dispersion. In 1992 the total output of refined kaolin was some 20 Mt ( 1 Mt ≡ 1 × 109 kg ) having a value which amounted to $US 2000 million. The paper industry is by far the major market for kaolin and it is sold on the basis of its white colour. The white colour is affected by the concentration of colour-bodies which are paramagnetic and which can be reduced by high-gradient magnetic separation. These separators were developed for the kaolin industry in the United States (Oder and Price 1973) and consisted of an iron-bound solenoid which provided a magnetic field within the solenoid of up to 2 T. The solenoidal space was filled with a fine ferromagnetic wire matrix which acted as a multiplicity of secondary poles occupying 5-10% of the solenoidal space. The radius of the wire is chosen to meet the needs of the separation process, and for kaolin type 430 stainless steel is used with a strand radius of commonly 70-80 mm. Field gradients as high as 0.1 T µ n−1 can be achieved. This kind of separator has been described by Marston et al (1971), and is shown in cross-section in figure G6.0.1. The use of a finely divided matrix was previously suggested by Frantz (1937) but not at magnetic fields much higher than the field needed to saturate the magnetization of the matrix. In this method a slurry, containing paramagnetic particles, is passed through the matrix which captures the particles that holds them while the magnetic field is on. Eventually the efficiency of the trapping process becomes reduced by the accumulation of the captured particles. These trapped particles can be released and the efficiency of the matrix restored by switching off the magnetic field or by withdrawing the matrix from the magnetic field and washing the particles from the matrix. Machines operating in this way are batch machines and many are used in the clay industry to brighten clay by reducing the iron content. The ironbound solenoids typically weigh 200-250 t with copper coils weighing about 60 t. A typical separator of this type, manufactured by Eriez Magnetics, is shown in figure G6.0.2. The power level required to generate 2 T within a 2 m, or sometimes larger, solenoidal space was approximately 0.5 MW but more recently values of 260 kW have been achieved. The switch off and on time for these systems is typically 75 s so, for high efficiency, the feed time must be appreciably greater than 150 s.

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Figure G6.0.1. Cross-section through the centre of a Marston (Marston et al 1971) high-gradient magnetic separator. In the older machines, the matrix, of type 430 stainless steel wool, was contained within a cylindrical cavity of 213 cm diameter and 51 cm deep. The copper coils provided a magnetic field of 2 T parallel to the axis of the cylindrical cavity containing the matrix. The slurry of kaolin was fed to the matrix through a number of holes (not shown) in the bottom pole cap, and after passing through the matrix, it passed for collection through the upper pole cap through a number of holes (not shown).

Figure G6.0.2. Shows a conventional magnetic separator of the Marston (Marston et al 1971) type. This is the model C-84 HGMS built by Eriez Magnetics, Erie, PA, USA. This HGMS, used to remove micrometre-sized particles of ilmenite and other iron oxides from kaolin clay, has an 2134 mm diameter canister with an operative height of 508 mm. It will process up to 1892 1 min−1 of kaolin slurry. The power supply is 2.1 m wide, 3 m long and 2.4 m high and weighs 9 t. At a maximum field strength of 2 T it provides 3535 A at 400 kW. The separator is 4.9 m square and 1.9 m high. It weighs 338 t. Copyright © 1998 IOP Publishing Ltd

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G6.0.2.1 Theory of high-gradient magnetic separation Magnetic separation is achieved by a combination of a magnetic field and a field gradient which generates a force on magnetizable particles such that paramagnetic and ferromagnetic particles move towards the higher-magnetic-field regions and the diamagnetic field particles move towards the lower-field regions. The force Fm on a particle is given in equation (G6.0.1). The fundamental element in the capture process is the interaction between a small magnetizable particle, usually paramagnetic, in a uniformly applied magnetic field (Watson 1973). Consider a ferromagnetic wire of radius a and a saturation magnetization Ms placed axially along the z axis as shown in figure G6.0.3. A uniform field H0 large enough to saturate the wire is applied in the x direction. Paramagnetic particles of susceptibility χ and volume Vp (=(4/3)πb 3 ) and density pp are carried past the wire by a fluid of viscosity η moving with a uniform velocity V0 in the negative x direction.

Figure G6.03. Basic filter configuration. A ferromagnetic wire of radius a and a saturation magnetization Ms placed axially along the z axis. A uniform field H0 large enough to saturate the wire is applied in the x direction. Paramagnetic particles of susceptibility χ and volume Vp (=(4/3)πb3 ) and density ρρ are carried past the wire by a fluid of viscosity η moving with a uniform velocity V0 at an angle α to the negative x direction. In this discussion α = 0.

The flow around the wire is treated in the hydrodynamic approximation in which the fluid can be considered frictionless. Equations can be derived describing the particle trajectories in terms of cylindrical coordinates ra , θ and za . These are related to x and y by x/a = ra cosθ and y/a = ra sin θ. The magnetic permeability of free space µ0 is given by µ0 = 4 × 10−7 H m−7. Far away from the wire the particles move at the same velocity as the fluid. It is desired to calculate the effective capturing area that the wire presents to the moving stream. The capturing area/unit length of wire is 2Rca where Rc is the capture radius. This is illustrated in figure G6.0.3. The equations of motion can be derived by setting the Stokes viscous drag FD on the particle to the magnetic force Fm .

where Vp is the particle velocity.

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where (Fm )r and (Fm )θ are the two nonzero components of the magnetic force Fm written in Stokes form. Here K = MS/2H0 is the coefficient which multiplies the short-range term and Vm is the magnetic velocity and is given by

where Ms and H0 are in A m−1, all lengths are in m and η is in units of Pa s. (Water at 20°C has η = 10−3 Pa s.) An examination of equation (G6.0.3) reveals that there are two regions on the wire that are attractive to paramagnetics and two regions which are repulsive. The equation of motion is obtained from

Far away from the wire the particles move at the same velocity as the fluid and it is desired to know the thickness of the band which will be captured by the wire or the effective capturing area the wire presents to the stream, 2Rc a/umit length of wire, where Rc is called the capture radius and is illustrated in figure G6.0.3. For our purposes when Vm /V0 is approximately 1 then Rc is given by

As capture proceeds the capture efficiency becomes reduced as the capture radius is reduced. If all the particles that were magnetically induced to collide with the wire were retained, the capture radius would have the form (Liu and Oak 1983, Luborsky and Drummond 1976)

where f is the relative volume of the captured material; that is, the volume of captured material divided by the volume of the wire. This expression, although approximate, is suitable for this discussion. The equation for Rc is plotted against f in figure G6.0.4 for a number of values of Vm /V0.

Figure G6.0.4. The capture radius Rc versus f for various values of Vm /V0 and fm a x according to the force-balance model (Liu and Oak 1983, Luborsky and Drummond 1976) is shown as a solid line and labelled in the legend as Rc ( fm a x ). Copyright © 1998 IOP Publishing Ltd

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The question of the stability of the captured material must now be considered. In order for a particle that has been captured to be retained the value of (Fm )θ, the tangential magnetic force, must be greater than the tangential fluid drag force (Fd )θ. Using this model the maximum value of f can be calculated and is denoted by fmax

fmax is plotted in figure G6.0.4 and may interpreted as the point on the Rc versus f curve beyond which further material cannot be collected since beyond that value of f, (Fm )θ < (Fd )θ . This means that the capture radius Rc is fairly constant as the captured material builds up to fm a x after which Rc = 0. This theory predicts that particles with Vm / V0 < 0.62 cannot be retained although they can strike the wire, but the probability of them striking the wire decreases with increasing f as 1/(4f + 1)). The filtering performance of the magnetic stainless steel wool has been evaluated in the form of deep beds where the wire wool occupies about 5% of the volume but for special applications knitted meshes are often used, as shown in figure G6.0.5. It was found that the number of particles coming out of the filter per unit volume of slurry, No u t , compared with the number of particles per unit volume of slurry going in, Ni n , for a filter of length L , is given by (Watson 1973)

where F is the fraction of space occupied by the matrix and where L0 (=3πa/4FRc ) is a characteristic length. This expression is only approximately correct when F or Rc is large due to random overlap of capturing areas in the filter. It is important to realize that this equation only applies to the initial application of the slurry to the filter because as the volume of magnetics builds up, the rate of particle capture decreases. The presence of the captured volume, V, alters the flow around the wire until at some particular volume of magnetics, Vm a x = F L Ac fm a x , where Ac is the cross-sectional area of the canister of length L. Further, if the Reynolds number is greater than 6, capture can occur on the downstream side of the wire as shown in figure G6.0.6 (Watson and Li 1992). The solution of the magnetic separator equations is very complex and has been studied in the literature. When the attraction between the matrix and the particles is strong, the separator equations can be simplified (Watson 1978)

where V0 is the velocity of the slurry, where Vp is the particle volume and where t is time. This equation can be used to estimate the separator performance. The high values of applied field available from superconducting magnets can be used to increase Vm and this can be used to either ( i ) increase extraction with high values of Vm /V0 or ( ii ) increase the processing rate keeping Vm / V0 constant. In HGMS the ferromagnetic wire occupies 5% of the space. In order to quantify the amount of material passing through the separator, it is convenient to use the volume of the canister less the volume of the matrix as the unit of volume Vc. The high-gradient magnetic separator process is a cyclical one; it has a feed part and a cleaning part, i.e. when the quality of the effluent is inadequate the matrix must be cleaned. In order to clean the canister it is usually necessary to reduce the field to zero and flush

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Figure G6.0.5. (a) For applications where smooth wires are needed ferromagnetic knitted wires are a useful matrix. The magnetic field should be applied to this section of mesh with an orientation perpendicular to the page, (b) Shows a section of a ferromagnetic 430 stainless steel pad used as a matrix for the treatment of kaolin clay. As normally used the pads are 95% porous and consist of wire strands with average cross-sectional areas between 2500 and 40000 µm2. (c) Shows a scanning electron micrograph of a typical length of the stainless steel matrix shown in (b). The long side is approximately 150 µm and the short side about 50 µm. (d) Shows a scanning electron micrograph of an expanded metal mesh matrix often used in the processing of minerals. The size of the holes and the strand dimensions are chosen for the particular separation process.

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Figure G6.0.6. The cross-section through a nickel wire 25 mm in diameter. The flow is horizontal left to right parallel to the applied magnetic field. Paramagnetic particles are captured on the upstream and downstream side of the wire. Reynolds number = 8.5 (Reynolds number = 2ρLbV0 /η where ρL is the fluid density and the other symbols have their usual meaning). The two regions of the wire attractive to paramagnetics are separated by two repulsive regions, as shown in the figure.

the captured material from the matrix. If no canister volumes of slurry can be fed before flushing, the processing rate P can be written

where ρ is the dry weight of material per unit volume of slurry, A is the cross-sectional area of the canister, D is the dead time, i.e. the time between stopping and restarting the feed, τ is the time for one canister volume = L/V0 where L is the length of the canister and R is the fraction of material passing through the separator. For the process to be efficient it is necessary that N0 » D/t. When using HGMS two possibilities exist: ( i ) systems which switch off the field to flush ( ii ) systems which leave the field on. G6.0.3 Low-Tc superconducting magnetic separators G6.0.3.1 Superconducting magnetic separators which switch on and off In this approach the superconducting solenoid which generates the magnetic field is switched off in order to clean the matrix. This kind of superconducting machine adopts the same solenoid configuration as the conventional iron-bound solenoid separator and, similarly to the conventional separators, it leads to a dead time of approximately 200 s giving a duty factor of ∼75%. The only company to make machines of this type is Eriez Magnetics, Inc. The projected electrical power cost inflation in the USA in the late 1970s, which was overestimated, appeared to make this superconducting separator competitive compared with the conventional machine. A processing cost comparison between a conventional separator and superconducting separators is shown in figure G6.0.7. Copyright © 1998 IOP Publishing Ltd

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Figure G6.0.7. Cost ($/tonne) plotted versus power cost ($/100 kWh). Compared are the processing cost of the conventional separator (Con. cost/tonne), the first superconducting separator (Sup. Cost 1986) and the improved larger-capacity reciprocating canister system. Costs include the capital cost of the machine depreciated over 10 years.

Eriez Magnetics and J M Huber Corporation of Wrens, GA, USA have produced a superconducting separator, shown in figure G6.0.8, which has operated on kaolin clay without problems since May 1986.

Figure G6.0.8. This photograph shows a superconducting HGMS built by Eriez Magnetics, Erie, PA, USA and installed at the plant of J M Huber where it has been on-line since 8 May 1986. The machine is of the Marston type but has superconducting windings which supply a field of 2 T to a 213.4 cm diameter canister containing fine ferromagnetic stainless steel wool. The 1000 1 liquid helium storage vessel is shown on the platform to the right of the separator. The separator itself weighs 250 t. Copyright © 1998 IOP Publishing Ltd

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This machine design involved the least technical risk but until recently the savings in power could not have justified the capital cost of the superconducting coils which simply replaced the conventional coils. The Eriez Magnetics system has the following characteristics: Canister Magnet size Magnet weight Ramp time Helium usage Liquefier capacity Compressor power Actual power Field strength

213.36 cm diameter × 51 cm deep 391.16 cm2 × 210.89 cm deep 227 t 1 min maximum 2 l per cycle at 1 min ramp time 20 l h−1 from gas at 300 K to 4.2 K 47.8 kW (nominal) 75 kW 2 T.

Based on the success of the first Eriez superconducting separator, J M Huber have installed a second 2 T superconducting separator, 3.05 m in diameter, which has a capacity twice that of the earlier separator. The processing cost per tonne is marginally cheaper than in the first superconducting system. G6.0.3.2 Performance of the early switch-on, switch-off systems From an operational point of view, these separators perform well provided the duty factor can be kept high, that is, N0/(N0 + 1 + D/τ ) > 0.75. This is the case for the processing of kaolin from Georgia, USA at 2 T or for cleaning up water pollution. This system becomes difficult when the following situations arise. ( i ) When V0 is increased with applied field, then τ is reduced and if τ is reduced to the situation where D/τ » 1, all the increase in the processing rate due to the increase in V0 is lost. This situation often occurs when large amounts of material are removed. This means, in practice, that fields above about 2.5 T become impractical. ( ii ) Another problem associated with an increase in V0 is due to the increasing frequency with which the field must be switched on and off as there is helium boil-off when switching occurs. With the Eriez machine this amounts to 2 1 of liquid helium per switch, which doubles the helium requirement, if the machine is switched on and off four times per hour as was usual with machines operating at 2 T on Georgia, USA kaolin. If this machine were operated on the same job but at 4 T the boil-off would be 161. The power costs and liquefier costs therefore increase rapidly as τ decreases. ( iii )If the machine has simply had coils replaced with superconducting coils the cost of the machine is considerably higher so that cost savings due to electrical power must at least be greater for economic viability. This is shown in figure G6.0.7. Clearly, for mineral processing, switch-on switch-off systems at high field are not economically viable. Other solutions must be sought in these cases because of the problems outlined above in ( i ), ( ii ) and ( iii ). It should be also pointed out that the machines that result from the alternative solution, to be discussed below, are much more attractive economically even in situations where switch-on switch-off systems are viable. If we are not discussing new machines, however, and we consider retrofitting conventional switchoff, switch-on systems, there are other alternatives to be considered, providing the problems raised in ( i ), ( ii ) and ( iii ) can be mitigated as discussed in the following section. G6.0.3.3 Improved superconducting machines which switch on and off In February 1987 Advanced Cryo Magnetics, Inc. were formed and they have attempted to tackle the problems associated with the switch-off, switch-on systems, enumerated in the previous section, and in Copyright © 1998 IOP Publishing Ltd

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particular, they have attempted to reduce the losses of liquid helium when the superconducting magnet is switched on and off. They have identified the problem as being caused by eddy currents generated in the vacuum vessel, liquid helium container and the superconducting magnet support when the magnetic field is switched on and off. They solve the problem as follows (Purcell et al 1991): the superconducting coil is positioned within the cryogenic containment vessel which consists of a liquid helium vessel which is a relatively thin stainless steel inner tube containing the liquid helium and the superconducting magnet. The stainless steel in this tube is so thin that the eddy current losses are very small. This thin tube is contained within a relatively thick outer stainless steel tube and is held in place by insulating spacers so that the inner and outer tubes do not touch each other. The thick outer tube supports the magnet but in order to reduce eddy currents it contains at least one electrically insulating joint so that the outer tube presents a high electrical resistance to the induced electromotive force generated by the changing field and thereby reduces the losses in the outer tube. This tube is contained within a shield at liquid nitrogen temperature at high vacuum and surrounded by superinsulation (Purcell et al 1991). The earlier switch-on switch-off systems were actually 30% higher in cost than the machines with copper coils due to the more sophisticated superconducting magnet, the cryogenics and above all the cost of a very large liquefier. In contrast Advanced Cryo Magnetics, Inc. claim that their machine is marginally cheaper than the machines with copper coils. In July 1992 Advanced Cryo Magnetics, Inc. signed an agreement with J M Huber to bring this improved superconducting technology to all of Huber’s copper magnets over the next three years. The first retrofit was completed in the first half of 1993. In November 1992, Advanced Cryo Magnetics, Inc. accepted a contract from Englehard to retrofit a magnet of approximately 3 m diameter at Mclntyre, Georgia, USA with a 2.5 T superconducting coil. Figure G6.0.9 shows a superconducting coil being wound at Advanced Cryo Magnetics, Inc. in San Diego, California, USA.

Figure G6.0.9. Shows a pancake coil of an Nb Ti superconductor being wound at Advanced Cryo Magnetics, Inc., San Diego, CA, USA. Advanced Cryo Magnetics, Inc. was formed in February 1987 to work in the area of the design and manufacture of superconducting magnets and cryogenic equipment. Copyright © 1998 IOP Publishing Ltd

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Figure G6.0.10. The first reciprocating superconducting HGMS at the laboratories of English Clays Levering Pochin Ltd, St Austell, Cornwall, UK. This machine operates in the vertical mode in contrast to the industrial machines which operate horizontally. This machine was equipped with one operational separation canister with two dummy canisters for magnetic balancing. The separator was operated up to and at its maximum magnetic field of 5 T.

G6.0.3.4 Reciprocating canister superconducting magnetic separators The superconducting separator with a reciprocating canister was pioneered in the 1970s by English Clays Levering Pochin ( ECLP—now English China Clays, plc, a large kaolin producer in England ) and the inception and promotion of the system was largely due to the vision of Professor N O Clark, who was Director of the ECLP Research and Development Department at that time. Figure G6.0.10 shows the first superconducting reciprocating canister system under test in the mid 1970s at the Research and Development Department of ECLP. In the reciprocating canister system ( RCS ) ( Windle 1975 ) one superconducting magnet uses two separate canisters. When one canister is separating the other is out of the field being cleaned. Then the clean canister is returned to the field while the other is moved out of the field for cleaning. The cycle then continues. The canister train must be magnetically balanced in order that the canisters can be moved in and out of the field without using large forces. This is achieved by arranging things so that when the transfer takes place the amount of iron in the coil remains constant. Under these conditions the superconducting magnet can be left at high field continuously. Consequently this means that a helium liquefier requiring less power can be used. The dead time can be reduced to a few seconds. Copyright © 1998 IOP Publishing Ltd

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In this situation feed times of less than one minute can easily be achieved while maintaining a high duty factor and fields as high as 5-8 T can be used. This means that the processing rate of the RCS can be a factor of 8-10 times higher per unit area of canister than the switch-on switch-off type of system, as shown in figure G6.0.7. For a given separation problem, the RCS is appreciably smaller in size and the low boil-off, resulting from the field not being switched, means a smaller helium liquefier requiring less power can be used or no helium liquefier in the latest developments to be described below. The construction of the pilot scale RCS took place after extensive trials with various types of clay originating from Cornwall, UK. The system was pilot scale in the sense that the superconducting magnet system was 12.7 cm clear bore with a maximum field of 5 T, as shown in figure G6.0.10. The helium liquefier was full scale, using turbo-expanders which were specially cut to provide 2 1 h−1 rather than the normal 18 1 h−1 of the machine with properly cut turbines. All aspects of the full-scale reciprocator were tested and the valves were operated under pressure in proper sequence for two years. Furthermore, Watson (1997) suggested a radial feed system which was particularly suited to the solenoidal geometry used in the separator and which gave a threefold increase in the production rate. This radial feed concept was developed and tested. Cyclical matrix compression tests were done which revealed that if the compressive stress on the matrix, during reciprocation, was too large the matrix did not recover elastically which left a radial feed system prone to channelling, i.e. some clay could bypass the matrix. In order to avoid this problem, the compressive stress was kept below the critical value by supporting the matrix. Some of this work has been described by Riley and Hocking (1981). Cryogenic Consultants Limited (CCL) have produced a mobile RCS system employing an internal closed-cycle refrigerator, with a power consumption of 14.7 kW, designed to run for six months between service. The magnet operates at 5 T in an active separation volume 26.6 cm diameter and 50 cm in length. The magnet is surrounded by an iron yoke to produce a low field around the system which allows the saturated matrices to be cleaned much closer to the magnet and consequently the canister train is reduced in length. The separator and control system, which weighs 5.8 t, rest on a bed of concrete 7.4 m long and 1.7 m wide. It is estimated that the off-balance forces on the canister train are 0.5 t. After extensive trials with their clays at the Magnetic Separation Laboratory in the Institute of Cryogenics, University of Southampton, UK, the clay company, Goonvean and Rostowrack, have purchased one of these systems which has run since it was installed early in 1989. On 16 December 1991, Carpco, Inc., Jacksonville, FL, USA acquired the Magnetic Separation Group from Cryogenic Consultants Limited. The group now runs under the name of Carpco SMS Ltd, Slough, Berkshire, UK. The superconducting separator above has been designated with the name and model number HGMS 5T/280/5/S. A second unit of this type was installed at the site of Dorfner, a china clay company in Germany in September 1992. Following this success, Carpco SMS Ltd developed a much larger machine operating at 5 T with the same reciprocating canister design, the Cryofilter HGMS 5T/460, shown in figure G6.0.11. This machine has a reciprocating tube 0.5 m in diameter. The machine has an overall height of 3.385 m, an overall length of 13.3 m and a width of 3 m and a weight of only 49 900 kg which greatly simplifies the transportation and installation at the site. The machine requires 1000 1 of liquid helium per year which is transferred yearly. No liquid nitrogen is required, but a small refrigerator is used running at 10 kW, which provides cooling at an intermediate temperature. If this refrigerator is switched off, there is a small increase in helium loss. The magnet operates in the persistent mode and so is constantly at 5 T even in the event of loss of electrical power at the plant, but it means production can be started as soon as power is restored. The power required in normal operation is 9.4 kW. The machine treats china clay at an approximate rate of 20-60 t h−1 (dry weight). It requires no special foundations and typically it can be processing china clay within a week of installation. The first machine of this type was delivered in October 1993 to Caulim di Amazonia (Cadam), a large Brazilian kaolin company. The second large machine was delivered to Amberger Kaolinwerke, Amberg, Germany in November 1993. A third separator of this type has been delivered to Theile Kaolin Company of Sandersville, GA, USA. The

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Figure G6.0.11. The Carpco SMS Ltd’s industrial, reciprocating superconducting separator, Cryofilter HGMS 5T/460, during its final acceptance tests prior to shipment. The Cryofilter’s modular design weight of less than 50 t means that shipping, assembly and installation are simplified. No special foundations are needed and clay is usually processed within a week of installation. The Cryofilter HGMS greatly reduces power costs, by up to 95% or more when compared with conventional HGMS units. The Cryofilter HGMS has a production rate of between 20 and 50 t h−1 of kaolin (dry weight). The power consumption under normal operation is 10 kW.

Cryofilter was installed in the second quarter of 1994 as part of a planned expansion of Theile’s mine site at Wrens, GA. As of January 1997, ten of these machines have been installed and three in are the process of manufacture. G6.0.3.5 Superconducting rotating drum magnetic separator A superconducting rotating drum separator, known by the acronym DESCOS (drum equipped superconducting ore separator), has been developed by KHD Humbolt Wedag AG, Cologne, Germany, and is shown in operation in figure G6.0.12(a). These superbly engineered systems have been in operation for a number of years. They can be operated on both wet and dry systems and the change from wet to dry operation can be accomplished without returning the system to room temperature by rotating the superconducting coils through an angle of 100°. The magnetic field is generated by four D-shaped superconducting coils with the circular section close to the drum surface. Because of their unusually high field intensity for a drum separator (with a field of 3.2 T and a gradient of 100 T m−1 at the drum surface in the separation zone and a field of 0.1 T on the drum surface on the opposite side of the drum to the separation zone) combined with a large effective range of the magnetic force, the DESCOS magnetic separators are suitable for separating coarse and fine-grained weakly magnetic compounds at rates of up to 180 t h−1. Follow-up models have been planned with magnetic fields of 5 T at the drum surface. The field will be generated by four saddle-shaped superconducting coils as shown in figure G6.0.12(b). The drum, which rotates at 40 revolutions per minute, is made of a plastic material and has an overall diameter of 1.2 m and can be up to 1.5 m long. To aid separation a ferromagnetic wire can be wound on the surface of the drum. There is no danger of the separating section being blocked by oversized material or strongly magnetic material contained in the feed. This eliminates the need for screening and scalping of the strongly magnetic material by a preliminary magnetic separation. One plant has been operating since 1988 on magnesite separation in Turkey. For particles between 20 and 200 mm the processing rate is 180 t h−1 with a drum 0.8 m long; unfortunately, the high cost of DESCOS is probably preventing very wide acceptance of this machine. Copyright © 1998 IOP Publishing Ltd

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Figure G6.0.12. (a) The superconducting drum separator, DESCOS, built by KHD Humbolt Deutz AG, Cologne, Germany, which has been operating at a mine site in Turkey since 1988. (b) The separation in a DESCOS unit is identical to that in conventional drum separators. By suitably positioning the swivel-type magnet system inside the drum, it can be used for wet or dry processes. The magnetic field on the drum covers one third of the complete drum circumference. The flux density is 2.8-3.2 T over an axial length of 1 m. 1—Magnetic coils; 2—radiation shield; 3—vacuum tank; 4—drum; 5—plain bearing; 6—helium supply; 7—vacuum line; 8—current supply.

G6.0.3.6 Summary of the situation for low-Tc superconducting magnetic separators Following the initial installations of superconducting separators based on iron circuits by Eriez Magnetics in the mid-1980s, the kaolin industry appears to have turned to the new reciprocating canister systems from Carpco SMS Ltd which offer many advantages in operation and installation. There also appears to Copyright © 1998 IOP Publishing Ltd

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be potential in the retrofitting of superconducting coils into the earlier Marston-type machines. For other minerals the rotating drum separator from KHD has been applied and at this point only one has been sold. The market is probably restricted by their high cost-to-benefit ratio.

G6.0.4 Application of high-Tc superconductors to magnetic separation G6.0.4.1 Introduction to high-Tc superconductors The discovery, in 1986, of high-temperature superconductivity by Bednorz and Müller in La—Ba— Cu—O stimulated an enormous amount of research throughout the world into these and related compounds. The discovery of the Y—Ba—Cu—O (YBCO) compounds with superconducting transition temperatures as high as 92 K created a very real possibility of practical devices operating in liquid nitrogen at 77 K which implies that the capital and running costs associated with the production and maintenance of the necessary conditions to operate these devices is considerably reduced, perhaps by as much as a factor of ten. The further discoveries of the Bi(Pb)—Ca—Sr—Cu—O (BSCCO) materials and the Tl—Ba—Ca—Cu—O (TBCCO) materials with values of Tc reaching 125 K made the possibility of the operation of large-scale superconducting devices at liquid nitrogen temperatures even more attractive because of their lower values of reduced temperature T/TC . Two major problems, weak links and weak flux pinning, have plagued high-Tc conductor and material development efforts from the beginning. Recent advances in mechanical processing of tape conductors based on BSCCO show great promise in overcoming weak links with low-temperature Jc values > 105 A cm−2 and extending to fields of 25 T. Unfortunately these materials show weak pinning at moderate temperatures for fields aligned along the c axis. This manifests itself in a Jc which decreases precipitously with magnetic field for T > 30 K ( Maley 1991). By contrast conductors based on YBCO exhibit much stronger pinning at higher temperatures with smaller sensitivity to the orientation of the magnetic field but remain severely limited by weak links which results in the value of Jc decreasing dramatically with the application of weak fields. For example in the untextured YBCO material a few tenths of a tesla can produce a reduction of Jc by a factor of 10-102 (Maley 1991). Using the powder-in-tube method Haldar et al (1992, 1993, 1995) have described silver-clad tapes of Bi-2223 material in ≥100 m lengths of mono and ≥800 m lengths of multifilament tape. The tapes were wound into pancake and layer coils using the ‘wind and react’ and the ‘react and wind’ methods. A ‘wind and react’ pancake coil generated a maximum self-field of 2.6 T, 1.8 T and 0.36 T at temperatures of 4.2 K (liquid helium), 27 K (liquid neon) and 77 K (liquid nitrogen) respectively. A layer-wound solenoidal coil using a ‘react and wind’ method has been constructed from a 200 m length of 37 filament tape on a 40 mm diameter tube with 16 turns per layer and with 70 layers. The resulting coil generated a central field of 0.1 T at 77 K which corresponded to 4190 ampere turns. A high-Tc superconducting magnet has been described by Lue et al (1994) based on a Bi-2223/Ag conductor manufactured with the metallic precursor process (Otto 1993). The magnet was constructed by a ‘react and wind’ technique using pancake coils. The winding had an inside diameter of 2.54 cm, an outside diameter of 8.7 cm and an overall length of 10.75 cm. The tape conductor was 0.15 mm thick with a width of 5.33 mm. Using a Hall probe in the centre of the magnet the maximum field measured at 77 K was 0.2 T. This coil produced a field of 1.1 T at 4.2 K. In a paper by Kitamura et al (1993) largely devoted to the production of Bi-2223, it is reported that a small coil was produced which generated a field of 1.5 T at 20 K using 96 m of Ag-sheathed tape so a field of 0.3 T can be expected at 77 K.

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G6.0.4.2 Provision of the magnetic field for the magnetic separator Most of the efforts of researchers and organizations interested in practical devices have been towards the production of wires of high-temperature superconductors. For many large-scale applications this is necessary although in these cases the economic advantage of using high-temperature superconductors is more limited as the wire-wound device will be of approximately the same cost as for a conventional superconducting device and as the cost of a large wire-wound superconducting system is usually dominated by the winding, which is 70-80% of the cost, the saving will only be in the cryogenic areas, such as refrigeration and the cryogenic envelope. However, there are other applications where a steady magnetic field is required and here there is the possibility of trapping the magnetic field in a hollow tube of superconductor (Watson 1988a, b, Wipf and Laquer 1989) or in a solid disc (Gotoh et al 1990, 1991) without the necessity of winding a coil. These devices are called flux tubes and are similar to solenoidal permanent magnets. As discussed above, operating at liquid nitrogen temperature allows the cryostat and the refrigeration costs to be reduced by a factor of ten. It can be estimated that the cost of manufacturing flux tubes, if they are made by fairly standard ceramics processing, may be down by a factor of 20 compared with the equivalent solenoidal wire-wound magnet. This means this stored magnetic field is less per unit volume by at least an order of magnitude than present-day field volumes. This must produce a significant change in the economics of many devices. For a superconducting separator it is possible to use current-carrying wires to supply the magnetic field, but designs will be considered which use flux tubes or discs to supply the magnetic field. G6.0.4.3 The superconducting flux tube and discs There are a number of reasons why the approach using flux tubes or a disc may be successful and one of the earliest applications of the high-temperature ceramic superconductors. The first reason relates to the development of high values of critical supercurrent Jc in disordered inhomogeneous materials. The perovskite based ceramic YBCO superconductors are heavily twinned, contain weak links (intergrain electron tunnelling barriers) and have properties that vary with crystalline orientation (Campbell et al 1987, Maley 1991). If one attempts to make superconducting wire from this material, the weak links reduce the critical current to impractically low values in weak fields. In the macroscopic tubes the same problem of weak links occurs; however, it seems more likely that a device of reasonably small size can be melt-textured more easily than if the slow melt-texturing process were applied to extremely long lengths of wire. This view is supported by the experimental results (Hikata 1990, Maley 1991). The likelihood of quickly producing a ceramic superconductor with high Jc , at least 5 ≥ 104 A cm−2, is very much better than for wire especially for YBCO. The second reason relates to the stability of the superconductors against flux jumping, a process which rapidly dissipates the stored magnetic energy in the superconductor in the form of heat which can damage the tube (Wipf and Laquer 1989). In the low-Tc superconductors the flux jumping problem is severe and flux jumping limits the amount of flux which can be trapped in a flux tube. For example, for Nb3Sn the most favourable temperature is 16 K where a field of the order of 0.9 T can be trapped (Wipf and Laquer 1989). A necessary condition for a flux jump to occur is that the adiabatic flux jump field BF J has been exceeded. BF J is the maximum field difference that can exist across a superconductor in the adiabatic limit, i.e. when Dm ≥ Dt h where Dm and Dt h are the magnetic and thermal diffusivities respectively. These processes have been studied by Swartz and Bean (1968), in the adiabatic limit; they obtained an expression for BF J

Neuringer and Shapira (1962) carried out a careful investigation to verify the Swartz and Bean criterion, under conditions approximately those of the adiabatic limit, and found the flux jumping behaviour was Copyright © 1998 IOP Publishing Ltd

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in good agreement with the predictions of equation (G6.0.13). The essential point is that the value of where Cp is the specific heat m−3 at constant pressure and the value of CP for the ceramic superconductors at 77 K is approximately 100 times greater than for Nb3Sn at 4.2 K (Thompson et at 1987). The value of BF J for YBa2Cu3O7-δ at 77 K calculated from equation (G6.0.13) is 14 T. Flux jumping in BSCCO-2212 has been investigated by Burgoyne and Watson (1994) which shows that the material is stable against flux jumping at 77 K but flux jumping is present at 4.2 K. BF J is actually independent of Jc , but Jc controls the thickness of the superconductor required to trap a particular value of field. A useful device can be constructed if the average value of Jc > 5 × 104 A cm−2 provided that Jc does not decrease too rapidly with field. For example if Jc = 5 × 104 A cm−2 then with a tube wall thickness of 1 cm approximately 6 T can be retained. There has been much recent progress in the production of melt-textured ( Jin et al 1988a, b) YBa2Cu3O7-δ materials ( Murakami 1992, Salama and Lee 1994, Weinstein et al 1994 ). Weinstein et al ( 1994 ) have reported single YBCO grains 2 cm in diameter being made in production quantities. The materials have values of Jc ∼ 104 A cm−2 for melt-textured material, Jc ∼ 4 × 104 A cm−2 after light-ion irradiation and Jc ∼ 8.5 × 104 A cm−2 after heavy-ion irradiation. Using 2 cm discs after light-ion irradiation made it possible to fabricate a minimagnet which trapped 2.25 T at 77 K, 5.3 T at 65 K and 7 T at 55 K (Weinstein et al 1994). These materials are considerably stronger than permanent magnet materials normally used in drum separators so that we are now able to consider the possibility of drum separators constructed with superconducting permanent magnets. G6.0.4.4 Charging the flux tube or a disc with magnetic field (a) Flux trapping followed by flux compression In the following analysis, it will be assumed that the critical current density Jc at a point in the superconductor is of the form

where T is the temperature and H is the local magnetic field assumed to be perpendicular to the critical current Jc . The simplest, but least versatile, method of trapping flux in the tube uses the fact that when a multiply-connected superconductor is cooled below its transition temperature in a magnetic field flux is trapped in the holes in the body. In particular if a tube is used and if the external field is reduced to zero after the tube has become superconducting, the tube will behave like a superconducting permanent magnet and the space inside the tube will contain a considerable magnetic field ( Watson 1968 ). Using the form for Jc versus H given in equation ( G6.0.14 ) and using a modified Bean critical state model, Watson has calculated the maximum trapped field HT(T) in a long cylinder of superconductor of external radius R0 and of wall thickness t ( for the case of a disc R0 = τ ) as follows

This method suffers from the disadvantage that a source of field must be available to supply a maximum field of HT over the period of time required to cool the superconductor from Tc to the operating temperature, for example 77 K. Furthermore, for a flux tube, if a superconducting mandrel is inserted into the hole a considerable flux compression can be achieved (Hildebrant et al 1962). Neglecting penetration of flux into the mandrel which is valid when α ( T ) γ ( T )τ, the ratio of the final field to the initial field is inversely proportional to the ratio of the initial area to the final area of the hole. If a cylindrical superconducting mandrel, of the same material as the cylinder, is chosen with an outer radius ( R - τ ) and a wall thickness τ and if 2( R - 2τ )2 = ( R - τ )2, the trapped flux will increase to approximately 2HT( T ) if γ ( T ) « 1. This occurs when but the disadvantage is that the working volume is reduced by approximately a factor of two. Copyright © 1998 IOP Publishing Ltd

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Swartz and Rosner ( 1962 ) in their study of flux trapping and flux compression in Nb3Sn cylinders found that the field strength which could be trapped in these cylinders was limited by flux jumping to between 0.6 to 1.6 T. A flux jump is a process in which the stored magnetic energy is released suddenly when an unstable flux configuration arises within the superconductor. The flux jumping process will be considered in more detail below as this has been one the major limitations to the practical use of flux tubes with low-Tc superconductors. However, it has been shown that flux jumping will not be a problem ( Burgoyne and Watson 1994 ) in the flux creep of high-Tc BSCCO superconductors because the thermal activation of flux lines gradually diminishes the trapped flux so that some method of restoring the trapped flux is required. Flux compression is also limited by the fact that the final high-field working volume is considerably less than the total volume required to generate it which is a considerable drawback. (b) Introducing flux into the tube using an external solenoid followed by flux compression In this method the superconducting tube is inside a solenoid, coaxial with the tube, which can magnetize the superconducting material up to a field Hm a x large enough for the flux to penetrate the wall thickness τ after which the applied field in the solenoid is turned off leaving flux trapped in the flux tube. If it is assumed that the tube has the same superconducting properties as the tube above in (a) and if the residual field is to be identical to HT in equation (G6.0.15), then Hm a x is given by

If γ ( T ) is small then HT ∼ α( T )τ and Hm a x ∼ 2α( T )τ = 2HT so in order to obtain a trapped flux of HT in the tube a field of twice HT must be applied with the solenoid; however this can be done quickly and it is not necessary to change the temperature. Flux compression can then be done as described in the section above. Again there is the difficulty that the final working volume is limited and there is the problem of flux creep if BSCCO material is used. (c) Charging the flux tube with a flux pump A more economical use of the superconductor is desirable for practical applications. Using a large working flux tube which can be charged with magnetic field by smaller devices known as flux pumps satisfies this condition and, in principle, large fields could be produced in large volumes from a small volume of superconductor configured as a flux pump (Hildebrant et al 1962). A cyclical device has been described by Hildebrandt et al (1962), as shown in figure G6.0.13, in which flux can be introduced into a pump tube through a flux gate and then compressed with a mandrel and passed through another flux gate into

Figure G6.0.13. A flux tube with a flux pump and superconducting mandrel. The mandrel has the same cross-sectional area as the pump volume. Copyright © 1998 IOP Publishing Ltd

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the working region. This process can be continued indefinitely, but even in the absence of flux jumping, the compression of flux will be limited by flux penetration into the mandrel but this point may be reached only at high field if Jc is high and only depends weakly on field up to the values of field to be trapped because penetration into the mandrel depends inversely on Jc, as discussed in (a). The flux gate, as shown in figure G6.0.14, is a region of the superconductor which can be switched thermally so that flux can be either trapped or passed through the gate. Flux pumping methods have been reviewed by van Beelen et al (1965). The mode of operation is as follows: ( i ) Open gate 1, gate 2 closed. Insert magnetized ferromagnetic mandrel into pump volume. ( ii ) Close gate 1 and open gate 2. ( iii )Remove ferromagnetic mandrel from the pump volume. ( iv )Insert superconducting mandrel into pump volume thereby squeezing the flux into the working volume. ( v ) Close gate 2, open gate 1. Flux has been transferred to the working volume. ( vi )Repeat cycle.

Figure G6.0.14. Flux gates A, B, C and D are tubes carrying liquid nitrogen (LIN). E and F are electric heaters. The gate is originally at 77 K and fully superconducting. Heat is supplied through E and F to destroy superconductivity and allow the passage of flux.

In figure G6.0.14, A, B, C and D are tubes carrying liquid nitrogen (LIN). E and F are electric heaters. The gate is originally at 77 K and fully superconducting. Heat is supplied through E and F to destroy superconductivity and allow the passage of flux. If the distance B-F is approximately 1 cm and if the wall thickness is 1 cm then approximately 37 W must be supplied to raise E-F to 91 K with A, B, C and D at 77 K. Actually to permit flux movement the increase in temperature can be much less, namely an increase to the temperature where the trapped flux is equal to Hc 2. This method is by far the best in its use of superconducting material as the working volume can be made relatively large compared with the pumping volume and the working volume does not depend on the degree of flux compression. (d) A superconducting reciprocation canister separator with a flux tube This method involves replacing the low-Tc coil shown in figure G6.0.11 by a flux tube. As discussed above, this approach has the advantage that the cryogenics are greatly simplified when the operating temperature is 77 K and the capital and running costs are greatly reduced. The system runs in the persistent mode. As the field in the flux tube decreases with time due to flux creep, the field is topped up using the flux pump. This approach, if it is developed, could provide the cheapest option for magnetic separation on a large scale. Copyright © 1998 IOP Publishing Ltd

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However, such an approach must wait for the development of large pieces of high-Tc superconducting materials with current-carrying capacity approaching that of present-day low-Tc superconductors at high field before this approach is viable. G6.0.5 Other designs of superconducting separators with high-Tc material G6.0.5.1 A superconducting reciprocation canister separator with a superconducting wire-wound coil This method involves replacing the low-Tc coil shown in figure G6.0.11 by a high-Tc coil. Again as discussed above, this approach has the advantage that the cryogenics are greatly simplified when the operating temperature is 77 K and the capital and running costs of the refrigeration system are greatly reduced. The system, when possible, runs in the persistent mode. However, such an approach must wait for the development of long lengths of high-Tc superconducting wire with the current-carrying capacity approaching that of present-day low-Tc superconductors at high field before this approach is viable. It is unlikely that the capital cost of the separator magnet will be reduced much below that of the equivalent low-Tc separators. However, the advantage of this approach is that the design is based on an operational low-Tc system which has proved reliable in industrial applications. G6.0.5.2 The construction of a superconducting separator with an iron return circuit (a) The magnetic circuit The magnet is an iron electromagnet with superconducting windings (Watson 1992). Magnetic separators with iron return circuits of the type shown in figure G6.0.15 using non-superconducting coils have been employed in the minerals industry for many years. Because of diminishing resources and a loss of grade in older mines, it is interesting to look at an application of superconductors where the coil provides extra

Figure G6.0.15. Preferred orientation of the magnet for magnetic separation. The canister is positioned in the gap between the pole pieces. Copyright © 1998 IOP Publishing Ltd

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ampere turns which would be extremely expensive for copper coils if the magnetic circuit were close to saturation. It might be interesting to remove the coils from older machines and replace them with superconducting coils and run them much more cheaply and much closer to saturation. It is proposed that the high-Tc superconducting coil is mounted on the iron return circuit as shown in figure G6.0.15. Usually in this type of system there are two coils; however here it is proposed to use one coil as it is much more convenient to construct one liquid nitrogen container than two. The coil is placed further away from the gap than is normal but because high-Tc superconductors are still quite sensitive to magnetic field, it is better to keep the coil away from the stray field from the gap. The canister containing the ferromagnetic stainless steel matrix is placed in the gap. The main advantages are the ability to feed the canister upwards, thereby avoiding problems with entrained air in the feed cycle: a uniform fluid distribution is easier to obtain when feeding in an upward direction and when the feed and flow are parallel the magnetic separation process is more effective. (b) The magnetic circuit for the separator Although the performance of the materials at 77 K is limited it is still possible, by magnetizing a magnetic circuit, to construct a magnetic separator with a useful separation capacity. In the following example of how high-Tc superconductors could be employed, the dimensions of the canister will be the same as that used in the magnetic treatment of blood components (Richards et al 1993, Roath et al 1990, Thomas et al 1993), namely 3 cm in diameter and 3 cm long. For dR C a value of 0.5 m is reasonable. Making a conservative estimate of the properties of BSCCO—Ag tape, for example, dimensions 25-100 m in length, 2-4 mm wide by 0.1-0.15 mm thick, 70-80% being BSCCO, the value of Ic , the critical current, in zero field is about 3 A cm−2 in a field of 0.1 T. Ic is approximately 0.3 A cm−2. It is also assumed that the current depends linearly on the field as in equation (G6.0.14). For the magnetic circuit we have that

where NC IC is the ampere turns provided by the superconducting coil, BA is the flux density in the separation canister with area AA and length dA , µA is the relative permeability of the contents of the canister and dR C is the length of the rest of the magnetic circuit with permeability µRC. Using the dimensions for the canister given above and where AA = 7.07 × 10-4 m2, NC IC versus BA can be calculated for the ball matrix and for the wire matrix. The canister should maximize the useful space in the gap and make the best compromise with the need to feed the suspension through the matrix as uniformly as possible. A section through the proposed canister is shown in figure G6.0.16. The best arrangement is to feed the canister from the side. The same condition is required for uniform feed in both cases, that is the pressure drop of the fluid in the space between the inlet and the distribution plate must be much less than the pressure drop through the distribution plate. As a separation matrix in the canister, which provides the secondary pole system, there are a number of choices. Medium or fine grade material, made from ferromagnetic 430 grade stainless steel wool, occupying about 5% of the space, is suitable but for microbiological separation the wire must be smooth. At fields B0 <1.2 T, a matrix consisting of small smooth ferromagnetic stainless steel balls is an extremely effective separation matrix. It is fairly clear that these two options have very different effects on the magnetic circuit as the two options have very different magnetic reluctances. (c) The ball matrix separator If a separation canister is filled with ferromagnetic spheres as a matrix (Watson and Watson 1983) with the same magnetic properties as the iron return circuit (Bleaney and Bleaney 1957), then, as randomly packed spheres occupy 66% of the space, we can write

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Figure G6.0.16. The cross-section over which the canister is fed is rectangular and is 30 mm by 24 mm. The thickness of the canister, the gap thickness, is 30 mm but the length of the canister must also incorporate the inlet and outlet distribution systems. In the figure these are shown as occupying a total of 6 mm so the effective separation length, in other words the length filled with matrix parallel to the feed velocity, is 24 mm.

where 0.66 is the packing factor for randomly packed spheres.

where Lc is the length of the coil having a total of Nc turns. Using the values of magnetic induction of iron given by Bleaney and Bleaney (1957), using equation (G6.0.19), Hc can be calculated as a function of the magnetic induction in the circuit. The current-carrying capacity IC ( HC ) can thus be determined. For the ball matrix separator, figure G6.0.17 shows that a magnetic induction of almost 1.5 T can be achieved with 100 turns of the BSCCO—Ag tape. A coil length of 10 cm and a tape width of 4 mm means that at least four layers are required. (d) Ferromagnetic stainless steel wire matrix In what follows it is assumed that the stainless steel wire has the same magnetic properties as the material of the magnetic circuit. Therefore as the filling factor of the wire in the canister is 5% we have

and equation (G6.0.17) becomes

and

The actual number of turns Nc for the wire compared with the value for the ball matrix is shown in figure G6.0.17. In order to reach an induction in the magnetic circuit of 1.5 T, twice as many turns are required for the wire matrix as for the ball matrix. A high-Tc coil of 240 turns of BSCCO—Ag tape can be used as a magnetic separator with a canister containing a wire wool matrix or a ball matrix and should allow a field up to 1.5 T with the wire matrix and 1.65 T with the ball matrix at 77 K: at lower temperatures saturation of the magnetic circuit can be achieved. Copyright © 1998 IOP Publishing Ltd

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Figure G6.0.17. The number of turns Nc required in the coil to produce the induction BA for the ball matrix and for the wire matrix.

G6.0.5.3 Drum separator with permanently magnetized superconducting magnets The construction of the superconducting drum separator is shown in figure G6.0.6(b) and it is possible that the superconducting coils can be replaced by superconducting permanent magnets as described above. In DESCOS the field at the drum surface is 2.8-3.2 T. For YBCO at 65 K, which is about the limit for subcooled liquid nitrogen, the trapped field was 5.3 T ( Weinstein et al 1994 ). It seems reasonable that the field distribution within DESCOS could be duplicated or surpassed with this material to give fields at the drum greatly in excess of 3.2 T at the drum surface. Further improvements can be achieved either by reducing the temperature or by irradiating the YBCO with heavy ions (Weinstein et al 1993, 1994). One factor must be carefully considered in the application of these materials as permanent magnets, namely flux creep. By the thermally activated processes it is possible for the strength of the trapped fields to decrease with time so that at some point it would be necessary to remagnetize the superconductors (Dew-Hughes 1988, Kes et al 1989). Measurements by Weinstein et al (1993) indicate that the maximum trapped field at time t , BT, max ( t ) is related to the maximum trapped field at t = 0 as follows

second-order terms were found to be small and were of the sign to reduce creep. With t in minutes, β ∼ 0.043 ± 0.009 so that a magnet which decreases by 13% in the first week decreases by a further 13% in the next 20 years. The stability may be improved in a number of ways, for example by waiting for or by activating the magnet to BT < BT, m a x , β can be reduced, very approximately (Weinstein et al 1991) β ≈ (BT /BT, m a x )2, and creep may also be reduced by operating at lower temperature; near 77 K β ≈ T. G6.0.6 Conclusions Commercial low-Tc superconductors have been successfully applied to the manufacture of HGMSs which, since 1984, have found increasing application in kaolin beneficiation for use in the paper and the ceramics Copyright © 1998 IOP Publishing Ltd

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industries. A new design, the reciprocating canister system superconducting separator, has taken an increasing share of the market since 1989. There also appears to be a growing business in retrofitting superconducting coils into the older HGMSs which used copper coils to supply the magnetic field. The prospect for high-Tc superconductors appears to be excellent, promising large cost reductions in the cryogenic installation in terms of cheaper components, such as Dewars and refrigrerators, with increased reliability. Furthermore, if only d.c. magnetic fields are required and superconducting permanent magnets can be used, either flux tubes or discs, further reductions in the capital cost can be expected as the costly processes of wire manufacture and coil winding can be avoided. References Bleaney B I and Bleaney B 1957 Electricity and Magnetism 1st edn (Oxford: Oxford University Press) Burgoyne J W and Watson J H P 1994 Flux jump instabilities in a bulk Bi2Sr2CaCu2O8 + x Cryogenics 34 507–12 Campbell A M et al 1987 Magnetisation and critical currents of high-Tc superconductors High Tc Superconductors and Potential Applications ed J Vilain and S Gregoli (Brussels: Commission of the European Communities) pp 91–2 Dew-Hughes D 1988 Model for flux creep in high-Tc superconductors Cryogenics 28 674–7 Frantz S F 1937 US Patent 2 074 085 Gotoh S, Murakami M, Koshizuka N and Tanaka S 1990 Magnetization study of YBaCuO prepared by quench and melt growth process Physica B 165&166 1379–80 Gotoh S, Murakami M, Fujimoto H and Koshizuka N 1991 Magnetic properties of superconducting permanent magnets of YBa2Cu3Ox Physica C 185–189 2499–500 Haldar P, Hoehn J G Jr, Rice J A and Motowidlo L R 1992 Enhancement in critical current density of Bi— Pb–Sr—Ca—Cu—O tapes by thermomechanical processing Appl. Phys. Lett. 60 495–8 Haldar P, Hoehn J G Jr, Rice J A, Motowidlo L R, Balachandran U, Youngdahl C A, Tkaczyk J E and Bednarcyzk N 1993 Fabrication and properties of high-Tc tapes and coils made from silver-clad Bi-2223 superconductors IEEE Trans. Appl. Supercond. AS-3 1127 Haldar P, Hoehn J G Jr, Iwasa Y, Lim H and Yunus M 1995 Development of Bi-2223 HTS high field coils and magnets IEEE Trans. Appl. Supercond. AS-5 512–5 Hikata T 1990 Magnetic field dependence of critical current density in Bi—Pb—Sr—Ca—Cu—O silver-sheathed wire Cryogenics 30 924–9 Hildebrant A F, Elleman D D, Whitmore F C and Simpkins R 1962 J. Appl. Phys. 33 2375–9 Jin S, Tieful T H, Sherwood R C, Davis M E, van Dover R B, Kommlott G W, Fastnacht R A and Keith H D 1988a High critical currents in Y—Ba—Cu—O superconductors Appl. Phys. Lett. 52 2074–6 Jin S, Tieful T H, Sherwood R C, van Dover R B, Davis M E, Kommlott G W and Fastnacht R A 1988b Phys. Rev. B 37 7850 Kes P H, Aarts J, van den Berg J, van der Beek C J and Mydosh J A 1989 Thermally assisted flux flow at small driving forces Supercond. Sci. Technol. 242–8 Kitamura T, Hasegawa T and Ogiwara H 1993 Design and fabrication of Bi-based superconducting coils IEEE Trans. Appl. Supercond. AS-3 939–41 Kolm H H 1971 US Patent 3 567 026 Liu Y A and Oak M J 1983 Studies in magnetochemical engineering part 2: theoretical development of a practical model for high-gradient magnetic separation AIChE J. 29 771–9 Luborsky F E and Drummond B J 1976 Build-up on particles on fibers in a high field—high gradient separator IEEE Trans. Magn. MAG-12 463–5 Lue J W, Dresner L, Schwenterly S W, Aized D, Campbell J M and Schwall R E 1994 Stability measurements on a 1 T high-temperature superconducting magnet IEEE Trans. Appl. Supercond. AS-5 230–3 Maley M P 1991 Overview of the status of and prospects for high-Tc wire and tape development J. Appl. Phys. 70 6189–93 Marston P G, Nolan J J and Lontai L M 1971 US Patent 3 627 678 Murakami M 1992 Processing of bulk YBaCuO Supercond. Sci. Technol. 5 185–203

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Neuringer L J and Shapira Y 1966 Nb-25% Zr in strong magnetic fields: magnetic, resistive and thermal behaviour Phys. Rev. 148 231–46 Oder R R and Price C R 1973 Brightness beneficiation of kaolin clays Tech. Assoc. Pulp Paper Ind. 56 75–8 Otto A 1993 Multifilament Bi-2223 tapes made by a metallic precursor route IEEE Trans. Appl. Supercond. AS-3 915 Purcell J R, Burnett S C and Creedon L R 1991 US Patent 5 019 247 Richards A J, Thomas T E, Roath O S, Watson J H P, Smith R J S and Lansdorp P M 1993 Improved high gradient magnetic separation for positive selection of human blood mononuclear cells using ordered wire filters J. Magn. Magn. Mater. 122 364–6 Riley P W and Hocking D 1981 A reciprocating canister superconducting magnetic separator IEEE Trans. Magn. MAG-17 3299–301 Roath S, Richards A, Smith R and Watson J H P 1990 High-gradient magnetic separation in blood and bone marrow processing J. Magn. Magn. Mater. 85 285–9 Salama K and Lee D F 1994 Progress in melt texturing of YBa2Cu3Ox Supercond. Sci. Technol. 7 177–93 Swartz P S and Rosner C H 1962 Characteristics and a new application of high field superconductors J. Appl. Phys. 33 2292–300 Swartz P S and Bean C P 1968 A model for magnetic instabilities in hard superconductors: the adiabatic critical state J. Appl. Phys. 39 4991–8 Thomas T E, Richards A J, Roath O S, Watson J H P, Smith R J S and Lansdorp P M 1993 Positive selection of human blood cells using improved high gradient magnetic separation filters J. Hematother. 2 297–30 Thompson J R, Christien D K, Sekula S T, Brynestad J and Kim Y C 1987 Magnetization studies of the high-Tc compound YBCO Proc. Cryogenic Engineering Conf. (New York: Plenum) van Beelen H, Arnold A J P T, Sypkens H A, van Braam Houckgeest J P, Ouboter d B, Beenakker J J M and Taconis K W 1965 Physica 31 413–43 Watson J H P 1968 Magnetization of synthetic filamentary superconductors. B. The dependence of the critical current density on temperature and magnetic field J. Appl. Phys. 39 3406–13 Watson J H P 1973 Magnetic filtration J. Appl. Phys 44 4209–13 Watson J H P 1976 Radial flow canister UK Patent 1 530 296 Watson J H P 1978 Approximate solutions of the magnetic separator equations IEEE Trans. Magn. MAG-14 240–5 Watson J H P 1988a High magnetic field production with ceramic superconductors Physica C 153–155 1401–2 Watson J H P 1988b High magnetic field production with superconductors at 77 K Proc. 12th Int. Cryogenic Engineering Conf. ed R G Scurlock (London: Butterworth) Watson J H P 1992 The design for a high-Tc superconducting magnetic separator Supercond. Sci. Technol. 5 694–702 Watson J H P and Li Z 1992 Theoretical and single-wire studies of vortex magnetic separation Minerals Eng. 5 1147–65 Watson J H P and Watson S J P 1983 The ball matrix magnetic separator IEEE Trans. Magn. MAG-19 2698–704 Weinstein R, Chen I-G, Liu J and Lau K 1991 Permanent magnets composed of high temperature superconductors J. Appl. Phys. 70 6501–3 Weinstein R et al 1993 Permanent magnets of high-Tc superconductors J. Appl. Phys. 73 6533–5 Weinstein R et al 1994 Progress in HTS trapped field magnets: Jc , area and applications Proc. 4th World Congress on Superconductivity (New York: Pergamon) Windle W 1975 UK Patents 1 469 765, 1 599 824, 1 599 825 Wipf S F and Laquer H L 1989 Superconducting permanent magnets IEEE Trans. Magn. MAG-25 1877–80

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G7 High-frequency cavities

W Weingarten

G7.0.1 Introduction High-energy physics has prospered for the last decades thanks to the progress in particle accelerator techniques. The quest was always for higher centre-of-mass energies and larger luminosities (event rate for unit cross-section). The most recent accomplishment has been the operation at centre-of-mass energies between 91 GeV and 2 TeV of the large electron-positron collider ( LEP ) at CERN, Stanford linear collider ( SLC ) at SLAC, Hera at DESY and the Tevatron at Fermilab. An increasingly demanding problem is the reduction of investment and operation costs of such large accelerators. For this reason the benefits of superconductivity are exploited for magnets (at Tevatron and Hera) and radiofrequency ( RF ) cavities (at LEP and Hera). Combining energies with affordable costs introduces different problems for proton and electron accelerators. Circular accelerators offer a cost-effective way of obtaining high energies because the particles traverse the accelerating sections many times. The energy limit is determined by the emission of synchrotron radiation, which increases with the fourth power of γ, the ratio of the particle energy over its rest mass. Accelerators for protons, a relatively massive particle, are still far away from this limit. Therefore future proton accelerators such as the Large Hadron Collider ( LHC ) at CERN will be circular machines. On the other hand, circular accelerators for electrons, a relatively light particle, emit large amounts of synchrotron radiation. An estimation shows ( Richter 1979 ) that future accelerators for electrons will be linear colliders like the pioneering SLC at Stanford. The reason is that the investment costs are proportional to the length (for linear colliders) or the circumference (for circular colliders). For a beam energy E and an accelerating gradient g , they are proportional to E/g and to respectively. Hence, above a certain energy a linear collider is cheaper. Since particles will traverse the accelerating section only once, it is essential to have the necessary high accelerating gradients. The other demanding need is to keep the mains power as low as possible for a given luminosity and particle energy; in other words, the efficiency of transforming mains power to beam power should be at a maximum. In principle, RF superconductivity should provide a technical solution for both constraints, that is a high accelerating gradient and high mains-to-beam-power conversion efficiency. The first idea for the use of superconducting cavities for accelerators was the proposal to construct a linear accelerator (linac) for electrons to perform electron-proton scattering coincidence experiments (Wilson 1963). The signal-to-noise ratio is proportional to the duty cycle, whch means that a continuous wave (CW), nonpulsed beam is the best choice. In addition a CW beam facilitates the task of beam Copyright © 1998 IOP Publishing Ltd

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control (no transient effects). These ideas have culminated in the construction and operation of so-called superconducting recyclotrons (Stanford HEPL, Urbana and the Continuous Electron Beam Accelerator Facility (CEBAF) in the USA and at Darmstadt in Germany). Basically these are superconducting linacs, for which the beam is recirculated several times to reduce the investment costs. Because of the excellent beam quality superconducting accelerators are also used for free electron laser (FEL) applications. G7.0.2 Technical motivation for the use of superconducting cavities G7.0.2.1 The mains-to-beam-power conversion efficiency Electrical power from the mains to the beam is transferred via the following chain: high-voltage power supply—klystron—waveguide—RF cavity—beam. For the time being the largest power loss has to be tolerated between the RF cavity and the beam. In order to establish the accelerating voltage V in the cavity, RF currents are induced at its surface generating the heat Pc. A shunt impedance R = V 2/( 2Pc ) is defined to account for these RF losses. R should be as large as possible, which can be achieved by a careful design of the resonator geometry. RF accelerating cavities manufactured from high-conductivity copper have shunt impedances of typically 10-15 MΩ for a 1 m long cavity. With an average beam current Ib and beam power Pb = V Ib (for the particle riding on the crest of the RF wave), the conversion efficiency is defined as

As an example, for the normal-conducting RF system of LEP at CERN (Geneva) with R = 43 MΩ, Ib = 6 mA, V = 3 MV, we get η ≈ 15%. The shunt impedance R of an RF cavity is related to its Q value (cf section G7.0.8.1) by R = (R/Q )Q. ( R/Q ) is independent of the RF frequency and is determined by the geometry of the cavity. The Q value for its part is inversely proportional to the surface resistance RS , Q = GRs−1, with G another geometrical constant (of typically 250Ω Hence, to have large shunt impedances we are faced with the development of RF cavities with surfaces of low surface resistance. The second factor (apart from the shape) which controls the shunt impedance is the metal, from which the resonator’s walls are manufactured. The dissipated power Pc in a normal-conducting cavity with a given length and gradient varies with frequency ω as PC ∼ ω −1/2 . That is why, in a conventional linac, higher frequencies are preferred. In contrast, in a superconducting cavity PC ∼ ω (in the ideal case, as will be shown later), favouring lower frequencies. But there are other factors which play a role in determining the operating frequency (cf sections G7.0.2.3 and G7.0.3.1). G7.0.2.2 The cryogenic efficiency It is the large ‘improvement factor’ of the shunt impedance (∼105 ) compared with conventional copper cavities that makes superconducting cavities so attractive. However, the dissipated power Q 2 = PC ( figure G7.0.1 ), has to be removed at cryogenic temperatures, at the boiling temperature under normal pressure of liquid helium, T2 = 4.2 K, or at even lower temperatures, depending on the RF frequency (cf section G7.0.3.1). Hence, the entropy current S = Q 2/T2 which, according to the second law of thermodynamics, in the ideal (reversible) case is Q 1/T1 at room temperature T1 ≈ 300 K. According to the first law of thermodynamics, the power

has to flow into the compressor. The Carnot efficiency ηc for a refrigerator is defined as

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Figure G7.0.1. Schematic power flow in a refrigerator.

For T1 = 300 K and T2 = 4.2 K , ηc = 1/70. The ‘thermodynamic efficiency’

is the ratio of the power needed to operate the compressor in the ideal case to the ‘real’ power The total cryogenic efficiency is

.

With ηt d ≈ 0.3 for large refrigerators the total cryogenic efficiency is ηc r = 4.5 × 10−3. Unavoidably, in a superconducting accelerator some power Pc r flows into the liquid He, even in the absence of RF (standby heat load of the cryostat). The efficiency η for a superconducting accelerator of RF-to-beam power conversion is then

As an example, for the superconducting cavity and cryostat for LEP with Pc = 56 W, Pb = 160 kW and Pc r = 25 W, we obtain a total efficiency of η = 90%, which is larger by a factor of six than for a conventional RF system. G7.0.2.3 Lower impedance of superconducting cavities Copper cavities are optimized for a high shunt impedance. This can ideally be achieved with several cylindrical cavities in series with small iris holes for the beam to pass through. Typical values for a 1 m long cavity are 10–15 MΩ. The total dissipated RF power to be removed gives the performance limit, because issues of low surface electric and magnetic fields are of minor importance. These criteria are taken into account in the design as shown schematically in figure G7.0.2 (top). Superconducting cavities, in contrast, are designed for low surface electric fields and other reasons (e.g. to avoid electron multipacting or to efficiently extract the power which the beam transfers into the higher order modes). The shunt impedance is sufficiently large thanks to the superconducting material. A typical value for a 1 m long cavity is 1 TΩ (at 4.5 K). Hence one can sacrifice a small amount of shunt impedance by choosing larger iris openings and a rounded shape (figure G7.0.2 (bottom)). The shunt impedance is only defined for the accelerating mode. In a more general sense the impedance of a cavity is a quantity which governs the interaction of the beam with the cavity. The impedance increases with length and decreases for a larger iris diameter (hence increases with frequency). As superconducting cavities have a larger field gradient (about a factor of four) and a larger iris diameter, they present more than a factor of ten lower impedance (for the same accelerating voltage) to the beam compared with normal-conducting cavities (Zotter 1985). Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.2. The shunt impedance is determined by the shape and the metal of the resonator’s wall. In a copper cavity the shunt impedance is maximized by a design with small beam holes and nose cones (top). A superconducting cavity has a priori a large shunt impedance thanks to the low wall losses. Therefore a small amount of shunt impedance can be sacrificed (larger iris holes) to better cope with other issues (beam cavity interaction through wake fields, peak surface fields, bottom).

The rise time of beam instabilities is inversely proportional to the product of impedance and current (beam wake field). The beam remains stable if the rise time is larger than the damping time of the synchrotron and betatron oscillations. There is a minimum tolerable rise time, and, consequently, the maximum current is inversely proportional to the impedance; hence the advantage of superconducting cavities over normal-conducting cavities. G7.0.3 Design and production G7.0.3.1 The operating frequency As we have seen, the figure of merit needed for the accelerating mode to have low RF dissipation is the shunt impedance R, which should be maximized. It is defined as the equivalent resistance for a parallel RF resonant circuit. The RF voltage amplitude of this resonant circuit is the accelerating voltage V, and the dissipated power of this resonant circuit is the losses Pc in the wall of the cavity

Without loss of generality, we want to maximize R per length L (which is called r )

where we have made use of the definition of r/Q = Ea2/(2ωU/L), Q = ωU/PC , V = Ea L , RS Q = G, the geometry factor. U is the stored energy, Ea the accelerating gradient, RSB C S the BCS part of the surface resistance, and Rs0 its residual part (cf sections G7.0.4.2 and G7.0.6.1). Now we shall look at the dependence on the frequency of the different factors. The BCS surface resistance goes with the square of the operating frequency, and the residual surface resistance is nearly independent of the operating frequency (cf figure G7.0.7 later in this chapter). Therefore, the condition RSB C S Rs0 is valid for frequencies larger than 3 GHz, and the condition RSB C S Rs0 is valid for frequencies smaller than 300 MHz

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To maximize r, we prefer lower frequencies for f ≥ 3 GHz and higher frequencies for f < 300 MHz. Therefore, both very low and very high frequencies are not useful, and superconducting accelerating structures are operated between 300 MHz and 3 GHz, approximately. This is strictly speaking only true for structures used for the acceleration of high-energy particles which are travelling at speeds close to the velocity of light ( β = 1 ). The corresponding gap width is between 5 and 50 cm. For particles moving much more slowly, the gap width in this frequency range would be unpractically small. Therefore lower frequencies are preferred (cf section G7.0.10). G7.0.3.2 Computer codes Computer codes (SUPERFISH, URMEL, URMEL-T, MAFIA, Corlett 1992) are sufficiently precise and form the basis of the first design of an accelerating structure. They allow the determination of the relevant parameters such as the resonant frequency, the inter-cavity coupling factor (for a multicell cavity), the R/Q value, the geometry factor and the electric and magnetic field enhancement factors (ratio of electric and magnetic surface fields/accelerating gradient). Figure G7.0.3 shows the distribution of the electric fields in a π-mode accelerating structure used for the e+e- collider LEP at CERN.

Figure G7.0.3. π-mode electric field line distribution of the LEP four-cell structure computed by SUPERFISH.

The precision obtained depends on the mesh size and the parameter to be calculated. For URMEL-T and N = 500–5000 mesh points, a typical computing time of 0.3–10 min is needed, respectively, and one obtains a precision of ∆ f/f0=1/( 3.5N ) for the resonant frequency. G7.0.3.3 Design of end cells Special care has to be taken in the design of the end cells. They are slightly different in geometry for two reasons. Firstly, the beam tube represents a perturbation in comparison with the inner cells and would lead to an ‘un-flat’ field distribution, if not properly compensated. Such a nonuniform field distribution represents a slightly lower R/Q value and, what is worse for a superconducting cavity, larger surface electric and magnetic fields for a given accelerating gradient (cf sections G7.0.5 and G7.0.6.3). The compensation is achieved by a modification of the cell shape of the end cells with regard to the inner cells. Secondly, the higher-order mode (HOM) field distribution has to be taken into account as well (cf section G7.0.8.2). The HOM couplers are in general attached to openings in the beam tubes and not to openings on the individual cells. The reason for this is that any perturbation of the rounded shape of the individual cells (favouring electron multipacting, cf section G7.0.6) should be avoided. Therefore the HOM must have a sufficiently large amplitude in the end cells so that they can be damped properly. This has to be assured by a proper shaping of the end cells, too. Copyright © 1998 IOP Publishing Ltd

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G7.0.3.4 Shaping, welding and surface processing After the cavity has been designed, in nearly all cases half-shells are produced from sheet metal (niobium or copper) by deep drawing or spinning on a lathe. They are joined together by electron-beam welding (with smooth welds). The whole cavity is etched, rinsed, coated with a thin niobium layer, if needed, as in the case of a copper cavity substrate, and tested with low-power RF (several 10–100 W ). With the aim of reducing the production cost, it has been shown that cavities can also be formed from a single tube by hydroforming (Hauviller 1989, Kirchgessner 1988) or from a single plate by spinning (Palmieri 1994). The hydroforming method for niobium is still in an early development stage. In all fabrication steps, clean working conditions are mandatory. If these conditions are not carefully respected, the performance of a superconducting cavity can be severely degraded. For example, small normal-conducting metal particles (mostly iron) of micrometre size could be deposited on the cavity surface and eventually be evaporated under the action of the RF magnetic field, thus creating a much larger zone of normal-conducting material on top of the superconducting surface. This might ultimately lead to a thermal quench (cf section G7.0.6.2). Small dust particles of submicrometre size could be deposited onto the cavity surface and emit electrons under the action of the electric field ( field emission electron loading, cf section G7.0.6.3 ). Particularly harmful are dust particles, if they remain on the copper surface of a cavity to be sputter-coated afterwards with a superconducting material like niobium. They create defects leading to poor adhesion and removal of heat. They can be heated up by the RF magnetic field up to the critical temperature of niobium, which will increase the RF losses. Even worse, they may be ultimately heated up beyond the melting temperature and cooled by radiation, and/or eventually produce electrons by thermionic emission. All these effects have been observed in superconducting cavities. The cleaning of the surfaces exposed to the RF field is achieved by rinsing or spraying with ultrapure agents such as water and/or alcohol, sometimes under high pressure or ultrasonic agitation (Saito et al 1990). The rinsing agents used have to comply with the high standards demanded by semiconductor manufacturing (table G7.0.1).

The assembly of ancillary equipment, e.g. the power coupler, probe antenna, HOM couplers, or the joining of individual cavities into a larger module also has to be executed in a clean room of class 100 or better (figure G7.0.4). We summarize the design criteria and state of the art (starting from the cavity proper and passing to the ancillary equipment and cryostat): ( i ) spherical shape with slightly inclined iris to facilitate the drainage of rinsing fluids Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.4. The four-cavity module for LEP being equipped with HOM and power couplers in a class 100 clean room. Courtesy CERCA.

( ii ) limited number of cells: careful computer-aided designing of cavity geometry imperative ( iii ) no holes in the cavity proper: couplers attached to the beam tube ( iv ) be aware of minimizing mechanical vibrations right from the beginning of the design ( v ) metal sheet forming ( vi ) rinsing to be done with clean and dust-free fluids ( vii ) compact coaxial antenna-type couplers for RF power input and HOM power damping ( viii )RF windows—if possible—between beam and insulation tank vacuum to minimize consequences of break ( ix ) all ancillary equipment to be cleaned according to the standards of cleaning the cavity ( x ) assembly under dust-free conditions ( xi ) frequency tuners without moving parts ( xii )cryostat gives access to vital parts without being forced to dismount it from the beam line. A gradient of 5–10 MV m−1 at Q = 3–5 × 109 has been obtained in industrial manufacturing of large niobium accelerating structures (National Laboratory for High Energy Physics (KEK), CEBAF, CERN, cf table 6 of Benvenuti et al 1996). G7.0.4 Surface resistance G7.0.4.1 The anomalous skin effect One might be tempted to manufacture an RF cavity from copper and reduce its surface resistance by operating it at cryogenic temperature, for example 4.2 K. For high-purity copper the increase in conductivity σ (residual resistivity ratio—RRR) is 5000, for oxygen-free high-conductivity (OFHC) copper it is still 100. Unfortunately, the surface resistance does not fully exploit this because it is subjected to the anomalous skin effect (figure G7.0.5). The skin effect in a general sense is created by surface currents in the metal which short-circuit the electric field parallel to the surface. In the anomalous skin effect only electrons whose free path ranges Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.5. The anomalous skin effect in a 500 MHz OFHC copper cavity: versus . The straight line and follows the normal skin effect in metals. For temperatures smaller than 65 K, Rs decreases more slowly with finally saturates.

within a surface layer of the effective penetration depth δe f f contribute to the shielding current. Hence, the effective density of electrons is reduced by the factor α (δe f f /l ) , with l the electron mean free path and α a phenomenological factor near unity (≈3/2). The effective conductivity is, therefore, given by

Introducing this into the formula for the skin depth (table G7.0.2), we obtain

Similarly, the effective surface resistance Rs e f f for the anomalous skin effect is

With equations (G7.0.10) and (G7.0.11) we end up with

which is, as l/σ = constant, independent of σ and exhibits the characteristic frequency dependence (∼ω 2/3 ) of the surface resistance in the anomalous limit. With typical values for a standard metal, σ ( Ω −1m−1 )=1.21 × 1015 × l (m) ( Pippard 1960 ), we obtain at 500 MHz, Rs e f f = 1.3 mΩ, which represents a gain of a factor of only four compared to room temperature—independent of the conductivity at low temperature. G7.0.4.2 The surface impedance of superconductors Another way to decrease the surface resistance is to make use of a superconducting surface. It is well known that the d.c. resistivity of a superconductor falls to zero below a critical temperature Tc . For RF, however, a nonzero surface resistance remains, which decreases with lower temperature. The two-fluid model of a superconductor is appropriate to understand this. Suppose there are two fluids of electrons in the superconductor, the normal-conducting ones with density nn and the superconducting ones with density ns . Electric fields, if there are any, will drive currents through the two fluids, as they do Copyright © 1998 IOP Publishing Ltd

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through two impedances in parallel. The familiar Maxwell equations for a metal have to be completed by the equations describing the material, a combination of the London equations and Ohm’s law (constitutive equations). In table G7.0.2 the equations for the two-fluid model are contrasted with those for a normal conductor. Omitting the detailed derivation, the surface resistance of a superconductor is

For T < Tc /2, nn can be approximated by a Boltzmann factor nn ∼ e −∆/κ T, with ∆ = 1.95kTc for niobiun ( Tc = 9.25 K ), thus taking into account the pairing energy 2∆ for breaking a Cooper pair. Hence, the surface resistance is RSSC = Aω 2 e −∆/k T , with A a function of the parameters of the normal conductor nn , l , m and υF . This result is roughly the correct one worked out from the BCS theory (Abrikosov et al 1959, Mattis and Bardeen 1958)

It is instructive to check the validity of the approximations made to derive the superconducting surface impedance in table G7.0.2, by using some numbers for the standard metal. With σn( 300K )=7.6 × 106(Ωm)−1, a typical value for niobium, σn e f f (4.2K)((Ωm)−1)=9.4 × 108/( f(GHz))1/3 ,nn ( 300 K ) = 6 × 1028 m−3, nS( 4.2 K ) ≈ nn( 300 K ), ∈0 µ0ω 2λ L2 = 2 × 10−13 ( f (GHz))2 « σn e f f µ0ω 2λ2L ≈ 3.5 × 10−3( f (GHz))2/3 « 1 . Figures G7.0.6 and G7.0.7 show the temperature and frequency dependence of the surface resistance, respectively, measured at Wuppertal University (Klein 1981, Lengeler et al 1985, Müller 1983). The ω 2 dependence of the surface resistance is a consequence of the frequency-independent penetration depth (Tinkham 1965). Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.6. The surface resistance of a 3 GHz niobium sheet cavity ( RRR = 40 ) versus Tc /T. The residual resistance is 4 nΩ.

Figure G7.0.7. The BCS surface resistance RSB C S of niobium versus frequency at 4.2 K (extrapolated to 1.8 K). The lowest residual surface resistance data are indicated at the bottom (Rs 0 , cf table G7.0.4 later in this chapter).

It is evident that superconductors with a larger pairing energy ∆ ( higher Tc ), should allow a smaller surface resistance at the same temperature, provided A’ is about the same. This is the reason why both the conventional and new high-Tc superconductors such as Nb3Sn and YBaCuO are also being investigated for RF applications. The surface resistance of commercially available niobium is given by RSBC S (nΩ ) = 105 ( f (GHz))2 exp(− 18/( T ( K )))/( T ( K )). For Nb3Sn-coated cavities (Tc = 18.1 K) we have RSBC S (nΩ ) = 105 ( f (GHz))2 exp(− 40/( T ( K )))/( T ( K )). For comparison, the surface resistance for copper at room temperature is Rs (mΩ ) = 7.8 ( f (GHz))1/2, 105 times larger than for niobium at 4.2 K and 350 MHz. Consequently, the shunt impedance is larger by the same factor for superconducting cavities. The two-fluid model, although not giving precise results, has the virtue of creating the correct frequency and temperature dependence and shows that the coefficient A’ is related to the conductivity of the normal-conducting electrons. Hence, for frequencies f < 10 GHz the approximations are justified. The wavevector κ is purely imaginary with |κ | = J/λL. The RF wave penetrates into the metal for a distance λL, which is independent of the frequency and equal to the d.c. penetration depth. These considerations can be refined for ‘dirty’ superconductors in a heuristic approach. The London equation

(cf table G7.0.2) is only correct for ‘London’ superconductors (hence the name ‘London’ penetration depth λL ) where the fields and currents vary slowly in space on the scale of the extension of the Cooper pairs ξ (coherence length): λL > ξ. A general penetration depth λ has to be defined. According to Pippard, equation (G7.0.16) has to be modified, by analogy with the anomalous skin effect, where the conductivity σ in Ohm’s law J = σ E is given by equation (G7.0.10). Hence, the replacement

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has to be done, such that

with

and ξ0 being the coherence length for l → ∞. Consequently, in equation (G7.0.14), λL is replaced by λL( ξ0 /ξ( l ))1/2, and we obtain

Two extreme cases may be considered: ( i ) l » ξ0 and ( ii ) l « ξ0. With σn ∼ l we get in case ( i ) RS ∼ l and in case ( ii )RS ∼ l −1/2, indicating a minimum of RS at l = ξ0/2 and an increase of RS for smaller and larger l ( figure G7.0.8 ). With l (nm) ≈ 2.7 × RRR, this minimum is for niobium (ξ0 = 38 nm) at RRR = 7 (not visible in figure G7.0.8).

Figure G7.0.8. The BCS surface resistance of niobium at 350 MHz (unpublished data) and 4.2 K versus RRR (the ratio of d.c. resistivity at 300 K and 4.2 K in the normal-conducting state). RRR is proportional to the mean free path l of the electrons. The line indicates calculations performed by Padamsee et al (1987) using a computer program by Halbritter (1970).

G7.0.5 The critical field of superconductors—RF case The maximum surface RF field is given by the condition that the magnetic fields starts to penetrate the superconducting metal. Deep inside the metal this should happen at the thermodynamic critical field Bc t h for type I and at the lower critical field Bc 1 for type II superconductors. For a smooth surface, however, there is a surface barrier, which prevents the magnetic field from entering deep inside the metal (Bean and Livingston 1964, de Gennes 1966, Matricon and St James 1967, Yogi et al 1977). This barrier disappears at the critical superheating field (also called the surface barrier field) Bs h . Bs h is larger than Bc t h and Bc l for type I and type II superconductors respectively. It approaches Bc t h for extreme type II superconductors. A few experimental results may illustrate this feature. Some Bs h , values for the superconductors Sn, In, Pb, Nb, Nb3Sn—typical materials studied in RF superconductivity—are given in table G7.0.3, to be compared with the maximum surface fields Be x p experimentally obtained so far. The first three metals (Sn, In, Pb) are type I superconductors, the last two (Nb, Nb3Sn) are of type II. It can be concluded that for lead, for example, Be x p o clearly exceeds Bc t h , and is very close to Bs h . For Sn, Be x p is identical to Bc t h for several temperatures, whereas for Nb (Hays and Padamsee 1995) and Nb3Sn the maximum obtained magnetic fields clearly exceed the lower critical field Bc 1. Copyright © 1998 IOP Publishing Ltd

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A series of experiments on small samples of Sn-In and In-Bi alloys indicates that the limiting field in RF is indeed Bs h for type I and type II superconductors ( Yogi et al 1977 ). If this is confirmed in the future, then very high magnetic fields can, in principle, be hoped for in superconducting accelerating cavities. With typically 4 mT RF surface magnetic field per MV m−1 accelerating gradient the ultimate limit for niobium is then 60 MV m−1 and for Nb3Sn it is 100 MV m−1. G7.0.6 Discrepancies between theory and experiment—anomalous losses In theory, accelerating fields of about 60 MV m−1 should be obtained in superconducting niobium cavities. By lowering the temperature of the liquid-He bath sufficiently, the RF losses should be negligibly small. At 350 MHz and 2 K, as an illustration, the superconducting LEP accelerating cavities should have a Q value of 8 × 1011 and dissipate only 30 W at 60 MV m−1 accelerating gradient. Nevertheless, experimentally obtained best results in these cavities are Q = 3 × 109 with about 200 W dissipated power at 10 MV m−1 accelerating gradient (figure G7.0.9). Unfortunately, several phenomena, in general not directly linked with RF superconductivity, make the theoretical goal hard to achieve. Diagnostic tools have been developed that allow the study of performance limits. The response of the cavity to an RF pulse can be analysed for diagnostics. Secondly, from the Q versus E curve information

Figure G7.0.9. Q versus the accelerating gradient at 4.2 K of a LEP-type niobium sputter-coated copper cavity at 352 MHz. Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.10. Temperature maps from a 500 MHz single-cell niobium cavity cooled down in different ambient magnetic fields ((a) Be x t , < 2 µ T, (b) 20 µ T, (c) 40 µ T ) at the same RF field amplitude. The degree of darkness increases with the temperature. The ambient field is directed perpendicular to the cavity axis and creates RF loss in two equatorial regions separated by 180° (dark), where the flux enters and where it leaves the cavity surface. In addition a ‘hot’ spot is visible near the ‘north’ pole (‘N’).

on the status of the cavity can be drawn. Thirdly, a very powerful diagnostics device is the recording of the surface temperature of the cavity at high field by temperature sensors (‘temperature mapping’ (Bernard et al 1981, Piel and Romijn 1980), figure G7.0.10). Performance limits go together with a specific mechanism of RF loss, which we will call ‘anomalous ’. Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.10. Continued.

These mechanisms are commonly observed, but they are untypical of a surface as described in the framework of the BCS theory, being free from impurities and contamination. Typical signatures of ‘anomalous loss’ are shown in figure G7.0.11. Residual loss is independent of the temperature. The physical origin is not well known in most cases; what could be identified is loss from dielectric materials and trapped magnetic flux. A ‘quench’ is often observed in niobium cavities made of industrial grade niobium without post-purification (typically B < 20 mT, which corresponds to an accelerating gradient of 5 MV m−1, cf section G7.0.6.2). It is caused by normal-conducting inclusions and other imperfections (welding beads, residuals from the cleaning treatment). They are heated up in the RF magnetic field and drive the superconducting metal nearby into the normal-conducting state. Then an instability shows up: the stored energy of the cavity is dumped in this normal-conducting region, and by this action the RF field in the cavity is abruptly decreased. At low RF field, the RF dissipation is again low enough that the niobium will go back into the superconducting state. As the RF coupling is adjusted for this situation, the cavity fills up again with RF energy until the cycle starts as described above ( ‘self-pulsing breakdown’ ). Electron multipacting ( from multiple impact ) is an electron multiplication phenomenon which can happen if the secondary-electron emission coefficient δ is larger than unity. As the electrons are resonant with the RF field amplitude, multipacting is visible at distinct field levels. If the electron trajectories are restricted to a point of the surface ( by electron kinematics in the RF field ), the currents can be sufficiently high to dissipate enough power to induce a breakdown in spite of the low impact energy (about 50–1500 eV). Properly cleaned niobium cavities ( low 8 ) of rounded shape have been shown to be virtually free of multipacting (Klein and Proch 1979). Field emission is another loss due to electron impact. The current is generated by the large electric surface field at point-like emitting sites (e.g. dust particles). In contrast to electron multipacting, the current is relatively small (∼nA to µA), but the impact energy is large (∼MeV and more). The current increases exponentially with the RF surface field amplitude, such that the gradient is limited even when the RF Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.11. Rs(Tc /T ) and Q( Ea ) curves that show typical performance limits.

power is increased. Here also, breakdown may occur. In addition, heat and large amounts of x-radiation are generated by bremsstrahlung of the electrons when stopped at the cavity wall. Nonquadratic loss is indicated by a nonhorizontal Q versus E curve. It is often observed in cavities coated with niobium or other materials (NbN, NbTiN, YBaCuO). It is hypothesized that the fraction of normal-conducting surface increases with the RF field amplitude at the expense of the superconducting surface (Weingarten 1996). Hysteretical Q versus E curves are observed for superconducting material in loose contact with the substrate (welding beads in niobium sheet-metal cavities or peeled-off niobium flakes in niobium sputter-coated cavities). When increasing the RF field amplitude, they switch to the normal-conducting state at a certain RF magnetic field amplitude, upon which the Q value abruptly goes down by a small amount, which corresponds to the tiny surface of the flakes. When decreasing the RF field amplitude, the SC metal in poor contact with the substrate remains in the normal-conducting state until the RF dissipation becomes so small that the cooling is sufficient to put them back into the superconducting state. G7.0.6.1 The residual surface resistance The residual surface resistance Rso lumps together all physical effects that create a deviation of the experimentally obtained surface resistance Rs from the one described by theory, RSB C S

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Hence, Rs 0 comprises a variety of effects, some of a trivial nature, others more fundamental. As the residual resistance is very small, its relevant physical parameters are difficult to control experimentally, but research work is being pursued along these lines. The challenge is to improve the technology of cavity manufacturing and to reduce fabrication flaws and surface contamination, such that the ‘trivial’ causes for residual loss are suppressed to the utmost. This in itself is not an easy task, as, for example, 1 ppm (part per million) contamination of iron, concentrated into large enough lumps, would already create the same RF loss as niobium at 4.2 K and 350 MHz. This stresses how important it is to avoid any contamination of a superconductor such as niobium during reduction, refinement, casting, furnace treatment, rolling, welding, polishing, rinsing and final assembly. A well-established mechanism contributing to the residual surface resistance is trapped magnetic flux caused by the presence of a residual static magnetic field B of various origins during cool-down (Gittleman and Rosenblum 1966, Pioszyk et al 1973, Vallet 1992). Owing to the complex geometry and the abundance of pinning centres in the wall of a superconducting cavity, a Meissner expulsion of the magnetic field at the transition from the normal-conducting to the superconducting state has not been observed. The magnetic flux is trapped. It splits up into N ‘fluxoids’ or ‘vortices’ (on a 1 m2 surface) of typical radius λ in which the magnetic field is Bc 1

Each fluxoid has a normal-conducting core with cross-section πξ 2 where ξ is the coherence length of the Cooper pairs. The additional surface resistance Rm due to trapped magnetic flux is

Rn is the surface resistance of the metal in the normal-conducting state. With the Ginzburg-Landau parameter κ = λ/ξ ∼ l for niobium, in order to have Rm < RS ≈ 10−5Rn , B/BC 2 < 10−5. This implies that for niobium the ambient magnetic field has to be shielded to B < 10−5 BC 2 = 105 × 240 mT = 2.4 µ T. Extreme type II superconductors (κ » 1) should be insensitive to trapped magnetic flux (as was observed in copper cavities coated with a thin niobium layer (Arnolds-Mayer and Weingarten 1987, Bernard et al 1992)). RF losses of a dielectric nature (which are proportional to E 2 and not to B 2 ) are also well established. They have been attributed to residuals from the final rinsing remaining on the surface (Bernard et al 1981). Other causes that may contribute to the residual loss will only be enumerated: rough surface, weak superconducting spots, imperfectly cooled islands of surface coatings, thermoelectrically induced currents producing trapped magnetic flux and others. The lowest residual surface resistances obtained so far at different frequencies are listed in table G7.0.4 and visualized in figure G7.0.7. G7.0.6.2 Surface defects of localized enhanced losses In X-band cavities of relatively small surface area (some square centimetres), exposed to an RF field, high surface magnetic fields were obtained more than two decades ago; however, these high fields could not be achieved in larger cavities for accelerator application at lower frequencies. It has long been suspected that small normal-conducting surface defects with much higher RF losses, relatively widely dispersed, prevent higher fields. If so, by statistical arguments, the probability of obtaining high surface fields is much larger in small single-cell cavities of higher frequency. Furthermore, the number of tests performed with small cavities at high frequency is also much larger, which made a test result with a very high surface field more probable. In some experiments at CERN, hot spots of enhanced RF loss causing breakdown were detected and localized by temperature mapping, then cut out and inspected under a scanning electron microscope Copyright © 1998 IOP Publishing Ltd

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(Padamsee et al 1983). It turned out that, for accelerating fields less than ∼8 MV m−1 (on reactor-grade niobium), a surface defect which had clearly induced the thermal breakdown could be attributed to every hot spot. They were welding ‘beads’ (welding holes, chemical residues, normal-conducting in clusions) of a diameter of typically 100 µm. Why do these defects cause a breakdown? Let us examine a defect, represented by a half-sphere with radius r0 of normal-conducting metal exposed to the RF field. It is embedded in a superconducting metal of wall thickness d and thermal conductivity ( 4.2 K ) λ ( figure G7.0.12 ( left )), cooled by liquid He at temperature TB . The defect represents a heat source of strength Q ( W ). Under certain assumptions (r0 « d , no temperature drop across the metal-liquid interface), the problem is equivalent to a circular symmetric one with a heat source of twice the strength Q, for which the heat conduction equation

is solved by

The heat source strength is

Figure G7.0.12. Modelling the thermal breakdown (quench) caused by a normal-conducting defect, producing heat at a rate Q. The real problem (left) can be modelled by a circular symmetrical one (right). Copyright © 1998 IOP Publishing Ltd

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where Rsn is the surface resistance of the normal conducting metal and H the local magnetic surface field. Hence

Breakdown will occur when the temperature at the normal-conducting-superconducting interface ( r = r0 ) approaches Tc . This gives a condition for the maximum RF field Hm a x (Padamsee 1983, Padamsee et al 1983)

With typical values λ = 10 W mK−1, TB = 1.8 K, Rsn = 12 mΩ, r0 = 10−4 m, we obtain Bm a x = µ0Hm a x = 20 mT, which corresponds to a 5 MV m−1 accelerating gradient, in qualitative agreement with the experimental results obtained a decade ago with reactor-grade niobium. Computer simulations have shown (Padamsee 1983) that for a defect radius r0 > 20 µm, λ ≤ 75 W mK−1, equation (G7.0.28) is correct within less than 20%. It becomes immediately clear that, to increase the maximum surface field Hm a x , the number and size of the surface defects has to be reduced, and, on the other hand, superconducting metal of increased thermal conductivity must be made available ( ‘thermal stabilization’ of defects). It was found (Schulze 1981) that the thermal conductivity of niobium is most severely affected by interstitial impurities like oxygen, nitrogen and carbon. These impurities could be considerably reduced by repeated furnace treatments and/or solid-state gettering by wrapping a foil of metal with a large affinity to these impurities around the cavity in a furnace at 1200–1400 °C. Yttrium and titanium were used as getter materials ( Kneisel 1988, Padamsee 1985, 1988 ). Another way has been chosen which is to coat a high-thermal-conductivity material like OFHC copper with a thin superconducting film of niobium or lead of ∼1– 2 µm thickness, sufficiently thick to carry the supercurrent. As an example, LEP-type cavities sputter-coated with niobium (Benvenuti 1992, Benvenuti et al 1984, 1985) exhibit maximum accelerating fields and Q values comparable with niobium sheet metal or even better (figure G7.0.9). Although occasional problems occur due to poor adhesion of the niobium film to the copper substrate at contamination at the copper-niobium interface, these cavities are not limited by a thermal breakdown. In addition, they show almost no extra RF loss due to trapped magnetic flux up to at least more than twice the earth’s magnetic field, significantly alleviating the task of obtaining a low residual surface resistance (cf section G7.0.6.1). G7.0.6.3 Field emission electron loading The less frequent the thermal breakdown of superconducting cavities becomes, the more other breakdown phenomena, typical of higher electric fields, such as electron emission, become important. Field emission of electrons from metallic surfaces is determined by the work function and, according to the Fowler-Nordheim law, should occur only at very large electric fields ( ∼GV m−1 ). The field emission current density J as a function of the local electric field E at the emission site is described by the Fowler-Nordheim theory

It is common practice to use for niobium as the work function Φ = 4.3 eV without a particular justification and E = β Es , Es being the macroscopic surface field, and m the electron mass. β is a phenomenological factor, called the ‘field enhancement factor’. J increases very rapidly above a certain Copyright © 1998 IOP Publishing Ltd

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threshold: for example between E = 1 GV m−1 and 2 GV m−1 the current density (for tungsten with Φ = 4.5 eV ) increases by 15 orders of magnitude. For a typical emitting surface of 10−12 m2 and E = 5 × 109 V m−1 the maximum RF current is ∼50 µA, which is in the typical range of instantaneous RF currents observed in superconducting cavities ( Romijn et al 1983 ). However, the corresponding electric surface fields observed are much lower. Temperature mapping (Bernard et al 1981, cf earlier in section G7.0.6) of superconducting cavities and d.c. studies on broad-area cathodes already showed field emission at around 10 MV m−1, which is a factor β = 100 lower than the threshold field. The physical explanation for β is controversial. It was also shown from temperature mapping that the electrons originate from isolated sites on the surface and that dust particles may be emitted. D.c. field emission experiments proved that these sites are micrometre-sized particles ( Niedermann et al 1986, 1990 ). There is no evidence to believe that field emission observed in superconducting RF cavities is of a different nature.

Figure G7.0.13. World data of peak surface electric field in accelerating cavities versus cavity surface area. The line gives a prediction of a statistical model.

Field emission electron loading in RF cavities can be described on a statistical basis ( Lyneis 1973, Padamsee et al 1985, 1993, Weingarten 1989 ). Figure G7.0.13 is taken from a talk by Padamsee and shows data on the surface peak fields versus the cavity surface area. The larger the surface exposed to a high RF electric field, the larger is the probability of finding an emitting site with a large β value and hence a low field emission onset field. Very large surface electric fields ( 113 MV m−1 ) have been obtained in a niobium two-cell cavity (Graber et al 1994a, b). This observation indicates a lower limit for the maximum electric surface field which can be obtained in a niobium cavity. In a specially designed cavity where the electric field was concentrated in a small area, even larger electric fields could be obtained (145 MV m−1, Moffat et al 1990). The high-field area was analysed in an x-ray dispersive scanning electron microscope and showed a variety of elements. It appears that the classical projection model for metallic particles and/or the nature of the cavity surface-particle interface can explain many instances of RF field emission. Considerable progress in reducing the density of emission sites has been made in recent years by heat treatment of cavities at T = 1400 °C, by improving the rinsing methods (rinsing under high pressure, Bernard et al 1991), and by injecting short pulses of large RF power into the cavity (Crawford 1995, Graber et al 1994a, b). 200 MV m−1 surface fields (d.c.) have been obtained on 1 cm2 large samples by ultraclean water rinsing, which is an excellent result also from the technical point of view (Mahner et al 1995). Copyright © 1998 IOP Publishing Ltd

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G7.0.7 Materials other than niobium As already mentioned, materials with a high enough Cooper pairing energy 2∆ would allow a considerable reduction of the BCS surface resistance. According to the BCS theory, ∆ is related to the critical temperature, ∆ = 3.5κTc . Therefore, superconductors with a higher critical temperature than niobium are being intensively studied. Among them are the well known ‘old’ high-Tc superconductors, such as the B1 and A15 compounds NbN and Nb3Sn ( Tc = 17 K and 18.5 K respectively), and since their discovery in 1986 (Bednorz and Müller 1986), also the ‘new’ high-Tc superconductors such as YBa2Cu3O7− δ ( Tc = 93 K ) ( Wu et al 1987 ). Thin coatings of these high-Tc materials on RF cavities are produced either by thermal diffusion out of the vapour phase ( Nb3Sn, NbN ), by sputter deposition ( NbN, NbTiN ), by evaporation ( NbN ) or by electrophoretical deposition in a high static magnetic field from an organic suspension of powdery superconducting material and sintering ( YBa2Cu3O7 − δ ) ( Hein et al 1990 ). The best results on RF cavities (made completely from high-Tc materials) are shown in table G7.0.5.

It is worthwhile evaluating the RF surface resistance of YBa2Cu3O7 − δ at 77 K which, from the point of view of mains power consumption, is the same as for niobium at 4.2 K (figure G7.0.14). Data on YBa2Cu3O7−δ were taken from a talk given by Müller (1989) and comprise bulk, textured, single-crystalline and thin-film material results. It can be seen that single-crystal and thin-film high-Tc oxide ceramic superconductors are competitive with niobium, in particular at higher frequencies for low-power applications. G7.0.8 Coupling to superconducting cavities G7.0.8.1 Coupling RF power into the cavity Most of the RF input couplers used are coaxial line couplers. Therefore, the coupling between the waveguide and cavity fields is produced by a time-varying electric field. It is a current (displacement current) which is injected by the coupler into the cavity ( Dôme 1992, Haebel 1992 ). A lumped circuit diagram as shown in figure G7.0.15 is used to model the resonator, the coupling network and the power transmission line. The resonator is represented by lumped elements in parallel. In this way we have a convenient analogy of the cavity accelerating voltage and the lumped circuit peak voltage (cf legend of figure G7.0.15). However, in this model, the coupling goes via a magnetic field. We know from waveguide theory that by a shift on the line of a quarter wavelength we go from maximum magnetic field to maximum electric field and vice versa. That is why we introduce a reference plane in the transmission line between the generator and the cavity a quarter wavelength ahead of the transformer. Hence we have an analogy between the fields in the lumped circuit model and the coaxial coupler tip. Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.14. The surface resistance of YBaCuO at 77 K. The surface resistance which is thermodynamically equivalent to niobium at 4.2 K is indicated by the line.

Figure G7.0.15. Lumped circuit representation of a superconducting cavity including an RF power source and beam. R—shunt impedance, R/Q—V 2/(ω0U ), Q0—unloaded Q factor, V—accelerating voltage, U—stored energy, ω0—resonant frequency, V±—reflected and forward voltage, Ib—beam current, Qe x t —external Q factor, Z0—line impedance, n —transformer ratio.

The application determines the coupling layout. A useful number which depends only on the coupling geometry is the ratio of the stored energy U and the radiated energy ∆U within one cycle (or the radiated power Pr a d ) of the coupling hole, which is the external Q value

We always assume, if not otherwise mentioned, that this energy is completely absorbed at a load outside (usually 50 Ω), and nothing is reflected back into the cavity. For most applications we want to have ‘critical coupling’ to the cavity (coupling factor β = 1)which

implies Qe x t , = Q0. This condition minimizes the reflected power (which is waste power). Copyright © 1998 IOP Publishing Ltd

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In addition, for tests of superconducting cavities in the laboratory, it minimizes the error made in determining the dissipated power in the cavity and hence the unloaded Q value. We have therefore a weak coupling or a high Qe x t = Q 0 = 109 to 1010. For a superconducting cavity operated in an accelerator, the input power is almost totally transferred to the beam. In the lumped circuit model of figure G7.0.15 the (d.c.) beam current Ib ( in phase with V and with a peak RF component 2Ib ) can be represented as a current source with a current flowing across a gap with the accelerating voltage V. Under critical coupling (no reflection), the generator current

splits up into the beam current 2Ib and the current on the cavity wall V/R, hence ( Qo » Qe x t )

For the LEP superconducting cavity with V0 = 10 MV (taking without loss of generality the synchronous phase angle φ = 0 ), Ib = 10 mA and ( R/Q ) = 232 Ω we get Qe x t ≈ 2 × 106. Figure G7.0.16 is a schematic diagram of an RF power coupler typical of storage ring cavities. Such a coupler was tested on a cold cavity up to an RF amplitude corresponding to 250 kW RF power ( Boussard 1996 ). The coaxial transmission line power couplers suffer from one-side electron multipacting on the outer conductor ( Haebel et al 1996 and references therein ), which may lead to vacuum bursts, arcing and even melting of metal. The culprits are condensed gases on the warm-cold transition of the outer conductor, which increase the secondary-emission coefficient. Therefore, after ‘conditioning’ ( operating the coupler with increasing RF power with the pressure low enough to avoid damage ), arcing reappeared at power levels for which it was absent previously ( ‘deconditioning’ ). If E2 is the electric field at the outer radius r2 , the resonances of multipacting of order n occur at certain values E2 n. On the other hand, for a given resonant field E21 the power transported is proportional to Z0, the characteristic line impedance. In a LEP coupler with Z0 = 50 Ω multipacting appears at 90 kW. In a coupler with Z0 = 75 Ω it can be pushed

Figure G7.0.16. A schematic diagram of an RF power coupler for storage ring cavities. Copyright © 1998 IOP Publishing Ltd

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to larger values of RF power (135 kW). The multipacting resonances are proportional to ( fr2 )2, f being the RF frequency. As the RF power transported is proportional to the square of the field, this resonance scales with the fourth power of the frequency. That is why for larger frequencies a larger RF power can be transmitted before multipacting will occur. The multipacting problem observed in the LEP power couplers could be solved by a modification of the geometry ( larger Zo = 75 Ω ), by applying a d.c. voltage ( 2500 V ) between inner and outer conductors (suppressing the multipacting resonance) and by baking the RF window prior to cool-down (which shortened considerably the conditioning needed for the coupler on a cold cavity). Superconducting cavities installed in large storage rings are optimized for an efficient transfer of RF power to beam power. In most cases large klystrons of more than 1 MW power are used as the RF source. The RF power is split by a waveguide distribution system into power levels of about 100 kW, which are well suited to the RF power input coupler window. This scheme of RF power distribution is relatively rigid, as small variations of the power flow and/or small variations of the coupling strength to the cavity cannot be completely avoided. In addition, a change in beam current would induce a change in the accelerating gradient, which is undesired. All these inconveniences may be avoided by a variable coupling scheme (Haebel 1992). Three stub tuners installed in the waveguide system serve the same purpose by matching the waveguide impedance to the cavity impedance (Dwersteg 1989). G7.0.8.2 Coupling RF power out of the cavity The beam-induced RF power What is highly welcome in the case of acceleration, the high shunt impedance of a superconducting cavity, can be difficult to master under other circumstances. An accelerating cavity usually resonates not only on the fundamental mode for acceleration, but also for HOMs. There are different families of HOMs, TM-and TE-like modes, of different rotational symmetry (monopole, dipole, quadrupole, etc). All these can be excited by RF currents in the cavity, provided that one of the Fourier components of the current coincides with the resonant frequency of one of the HOMs and the current vector has a component parallel to the HOM’s electric field vector. High fields (beam wake fields) can be generated in this case, exercising a force on the beam, leading to beam instabilities and even beam loss. On the other hand, the beam loses energy to the electromagnetic energy of the HOMs, which renders the transfer of RF power to beam power less effective. Or, which is as bad, the high fields may cause damage to the cavity by, for instance, a thermal breakdown (‘quench’), driving the cavity surface normal conducting in an uncontrolled way. Obviously, the build-up of HOM fields has to be avoided. We want to calculate the steady-state HOM voltage induced by the periodic passage of bunches (repetition time Tb ) that constitute the beam ( Wilson 1982 ). We assume that the bunches are infinitely short and that each of them carries a charge q. Suppose a HOM of the cavity is excited to a steady state with the voltage amplitude V0. The bunch charge q does work W at the cavity voltage W = qV0. Therefore, by energy conservation, the stored energy U = 1/2CV02 is incremented by the amount ∆U = CV0∆V0 = ω0CV0∆V0 /ω0 ) = V0∆V0 /(ω0( R/Q )) = qV0 , which leads to

In the absence of an external drive generator a voltage V0 in the cavity decays as

Suppose that immediately after the passage of the bunch the HOM voltage amplitude in the cavity is V0 + ∆V0. Until immediately prior to the passage of the next bunch it will decay and turn its phase according to equation (G7.0.35), and, by supposition, equals V0

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After a short calculation we get

As the average voltage is 〈V0 〉 = V0 + ∆V0/2, we end up with where

With the average beam current 〈Ib 〉 = q/Tb , the RF current at the HOM frequency is for sufficiently short bunches (in the δ −pulse limit) 2〈Ib 〉. The deposited average power PH O M into one HOM of the cavity is

‘Re’ indicates the real part and the asterisk marks the complex conjugate number. If the subsequent bunch does not ‘see’ any voltage amplitude left behind from the previous one, QL « ω0Tb , F = 1

On the other hand, in the ‘worst’ case, if the subsequent bunch ‘sees’ all of the voltage amplitude left behind from the previous bunch, ω0Tb « QL , and exp( iω0Tb ), we have F = 4QL/(ω0Tb ), hence

PH O M is dissipated partly in the resistor Z0 , on the exterior of the cavity, partly in R , inside the cavity

Hence

and

The HOM power can attain rather high values and requires dissipation into a room-temperature load. We see that, by a careful design of the HOM coupler, Qe x t « Q0 , PC can be kept tolerably small: Pr a d ≈ PH O M , PC ≈ PH O M Qe x t /Q0. Damping the beam-induced RF power Damping of the beam-induced voltage at HOM cavity frequencies is accomplished by especially designed antennas or waveguides that do not extract RF power in the fundamental mode but only in the HOMs ( Haebel 1992 and references therein ). An antenna-type coupler is shown, for instance, in figure G7.0.17. Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.17. A HOM coupler developed for the superconducting cavities in LEP.

The electric cavity field supplies a (displacement) current to the coupler which, for the fundamental mode frequency, is short-circuited by a series resonator (notch filter), realized by the U-shaped lower part (in the form of a rod-like inductance) and a small gap to ground (capacitance). There is consequently no current to the load resistor outside, and hence no dissipation. At frequencies above the fundamental mode, the notch filter becomes an inductive reactance, and a current flows through the outside load resistor (at room temperature) causing dissipation. By a careful layout of the microwave circuit between the antenna tip and load, the HOMs of the cavity can be sufficiently damped over a frequency interval determined by the particular application. There are applications, as in modern multipurpose injectors (Super Proton Synchrotron (SPS) at CERN), where an acceleration cycle of e− and e+ is interleaved with a cycle of a proton beam passage. In this case, the fundamental mode has to be damped during the passage of the proton beam. An elegant way is to use RF feedback ( Boussard et al 1989b), where the electromagnetic field in the cavity is the replica of a reference signal. During the acceleration cycle, the reference signal corresponds to the accelerating voltage, during the passage of the proton beam, the reference signal is zero. The proton-beam-induced voltage is thus cancelled by the injection of RF power into the cavity. As in all feedback systems, oscillations may occur and the limit of stability has to be worked out. G7.0.9 Tuning superconducting cavities Tuning of an accelerating cavity has two objectives, dynamic tuning during operation and static tuning to obtain a sufficiently ‘flat’ field distribution. The cavity resonance frequency has to be in perfect accordance with the revolution frequency of the beam. This is generally accomplished by a frequency tuner that adjusts the resonance frequency in the vicinity of a stable master generator frequency. The frequency tuner makes use of the fact that, according to Slater’s theorem, a deformation of the cavity shape may change the electric energy stored in a different way from the way in which the magnetic energy stored is changing. To make them equal again, the Copyright © 1998 IOP Publishing Ltd

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resonance frequency will change. Of course, external forces may also alter the cavity shape and hence the resonance frequency, which has to be readjusted by the tuner. Particularly cumbersome can be mechanical oscillations of the cavity driven by pressure oscillations of the helium gas or ponderomotive oscillations driven by the RF field (Ceperley 1972, Karliner et al 1970). Therefore, the tuner feedback circuit has to react fast enough without causing oscillations. Dynamic tuning requires readjustment of the cavity tune according to the beam current and phase (compensation of ‘ beam loading ’ (Boussard 1985)). The tuner acts such that the cavity with a passing beam presents a real load to the RF generator (pure resistance). This condition is the best operating point for the power generator, because the generator power is entirely converted into beam acceleration and cavity losses (which are very small for a superconducting cavity). The accelerating cavities can be tuned by a mechanical change of the total length, accomplished either by an electric motor or, to avoid jamming at low temperatures, by thermal and/or magnetostrictive action on a supporting bar of ferromagnetic material such as nickel. For reasons of economy, the accelerating cavity usually consists of several single cells electromagnetically coupled together with one feed for RF power. If the field excitation in the different single cells is very different, for a fixed total accelerating voltage there is one single-cell with high field excitation. As the losses and the probability of defects causing a quench increase nonlinearly with field, the performance of the cavity is inferior to that of a cavity with a flat field distribution. Therefore, the cavity needs to be plastically tuned to achieve field flatness, mostly realized by inelastic deformation of the relevant single cells, which is not basically different from what is done for a normal-conducting cavity. G7.0.10 Special features of heavy-ion resonators Proton and heavy-ion accelerators are mostly operated for nuclear physics research. The projectile nucleus must have enough kinetic energy to overcome the Coulomb barrier of the target nucleus, in order to come close enough for the nuclear force to be felt by the two nuclei. The minimum kinetic energy T to overcome the Coulomb barrier is

with r0 = 1.3 × 10−15 m as the nuclear radius, A1 and A2 the atomic masses of the target and projectile nuclei, respectively, and Z1 and Z2 the nuclear charge numbers. For illustration, the α particles which hit the gold foil in the classical Rutherford scattering experiment need a kinetic energy of

This kinetic energy corresponds to

D.c. high-voltage generators are used to accelerate protons and heavy ions. In such an accelerator, all the RF voltage gain of the injected particles is independent of the kinetic energy (or β values). Unfortunately, the accelerating voltages obtained are relatively small (V < 10 MV ) compared with those obtained in RF resonators. For RF acceleration, low RF frequencies are favoured because they still offer an efficient acceleration for a large range of β values. However, the structure may become impractically large. At high RF Copyright © 1998 IOP Publishing Ltd

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frequencies, small gaps are needed, which soon also become impractical from the point of view of mechanical tolerances. Superconducting structures offer an economic way of accelerating protons and heavy ions with β « 1. They are used as boosters for low-current CW tandem accelerators. One may also replace tandem accelerators by an electron cyclotron resonance ( ECR ) source, a superconducting injector and an RF quadrupole ( RFQ ) structure ( Delayen and Shepard 1990 ). Proton and heavy-ion accelerators are mostly based on low-frequency, individually powered and phase-controlled single-cell accelerating structures, chained one after the other in a cryostat. This is a versatile concept for efficiently accelerating ions of a large mass and velocity range (Bellinger 1977, 1986), which offers the possibility of interposing beam-focusing elements. The features of these structures are quite different from electron structures for β = 1 particles. The lowest frequency is determined by the lateral size of the structure, approximately equal to λ/4. The larger β is, the larger the frequency tolerated can be. The highest frequency would be given by the minimum gap length, approximately equal to β λ/2. At the low-frequency end, a lateral size of 1.5 m, for example, and at the high-frequency end, a gap size of 1 cm, for example, would be tolerable. These wavelengths would correspond to frequencies of 50 MHz and 1.5 GHz for β = 0.1 particles respectively. As the particles take a certain time to cross the gap, they do not ‘ see ’ a constant gradient during their passage. The average energy gain decreases with the transit time. The so-called transit-time factor fT is defined as the ratio of actual energy gain to the (hypothetical) energy gain if no transit-time effects were present. If one has to design heavy-ion structures, several aspects are important and sometimes contradictory. A comprehensive review has been given elsewhere (Bollinger 1977, 1986, Storm 1993). Their performance is generally limited by field emission (as are modern β = 1 structures). Therefore, the peak electric field should be minimized in order to bring down the threshold for electron emission. There is less concern for the peak magnetic field and the shunt impedance. A low frequency is preferred as a more efficient acceleration can be obtained for a larger range of β values. Furthermore, the stored energy is larger, which is an advantage for fighting perturbations of the resonant frequency due to environmental drive sources (‘microphonics’). The fast frequency tuning is usually done by feeding reactive power Pr into the cavity, which will induce a frequency shift ∆ω according to the formula (Markovich et al 1987)

where U is the stored energy. One has to find a compromise between a broad transit-time factor fT (small number of gaps) and a large energy gain (large number of gaps). The loading structure pushes the peak electric field up and has to be designed for good mechanical rigidity (to minimize microphonics). Multipacting has to be avoided by choosing correct gap distance to avoid two-point multipacting and by confirming the absence of resonant electron trajectories by computer simulations. Materials used are niobium of various provenances (sheet metal, clad or sputter-deposited on copper) and lead (electroplated on copper). The design of structures for proton and heavy-ion accelerators varies greatly. On one hand there is the pioneering superconducting heavy-ion accelerator ATLAS at Argonne, whose first stage became operational in 1978. It is based on ‘quarter-wave’ and ‘split-ring’ structures. It has accumulated more than 100 000 h beam time. It is still being upgraded by replacing the injector by a very low velocity superconducting linac capable of providing increased beam currents and ions up to the mass of uranium (Shepard 1993) On the other hand there was the pioneering proton linac developed at KfK Karlsruhe (Germany). It consisted Copyright © 1998 IOP Publishing Ltd

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of helix structures which were compact, but initially suffered from ponderomotive oscillations. Once this problem was solved, it was operated reliably (Cauvin and Fouan 1989 and references therein). Figure G7.0.18 shows a typical quarter-wave resonator and an ensemble of heavy-ion structures before assembly in the accelerator. A completely different approach to accelerate heavy ions is being pursued at the Technical University of Munich (Trinks et al 1986). A separate orbit cyclotron is under construction with superconducting sector magnets and superconducting sector cavities.

Figure G7.0.18. Heavy-ion resonators at JAERI (Japan): top: a schematic diagram; bottom: four resonators being assembled in their cryostat. Reproduced by permission of the Japan Atomic Energy Research Institute.

G7.0.11 Technological achievements for accelerating cavities What finally matters is the performance of the accelerating structure in the accelerator itself. It is here that the achievements of the technology of RF superconductivity should pay off and have to be assessed. Table G7.0.6 gives the performance of RF structures being permanently operated in electron accelerators. In several laboratories accelerating gradients between 5 and 10 MV m−1 were obtained in structures of 1 m length and more. There is increasing evidence that structures, after being installed in the accelerator, do not degrade, although examples demonstrating the opposite can also be given. Evidence is also increasing that structures keep the performance of the laboratory test, after being equipped with additional components such as couplers and moved to the accelerator. The working principle of the acceleration of electron beams with superconducting cavities has been in use for a long time. At KEK, 32 five-cell 500 MHz cavities delivered an additional voltage of 225 MV to the TRISTAN beams of 10 mA each. It was the first machine to use superconducting cavities on a large scale. At CERN, a LEP prototype cavity has been operated in the SPS for more than 10000 h without Copyright © 1998 IOP Publishing Ltd

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any irreversible degradation. DESY installed 16 cavities in Hera. The largest system of superconducting cavities actually in operation is at CEBAF (figure G7.0.19), which is a CW-continuous-beam recirculating accelerator with 4 GeV beam energy. The cavities are made of niobium sheet, based on a design from Cornell University, and are operated at 1.8 K. The average Q value and average gradient are 5 × 109 and 6 MV m−1, respectively. Since 1996 the largest superconducting RF system has been that of LEP at CERN, with an installed voltage from superconducting cavities of 1600 MV, bringing the energy per beam up to 80.5 GeV, the threshold for the W± pair production. In its final stage the corresponding parameters will be about 2700 MV and 97 GeV, respectively.

G7.0.12 Conclusion and outlook The big advance of the last decade in RF superconductivity for electron accelerators was the mastering of the industrial fabrication and putting into operation of larger numbers of structures with gradients between 5 and 10 MV m−1 and for beam currents roughly up to the 100 mA range. Existing and new accelerators are already based completely, or to a large extent, on this technology. Until now, this technology has cumulated in the construction of low- to intermediate-intensity electron storage rings and recirculating linacs, but there are already prototype developments for high-intensity-beam applications (particle ‘factories’, proton storage rings and proton linacs). Superconducting RF cavities are mainly used to provide economically RF voltage. At Cornell, a mono-cell 500 MHz cavity was operated in the Cornell Electron Storage Ring (CESR) storage ring with 155 kW RF power transmitted to a beam of 118 mA and 4 kW of HOM power extracted to a room-temperature ferrite load (Kirchgessner et al 1996). Superconducting RF cavities are used to keep the total impedance ‘seen’ by the beam tolerable (cf section G7.0.2.3). The beam current and hence the luminosity can be made sufficiently large. In the SPS at CERN, a 400 MHz superconducting cavity was tested as a prototype of the future LHC RF system with a circulating proton beam of 60 mA at about half (5 MV m-1) of the design gradient. Superconducting RF cavities alleviate the ‘beam loading’: thanks to the large RF voltage and stored energy the induced phase modulation from unavoidable gaps in the multibunch beam can be kept Copyright © 1998 IOP Publishing Ltd

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Figure G7.0.19. A schematic diagram of the CEBAF superconducting RF accelerator, Newport News, VA, now online for experiments. The electrons are accelerated to the 4 GeV design energy in successive passes through two 0.4 GeV anti-parallel linacs linked by five recirculation arcs. The accelerator serves three experimental areas. Courtesy of CERN Courier.

Figure G7.0.20. The kilowatt demonstrator FEL, with beam recirculation, energy recovery and infrared and ultraviolet output. Courtesy of CERN Courier.

sufficiently small. Otherwise such a modulation would lead to an intolerable shift in the interaction point of the circulating proton beams ( Boussard and Rödel 1993 ). CEBAF is a focal site for an FEL driven by superconducting structures of 1500 MHz ( figure G7.0.20 ). The application is surface processing and micromachining with ultrashort light pulses of high power and brightness, which interests industry and research laboratories. This activity was a spin-off from the pioneering operation of an FEL with the high-quality CW beam of the Stanford superconducting linac accelerator ( Deacon et al 1977 ). The low RF losses of superconducting RF cavities allow CW operation and a very low energy spread ( 2.8 × 10−5 at CEBAF ), crucial for FEL operation. Another superconducting cavity-based FEL activity is pursued at Darmstadt University ( Döbert et al 1996 ). Another application is directed at high-gradient accelerators ( TeV linear colliders ( Brinkmann 1994 )). A test facility for the superconducting version ( TeV electron superconducting linear accelerator, TESLA) is proposed at DESY, Hamburg. In the first phase, a 500 MeV superconducting electron accelerator with an accelerating gradient of 25 MV m−1 at 1.3 GHz and a Q value of 5x 109 is proposed. Ultimately, many tens Copyright © 1998 IOP Publishing Ltd

References

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of kilometres of superconducting RF structures are envisaged, which will make economical production methods of the structures and the cryostats mandatory. Thanks to the low RF losses superconducting cavities allow a gentle build-up of the accelerating gradient and larger pulse duration. The luminosity is achieved by a large beam power (high bunch charge and many bunches per pulse), which allows less stringent constraints on the beam size. As we have seen, superconducting RF technology is still far from its theoretically possible limits, leaving room for considerable improvement of performance in the future. Acknowledgments I would like to thank all my colleagues from CERN, in particular D Boussard and E Haebel, for their help and critical remarks. It was always a pleasure to have discussions with them. References Abrikosov A A, Gorkov L P and Khalatnikov I M 1959 A superconductor in a high frequency field Sov. Phys.-JETP 8 182 Allen M A, Farkas Z D, Hogg H A, Hoyt E W and Wilson P W 1971 Superconducting niobium cavity measurements at SLAC IEEE Trans. Nucl. Sci. NS-18 168 Arnolds-Mayer G and Chiaveri E 1988 On a 500 MHz single cell cavity with Nb3Sn surface Proc. 3rd Workshop on RF Superconductivity (Argonne National Laboratory, USA) p 491 Arnolds-Mayer G and Weingarten W 1987 Comparative measurements on niobium sheet and sputter coated cavities IEEE Trans. Magn. MAG-23 1620 Auerhammer J, Eichhorn R, Genz H, Gräf H D, Hahn R, Hampel T, Hofmann C, Horn J, Lüttge C, Richter A, Rietdorf T, Rüihl K, Schardt P, Schlott V, Spamer E, Slaschek A, Stiller A, Thomas F, Titze O, Töpper J, Wesp T, Weise H, Wiencken M and Winkler T 1994 Progress and status of the S-DALINAC Proc. 6th Workshop on RF Superconductivity ( VA, USA: CEBAF Newport News ) p 1203 Axel P, Cardman L S, Hanson A O, Harlan J R, Hoffswell R A, Jamnik D, Sutton D C, Taylor R H and Young L M 1977 Status of MUSL-2, the second microtron using a superconducting linac IEEE Trans. Nucl. Sci. NS-24 1133 Bean C P and Livingston J D 1964 Surface barrier in type-II superconductors Phys. Rev. Lett. 12 14 Bednorz J G and Müller K H, 1986 Possible high-TC superconductivity in the Ba-La-Cu-O system. Z. Phys. B 6 4189 Benvenuti C 1992 Superconducting coatings for accelerating RF cavities: past, present, future Proc. 5th Workshop on RF Superconductivity ( Hamburg: DESY ) p 189; Part. Accel. 40 43 Benvenuti C, Bernard Ph, Bloess D, Chiaveri E, Hauviller C and Weingarten W 1996 Various methods of manufacturing superconducting accelerating cavities Adv. Cryogen. Eng. 41 885 Benvenuti C, Calatroni S, Hauer M, Minestrini M, Orlandi G and Weingarten W 1992 (NbTi)N and NbTi coatings for superconducting accelerator cavities Proc. 5th Workshop on RF Superconductivity ( Hamburg: DESY) p 518 Benvenuti C, Circelli N and Hauer M 1984 Niobium films for superconducting accelerating cavities Appl. Phys. Lett. 45 583 Benvenuti C, Circelli N, Hauer M and Weingarten W 1985 Superconducting 500 MHz accelerating copper cavities sputter-coated with niobium films IEEE Trans. Magn. MAG-21 153 Bernard Ph, Bloess D, Flynn T, Hauviller C and Weingarten W 1992 Superconducting niobium sputter-coated copper cavities at 1500 MHz Proc. 3rd Eur. Particle Accelerator Conf. ( Berlin 1992 ) ed H Henke, H Homeyer and Ch Petit-Jean-Genaz (Gif-sur-Yvette: Editions Frontieres) p 1269 Bernard Ph, Bloess D, Hartung W, Hauviller C, Weingarten W, Bosland P and Martignac J 1991 Superconducting niobium sputter-coated copper cavities at 1500 MHz Proc. 5th Workshop on RF Superconductivity ( Hamburg: DESY ) p 487 Bernard Ph, Cavallari G, Chiaveri E, Haebel E, Heinrichs H, Lengeler H, Picasso E, Piciarelli V, Tückmantel J and Piel H 1981 Experiments with the CERN superconducting 500 MHz cavity Nucl. Instrum. Methods 190 257 Bollinger L M 1977 Superconducting heavy ion linacs IEEE Trans. Nucl. Sci. NS-24 1076 Bollinger L M 1986 Superconducting linear accelerators for heavy ions Annu. Rev. Nucl. Part. Sci. 36 475

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Boussard D 1985 Control of cavities with high beam loading IEEE Trans. Nucl. Sci. NS-32 1852 Boussard D 1996 Summing-up session: performance of RF Proc. 6th Workshop on LEP Performance (Chamonix, 1996) ed J Poole (Geneva: CERN) p 197 Boussard D, Cavallari G, Kindermann H P, Passardi G, Stierlin R, Tückmantel J and Weingarten W 1989 Long-term test of a LEP prototype superconducting cavity in the CERN SPS Proc. Particle Accelerator Conf. (Chicago, 1989) ed F Bennett and J Kopta (New York: IEEE) p 1783 Boussard D, Kindermann H P and Rossi V 1989 RF feedback applied to a multicell superconducting cavity Proc. Eur. Particle Accelerator Conf. (Rome, 1988) ed S Tazzari (Singapore: World Scientific) p 985 Boussard D and Rodel V 1993 The LHC RF system Proc. XVth Int. Conf. High Energy Accel. (Hamburg, 1992) ed J Rossbach Int. J. Mod. Phys. A (Proc. Suppl.) 2B 754 Brinkmann B R 1994 Low frequency linear colliders Proc. 1994 Eur. Particle Accelerator Conf. (London, 1994) ed V Suller and Ch Petit-Jean-Genaz (Singapore: World Scientific) p 363 Bruynseraede Y, Gorle D, Leroy D and Morignot P 1971 Surface resistance measurements in TE011-mode cavities of superconducting indium, lead and an indium-lead alloy at low and high RF magnetic fields Physica 54 137 Calarco J R, McAshen M S, Schwettman H A, Smith T I, Turneaure J P and Yearian M R 1977 Initial performance of the Stanford Recyclotron IEEE Trans. Nucl. Sci. NS-24 1091 Cauvin B and Fouan J P 1990 Status report on the SACLAY heavy ion superconducting linac Proc. 4th Workshop on RF Superconductivity (Tsukuba, Japan: KEK) p 175 Cavallari G, Arnaud C, Barranco-Luque M, Benvenuti C, Bernard Ph, Bloess D, Boussard D, Brown P, Calatroni S, Chiaveri E, Ciapala E, Erdt W, Frandsen P, Genesio F, Geschonke G, Gusewell D, Haebel E, Hartung W, Hauviller C, Hilleret N, Kindermann H-P, Orlandi G, Passardi G, Peschardt E, Rödel V, Tückmantel J, Weingarten W and Winkler G 1992 Status report on SC cavities at CERN Proc. 5th Workshop on RF Superconductivity (Hamburg: DESY) p 23 Ceperley P H 1972 Ponderomotive oscillations in a superconducting helical resonator IEEE Trans. Nucl. Sci. NS-19 217 Corlett J N 1992 Experience with cavity design programs Proc. CERN Accelerator School (Oxford, 1991) ed S Turner (Geneva: CERN) (CERN Yellow Report CERN 92–03 301, and references therein) Crawford C, Graber J, Hays T, Kirchgessner J, Matheisen A, Moller W-D, Padamsee H, Pekeler M, Schmüser P and Tigner M 1995 High gradients in linear collider superconducting accelerator cavities by high pulsed power to suppress field emission Part. Ace. 49 1 Dasbach D, Müller G, Peiniger M, Piel H and Roth R W 1989 Nb3Sn coatings of high purity Nb cavities IEEE Trans. Magn. MAG-25 1862 Deacon D A, Elias L R, Madey J M J, Ramian G J, Schwettman H A and Smith T I 1977 First operation of a free-electron laser Phys. Rev. Lett. 38 892 De Gennes P G 1966 Superconductivity of Metals and Alloys (New York: Benjamin) (Reprint Addison Wesley 1989); an excellent physically motivated treatment Delayen J R and Shepard K W 1990 Tests of a superconducting rf quadrupole device Appl. Phys. Lett. 57 514 Dobert S et al 1996 Laboratory talk Darmstadt Technical University Proc. 7th Workshop on RF Superconductivity Dôme G 1992 RF theory Proc. CERN Accelerator School (Oxford, 1991) ed S Turner (Geneva: CERN) (CERN Yellow Report CERN 92–03 1) Dwersteg B 1989 SC-cavity operation via WG-transformer Proc. 4th Workshop on RF Superconductivity (Tsukuba, Japan: KEK) p 593 Fabbricatore P, Fernandes P, Gualco G C and Musenich R 1989 Superconducting properties of B1 nitrides films obtained by gas-metal reaction for RF application IEEE Trans. Magn. MAG-25 867 Furuya T, Akai K, Asano K, Hara K, Hosoyama K, Kabe A, Kojima Y, Kubo K, Mitsunobu S, Nakai H, Noguchi S, Saito K, Sakamoto T, Tajima T and Takahashi T 1993 Activities of RF superconductivity at KEK Proc. 6th Workshop on RF Superconductivity p 131 Geschonke G 1996 Experience with superconducting RF cavities in LEP Proc. 7th Workshop on RF Superconductivity (France: Saclay) p 143; Part. Accel. 53 315 Gittleman J I and Rosenblum B 1966 Radio-frequency resistance in the mixed state for subcritical currents Phys. Rev. Lett. 16 734

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Graber J, Crawford C, Kirchgessner J, Padamsee H, Rubin D and Schmüser P 1994 Reduction of field emission in superconducting cavities with high power pulsed RF Nucl. Instrum. Methods A 350 572 Graber J, Kirchgessner J, Moffat D, Knobloch J, Padamsee H and Rubin D 1994 Microscopic investigation of high gradient superconducting cavities after reduction of field emission Nucl. Iustrum. Methods A 350 582 Grander H 1996 CEBAF commissioning and future plans Proc. Particle Accelerator Conf. ( Dallas, 1995 ) (Piscataway, NJ: IEEE) Haebel E 1991 Couplers, tutorial and update Proc. 5th Workshop on RF Superconductivity (Hamburg: DESY) p 334; Part. Accel. 40 141 Haebel E, Kindermann H-P, Stirbet M, Veshcherevich V and Wyss C 1996 Gas condensation on cold surfaces, a source of multipacting discharges in the LEP2 power coupler Proc. 7th Workshop on RF Superconductivity Halbritter J 1970 Comparison between measured and calculated RF losses in the superconducting state Z Phys. 238 466 Hauviller C 1989 Fully hydroformed RF cavities Proc. Part. Ace. Conf. ( Chicago, 1989 ) ed F Bennett and J Kopta (New York: IEEE) p 485 Hayes T and Padamsee H 1995 Response of superconducting cavities to high peak power Proc. Particle Accelerator Conf. ( Dallas, TX, 1995 ) (Piscataway, NJ: IEEE) p 1617 Hein M, Kraut S, Mahner E, Müller G, Opie D, Piel H, Ponto L, Wehler D, Becks M, Klein U and Peiniger M 1990 Electromagnetic properties of electrophoretic YBa2Cu3O7− δ films J. Supercond. 3 323 Hillenbrand B, Martens H, Pfister H, Schnitzke K and Uzel Y 1977 Superconducting Nb3Sn cavities with high microwave qualities IEEE Trans. Magn. MAG-13 491 Karliner M M, Petrov V M and Shekhtman I A 1970 Vibration of the walls of a cavity resonator under ponderomotive forces in the presence of feedback Sov. Phys.-Tech. Phys. 14 1041 Kirchgessner J L 1988 Forming and welding of niobium for superconducting cavities Proc. 3rd Workshop on RF Superconductivity (Argonne National Laboratory, USA) p 533 Kirchgessner J et al 1996 Laboratory talk Cornell University Proc. 7th Workshop on RF Superconductivity (France: Saclay) p 35 Klein U 1981 Untersuchungen zu Feldbegrenzungsphänomenen und Oberflächenwiderständen von Supraleitenden Resonatoren Thesis Wuppertal University WUB-DI 81–2 Klein U and Proch D 1979 Multipacting in superconducting RF structures Proc. Conf. on Future Possibilities for Electron Accelerators ( Charlottesville, VA, 1979 ) ed J S McCarthy and R R Whitney (Charlottesville, VA: University of Virginia) pp N1–17 Kneisel P 1988 Use of the titanium solid state gettering process for the improvement of the performance of superconducting r.f. cavities J. Less Common Met. 139 179 Kneisel P, Stoltz O and Halbritter J 1972 On the variation of RF surface resistance with field strength in anodized niobium cavities Proc. Applied Superconductivity Conf. (Annapolis, MD, 1972) (New York: IEEE) p 657 Lengeler H, Weingarten W, Muller G and Piel H 1985 Superconducting niobium cavities of improved thermal conductivity IEEE Trans. Magn. MAG-21 1014 Lyneis C, Kojima Y, Turneaure J P and Viet Nguyen Tuong 1973 Electron loading in L- and S-band superconducting niobium cavities IEEE Trans. Nucl. Sci. NS-20 101 Mahner E, Müller G, Piel H and Pupeter N 1995 Reduced field emission of niobium and copper cathodes J. Vac. Sci. Technol. B 13 607 Markovich P M, Shepard K W and Zinkann G P 1987 Status of RF superconductivity at Argonne National Laboratory Proc. 3rd Workshop on RF Superconductivity (Argonne National Laboratory, USA) p 435 Matricon J and St James D 1967 Superheating fields in superconductors Phys. Lett. 24A 241 Mattis J D and Bardeen J 1958 Theory of the anomalous skin effect in normal and superconducting metals Phys. Rev. 111 412 Minehara E, Nagai R and Takeuchi M 1989 Fabrication and RF properties of high-Tc superconducting microwave passive elements Proc. 4th Workshop on RF Superconductivity (Tsukuba, Japan: KEK) p 695 Moffat D, Flynn T, Kirchgessner J, Padamsee H, Rubin D, Sears J and Shu Q 1990 Superconducting niobium RF cavities designed to attain high surface electric fields Proc. 4th Workshop on RF Superconductivity (Tsukuba, Japan: KEK) p 445

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Muller G 1983 Supraleitende Niobresonatoren im Millimeterwellenbereich Thesis Wuppertal University WUB-DI-83–1 Muller G 1989 Microwave properties of high-Tc oxide superconductors Proc. 4th Workshop on RF Superconductivity p267 Niedermann Ph, Renner Ch, Kent A D and Fischer ∅ 1990 Study of field-emitting microstructures using a scanning tunneling microscope J. Vac. Sci. Technol. A 8 594 Niedermann Ph, Sankarraman N, Noer R J and Fischer ∅ 1986 Field emission from broad-area niobium cathodes: effects of high-temperature treatment J. Appl. Phys. 59 892 Padamsee H 1983 Calculation for breakdown induced by ‘large defects’ in superconducting niobium cavities IEEE Trans. Magn. MAG-19 1322 Padamsee H 1985 A new purification technique for improving the thermal conductivity of superconducting Nb microwave cavities IEEE Trans. Magn. MAG-21 1007 Padamsee H 1988 High purity niobium for superconducting accelerator cavities J. Less-Common Met. 139 167 Padamsee H, Banner M, Kirchgessner J, Tigner M and Sundelin R, 1979 Muffin-tin cavities at X-band for linear accelerator application IEEE Trans. Magn. MAG-15 602 Padamsee H, Green K, Jost W and Wright B 1985 Cornell University unpublished note Padamsee H, Green K, Jost W and Wright B 1993 A statistical model for field emission in superconducting cavities Proc. IEEE Particle Accelerator Conf. ( Washington, DC, 1993 ) (New York: IEEE) p 998 Padamsee H, Smathers D, Marsh R and Van Daran B 1987 Advances in production of high purity Nb for RF superconductivity IEEE Trans. Magn. MAG-23 1607 Padamsee H, Tückmantel J and Weingarten W 1983 Characterization of surface defects in niobium microwave cavities IEEE Trans. Magn. MAG-19 1308 Palmieri V, Preciso R, Ruzinov V L, Stark S Y and Kulik I I 1994 A new method for forming seamless 1.5 GHz multicell cavities starting from planar disks Proc. 1994 Eur. Particle Accelerator Conf. ( London, 1994 ) ed V Suller and Ch Petit-Jean-Genaz (Singapore: World Scientific) p 2212 Peiniger M, Hein N, Klein N, Muller G, Piel H and Thuns P 1987 Work on Nb3Sn cavities at Wuppertal Proc. 3rd Workshop on RF Superconductivity (Argonne National Laboratory, USA) p 503 Philipp A 1982 Thesis Kernforschungszentrum Karlsruhe ( Report KfK-3268 ) Piel H and Romijn R 1980 Temperature mapping on a superconducting RF cavity in subcooled helium Report CERN EF/RF/80–3 Pioszyk B, Kneisel P, Stoltz O and Halbritter J 1973 Investigations of additional losses in superconducting niobium cavities due to frozen-in flux IEEE Trans. Nucl. Sci. NS-20 108 Pippard A B 1960 Experimental analysis of the electronic structure of metals Rep. Prog. Phys. 23 176 Proch D, Dwersteg B, Kreps G, Matheisen A, Möller W-D, Renken D, Sekutowicz J and Singer W 1993 Laboratory report DESY Proc. 6th Workshop on RF Superconductivity (VA, USA: CEBAF Newport News) p 77 Richter B 1979 The next generation of accelerators IEEE Trans. Nucl. Sci. NS-26 4261 Romijn R, Weingarten W and Piel H 1983 Calibration of the scanning thermometer resistor system for a superconducting accelerating cavity IEEE Trans. Magn. MAG-19 1318 Saito K, Kojima Y, Furuya T, Mitsunobu S, Noguchi S, Hosoyama K, Nakazato T, Tajima T, Asano K, Inoue K, lino Y, Nomura K and Takeuchi K 1990 R + D of superconducting cavities at KEK Proc. 4th Workshop on RF Superconductivity (Tsukuba, Japan: KEK) p 635 Schnitzke K, Martens H, Hillenbrand B and Diepers H 1973 TE011 X-band niobium cavity with critical magnetic flux density higher than Bc 1 Phys. Lett. 45A 241 Schulze K 1981 Preparation and characterization of ultra-high-purity niobium J. Met. 33 33 Shepard K W, Added N, Clifft B E, Crandall K, Givens J, Kedzic M, Potter J, Potokuchi P and Roy A 1993 Superconducting RF development at ATLAS Proc. 6th Workshop on RF Superconductivity p 1 Storm D W 1993 Review of low-beta superconducting structures Proc. 6th Workshop on RF Superconductivity p 216 Tinkham M 1965 Superconductivity (New York: Gordon and Breach); 1975 Introduction to Superconductivity (Malabar, FL: Krieger); two excellent textbooks on superconductivity Trinks U, Assmann W and Hinderer G 1986 The tritron: a superconducting separated-orbit cyclotron Nucl. Instrun. Methods A 244 273 Turneaure J P 1967 Thesis Hansen Laboratory, Stanford University, CA

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Further reading

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Turneaure J P 1971 Measurements on superconducting Nb prototype structures at 1300 MHz IEEE Trans. Nucl. Sci. NS-18 166 Turneaure J P 1972 The status of superconductivity for RF applications: Proc. Applied Superconductivity Conf. (Annapolis, MD, 1972) (New York: IEEE) p 621 Turneaure J P and Viet Nguyen Tuong 1970 Superconducting Nb TM010 mode electron-beam-welded cavities Appl. Phys. Lett. 16 333 Vallet C, Boloré M, Bonin B, Charrier J P, Daillant B, Gratadour J, Koechlin F and Safa H 1992 Flux trapping in superconducting cavities Proc. 3rd Eur. Particle Accelerator Conf. ( Berlin, 1992 ) ed H Henke, H Homeyer and Ch Petit-Jean-Genaz (Gif-sur-Yvette: Editions Frontieres) p 1295 Weingarten W 1989 On electrical breakdown in superconducting accelerating cavities IEEE Trans. Electr. Insul. EI-24 1005 Weingarten W 1996 Progress in thin film techniques Proc. 7th Workshop on RF Superconductivity (France: Saclay) p 129; Part. Accel. 53 199 Wilson P 1963 Investigation of the Q of a superconducting microwave cavity Nucl. Instrum. Methods 20 336 Wilson P 1982 High energy electron linacs: application to storage ring RF systems and linear colliders Proc. Fermilab Summer School (Batavia, 1981) Physics of High Energy Particle Accelerators ( AIP Conf. Proc. 87 ) ed R A Carigan, F R Huson and M Month (New York: AIP) p 452 Wu M K, Ashburn J R, Torng C J, Hor P H, Meng R L, Gao L, Huang Z J, Wang Y Q and Chu C W 1987 Superconductivity at 93 K in a new mixed phase Y-Ba-Cu-O compound system at ambient pressure Phys. Rev. Lett. 58 908 Yogi T, Dick G J and Mercereau J E 1977 Critical rf magnetic fields for some type-I and type-II superconductors Phys. Rev. Lett. 39 826 Zotter B 1985 Transverse instabilities due to wall impedances in storage rings IEEE Trans. Nucl. Sci. NS-32 2191

Further reading Bonin B (ed) Proc. 7th Workshop on RF Superconductivity ( Gif-sur-Yvette, 1995 ) ( France: Saclay ); the invited talks are published in 1996 Part. Accel. 53 Hartwig W and Passow C 1975 RF superconducting devices Applied Superconductivity II 2nd edn, ed V L Newhouse ( New York: Academic ) Kojima Y (ed) Proc. 4th Workshop on RF Superconductivity ( Tsukuba, 1989 ) ( Tsukuba: KEK ) Kuntze M (ed) Proc. 1st Workshop on RF Superconductivity ( Karlsruhe, 1980 ) ( Report KfK 3019, Karlsruhe ) Lengeler H (ed) Proc. 2nd Workshop on RF Superconductivity ( 1984 ) (Geneva: CERN) Lengeler H 1989 Superconducting cavities Proc. CERN Accelerator School ( Hamburg, 1988 ) ed S Turner (Geneva: CERN) ( CERN Yellow Report CERN 89–04 197 ) Orlando T P and Delin K A 1991 Foundations of Applied Superconductivity (New York: Addison-Wesley); a well structured introductory textbook developed from an advanced undergraduate course in superconductivity developed at MIT with an emphasis on applications Padamsee H 1988 Superconducting structures for electron accelerators J. Supercond. 4 377 Padamsee H 1992 Superconducting RF Proc. US Particle Accelerator School ( BNL Upton, 1989 ) ( AIP Conf. Proc. 249 ) ed M Month and M Dienes (New York: AIP) p 1402 Padamsee H and Knobloch J 1994 Issues in superconducting RF technology Proc. Joint US-CERN-Japan Int. Sch. on Frontiers of Accelerator Technology ( Maui, HI, 3–9 November 1994 ) ed S I Kurokawa et al (Singapore: World Scientific) pp 101 Padamsee H, Shepard K W and Sundelin R 1993 Physics and accelerator applications of RF superconductivity Annu. Rev. Nucl. Part. Sci. 43 635 Piel H 1989 Superconducting cavities Proc. CERN Accelerator School ( Hamburg, 1988 ) ed S Turner (Geneva: CERN) ( CERN Yellow Report CERN 89–04 149 ) Pierce J 1974 Superconducting microwave resonators Methods of Experimental Physics 11, Solid State Physics ed L Marton (New York: Academic) Proch D (ed) Proc. 5th Workshop on RF Superconductivity (Hamburg, 1991); the invited talks are published in 1992 Part. Accel. 40

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Ramo S, Whinnery J R and Van Duzer T 1965 Fields and Waves in Communication Electronics ( New York: Wiley ); an excellent pedagogically oriented introduction to electromagnetic fields, waveguides, cavities, based on a course for engineers at a senior level Septier A and Viet Nguyen Tuong 1977 Microwave applications of superconducting materials J. Phys. E: Sci. Instrum. 10 1193 Shepard K W (ed) Proc. 3rd Workshop on RF Superconductivity ( Argonne, IL, 1987 ) ( Report ANL-PHY-88–1, Argonne ) Sundelin R (ed) Proc. 6th Workshop on RF Superconductivity ( Newport News, VA, 1993 ) (Newport News: CEBAF) Tigner M and Padamsee H 1983 Superconducting microwave cavities in accelerators for particle physics Physics of High Energy Particle Accelerators ( AIP Conf. Proc. 105 ) ed M Month (New York: AIP) Turneaure J P, Halbritter J and Schwettman H A 1991 The surface impedance of superconductors and normal conductors: the Mattis-Bardeen theory J. Supercond. 4 341 Weingarten W 1992 Superconducting cavities Proc. CERN Accelerator School ( Oxford, 1991 ) ed S Turner (Geneva: CERN) (CERN Yellow Report CERN 92–03 318) Weingarten W 1994 Superconducting cavities—basics Proc. Joint US-CERN-Japan Int. Sch. on Frontiers of Accelerator Technology ( Maui, HI, 3–9 November 1994 ) ed S I Kurokawa et al (Singapore: World Scientific) pp 311 Wilson E (ed) Proc. CERN Accelerator School ( Hamburg, 1995 ) (Geneva: CERN) ( CERN Yellow Report at press ); these are the proceedings of a course dedicated to superconductivity in particle accelerators

Copyright © 1998 IOP Publishing Ltd

G8 A superconducting transportation system

E Suzuki, S Fujiwara, K Sawada and Y Nakamichi

G8.0.1 Introduction G8.0.1.1 Why did Japan develop a high-speed Maglev? The Tokyo-Osaka corridor occupies only 10% of Japan’s area, but accounts for about 50% of GDP (gross domestic product), and it is the heart of Japanese economic activities. Therefore, there are huge volumes of passenger movement through this corridor, and the main traffic mode in this area is the Tokaido Shinkansen bullet train which opened in 1964, and holds a far superior position to highway and airline services. The Tokaido Shinkasen runs at a maximum speed of 270 km h−1, covering the distance between Tokyo and Osaka within 150 min, and carrying 140 million passengers a year, or more than 42 billion passenger km a year, which means that a mere 515 km double-track line carries more than all of the British Railway lines do. One hundred and forty trains depart from Tokyo central station a day. Each train consists of 16 passenger cars. It is 400 m long, and can carry 1300 passengers. In one hour of the peak prime time 11 trains depart from Tokyo, and this is the limit of the traffic capacity of the Tokaido Shinkansen, but there is so much demand that not all passengers can get reservation tickets in some cases. The traffic volume is still increasing year by year. In order to improve this situation, the traffic capacity should be increased. Japan needs another high-speed line between Tokyo and Osaka. What system should be introduced? JNR (Japanese National Railways) had investigated various high-speed train systems for many years, and from the viewpoint of speed, safety, maintenance, pollution and future prospects, they decided to develop the superconductive Maglev. G8.0.1.2 The merits of the superconducting Maglev In the superconducting magnetic levitation (Maglev) system, all vehicles have superconducting magnets. On the guideway two types of ground coil are installed: propulsion coils and levitation coils. The vehicle is propelled, levitated and guided by the electromagnetic force acting between the on-board magnets and the ground coils. The superconducting Maglev system is very different from conventional high-speed rail systems and has many advantages as follows. Copyright © 1998 IOP Publishing Ltd

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(a) Speed In the conventional train, motors on the car rotate steel wheels and they push the rails, thereby the train achieves a propulsion force. Therefore, the friction between the wheels and the rails is indispensable. At low speed this friction is large, but it decreases with the speed. In contrast, the drag force of the train increases with the speed, so beyond a certain speed, the train cannot accelerate any more even if it is equipped with very powerful motors. The critical speed varies according to whether the rail conditions are wet or dry, clean or foul, etc. In the superconducting Maglev, the train is propelled magnetically by a linear motor, so it can get the propulsion force at any speed. As linear motor coils are installed along the guideway, and the propulsion power is supplied to these coils, even in a high-speed system in which high power is required, the Maglev vehicle is designed to be compact and very light. So 500 km h−1 commercial operation will be attained easily. (b) Safety Owing to its guideway structure, the Maglev vehicle will never be derailed. As the Maglev vehicle is supported by many levitation coils acting at a time, vehicles are influenced only slightly even if one coil happens to be out of order. Also, the levitation and guidance characteristics are stable even at very high speed. So, the superconducting Maglev is a quite safe transportation system. (c) Maintenance In conventional high-speed train systems, the maintenance of tracks and trolley wires is a big problem, but in the superconducting Maglev, a very light vehicle is levitated by many coils acting at a time and it is not in contact with anything. As for a trolley, the superconducting Maglev uses none at all. Therefore Maglev tracks require only a small maintenance cost. (d) Pollution An electric high-speed train is far superior to an automobile or aeroplane from the viewpoint of pollution. The superconducting Maglev is even better than electric high-speed trains, because of its smaller environmental noise and lower vibration levels. As for noises, the greatest one in the Shinkansen is the noise caused by the power collection from the trolley, and Maglev has no trolleys. In addition a very light vehicle causes only a slight environmental vibration. (e) Future prospects The superconducting Maglev has a huge potential as a high-speed transportation system. It is only the aero-drag force that limits the Maglev cruising speed to 500 km h−1, So, in the future, Maglev could run at more than 1000 km h−1 in a vacuum tube tunnel, and will cover the distance between the downtown areas of big cities in quite a short time. Moreover, when high-temperature superconducting materials become stable enough, the liquid-helium-cooled superconducting magnet on board could be replaced by a liquid-nitrogen-cooled one, and the superconducting Maglev will become an easier-to-operate system. G8.0.2 Principle and characteristics of the superconducting Maglev G8.0.2. 1 Principle The superconducting Maglev car is driven by a linear synchronous motor composed of the superconducting magnet on board and armature coils on the guideway. It is levitated above the ground by the electromagnetic force acting between the superconducting magnet and the levitation coil on the ground (Powell and Danby 1966).

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(a) Propulsion The propulsion system of the vehicle is a linear synchronous motor, whose field magnet is a superconducting magnet on board and whose armature coils are three-phase coils arranged on the guideway facing the on-board magnet. These three-phase coils are connected in series over a certain length and make a feeding section. Each section is connected to a feeder through a section switch. Both the field and the armature coils have an air core and the former is arranged on both sides of the bogie and the latter is arranged vertically on the guideway as shown in figure G8.0.1.

Figure G8.0.1. Guideway coils of the Miyazaki test track in Japan.

The principle of this linear motor is the same as that of a conventional rotary synchronous motor. Propulsive force is obtained by the armature current flowing synchronously with the vehicle motion. The frequency and the phase of the armature current are determined to coincide with the induced voltage using a continuous position detector for the field magnet. The propulsive force of a linear synchronous motor is proportional to the current amplitude and independent of vehicle speed with a constant phase angle. Thus this type of linear motor has good performance characteristics at high speed with a superconducting field magnet of high field. Generally a linear synchronous motor has better characteristics than a linear induction motor under the condition of large air gap. As the superconducting linear synchronous motor has a high-field magnet and a large pole pitch, it gives good performance at a gap of about 20 cm. As this type of motor has no iron core, electromagnetic forces act directly on the coil. The support of the motor or its mechanical strength must be designed considering those forces. The number of armature coils is large, and it is necessary to make their structure simple. Then a higher-harmonic magnetic field is generated the influence of which must be seriously considered. (b) Levitation The levitation system is composed of a superconducting magnet on board and short-circuited levitation coils on the guideway. There is another method by which metal sheet is arranged on the guideway instead of the short-circuited coils. Either system is the same in principle. When a vehicle moves, a current is induced in the guideway levitation coil. If there is an inductance in the circuit, the current lags behind the induced voltage, and a mean levitation force is generated. An example of a levitation coil is shown in figures G8.0.1 and G8.0.2 arranged on a horizontal plane. If the circuit has only a resistance Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.2. The basic coil arrangement for levitation: (a) arrangement in a cross-section; (b) relative position.

Figure G8.0.3. Example of levitation force and drag force characteristics.

component, the mean levitation force is zero and a magnetic drag force (running resistance) is generated. In contrast, assuming only inductance in the circuit, the magnetic drag force is zero. A normal levitation coil generates both a levitation force and a magnetic drag force, and an example of their speed dependence is shown in figure G8.0.3. This characteristic is obtained with a constant distance maintained between the superconducting magnet and the levitation coils. Because this principle uses induced current in the guideway coil, levitation force is zero at zero velocity. Levitation force increases with the vehicle speed, and over a certain speed it increases only to a small extent. The vehicle takes off over a certain velocity which we call take-off velocity, and below that velocity the vehicle needs auxiliary wheels. Magnetic drag force has a maximum value at a certain low speed, over which speed it decreases in inverse proportion to the speed. This means that the loss in all the levitation coils is almost constant, independent of speed. Besides, the levitation system generates a lateral force and such a force must be considered. (c) Guidance The guidance system operates on the same principle as the levitation system. However, a levitation system must always generate a levitation force which supports the vehicle weight, but a guidance system must generate a guidance force only when the vehicle displaces laterally. Thus the circuit is usually formed with both sides connected to guideway coils as shown in figure G8.0.4. No net linked flux exists when the vehicle travels in the centre and no current is induced. Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.4. The principle of guidance: (a) principle of guidance; (b) coil arrangement in a cross-section.

G8.0.2.2 Constitution of the levitation system The drag ratio is an index of the levitation system (Powell and Danby 1966). To make the drag ratio great, the loss in the guideway coil must be decreased. Therefore the magnetomotive force in the guideway coil must be small. Assuming inductance L, resistance R of the guideway coil and mutual inductance between superconducting coils and guideway coils M (the amplitude of the mutual inductance during vehicle travel), the drag ratio is represented as drag ratio = ( vL/R ) ( ∂ M/∂ z )( 1/M). From this equation it is known that the mutual inductance is desirably small and its derivative in the vertical direction is desirably large. A method called null flux satisfies these conditions, and such a circuit is illustrated in figure G8.0.5 (Fujiwara and Fujimoto 1989). If the centre position of the superconducting magnet of this system agrees with the centre of the levitation, the net mutual inductance is zero. The mutual inductance is proportional to the vertical displacement of the magnet. Adopting this system the drag ratio can be doubled compared with the normal flux system (the levitation coil is arranged as shown in figure G8.0.1).

Figure G8.0.5. The null-flux levitation coil arrangement.

As guideway coils are arranged over the whole length, it is necessary to decrease the number of coils. Combined guidance and propulsion systems and combined levitation and guidance systems have been developed. Moreover all the functions of propulsion, levitation and guidance can be integrated with a circuit ( Murai and Fujiwara 1993 ). The construction of the coil is determined by considering the difficulty of attaching the coils to the guideway, the difficulty of electric insulation or the scale of the system. Copyright © 1998 IOP Publishing Ltd

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The levitation coil must satisfy the electromagnetic characteristics, to stand the force generated on the coil conductor and to endure long-term use. The construction cost must not be high, because the number of coils is large. To decrease the loss it is desirable that the electric resistance be small and that the time constant be large. To satisfy these conditions the conductor cross-section and the conductor space factor in the coil cross-section must be large. As induced voltage in the levitation coil is low, the electric insulation is simple. The material for the support of the coil must be nonconductive. High polymers, for example sheet-moulded compounds or epoxy resins, are used in the coils for the Miyazaki test track to strengthen and to fasten them with bolts. G8.0.3 History of the development of the Japanese Maglev G8.0.3.1 The early stage of development JNR started the study for the levitated railway system in 1962. From among the many candidates, JNR selected the superconducting Maglev system for the reasons mentioned in section G8.0.1. The technical feasibility of this system was demonstrated with a model vehicle ML 100 on the 480 m track in the Railway Technical Research Institute ( RTRI ) in 1972. ML100 was 7.0 m long, 3.5 t in weight and had four seats (figure G8.0.6). It ran at a maximum speed of 60 km h−1. Its smooth and silent run surprised the spectators. ML100 was propelled by a long stator linear induction motor ( LIM ). A LIM is quite easy to control, but its characteristics are not so suitable for the high-speed Maglev. So JNR decided to adopt a long-stator linear synchronous motor ( LSM ). The efficiency and power factor of LSMs are superior to those of LIMs and they permit a large gap between the on-board magnets and the propulsion ground coils. In 1975, the track was reconstructed and adapted to an LSM propulsion system and a new vehicle ML100A was made. At the same time a combined propulsion-guidance system was adopted, which had

Figure G8.0.6. ML100. Copyright © 1998 IOP Publishing Ltd

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been invented a few years earlier. This was a scaled-down model of the next Miyazaki test track, and ML100A made quite a stable and smooth run. Encouraged by this early success, the practical development of Maglev transport was started with the construction of the large-scale test tracks. G8.0.3.2 Miyazaki test track and ML-500 The new test track was constructed in the Miyazaki prefecture in Kyushu island as shown in figure G8.0.7. The first section of 1.3 km was completed in 1977 and then the track was extended in three stages. In 1979 it attained a total length of 7 km. Almost all of the track profile is level, but there is a downhill section with a gradient of 5/1000 at about 1 km from the test centre. The horizontal plan has a curve with a radius of 10000 m. The current to the ground propulsion coils was supplied by a frequency converter system, which consists of a motor-generator set and two 10 MV A cycloconverters.

Figure G8.0.7. The Miyazaki test track.

The practical tests started in July 1977 with the ML-500 vehicle on the inverted-T-shaped guideway. The basic characteristics of propulsion and levitation with superconducting magnets were tested in highspeed operation with the ML-500 vehicle shown in figure G8.0.8. ML-500 was 13.5 m long, 10.0 t in weight and superconducting magnets occupied half of it. The inverted-T-shaped guideway was utilized for this vehicle because the propulsive and braking forces should be applied as near as possible to the centre of mass of the vehicle, due to the high acceleration and braking required for a 500 km h−1 run along a 7 km long test track. An L-shaped superconducting magnet was used, which contained four superconducting coils. The two vertical ones were for the propulsion and guidance and the two horizontal ones for the levitation. A helium refrigerator and compressor were installed on the ground and liquid helium was supplied to the cryostats before run operation. ML-500 showed that the dynamics of the superconducting Maglev were stable from low speeds Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.8. ML-500.

Figure G8.0.9. The speed-distance curve of test vehicle ML-500.

to very high speeds. It ran many times at more than 500 km h−1, and recorded 517 km h−1 on 21 December 1979. Figure G8.0.9 shows its run curve. G8.0.3.3 MLU001 As a model of the practical Maglev train for passenger transport, and for the operational tests of dynamics on coupled vehicles, a three-vehicle train MLU001 was introduced in 1980 (figure G8.0.10). The guideway Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.10. MLU001.

cross-section was converted into a U-shape to increase the cabin space. Propulsion coils on the centre beam of the inverted-T-shaped guideway were removed to the newly built side walls. MLU001 had 32 seats for passengers and a manned operation was carried out successfully. Four I-shaped superconducting magnets were installed vertically on the truck of each vehicle. The superconducting magnets, each of which contained two superconducting coils, provided the lifting, lateral guiding and propulsive forces. The whole cryogenic system was installed on board. Both concentrated and distributed systems were studied. Through the operational tests with these pieces of equipment, the reliability of the on-board cryogenic system was improved. As the power supply system had been designed for ML-500 and did not have enough power to make the three-coupled train run fast, new cycloconverters were introduced. Two sets of new 16 MV A cycloconverters did not require a motor-generator, and could be connected directly to the power line. Accompanied by this new power supply system, the propulsion coils’ insulation was reinforced. MLU001 showed that the coupled Maglev vehicles’ dynamics were quite stable. It recorded 352 km h−1 with unmanned three-coupled vehicles, and 400.8 km h-1 with manned two-coupled vehicles. In spring 1987 JNR was changed from being a government enterprise into six private passenger railway companies and one freight rail company, the so-called ‘JR-group’. The Railway Technical Research Institute has taken over the Maglev development project. Just before this reorganization, a new Maglev test vehicle MLU002 was completed and in May of that year its running tests were started (figure G8.0.11). MLU002 consisted of one body and two trucks. The trucks were arranged not continuously but intermittently, similarly to the future revenue service vehicle. On both sides of each truck, I-shaped superconducting magnets were installed vertically as in the MLU001, but they were lighter and their superconducting magnets were more powerful than the ones of MLU001. Owing to its strong magnets and light body, the MLU002 vehicle could accommodate 44 passengers. More than 15 000 people enjoyed the high-acceleration and high-speed run of MLU002. Besides these trial rides, various tests were carried out as follows: Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.11. MLU002.

( i ) a speed-up test ( ii ) a dynamic braking and mechanical braking test ( iii ) a confirmation test of the inductive power collecting system ( iv ) an on-board refrigeration system test ( v ) a guideway irregularity test ( vi ) an aerodynamics test. The levitation coils were installed originally on the bottom of the U-shaped guideway, and the vehicle was levitated by the repulsive magnetic force between the coils and the on-board magnets. Later, the null-flux levitation system was invented. In this system, the ‘eight’-shaped levitation coils are installed on the side walls of the guideway, and these coils provide not only a lift force but also a lateral guide force. As this system had many advantages over the original one, it was introduced on the Miyazaki test track and tested. MLU002’s planned maximum speed was 420 kmh−1 on the 7 km long Miyazaki test track, but its superconducting magnets were not stable enough, so it could not reach that speed. Unfortunately in the punctured-tyre-detecting test, a broken tyre caused a rotating lock of the tyre-wheel made of magnesium alloy, which caught fire by friction and finally MLU002 was burnt out on 3 October 1991. This fire accident was not caused by the basic design of the superconducting Maglev, but warned us severely against the use of flammable materials. Copyright © 1998 IOP Publishing Ltd

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G8.0.3.4 MLU002N Several years after JNR’s reformation, the plan to construct a large-scale Maglev test line materialized in the form of the Yamanashi test line. As there were still many tests to be carried out at the Miyazaki test track for designing the Yamanashi test line, a new test vehicle MLU002N was developed (figure G8.0.12). This vehicle is a copy of MLU002, but its materials were selected carefully, and it is equipped with the aerodynamic brakes which will be adopted for the emergency brake in the Yamanashi vehicles (figure G8.0.13).

Figure G8.0.12. MLU002N.

Figure G8.0.13. The aerodynamic brake of MLU002N.

MLU002N has been operating since the beginning of 1993. About one year after it started it registered 431 km h−1 in an unmanned run (figure G8.0.14). In January 1995 it set a new speed record of 411 km h−1 in a manned run. This was attained because of the highly reliable new superconducting magnets. G8.0.4 Superconducting magnet and refrigeration systems G8.0.4.1 The requirements for a superconducting magnet for the Maglev The superconducting magnet is the most important component of a Maglev vehicle and plays the role of wheels in a conventional railway. All forces of propulsion, levitation and guidance work on the superconducting coil. The required concepts for designing a superconducting magnet of Maglev are as follows. ( i ) It has a high magnetomotive force and high stability in order to generate effectively the forces of propulsion, levitation and guidance which are necessary for high-speed running. The magnet naturally Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.14. Run curve of MLU002N (maximum 431 km h−1 ).

suffers vibration from being mounted on the vehicle and it must have enough strength and dura bility to endure this mechanical vibration. ( ii ) It weighs as little as possible considering that it has to be mounted on the vehicle. ( iii ) The magnet on the Maglev vehicle works independently of the ground facilities. As a result, the superconducting coil operates in the persistent current mode and the persistent current switch is indispensable to this. There are problems for this switch such as how to ease the operation in addition to holding high stability or how to reduce the heat generation in energizing. ( iv ) The running performance of Maglev depends upon the distance between the superconducting coil and the ground coils. In order to utilize effectively the flux from the superconducting magnet, it is desirable to shorten this distance. However, this complicates the inner structures of the magnet, which makes it difficult to keep the thermal distance great enough to insulate them. So it is important to set the outer vessel and the superconducting coil as close as possible to each other. ( v ) The heat leakage into the region of extreme low temperature is to be reduced, which leads to a smaller capacity of the on-board refrigerator. Once the magnet is cooled, the extreme low-tem perature state is to be held by the on-board refrigerator over the long term. So it is necessary to certify the reliability and durability of the magnet and the refrigerator which work continuously for 10000 h. ( vi ) It is most important to maintain the stability of the superconducting coil under the influence of various disturbances during operation and to reduce the heat generation due to these disturbances. ( vii )It is necessary to simplify the operation of the superconducting magnet and the refrigerator in the revenue service. In particular, tasks involving replenishing the liquid helium into the on-board magnet from the ground facilities are to be avoided.

G8.0.4.2 The structure of the superconducting magnet Figure G8.0.15 is a schematic view of the superconducting magnet which is to be used for future Maglev vehicles. This superconducting magnet is divided into the lower part and the upper part. The lower part contains the coil units, including four superconducting coils. The upper part is a liquid-helium tank on which the on-board refrigerator is installed. The helium gas evaporated in the coil units is re-liquefied by this refrigerator after compression by the compressor. Thus the quantity of liquid helium can be recovered without replenishment from the ground back-up facilities. Each superconducting coil itself is arranged Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.15. A schematic view of the superconducting magnet for Maglev.

vertically and maintains its superconductivity by immersion under the liquid helium supplied from the helium tank ( Nakao et al 1989, Tsuchishima and Herai 1991, Suzuki et al 1992 ). The superconducting magnet consists of the following elements. (a) Superconducting coil Figure G8.0.16 shows the structure of the superconducting coil. The superconducting coils are designed in racetrack form because they must support the electromagnetic forces. A levitation force works on the straight lines of these coils and a propulsion force acts on the arc sides. The superconducting coils in the magnet are integrated by being wound tightly and impregnated with epoxy resin, using superconducting wires of Nb-Ti with a rectangular cross-section of about 1 mm height and 2 mm width. The rated magnetomotive force of the magnet used for the vehicle at the Miyazaki Maglev test track is 700 kA and the maximum magnetic field strength is about 5 T. (b) Inner vessel The superconducting coil is held tightly inside the inner vessel which has a housing structure to prevent it from deforming itself into a circular shape under a hoop force. Filling the narrow space between the inner vessel and the coil with liquid helium, it is possible to make the coil superconductive at an extremely low temperature (4.2–4.5 K). As shown in figure G8.0.16 the inner vessel and superconducting coil are fixed with fasteners. Thus tightly pressed together, they are prevented from slipping against each other even if mechanical vibration or deformation of a coil occurs. This structure is one of the most important parts influencing the performance of the coil. (c) Power lead For energization or de-energization of the superconducting coil, it is necessary to connect the magnet with the power supply. The power leads which are permanently installed between the regions of extreme low temperature and room temperature have to satisfy both low heat conductivity and high electric conductivity. Thus the total heat leakage into the inner vessel of the magnet depends upon the performance of these power leads. The power leads are designed to endure an energizing current of about 500–600 A when they are cooled by a small amount of the cold gas evaporated in the magnet. Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.16. The structure of the superconducting coil.

(d) Persistent current switch The persistent current switch is attached to the superconducting coil and used in energizing or de-energizing it. This switch is an indispensable element for keeping the persistent current mode of the superconducting coil on the Maglev vehicle. On energization, we raise the current to the stated value of the superconducting coil from the power supply with the switch off, and then lowering the current from the power supply to zero with the switch on, we can get persistent current mode. In the early development stage of the Maglev vehicle, two kinds of persistent current switch were used. One was a mechanical type using a metal contact which touches in the ‘on’ state and separates in the ‘off ’ state. The other was a thermally controlled type whose off-on operation was effected by heating or cooling of the superconducting wire. After some inspection and testing, only the latter type has come to be adopted for our Maglev vehicle. In order to save the operating time of energization or de-energization and to reduce the Joule heat generated in the switch, the resistance in the ‘off ’ state of the persistent current switch should be made as large as possible. Now switches with 50Ω resistance are installed in the superconducting magnet of the MLU002N test vehicle. (e) Supporting columns All forces such as propulsion, levitation and guidance work on the superconducting coil. The supporting columns play the important role of transferring these forces to the frame of the bogie supporting the superconducting magnet. The requirements for these columns are naturally to have enough strength and to minimize the heat leakage into the region at liquid-helium temperature. The supporting columns are composed of several units set in a concentric circle for the purpose of forming a long heat conduction path in a narrow space. To satisfy the requirements of strength, low thermal conductivity and lightness, fibre-reinforced plastics such as GFRP, CFRP and A1FRP are used to constitute these columns. (f) Outer vessel and radiation shield plate For the purpose of preventing the heat leakage into the region at liquid-helium temperature, the radiation shield plate surrounds the superconducting coils completely and shields them from the radiant heat by feeding liquid nitrogen onto the plate. The outer vessel encloses this plate and holds a vacuum inside it. The outer vessel and radiation shield plate are ordinarily made of aluminium because of the requirements of lightness and in order to prevent the inner vessel from being heated by magnetic fluctuation from the ground coils. Copyright © 1998 IOP Publishing Ltd

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G8.0.4.3 Electromagnetic disturbances by ground coils The superconducting magnet which is installed on a Maglev vehicle endures quasi-static forces such as levitation, guidance and propulsion which vary with the sway of the vehicle. It also suffers from the influence of various disturbances in running. The main disturbance is the magnetic fluctuation from the ground coils which are arranged at intervals. The previous magnets on MLU002 were plagued with quenching troubles due to these electromagnetic fluctuations. Formerly the influence of these disturbances on the magnet was regarded as a.c. loss due to variations in the remaining magnetic field which is not shielded by the outer vessel and the thermal radiation shield plate which have high electrical conductivity. It was an urgent problem to make clear the dynamic behaviour and the characteristics of heating in the magnet under these continuous disturbances in practical use. So simulating facilities were constructed which could simulate a real disturbance on the magnet during running and the evaporation rate of liquid helium in a vibrating magnet was estimated. As a result of testing, it was found that an extreme increase of heat load on the inner vessel, which was different from the a.c. loss, was caused by the fluctuation of magnetic field at a particular frequency under utilization of these facilities (Nakashima et al 1993, Suzuki 1994). The cause of these phenomena was investigated from various aspects. As a result, it has been revealed that the superconducting magnet vibrates due to the disturbances and causes relative displacements between the radiation shield plate (or outer vessel) and the superconducting coil. Then an eddy current is induced on the shield plate because of these relative displacements. The eddy current on the shield plate causes a magnetic fluctuation on the inner vessel housing the superconducting coil and induces an eddy current on the inner vessel again. Finally heating occurs due to this eddy current and evaporates the liquid helium in the inner vessel (Tsuchishima et al 1994). G8.0.4.4 The on-board refrigeration system (a) The aim of an on-board refrigeration system The on-board refrigeration system becomes the most important item when deciding the total construction of the Maglev vehicle including the superconducting magnet. This system aims to keep liquid helium on board over a long time without periodic replenishment from the ground facilities. Figure G8.0.17 shows a flow diagram of the cryogenic system of a superconducting magnet. The refrigeration system is mainly composed of a helium refrigerator, a helium compressor, a helium gas reservoir tank and a control unit. These make a closed-cycle gas system. The on-board refrigerator is connected directly to the liquid-helium tank of the magnet. The helium compressor is set on the truck of the vehicle. The requirements for this refrigeration system are as follows. ( i ) Steady state re-liquefaction. The helium evaporated by the heat leaks and vibrations in running will be re-liquefied steadily by the on-board refrigerator and compressor. ( ii ) Characteristics during energization or de-energization of the magnet. In order to inspect or maintain the Maglev vehicle, the superconducting magnet may often be energized or de-energized. In this operation, which is supposed to take place once a day, the liquid-helium level in the tank will decrease temporarily due to excessive heat loss generated in the magnet. The gaseous helium evaporated due to this heat loss will be stored in the buffer tank by a control unit detecting the rise of pressure at the inlet of the compressor. The stored helium gas in the buffer tank will be re-liquefied by the redundant cooling capacity of the refrigerator at night. So the quantity of liquid helium can be restored by next morning. This means that operation of the cryogenic system without the periodic supply of liquid helium from the ground back-up systems is possible even under the energization and de-energization of the magnet daily.

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Figure G8.0.17. Gas flow diagram of the cryogenic system.

( iii )Minimization of the influence of failures. When some failure of the on-board refrigerator occurs, or the heat load temporarily exceeds the capacity of the refrigerator, it is desirable that the influence be small and the functions of the superconducting magnet continue as long as possible ( Nakashima and Herai 1989 ). So a direct cooling system was adopted as shown in figure G8.0.17. As the helium gas in the refrigerator and that in the magnet are the same in this system, the on-board refrigerator operates as both a liquefier and a refrigerator depending on the heat load of the superconducting magnet. This works automatically without any additional manual operation. Furthermore, this system has the advantage that when excessive temporary heat load exceeding the refrigerator capacity arises, the rate of evaporation of liquid helium in a magnet can be reduced to far less than the calculations based on the difference between the two, and that the recovery to a normal re-liquefaction rate can be quick when the heat load returns to normal. This is because the gas flowing into the high-pressure line is cooled in the heat exchanger of the refrigerator with recovery of the sensible heat of the evaporated gas. Figure G8.0.18 shows the effect of energizing the magnet with a current sweeping rate of 10 A s−1 with the refrigerator under operation. As shown in the figure, the helium gas produced in the liquid-helium tank is smoothly transferred to the buffer tank through pressure-control valves. The vaporized volumes of liquid helium under energization and de-energization are 1.72 1 and 1.87 1 respectively. This total amount of liquid helium is far less than the amount which the refrigerator can re-liquefy in several hours at night. (b) Constitution of the refrigeration system The components of the refrigeration system must satisfy extremely severe requirements such as limited size, weight and power consumption. The cooling capacity of the refrigerator is expected to exceed the heat load ( 3-5 W ) in the superconducting magnet and so far a refrigerator having a refrigeration capacity of 5-8 W at 4 K has been developed, in which a Joule-Thomson ( JT ) loop is associated with a precooling refrigeration cycle. Three types of on-board 4 K refrigerator with a JT loop have been developed for Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.18. Performance of the magnet during energization.

the Maglev vehicle. These are the Claude-cycle refrigerator, the (reversed) Stirling/JT cycle and the Gifford-McMahon (GM)/JT cycle. In spite of many efforts made to develop new automatic valve mechanisms and a laminated heat exchanger for the Claude-cycle refrigerator, the reliability of this refrigerator has not reached a satisfactory level. As a result, nowadays efforts are concentrated on developing the other two types of refrigerator (Terai et al 1993). Figure G8.0.19 shows the refrigeration system for each refrigerating cycle. ( i ) Gifford-McMahon (GM) cycle refrigerator. The GM refrigerator is the most common one as it is widely applied to magnetic resonance imaging (MRI), cryopumps and many laboratory systems because of its remarkably high reliability which is a consequence of its simple drive mechanism and few moving parts. Nevertheless, until recently it had been impossible to use it in a Maglev vehicle because of its low efficiency. However, recent improvements in the capacity and efficiency of the GM refrigerator over the last few years, by using magnetic compounds as regenerator materials, better drive methods, etc, are notable. As a result, the refrigeration capacity (or efficiency) has improved from 5 W to 8 W (from 0.6 × 10−3 to 1 × 10−3 ) with 8 kW input power. This GM refrigerator is directly connected to the liquid-helium tank of the magnet. The compressor to be set on the truck of the vehicle is a scroll-type one lubricated with oil. The compact oil-separation system at the exhaust of this compressor is achieved by common technology used in other fields (coalescer and charcoal). ( ii ) Stirling refrigerator. A Stirling refrigerator has theoretically an extremly high efficiency. This refrigerator provides a cooling capacity of over 10 W with 8 kW power input (or efficiency over 1.3 × 10−3 ) by the adoption of some magnetic compounds in the regenerator and by a modification of the refrigerator mechanism, etc. Both the Stirling refrigerators and the compressor will be mounted Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.19. The on-board refrigeration system of the Maglev.

together on the truck of the vehicle. A flexible transfer tube, through which the cooled helium gas flows, will connect the refrigerator with the liquid-helium tank of the magnet. A continuous operating time of 10000 h for these refrigerators has been confirmed. Moreover it is intended that research and development to improve reliability in continuous operation and to increase the mean time between failures (MTBF) under various severe environmental conditions such as vehicle vibration, electromagnetic disturbances and contamination of gas will continue.

G8.0.5 Power supply system for the Maglev In order to drive an LSM with the primary side which consists of superconducting magnetic coils aboard the vehicle and rows of propulsion coils (armature coils) on the ground, a power supply system is required. The power supply system is composed of not only a power converter but also many pieces of equipment such as a position detector. In this section, the following items in the power-supply system for driving the LSM are outlined (Ikeda et al 1988, Nakamichi et al 1992, Nakamura et al 1985): (i) ( ii ) ( iii ) ( iv )

hardware composition propulsion control system power feeding control system power converter.

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Figure G8.0.20. Composition of LSM propulsion control equipment.

G8.0.5.1 Hardware composition The composition of the LSM propulsion control system is illustrated in figure G8.0.20. The functions of typical devices are as follows. (a) Position detector The detection of vehicle position is performed by processing the induced signal in transposed inductive wires in which a radio signal transmitted from the vehicle is to be induced. Those inductive wires transposed consist of six pairs of receiving loops whose transposing cycle is equal to that of the propulsion coils on the ground. (b) Central controller The central controller installed at the testing centre presets testing conditions for running the vehicle and performs data processing for vehicle operation. For LSM propulsion control, the central controller calculates the vehicle speed and the distance based on a position detection signal transmitted from the vehicle, and gives to the total controller at the substation an appropriate thrust (current) that causes the vehicle to run according to a preset speed pattern. (c) Total controller The total controller installed at the substation exchanges signals with the testing centre and controls the power converters together with the section switchgear controller. (d) Leakage co-axial cable The technology related to leakage co-axial cable (LCX) has already been completed so that LCX is used for telemetry or communication between the vehicle and the testing centre on the ground. For instance, the brake command from the testing centre or the information on the vehicle are transmitted through the LCX. (e) Power converter The power converter for driving the LSM plays the most important role, because the Maglev system requires a large capacity of power converters which supply the power to the propulsion coils (armature Copyright © 1998 IOP Publishing Ltd

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coils). A power converter for driving the LSM has to meet two requirements. One is to be able to change the output frequency d.c. to a high frequency (around commercial frequency). The other is to be able to generate high voltages so that it is possible to supply the necessary three-phase sinusoidal currents even in the presence of an electromotive force of high voltage. As power converters satisfying these requirements, a cycloconverter, which is a direct conversion type, and a pulse width modulation (PWM) inverter have been developed. At the early stage of developing the power converters for driving the LSM, a cycloconverter was considered. However, recent remarkable technical progress concerning the high-power gate turn-off thyristor (GTO thyristor) has stimulated tremendous growth in the research of variable a.c. drive. This growth has also contributed to the development of power converters for driving LSMs in which a PWM inverter with large capacity whose switching element is the GTO has been implemented as a power converter for driving the LSM. G8.0.5.2 LSM propulsion control system The LSM propulsion control block diagram is given in figure G8.0.21. The control system functions are largely classified into current control, synchronization control and thrust calculation of speed control.

Figure G8.0.21. Block diagram of the LSM propulsion control system.

The current control is performed in the loop indicated by the broken lines. The LSM propulsion coil current Im is controlled in accordance with ip . The ip is a sine wave produced by multiplying the peak value of current (Ip) and the unit sine wave. Therefore, ip is expressed by the following equation ip (output current reference) = Ip sin(ω t + δ ). A unit sine wave (sin(ωt + δ)) is given by the synchronization control which produces the propulsion coil current synchronized with the on-board field phase. On the other hand, Ip is the command for the peak value of current to be transmitted from the testing centre, which is calculated by the thrust control to determine the thrust or regenerative braking force required to move the vehicle at a desired speed. Copyright © 1998 IOP Publishing Ltd

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In this control system, for instance, the compensation for electromotive force acting as an external disturbance on the control system (compensation of Em ) is done so that current and synchronization control have an especially quick response to ensure smooth vehicle propulsion. Synchronization control applies the principle of the PLL (phase locked loop) so that a smooth sine-wave pattern can be produced even when the position signal is more or less disturbed. G8.0.5.3 Power feeding control system The propulsion coils on the guideway are grouped into two or three sections as shown in figures G8.0.22 and G8.0.23 which are called a duplicate feeding system and a triplex feeding system respectively. For instance, the section groups of the duplicate power converter feeding system are connected to a pair of power converters via section switchgears. As shown in these figures, the control of section switchgears by the section controller makes it possible for the power converter to supply power only to the section in which the vehicle is passing. When the vehicle runs through two adjoining sections, two (in the case of figure G8.0.22) or three (in the case of figure G8.0.23) power converters feed each section to eliminate thrust fluctuations. This feeding control system makes it possible to efficiently feed the power to the propulsion coils without causing thrust fluctuations as the vehicle runs along the guideway.

Figure G8.0.22. A duplicate feeding system.

Figures G8.0.22 and G8.0.23 also show the operating conditions of the section switchgears and power converters when the vehicle moves from point 1 to point 4. Vehicle position detection for the feeding section switchgear control is performed by the transposed inductive wires. As the vehicle moves on, the section switchgears are controlled so that power is supplied only to the feeding section in which the vehicle is running and to the one into which the vehicle is about to go. The section switchgear opens and closes according to the position of the vehicle. Therefore, the section switchgear makes opening and closing operations at high frequency and accordingly it is required Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.23. A triplex feeding system.

to be highly reliable. The section switchgear is opened after detecting the null current due to the power converter which is controlled so as not to supply the output current. G8.0.5.4 Power converter As is well known, a power converter with large capacity is needed to generate variable-voltage variable-frequency power for driving the LSM. The three typical power converters for driving LSMs shown in figure G8.0.24 can be listed in chronological order of development. (a) Noncirculating-current-type cycloconverter The noncirculating-current-type cycloconverter needs a zero current interval when the direction of output current is changed. This requirement gives the output current a large distortion factor. The reactive power Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.24. Main circuit scheme of a power converter for driving an LSM.

and harmonic current due to the operating principle of the noncirculting-current-type cycloconverter are also generated at the input side, and the output frequency is usually within one third of the input frequency. The insertion of an M/G (motor-generator) between the utility system and the input of the noncirculating-current-type cycloconverter makes it possible to increase the output frequency and to eliminate the above-mentioned undesirable factors such as the reactive power. However, the M/G has the disadvantages of large power loss and complicated maintenance. (b) Circulating-current-type cycloconverter The circulating-current-type cycloconverter has two smoothing d.c. reactors ( Lo ) which connect positive converter CON-P and negative converter CON-N. The neutral points of the smoothing d.c. reactors are connected to the load (propulsion coil) via a feeding cable. The smoothing d.c. reactors contribute to the simultaneous operation of the positive converter and the negative converter. In the case of the noncirculating-current-type cycloconverter, simultaneous operation is impossible and consequently the zero current interval mentioned above is required. The output current is determined by the average voltage of the positive converter and the negative converter. The circulating current arising from the simultaneous operation of the positive converter and the negative converter is also determined by the voltage difference between the positive converter and the negative converter. Therefore, the output current and the circulating current are supplied by controlling the average voltage and the voltage difference respectively. The circulating current can also be utilized for regulating the input reactive power continuously. The cycloconverter generates a lag reactive power Copyright © 1998 IOP Publishing Ltd

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arising from the output current and the circulating current. In addition, at the input side of the cycloconverter, several phase-advanced capacitors, which produce a lead reactive power, are connected to the input terminals through thyristor switches so that the lag reactive power is cancelled. The operation of the thyristor switches for the phase-advanced capacitors depends on the operating condition of the cycloconverter. The circulating current is controlled such that the lead reactive power by the phase-advanced capacitors and the lag reactive power by the cycloconverter cancel each other. Consequently, the input power factor can be kept at unity. (c) PWM inverter The PWM inverter system consists of a rectifier and an inverter which are connected with a d.c. link. The d.c. link, which is not seen in a direct conversion type such as a cycloconverter, performs the function of suppressing the cross influence between the load (LSM) and the utility system. As PWM inverters seem to be the most popular these days, a concrete example of one of them is shown in figure G8.0.25. The main circuit of a PWM inverter is exemplified in figure G8.0.25 which shows the one installed at the Miyazaki test track for the JR Maglev. The PWM inverter for driving the LSM basically consists of multiple inverters with three phases which are connected in series by each output transformer. Although the cascade connection of the transformers shown in figure G8.0.25 serves to realize high output voltage with fewer harmonics, it is impossible for the inverter with an output transformer to supply the necessary power with low frequency to the LSM.

Figure G8.0.25. The main circuit of a PWM inverter.

Consequently, the composition of the main circuit of the inverter needs to be modified for low output frequency. The switches (SWL, SWH) shown in figure G8.0.25 enable the main circuit layout to be changed in accordance with table G8.0.1. While the output frequency is low, one of the unit inverters acts as a half-bridge inverter (HB) which does not need the output transformer, the other unit inverter acting as a full bridge inverter (FB). In the Copyright © 1998 IOP Publishing Ltd

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case of a high-speed circuit (high frequency), the two unit inverters in figure G8.0.25 act as a full bridge inverter. In addition to PWM function, the output controller in the inverter has the function of supplying the projected current to the LSM and of stabilizing the transformer operation. The former function contributes to current control and the latter to output voltage assignment control and asymmetrical magnetization suppression control. Figure G8.0.26 shows a block diagram of the output control. The output current controller supplies output current to the LSM corresponding to the output current reference given from the speed controller. The output voltage assignment controller controls the rate of the output voltage from FB and HB. For instance, the output voltage is wholly assigned to HB when the output frequency is extremely low. On the other hand, two FBs are assigned so that each supplies half the necessary output voltage. The asymmetrical magnetization suppression controller detects the occurrence of asymmetrical magnetization and prevents the iron core of transformers from being asymmetrically magnetized. The asymmetrical magnetization in the output transformer, which may lead to magnetic saturation, arises from a very low frequency or the PWM control.

Figure G8.0.26. A block diagram of the output control.

G8.0.5.5 Propulsion coil The power converters feed the propulsion coils on the guideway, and the fed coils act as magnets. The interaction between the propulsion coils and the on-board superconducting magnets produces a strong propulsive force. Copyright © 1998 IOP Publishing Ltd

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Superconducting magnets pull and push the propulsion coils with strong magnetic force when they move over the coils. So the propulsion coils must be mechanically strong enough to endure these frequent forces. Furthermore, the electromotive force generated in a propulsion coil increases with the speed, and is proportional to the number of trucks of the train, so the propulsion coils used in a high-speed and large-transportation-capacity system are exposed to high voltage. On the other hand, as propulsion coils are installed along the whole track, enormous numbers of coils are required for the Maglev line, so they must be made as cheap as possible. The original propulsion coils in the Miyazaki test track were made of alminium windings and SMC (sheet-moulding compound). SMC is a kind of glass-fibre-reinforced plastic. The coils were installed every 120° electrical angle, composed of a three-phase armature as shown in figure G8.0.27(a). They were a little shorter than the 120° electrical angle, and they were installed on the side walls in a single layer.

Figure G8.0.27. The effect of the double-layered structure.

Later, it was revealed that a large harmonic magnetic field produced by the single-layered propulsion coils caused a considerable temperature rise in the superconducting magnets. This does not matter for a short-run-time operation like the Miyazaki test vehicle, but for a long revenue-service line, the operation will be difficult if the heat generated inside the superconducting magnet is larger than the capacity of the on-board refrigerator. To reduce the harmonic magnetic field considerably, a double-layered structure is adopted in the Yamanashi test line, in which one propulsion coil covers a 180° electrical angle, and coils are installed every 120° pitch along the side wall. The composite magnetic field produced by those double-layered coils is as shown in figure G8.0.27(b). The harmful change in magnetic field is reduced remarkably. So far the propulsion coils at the Miyazaki test track have been made of an SMC mould and the double-layered propulsion coils have been made of an epoxy resin mould, which will be applicable to the Copyright © 1998 IOP Publishing Ltd

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high-voltage coils in future systems. To confirm this new structure, double-layered epoxy-resin propulsion coils were partially introduced in the Miyazaki test track and tested (figure G8.0.28).

Figure G8.0.28. A photograph of the coil installation.

G8.0.6 The basic characteristics of the running of the Maglev The running tests of the superconducting Maglev have been carried out over a long period at the Miyazaki test track. Obtained electromagnetic data agreed well with the calculated data. Some of those data are introduced as follows. The levitation force acting on the vehicle is equal to the gravitational force when the vehicle is running. Accordingly the levitation height is measured and compared with the calculation. Meanwhile the reaction of the levitation force can be measured at the guideway levitation coil. The main values of the coils at the Miyazaki test track are shown in table G8.0.2. The guideway is about 7 km long and its cross-section is U shaped. The normal-flux levitation coils are laid on the horizontal plane, and the null-flux levitation coils are attached to the side wall at a part of the guideway,

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instead of the normal-flux coils. The tested vehicles are MLU001 and MLU002. MLU001 has only one bogie which has eight superconducting coils and is about 10 t in weight. MLU002 had two bogies which had six superconducting coils each and the vehicle was about 17 t in weight. The superconducting magnets are attached on both sides of the bogie vertically. The vertical position of the bogie is detected with four optical sensors on the bogie measuring the distance from the guideway surface, and the lateral position is detected with two optical sensors. From those data the position of the bogie centre is calculated. An example of the current and forces in the normal-flux levitation coil is shown in figure G8.0.29, where eight superconducting coils pass over the levitation coil. The levitation force is positive irrespective of the current direction. The magnetic drag force ranges from a negative value to a positive value, and the mean value is a net drag force. Figure G8.0.30 shows the levitation height versus speed, which represents the distance between the coil centre of the levitation coil and the base side centre of the superconducting coil. The measured levitation height increases with an increase of vehicle speed and agrees well with the calculated value. The levitation height of the rear bogie is influenced by the residual current generated at the front bogie.

Figure G8.0.29. Current and force waveforms in the levitation coil.

Figure G8.0.30. The relation between levitation height and speed.

Figure G8.0.31 shows an example of the balanced displacement of the bogie at the section of the null-flux levitation coils. This displacement is the height between the centres of the superconducting coil and levitation coil. The measured displacement decreases with an increase of vehicle speed, and agrees well with the calculated value. Levitation force versus displacement is shown in figure G8.0.32. As the drag ratio, known from section G8.0.2, is large in the neighbourhood of zero displacement, the acting point is set where the displacement is near zero. Meanwhile the stiffness, which is the slope of the line in the figure, grows large and is not suitable for good ride quality. One of the features of the superconducting levitation system is a large clearance between the vehicle Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.31. The relation between vertical displacement and speed.

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Figure G8.0.32. The relation between vertical displacement and levitation force.

and the guideway. The response of the vehicle MLU001 as it passes over a guideway irregularity of 3 cm vertical humps is shown in figure G8.0.33 (Yoshioka and Miyamoto 1986). The vehicle passed over the humps safely without contact, because the clearance is about 10 cm. G8.0.7 Planning prospects in the future G8.0.7.1 Construction of a new Maglev test line The superconducting Maglev system in Japan has advanced from basic principles to its technical realization on the Miyazaki Maglev test track, but in order to introduce a Maglev system as a commercial transport means, it was considered necessary to construct a new long test line, which enables us to conduct tests in nearly the same conditions as in the revenue service. In 1990, the Japanese Ministry of Transport ( MOT ) approved the ‘Basic Technology Development Plan’ and ‘Yamanashi Test Line Construction Plan’, and authorized the Yamanashi test line as a nationally funded project. The following are the main items to be confirmed in the running tests on the new test line: ( i ) stable running of levitated trains at high speed with safety, comfort and environmental preservation; ( ii ) reliability and durability of various facilities including superconducting magnets and ground coils, and establishment of standards on maintenance; ( iii )structural standards specifying a minimum radius of curvature, the steepest gradient, etc, of the guideway; ( iv )mutual influence acting between two trains when they pass each other on the guideway; ( v ) running performance of a vehicle related to the cross-section of a tunnel and to the pressure fluctuation at the entrance and exit of the tunnel; ( vi )control system of the substation operation for multiple trains. A project team consisting of the Railway Technical Research Institute, the Central Japan Railway Company and the Japanese Railway Construction Public Corporation has now been formed to design and construct the new test line. Figure G8.0.34 shows the planned route of the Yamanashi test line. The new line is 42.8 km long including about 35 km of tunnel sections, partially with double tracks. It also has some sections with a maximum gradient of 4% and with a minimum radius of curvature of 8000 m. Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.33. The motion of a vehicle passing over irregularities.

Although progress has been delayed by about two years because of difficulties in a right-of-way purchase, it has been decided to begin the test run in 1997 on a priority section 18.4 km in length. The first set of trains was delivered to the depot in 1995. The construction of the guideway, the substation and the Test Center were completed in 1996. G8.0.7.2 Configuration of new Maglev trains The configuration of the new Maglev vehicles is shown in figure G8.0.35. There are two sets of trains which will run at a speed of over 500 km h−1 ( maximum speed 550 km h−1 ) on the Yamanashi test line. One of these trains is a three-car unit and the other a four-car unit. The car is of articulated bogie type having each truck with the superconducting magnets at its end away from the passengers’ seats to avoid the influence of magnetic field on passengers and personal effects. The height and cross-sectional area of a car are decreased to reduce the running resistance in the air and the size of tunnels. Two types of nose shape have been designed to decrease the aerodynamic influence (i.e. drag, noise Copyright © 1998 IOP Publishing Ltd

Planning prospects in the future

Figure G8.0.34. The planned route of the Yamanashi test line.

Figure G8.0.35. The configuration of the new Maglev vehicle. Copyright © 1998 IOP Publishing Ltd

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and micro-pressure waves in tunnels) for the lead cars of the train for the new test line. One type is called the ‘double-cusp’ style and the other is the ‘aero-wedge’ style. Brake systems in the Maglev system are classified roughly into electric (regenerative brake and rheostatic brake) and mechanical types (aerodynamic brake, wheel disc brake and landing frictional brake). The regenerative brake is primarily adopted because it can select the degree of deceleration freely. The other brakes are supposed to act in an emergency accident when the regenerative brake fails. G8.0.7.3 Superconducting magnet for a new test car Figure G8.0.36 shows a superconducting magnet to be mounted on the vehicle of the new test line. Each magnet weighs about 1.4 t and they are attached to the truck in one pair at the left and right sides. The levitative load for each magnet increases because of the decrease in the number of bogies of a new car having an articulated bogie system. This new magnet is of the vertical type consisting of a helium tank in which the on-board refrigerator is built, and four superconducting coils. The superconducting coils are in the form of a racetrack made of Nb-Ti alloy wires featuring a 700 kA magnetomotive force, 4.23 T maximum empirical magnetic field and 1350 mm pole pitch. The superconducting magnets are exposed to large external disturbances during running. A great amount of effort has been put into suppressing the increase in heat load by the magnetic fluctuation and keeping the stability of superconductivity under various disturbances. Many improvements on the structure of the superconducting magnets have been tried. The newest magnet has proved that the influence on heat load and stability have been very much reduced (Suzuki et al 1993).

Figure G8.0.36. The superconducting magnet for the new Maglev.

G8.0.7.4 Power supply system In the new test line, a PWM inverter with less restriction on the output frequency is employed for the power conversion system. The introduction of a three-power converter system having three groups of supply systems (figure G8.0.37) is being considered. This system has the advantage that its installation capacity is only three quarters of that of the two-power converter system, and even if one group fails, the Copyright © 1998 IOP Publishing Ltd

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Figure G8.0.37. The power supply system for the new Maglev.

other two groups can take over to continue the run. Control of power and car operation can be accomplished by processing the positional information received via the transposed inductive wire stretched along the guideway. G8.0.7.5 Structure of the guideway The guideway of the Maglev is a facility made of reinforced concrete, in the shape of a U, on which the ground coils are attached. The side wall of a guideway plays the important role of supporting the levitated vehicle during high-speed running and a very precise accuracy in the roughness of the surfaces of this

Figure G8.0.38. Configuration of ground coils for the new Maglev. Copyright © 1998 IOP Publishing Ltd

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wall is required in order to achieve good ride comfort. Three methods of attaching the ground coils on a side wall have been adopted in constructing the guideway for the new Maglev test line, and we intend to estimate the characteristics of precision, such as the deformation over time, the cost of production, etc, upon completion. Ground coils consist of coils in the shape of a racetrack for propulsion and coils in the shape of a figure of eight for levitation and guidance. Most of the propulsion coils in the new test line are arranged in a double layer overlapping each half of the coil to reduce heat load on the superconducting magnet as shown in figure G8.0.38. The levitation coils are attached to the propulsion coils and the opposite coils on both side walls are also connected via null-flux wire to guide the vehicle. References Fujiwara S and Fujimoto T 1989 Characteristics of the combined levitation and guidance system using ground coil on the side wall of the guideway 11th Int. Conf. on Magnetically Levitated Systems and Linear Drives pp 241–4 Ikeda H, Kawaguchi I, Outake T, Tanaka S and Wada J 1988 Power conversion system for Maglev vehicle MLU002 10th Int. Conf. on Magnetically Levitated Systems ( Maglev’88 ) pp 327–36 Murai T and Fujiwara S 1993 Characteristics of linear synchronous motor combined propulsion, levitation and guidance STECH’93 ( Yokohama, 1993 ) pp 58–63 Nakamichi Y, Okui A, Kaga S and Ikeda H 1992 Characteristics of PWM inverter with large capacity for driving LSM Symp. on Power Electronics, Electrical Drives, Advanced Electrical Motors ( Positano, 1992 ) pp 193–8 Nakamura K, Koike S, Tatsumi T, Maki N, Nakamichi Y and Nishi S 1985 LSM propulsion system of the Miyazaki Maglev test track Int. Conf. on Maglev Transport’85 ( Tokyo, 1985 ) pp 91–8 Nakao H, Yamaji M and Kurosawa K 1989 New type superconducting magnet for EDS system 11th Int. Conf. on Magnetically Levitated System Linear Drives pp 229–34 Nakashima H and Herai T 1989 The cryogenic system for magnetic levitation vehicles 11th Int. Conf. on Magnetically Levitated System Linear Drives pp 235–9 Nakashima H, Terai M, Shibata M, Yamaji M and Jizo Y 1993 Superconducting magnet and refrigeration system for Maglev vehicle 13th Int. Conf. on Magnetically Levitated Systems and Linear Drives ( Argonne, IL ) pp 160–4 Powell J R and Danby G R 1966 High-speed transport by magnetically suspended trains ASMEPaper66WA/RR-5.1–11 Suzuki E 1994 Heating phenomena in the superconducting magnet of Maglev caused by electro-magnetic vibration Cryogen. Eng. 29 495–503 (in Japanese) Suzuki E, Tsuchishima H, Herai T and Kishikawa A 1992 Construction and experimental results of the new type superconducting magnet Applied Superconductivity Conference ( Chicago, 1992 ) Suzuki E, Tsuchishima H, Terai M, Takizawa T, Yamaji M and Jizo Y 1993 Superconducting magnet for Maglev train Int. Conf. on Speedup Technology for Railway and Maglev Vehicles ( Yokohama, 1993 ) pp 352–7 Terai M, Mizutani T, Yamane T, Fujimoto Y, Fujimoto S, Suzuki M and Mita H 1993 On-board refrigeration system for Maglev vehicle Int. Conf. on Speedup Technology for Railway and Maglev Vehicles ( Yokohama, 1993 ) pp 364–8 Tsuchishima H and Herai T 1991 Superconducting magnet and on-board refrigeration system on Japanese Maglev vehicle IEEE Trans. Magn. MAG-27 2272–5 Tsuchishima H, Suzuki E and Terai M 1994 Development of new superconducting magnets for Yamanashi test line RTRI Rep. 8 17–22 (in Japanese) Yoshioka H and Miyamoto M 1986 Dynamic characteristics of Maglev vehicle MLU001 Int. Conf. on Maglev and Linear Drives pp 89–94

Further reading Ogiwara H 1986 Applied Superconductivity 1st edn ( Japan: Nikkan Kogyo Shinbun ) ( in Japanese )

Copyright © 1998 IOP Publishing Ltd

G9 Superconducting magnetic bearings

T A Coombs

G9.0.1 Introduction The discovery by Bednorz and Muller in 1986 of the first ceramic cuprate oxide superconductor, a new compound based on lanthanum-barium-copper oxide, and soon after of the cuprate yttrium-barium-copper oxide ( YBa2Cu3O7-x-YBCO ) by Wu et al (1987), working at Houston University, initiated a worldwide research effort into the search for new high-temperature superconductors. Since then other cuprate-based superconductors such as Tl2Ba2Ca2Cu3O10-x ( TBCCO ) and Bi2S2C2Cu3O10-x ( BSCCO ) have been discovered. The most appropriate of these materials for use in magnetic bearings is YBCO. BSCCO, for example, is ideal for making wires and tapes as it can be textured so that current will pass easily from grain to grain in a wire. However, the grains are smaller than in YBCO and its critical current is adversely affected by applied magnetic fields. TBCCO has a much higher critical current in an applied magnetic field than BSCCO but it still suffers from the problem of granularity. YBCO may be grown in large grains and these grains have high critical currents (even at the boiling point of nitrogen). The effective magnetization is a function of both the grain size and the critical current, thus the higher each is the stronger the magnet which can be made. The new high-temperature materials have three advantages over the low-temperature superconductors. The first and most obvious is the operating temperature. Higher temperatures mean lower operating costs, there is less requirement for expensive cryostats making the systems potentially less bulky and nitrogen is a much cheaper cryogen than, for example, helium. The second advantage is control. Although very high-field magnets can be made from low-temperature superconductors the field cannot be changed rapidly as the superconductors will quench. Finally (unlike YBCO) low-temperature superconductors cannot be made in large stable lumps.

G9.0.2 General principles This chapter outlines the magnetic properties of YBCO which are relevant to the design and construction of magnetic bearings. It then describes the effects of dynamic properties on the design of magnetic bearings and examines specific problems such as flux creep, stiffness and vibrations. Copyright © 1998 IOP Publishing Ltd

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G9.0.2.1 Magnetic properties (a) Diamagnetism If a magnet is suspended above a superconductor stable levitation may be achieved. This would not be possible if the superconductor were replaced by a coil in which constant current flowed, or by a soft ferromagnetic material in which the relative permeability µr > 1 or indeed by another permanent magnet (Braunbeck 1939). The superconductor magnet arrangement is stable because the superconductor is diamagnetic, i.e. µr < 1 and the susceptibility is negative. The twin conditions for stable levitation of an object in a static field are that the total force is zero and further that the divergence of the force ( ∇F ) is negative. Further if F is an irrotational field then F = ( x, y, z )−∇ψ ( x, y, z ) where ψ is a potential. From these conditions it can be shown that the force experienced by a dielectric body of volume V in an electrostatic field E0 is given by Fe = 1/2 ( ε − ε 0 )V ∇E 02. Similarly the force experienced by a magnetic body of volume V in an applied magnetic field H0 is given by Fm = 1/2 ( µ − µ 0 )V ∇H 02. For stability the divergence of Fm or Fe must be negative. In static fields the divergence of ∇E02 and ∇H02 cannot be negative and neither can (ε-ε0 ). However, if diamagnetic materials are present ( µr < 1) then ( µ - µ0 ) can be negative and stable levitation can be achieved without active control. Although a material parameter, we can define an effective µ or χ for a given superconducting sample in terms of total moment and external field. Below Hc (type I superconductors) and Hc l (type 2 superconductors) there is perfect diamagnetism i.e. the susceptibility is −1. Above Hc l the susceptibility depends on the amount of flux penetration. Physically the superconductor tries to prevent the magnetic flux from entering the body of the sample and an ‘image’ of the magnet is created in the superconductor (figure G9.0.1). The magnet to ‘ image ’ magnet interaction produces a levitation force. The above argument which has been developed for static fields in fact explains why stable levitation cannot be achieved for nondiamagnetic materials. It does not, however, fully explain levitation by superconductors. If a bowl-shaped superconductor were fabricated then a magnet could be placed in the bowl and stable levitation achieved. If, however, the superconductor were flat then there would be no resistance and therefore no restoring force to lateral motion of the magnet, the magnet could ‘slide off the superconductor. The reason why it does not is that in a type II superconductor where flux has entered the body of the sample the flux is pinned. Motion of the magnet would require motion of the flux (the image magnet in the superconductor would also try to move) and the pins resist that motion and hence provide a restoring force. Also crucial to levitation is the maximum B field available from the image magnet and this is a function of the magnitude of the currents flowing (critical currents) and the length scale over which they flow. Critical currents and flux pinning are examined in more detail in the next section. (b) Critical currents and flux pinning The critical state or Bean model may be used to predict the total magnetic moment in type II superconductors. For example, when applied to a large cylindrical grain of YBCO, the model predicts that the total magnetic moment (m ) divided by the volume, that is the ‘effective’ magnetization Me f f , is a function of Jc and the grain size d.

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Figure G9.0.1. Magnet interacting with ‘image’ magnet in superconductor.

Hence the magnetization is defined as the total moment divided by the volume (M cannot be defined on a local scale for a superconductor). In YBCO there is strong pinning and hence a high Jc. Whilst there is intrinsic pinning due to, for example, superconducting and chemical phase variations, crystal dislocations and grain boundaries, a large contribution is provided by normal inclusions such as precipitates of Y2BaCuO5 (211). In addition YBCO may be grown in large grains and hence there is a large diameter (d) with a correspondingly large magnetization. The relationship given in equation (G9.0.1) was shown experimentally in a joint project between Bell Labs and Cornell University (Chang et al 1992). In this work S Jin of Bell Labs prepared a set of disc-like YBCO specimens with different average grain sizes ranging from 4µm to 410 µm. Measurements of the ‘effective’ magnetization showed a direct correlation with grain size. The same experiment also showed a monotonic increase in the levitation force with grain size. The experiments were carried out using permanent magnets of diameters 1.6, 3.2 and 6.4 mm. The constant of proportionality depends on the geometry and may be calculated by summing the dipole moments of individual current loops, as follows

This expression is derived from the expression for the magnetic dipole moment of a current loop. This is simply the current in the loop ( i ) times the area of the loop ( A ). Equation (G9.0.2) assumes a number of current loops and sums the contribution of each. Assuming a square geometry then this reduces to the following integral

where w is the width of the sample and t is its thickness, or alternatively in circular geometry for a sample Copyright © 1998 IOP Publishing Ltd

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diameter d and thickness t

Given that the ‘effective’ magnetization is the moment divided by the sample volume then both equations (G9.0.3) and (G9.0.4) reduce to the same expression (where d is the diameter of the circular section and the width of the square section) Hence the ‘effective’ magnetization (for the simple geometries given) is a function of both the critical current density and the diameter or width of the grain. This relationship shows the maximum possible magnetization of the sample. Under these conditions there are currents flowing in the whole of the sample. The sample is said to have been fully penetrated. If a superconductor is cooled in the absence of a magnetic flux then when a permanent magnet is brought up to it will repel the magnetic flux (see section G9.0.2.1( a )) and hence provide a levitation force. A superconductor may itself act as if it were a permanent magnet. If it is cooled in the presence of a magnetic field ( >Hc 1 ) then magnetic flux will be trapped inside the body of the superconductor and remain when the external field is removed. Even if there is no initial magnetic field then when an external field (above Hc 1 ) is applied flux will be driven into the sample and become trapped. Superconducting magnets have been reported in which the trapped flux is far in excess of that available from conventional permanent magnets. For example, Liu et al (1995) who use a system of ion bombardment to induce columnar defects which act as pinning centres and thereby trap the flux have reported flux densities observed immediately after activation of 2.6 T at 76.5 K, 7.5 T at 59 K and 8.34 T at 54 K. At 77 K the maximum field which may be trapped in YBCO is of the order of 5 T. This limit is dictated not by the upper critical field ( Hc 2 ) but by the so-called irreversibility line. At applied fields above the irreversibility line flux flow starts occurring, this causes losses and hence ‘electrical resistance’. The sample may still be said to be superconducting on a microscopic scale but not on a macroscopic scale. In addition this article restricts itself to the use of bulk materials. Superconducting coils can be made, but YBCO, although it has very high intra-grain critical currents ( Jc ), has very low inter-grain critical currents (two orders of magnitude less). Thus unless the wire is one single grain it can only carry very small currents and the field available is correspondingly small. Another of the high-temperature superconductors BSCCO also has low intra-grain critical currents but it can be textured which reduces the problem and wires can and are being made of it. However, the irreversibility line for BSCCO is very low at 77 K ( about 40 mT ). The irreversibility line is much higher at lower temperatures and it may prove economic in the future to wind coils from BSCCO and to use them at 30 K (which can be reached with a single-stage refrigeration cycle). (c) Levitation force By considering the total energy stored in the magnetic field it is possible to calculate the force ( F ) between two coaxial cylinders with a field B in the gap between them. In free space the energy is

The magnetic pressure is then determined by considering two semi-infinite cylinders with currents in the same sense and considering the work done to pull them apart. The work done by a force F moving through a distance dL is given by the change in the energy in the air gap ( Hayt 1981 )

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where A is the cross-sectional area of the cylinders. Thus the attractive force between two cylinders with a field B in the gap between the two cylinders is given by

The above expression has been developed for magnets in attraction. The same expression applies for magnets in repulsion. If the current in one cylinder is then reversed the forces have the same magnitude but are in the opposite direction (that is repulsive). For a field of 1 T we can expect to bear loads of up to about 4 × 105 N m−2; for a sample which is 16 mm in diameter 80 N; and for a 35 mm sample 384 N. It is also worth noting that although the above expression is given in terms of the area ( A ) of the bearing, B Jc is the maximum force per unit volume and an important calculation is the depth of penetration (t ) as was given in equations (G9.0.3) and (G9.0.4). In addition if the superconductor is to be used as a permanent magnet then in order to prevent demagnetization the applied fields must be much less than Jc d. G9.0.2.2 Dynamic properties and their effects on bearing design (a) Flux creep Also of importance when considering the use of YBCO as a component in a magnetic bearing is the phenomenon of flux creep ( Riise et al 1992 ). This is especially true if bearings are constructed in which the ‘permanent’ magnet is made of YBCO rather than being a rare-earth magnet. The mean flux density in a sample creeps over time according to the following logarithmic relationship ( Moon and Chang 1994 ).

where β is a constant of proportionality. This relationship means that although the initial rate of flux loss may be quite rapid the rate of decay reduces progressively. Moon and Chang ( 1994 ) report that tests performed by Wienstein et al (1995) at the University of Houston showed a 13% drop in trapped field in YBCO over 1 week which means that it would take a further 19 years for the next 13% to be lost. (b) Stiffness If magnetic bearings are to replace rolling element bearings then one of the major problems which has to be addressed is the stiffness. In general stiffness depends on geometry and is most usefully presented in the form of the force developed by a suitable dimension such as the radius of the superconductor. Ma et al (1992) give the following equation for stiffness for a circular section of superconductor

Typical values are: 2R = 0.6 cm ( grain size ) B = 0.3 T d = 0.5 cm ( thickness ) Jc = 104 A cm12. These values give an overall stiffness of 0.9 N mm−1 which is sufficient for some applications but is a factor of 105 less than that required for, for example, a thrust bearing in a gas turbine. Whilst these figures are pessimistic, even a doubling of the grain size, the flux density and the thickness, coupled Copyright © 1998 IOP Publishing Ltd

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with an improvement in Jc to 105 A cm−2, would still leave the stiffness a factor of 1000 down on the requirement. (c) Hysteresis Figure G9.0.2 shows a typical force-displacement curve for a circular magnet suspended above a superconductor which has been cooled in the absence of magnetic flux. There are two types of hysteresis loop shown in the figure. The so-called major hysteresis loop is developed when a superconductor is cooled in the absence of magnetic field. It is brought up to the magnet (from a long distance away) and then withdrawn. If at any stage of a major hysteresis loop the direction of travel is reversed for a small distance then a minor hysteresis loop develops.

Figure G9.0.2. A typical force-displacement curve showing both major and minor hysteresis loops.

This is potentially a problem if the bearing is to be subjected to cyclic loading as the force developed at a certain bearing gap will be dependent on the bearing history. The effects can be mitigated by restricting the magnitude of the cyclic loading so that only minor hysteresis loops are followed. Minor hysteresis loops are repeatable but problems are encountered if the loading pattern exceeds a minor hysteresis loop as is shown in the next section. (d) Dynamic loading and vibrations Experiments have shown that vibrations can cause the bearing gap to decay. This decay can be either positive or negative depending on the history of the bearing. This effect has been observed by various groups which include Nemoshkalenko et al (1990), Terentiev and Kuznetsov (1990, 1992), Hikihara and Moon (1994, 1995) and Coombs and Campbell (1996) working in the IRC in Superconductivity at Cambridge. Nemoshkalenko noted in 1990 that the amplitude of response to induced vibrations was frequency dependent. Terentiev and Kuznetsov (1992) obtained very similar results to those obtained at the IRC and noted the existence of a threshold input below which gap decay did not occur and also, crucially, that the threshold was frequency dependent. The results obtained are entirely consistent with the physical model presented below. This effect is over and above that which might be attributed to thermally activated flux creep. In the absence of vibrations although gap decay does occur the characteristics are different chiefly in a much slower rate of gap decay. Sample traces are shown in figure G9.0.3 which shows the variation of bearing gap with time. The decay is actually a function of the number of cycles applied and hence only indirectly a function of time. Although trace (b) where there are no vibrations shows some decay the trace is clearly different in both magnitude and shape from trace (a) which was taken with applied vibrations. The diagram also shows that the decay can be either positive or negative, i.e. lead to the bearing collapsing or to the gap increasing as is shown in trace (c). The effect of vibrations is to induce a cyclic loading on the bearing. Under the influence of cyclic loading a bearing may collapse as is shown in figure G9.0.3(a). The physical mechanism is a guided random Copyright © 1998 IOP Publishing Ltd

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Figure G9.0.3. Decay profiles.

walk mechanism in which repeated pitching of the bearing causes the magnet to advance irreversibly towards the superconductor. This is well described by Brandt (1988) who has shown the existence of a range of stable levitation positions at a single given force for the configuration described here. The effect of vibrations is to advance through these positions and alter the bearing gap. In the force-displacement curve of figure G9.0.2, the superconductor is initially levitated at a position corresponding to point A. When load is increased the superconductor moves to point B. When the load is again relaxed, however, the superconductor does not return to point A but instead moves to point C which is at a somewhat smaller gap than represented by point A. If the load were then increased back to that corresponding to point B then the superconductor would simply oscillate between points B and C and the bearing gap would not decrease any further. However, the effect of vibrations is to induce oscillations in the system and the amplitude of that response is dictated by the dynamic magnifier (which is a function of the stimulating frequency over the resonant frequency and the damping). If that response lies outside the force-displacement envelope (the major hysteresis loop) then on the second loop the bearing will move to some point past C and the bearing gap decrease. However, in moving up the curve the stiffness is increased so that (if the driving frequency Copyright © 1998 IOP Publishing Ltd

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is initially greater than the resonant frequency) the resonant frequency can move closer to the driving frequency. This will increase the amplitude of the response to a greater value than before so that again it exceeds the minor hysteresis loop and the gap will decrease further. This continues until the vibrations are at an amplitude (dictated by the dynamic magnifier) which is entirely within the force-displacement envelope. If such a condition does not occur then the equilibrium point will continue to move and the bearing will collapse altogether as occurs in figure G9.0.3(a). (e) Vibration model A superconducting bearing may be represented by a spring damper system in which under small oscillations there is very high Q ( low damping ) ( figure G9.0.4 ). Within a minor hysteresis loop the system behaviour may be describe by the following simple second-order differential equation.

where m is the mass of the system which leads to a force proportional to the acceleration of the system. λ is the coefficient of viscous damping which leads to a force which is proportional to the velocity of the system. Finally s is the spring constant which leads to a force proportional to the displacement of the system.

Figure G9.0.4. The spring damper model.

This system has a natural frequency. When the bearing is vibrated the rate at which energy will be input into the system will be a function of the ratio of the frequency of the vibration relative to the natural frequency of the system. The dynamic magnification of the output relative to the input is given by the following equation (where Θi is the input and Θ0 is the output)

Thus the magnitude of the response depends on c (the damping factor). If c is zero then when ω = ωn the response is infinite. In general when c is small the dynamic magnification increases as ω increases from zero, slowly at first but then very rapidly as ω approaches the natural frequency. Beyond the natural frequency the response initially falls off rapidly and then tends to zero as ω tends to infinity. Vibrations will build up to a steady-state value dictated by the dynamic magnifier. During this time the sample will move, since its equilibrium position (the stable levitation height) depends on the maximum travel, and the bearing gap reduces. Once the steady-state value has been reached, then the gap will not change any more. A detailed description of this effect is given by Brandt (1988) who models the behaviour inside a minor hysteresis loop using the simple spring damper model given above. Both c and ω in equation (G9.0.12) may be calculated from measurements of the amplitude and frequency of response under forced vibrations in an experiment such as that performed by Takeda et al (1994) where the superconductor is supported on Copyright © 1998 IOP Publishing Ltd

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an elastic beam, or alternatively from the response to a step input when under active control by a method such as that described in section G9.0.3.2. In order to determine whether the system will advance we simply plot the steady-state response (as given by equation (G9.0.12)) on the same graph as the superconductor-magnet force-displacement envelope. If the steady-state response at a particular equilibrium position lies inside the force-displacement envelope then that is a stable equilibrium point and the bearing gap will not change. If, however, the steady-state response lies outside the envelope then the bearing gap will change, and continue to change, until the steady-state response is once more inside the force-displacement envelope or the bearing has collapsed. Using this method it is also possible to predict that under certain circumstances the bearing gap will actually increase. Figure G9.0.5 illustrates the method.

Figure G9.0.5. Steady-state response and force-displacement graphs.

In figure G9.0.5 the initial equilibrium point is shown dotted. The weight supported ( mg ) is shown by a chain line and the steady-state response curves are shown in bold. In the first case the bearing will simply collapse because everywhere the steady-state curves lie outside the force-displacement envelope. In the second case the gap will actually increase; the lower steady-state curve goes outside the envelope but here the gradient of the envelope is positive. In this case, therefore, the additional energy of vibration is dissipated in increasing the gap. G9.0.3 Bearing configurations This section examines the different configurations which may be used for magnetic bearings. Jayawant (1988) summarized all known possible methods of magnetic suspension or ‘levitation’: ( i ) suspension with controlled electromagnetics Copyright © 1998 IOP Publishing Ltd

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( ii ) suspension or levitation with permanent magnets or ferromagnetic materials ( iii ) levitation with superconductors ( iv ) levitation by repulsion forces produced by eddy currents induced in conducting materials ( v ) levitation with diamagnetic materials ( vi ) levitation with forces acting on current conductors situated in magnetic fields ( vii ) suspension by a tuned LCR circuit and electrostatic force of attraction (between two plates) ( viii )suspension by a tuned LCR circuit and magnetic force of attraction between an electromagnet and a ferromagnetic body ( ix ) mixed µ system of levitation. From this list the first three are by far the most significant in terms of construction of practical magnetic bearings. Suspension by eddy currents is also of interest (especially as the magnitude of trapped fields available from superconductors increases) as it is not limited by the saturation flux density of iron. Although superconductors behave as diamagnets, a separate section has been allowed for the materials such as bismuth and graphite which are themselves diamagnetic. However, these materials are so weakly diamagnetic that only small pieces can be levitated. The mixed µ system of levitation given as class ( ix ) above refers to the experiments performed by Homer et al (1977) where objects were stably suspended in the fields provided by superconducting coils and which were the forerunners of the experiments now being done on superconducting bearings. Figure G9.0.6 shows examples of the arrangements that were used. The rest of this section will confine itself to magnetic bearings which make use of superconductors. Given below is a list of the currently known configurations which use superconductors. The list is a generic list and all the arrangements given could be used in either a journal or a thrust bearing. G9.0.3.1 Passive devices In principle a superconducting bearing may be constructed using a ring or disc permanent magnet together with a superconductor. In practice this has proved to have three significant drawbacks: ( i ) magnetic flux density B: the B field is a function of the magnet strength and thus the very high fields potentially available from superconductors are not used ( ii ) gap creep: the gap in the bearing is a function of the history of the bearing, thus the gap may vary due to flux creep, large excursions from the steady-state loads and vibration ( iii )stiffness: the stiffness of the bearing is very low when compared with that of a rolling element bearing. This may actually be an advantage in certain situations. It will, for example, allow a shaft to rotate about its centre of gravity rather than the centre of the bearing. It is not an advantage when the bearing has to cope with large transitory loads whilst maintaining a nearly constant gap. Various configurations have been proposed to overcome one or all of these problems. A list of the known current designs follows. (a) ‘Monopole’ magnet to superconductor This is the most basic of the possible arrangements ( figure G9.0.7 ). Typically it would consist of a section or sections of bulk superconductor on the stator and a permanent magnet, polarized perpendicular to its face on the rotor. The superconductor is cooled away from the permanent magnet that is in zero field. When the magnet is brought up to the superconductor a repulsive force is developed. A typical force-displacement curve was given in figure G9.0.2. The characteristics are •

simplest possible arrangement, consisting of a ring or disc magnet polarized perpendicular to its face and mounted on the rotor: the superconductor is mounted on the stator

•

works in repulsion

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Figure G9.0.6. Mixed µ levitation: (a) iron disc suspended near a superconducting sphere; (b) iron disc suspended inside a superconducting magnetic flux screen; (c) iron body suspended between two constant-flux coils.

• limited to the B field provided by the magnet.

(b) Superconductor to superconductor One of the drawbacks of the magnet-superconductor arrangement is that it is limited by the B field of the magnet. Bulk samples of YBCO have the potential to trap large fields far in excess of those available from rare-earth magnets and above the saturation flux density of iron (about 1.8-2 T). This limitation may Copyright © 1998 IOP Publishing Ltd

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Figure G9.0.7. A ‘monopole’ superconductor bearing.

magnet

to

Figure G9.0.8. A superconductor-superconductor bearing.

be overcome by placing a superconductor on both the stator and the rotor and using each as ‘permanent’ magnets (G9.0.8). The characteristics of this configuration are: • • •

in this arrangement there is a superconductor on both the stator and the rotor; both may be magnetized and it may work in attraction or repulsion a requirement to cool the rotor as well as the stator field changes in one superconductor may cause changes in the trapped field of the other superconductor (and vice versa).

(c) Eddy current An alternative configuration would be to employ a superconducting magnet in conjunction with a copper disc (figure G9.0.9). When the disc is rotated above the superconducting magnet eddy currents are induced in the copper disc and a levitation force is generated. Little work has been carried out on this type of bearing as (in the absence of a B field greater than that available from conventional magnets) it has no clear advantages over an electromagnetic bearing.

Figure G9.0.9. An eddy current bearing.

The characteristics of this arrangement are: •

superconductor is acting as a magnet and is mounted on the stator; it interacts with a conducting disc mounted on the rotor and eddy currents are induced in the disc to induce an ‘image magnet’ in the disc

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• works in repulsion • limited only by the B field which can be trapped in the superconductor • only works when spinning.

(d) Superconductor to iron Another method of utilizing the very high fields available from superconductors would be to use them as one would in a true permanent magnet bearing and incorporate them into a magnetic circuit by putting iron on the rotor (G9.0.10). The maximum force available would be a function of the B field available from the superconductor and the saturation flux density of the iron. Although there may be some self-stabilizing due to the superconductor it is unlikely that this would be a stand-alone bearing and would therefore have to be used in conjunction with another bearing or bearings.

Figure G9.0.10. A superconductor-iron bearing.

The characteristics of this arrangement are: • • •

the superconductor acts as a magnet; it interacts with the iron rotor to produce an attractive force may require active control to work on its own or it could use the Coombs bearing method or could work in conjunction with another controlled bearing the available force is a function of the saturation flux density of iron and the B field trapped in the super conductor.

(e) Multipole magnet to superconductor Another problem with the simple arrangement given at the beginning of this section is the low stiffness. Stiffness can be increased by reducing the size of the pole pieces. Hence by using multipole arrangements the stiffness can be increased (figure G9.0.11). The characteristics c this arrangement are: • • • •

the same principle as the monopole arrangement except that the stiffness is increased by using a com pound magnet which has multiple opposed poles: this uses the principle that the stiffness will be a function of the inverse of the area of the pole pieces it has a higher stiffness than the monopole arrangement it works in repulsion it is limited to the B field provided by the magnets.

(f) Hybrid superconducting magnetic bearing (HSMB) In all the arrangements given above the superconductor is providing an axial levitation force. In addition it also provides a lateral restoring force which prevents excursions. In order to prevent gap drift and loss of levitation force due to flux creep in the superconductor the arrangement given in this section uses Copyright © 1998 IOP Publishing Ltd

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Figure G9.0.11. A multipole magnet-superconductor bearing.

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Figure G9.0.12. Hybrid superconducting magnetic bearing (HSMB).

magnets to provide the levitation force and the superconductor is present only to provide lateral stability against excursions (McMichael et al 1992). McMichael et al’s arrangement is shown schematically in figure G9.0.12. The characteristics of this arrangement are: • • • •

it has two opposed permanent magnets to provide the levitation force: a section of superconductor placed between the magnets provides stability it works in repulsion it is limited to the B fields provided by the magnets the bearing gap should no longer be dependent on the bearing history but will just be a function of the magnet field strength and the load applied.

G9.0.3.2 Active designs Since a superconducting bearing of the types given in section G9.0.3.1 has inherently low stiffness it is possible to produce designs of bearings in which the stiffness is provided by an active control system. There are three known methods for incorporating active control. The simplest method is to use a hybrid design in which there are effectively two bearings. One, the superconducting bearing, provides the steady-state levitation force; the other, an electromagnetic bearing, provides the stiffness. Another method is to build an electromagnetic bearing in which either bulk superconductors replace permanent magnets or superconducting wire is used for the coils (BSCCO see section G9.0.2.1(b)). The final method uses changes in the geometry to react to changes in loads. The electromagnetic bearing and the variable geometry bearing are described in more detail below. (a) Electromagnetic bearing This uses superconductors to support steady-state loads and a conventional or superconducting coil for active control. Its characteristics are: •

various designs already exist which use permanent magnets to bear the steady-state loads; conceivably the magnets could be replaced by superconducting magnets

•

reduced steady state power consumption: a major problem with electromagnetic bearings using coils is the ‘over capacity’ required due to the highly inductive nature of the coils. The rate at which the transient loads are applied as well as their actual magnitude is a major consideration in the design of the power amplifiers. In applications where there are rapidly changing and large (in comparison to the steady state) transient loads these loads dominate in the design of the bearing.

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(b) Variable geometry bearing (the Coombs bearing) In order to provide active control a method must be devised whereby active elements may be included in the system which will respond to changes in load. In an electromagnetic bearing active control is achieved by changing the current in the coils and hence changing the magnetic flux density. In bulk materials where the magnetic field is produced by persistent currents trapped in the superconductor this option is not available. However, the available force is a function of the flux density and the area over which it is applied. Thus even if the flux density remains constant then changes in available force may be made by changing the active area of the bearing, i.e. increasing the area of overlap between the magnetic components (see figure G9.0.13).

Figure G9.0.13. The basic principle of the variable geometry bearing.

Using this method active control may be achieved using bulk materials and the inherently low stiffness of the materials is no longer a problem. The characteristics of this method are as follows: •

•

This design utilizes the superconductors to balance both the steady-state loads and the dynamic loads. It does this by changing the geometry of the bearing in response to changing loads. It operates essentially as a magnetic lever. Actuation forces are applied perpendicular to the loading axis. This has the advantage of reducing the actuation forces required and hence the associated power electronics (see above). It is essentially working as a hybrid electromagnetic bearing although the actuation does not necessarily have to be produced by electromagnets. The major disadvantage of this configuration is that it has moving parts.

Any of the configurations given under section G9.0.3.1 (passive devices) may be incorporated into a variable geometry (Coombs) bearing.

G9.0.3.3 Examples of superconducting bearings There are many groups worldwide who are working on the design and construction of superconducting magnetic bearings. Given below are some examples of their work. (a) ISTEC In 1991, the Japanese succeeded in developing a superconducting magnetic bearing which could support a 2.4 kg rotor rotating at 30000 rev min−1 (Fukuyama et al 1991). The superconducting material used was melt-powder-melt-grown (MPMG) YBaCuO. Two sections of superconductor were used which were machined into rings of dimensions φ 40 × φ 65 × T 20 by grinding. The operating temperature of the superconductor was 77 K and the coolant used was liquid nitrogen which was constantly circulated over the superconductor. Figure G9.0.14 below shows the design of the superconducting magnetic bearing (Fukuyama et al 1991, Takaichi et al 1992). Two superconducting rings are used and these each act as a combined journal and thrust bearing. The principle used is repulsion and the two thrust bearings act in opposition, thus one Copyright © 1998 IOP Publishing Ltd

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Figure G9.0.14. A superconducting bearing test rig.

bearing provides a restorative force vertically upwards and the other vertically downwards. The journal bearings provide axial stability. The superconductors are field cooled, that is, they are cooled in the presence of a magnetic flux. The magnetic flux is provided by a conventional permanent magnet (Nd-Fe-B) with a surface flux density of 0.45 T. Incorporated in the design are top and bottom centring devices. These are provided so that the shaft is in the correct position when the superconductors are cooled. The bearing is actuated by a high-frequency motor. (b) Houston In 1992 McMichael proposed the HSMB (McMichael et al 1992). In this arrangement the superconductor is used as the filling in a permanent magnet sandwich in order to provide stability. In his paper McMichael made the following points. ( i ) Superconducting magnetic bearings in which the superconductor is cooled in the absence of magnetic field yield limited levitation and relatively low magnetic stiffness. In addition force creep occurs over long periods of time. ( ii )Superconducting magnetic bearings in which the superconductor is cooled in the presence of magnetic field (i.e. in situ) show negligible static levitation forces. They do, however, have much higher magnetic stiffness and, so long as the superconductor is not displaced from its original position, negligible force creep. Copyright © 1998 IOP Publishing Ltd

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Figure G9.0.15. The HSMB prototype.

McMichael claims that the HSMB allows for much greater stiffness and maintains a much higher static load lifting capacity than simple magnet-superconductor arrangements. In order to demonstrate this he constructed the arrangement shown in figure G9.0.15 in which the HSMB is present in the form of both a journal bearing and a thrust bearing. G9.0.4 Applications G9.0.4.1 Passive devices This chapter has highlighted the advantages and disadvantages of the various forms of superconducting magnetic bearings. The disadvantages are flux creep, relatively low stiffness and the effect of vibrations on the bearing gap (Coombs and Campbell 1995, 1996). However, both flux creep and the effect of vibrations may be accounted for in the bearing design and the advantages presented by a very low-friction device which requires no active control and little Copyright © 1998 IOP Publishing Ltd

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maintenance are considerable. Consequently there is considerable interest in finding applications for these devices. One such application is that of flywheel energy storage (Bornemann et al 1994, Chen et al 1994, Hull et al 1994). In this application energy is stored in the rotation of a flywheel. Hull et al (1994) provide the following calculation for the required coefficient of friction µ in order to make such a device economically viable

where dE/dt is the rate of energy loss, E is the stored energy, g is the acceleration due to gravity and υ is the velocity (for his calculation Hull assumes that all of the mass is concentrated at the rim of the flywheel). He assumes that dE/dt/E must be less than 0.1% an hour and that υ the rim velocity is 1000 m s−1. This gives µ < 1.4 × 10−5. He then assumes that at 77 K the refrigeration plant is operating at 30% of the theoretical maximum and that therefore 14 units of mechanical energy are required to remove one unit of heat at 77 K. Thus µ must be reduced by a factor of 14 giving an overall µ of < 1.0 × 10− 6. In tests using a 0.32 kg rotor in a vacuum Hull et al (1994) claim a µ of 3 × 10−7. Chen et al (1994) who use a bearing of the second type (i.e. superconductors provided for stability) do not provide figures for µ but have calculated energy loss per hour. With a 19 kg rotor spinning down from 740 rev min−1 the energy loss was found to be 13% per hour and at 2000 rev min-1 per hour the energy loss is 5% per hour. Both of these figures are somewhat greater than Hull et al ’s requirement of 0.1% per hour although 2000 rev min−1 translates into a rim velocity of only 5 m s−1 (as opposed to Hull et al ’s 2000 m s−1). Neither system approaches the energy storage capacity of 5 MW h−1 which is likely to be required (Hull et al 1994). However, this continues to be a major potential application of passive superconducting bearings. There are many other applications where magnetic bearings could be used (Moon and Chang 1994). These are: • gyroscopes • high-speed machine tools • angular momentum wheels for spacecraft • rotary scanners for optical and infrared devices • centrifuges, micromachine bearings • cryocooler turbines • cryopumps, aircraft engine bearings • gas turbines for power generation • high-speed spindles for textile manufacturing • computer disk storage devices.

A report prepared by John Hull at Argonne (Hull 1992) highlights the impact high-temperature superconductors may have on Maglev transportation. He lists levitation magnets, magnetic field shielding of passenger compartments, superconducting magnetic energy storage (SMES) and flywheels for energy storage. In general the applications will be limited to those in which there is a high added value such as a gas turbine. There are considerable advantages to be gained from the replacement of conventional rolling element bearings in a gas turbine by magnetic bearings. These are principally in terms of efficiency from higher operation speeds, smaller blade gaps (with active control) and (potentially) weight savings by removing the need for the oil cooling system. However, due to the low stiffness and the vulnerability to cyclic loading a passive bearing is unlikely to be satisfactory. Copyright © 1998 IOP Publishing Ltd

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G9.0.4.2 Active devices While a passive device is limited by stiffness and levitation drift an active device would at least in theory have the same range of applications as an equivalent electromagnetic bearing. Two major problems which have restricted the use of electromagnetic bearings are size and weight in comparison with the size and weight of a rolling element bearing. The size of an electromagnetic bearing is dictated by the loads which it has to carry and the flux densities which can be achieved in the magnetic materials used in both the stator and the rotor. However, since magnetic materials are used the flux densities are limited to < 2 T. A superconducting bearing which either used superconductors on both the stator and the rotor or employed the eddy current principle outlined earlier would not be restricted by this limit but would instead be restricted by the fields which can be developed in the superconductors. This has already been demonstrated to be far in excess of 2 T (Liu et al 1995). Since the magnetic pressure and hence the force developed increases with the square of the field, superconducting bearings may be constructed in which the volume of superconductor required is much smaller than the volume of the magnetic components in an electromagnetic bearing. However, this needs to be offset against the requirement for cooling of the superconducting bearing before a direct comparison may be made. There is a requirement for the replacement of rolling element bearings, in applications such as turbines, where the thrust-to-weight ratio may be increased by increasing the turbine operating speeds and by reducing blade tip clearances. In addition the incorporation of noncontact magnetic bearings would remove the need for an oil cooling system (needed for rolling element bearings) and the associated and costly bearing compartment seals. It is possible that one of the first uses for bulk superconducting materials such as YBCO will be in a magnetic bearing in just such an application. The choice of configuration, given in section G9.0.3, will be highly application specific. For example it is worth noting that the superconductor does not require iron to develop the B fields. Thus, if, for example, a design is arrived at which uses permanent magnets in conjunction with superconductors the field will be limited by the B field of the magnets. It will therefore use a similar field to that of a conventional electromagnetic bearing. However, only the active areas of the superconducting and the conventional magnetic bearings will be the same. The volume and weight of the conventional bearing will be dictated by the iron and the coils; that of the superconducting bearing will be driven by the size of the cryostat required.

References Bednorz J G and Müller K A 1986 Temperature and magnetic field dependence of the critical current on polycrystalline Ba2 - yCu3Oy Z. Phys. 64 189 Bornemann H J, Ritter T, Urban C, Zaitsev O, Weber K and and Rietschel H 1994 Low friction in a flywheel system with passive supercondcuting magnetic bearings Appl. Supercond. 2 439–47 Brandt E H 1988 Friction in levitated superconductors Appl. Phys. Lett. 53 1554–6 Braunbeck W 1939 Free suspension of objects by electric and magnetic fields Z Phys. 112 753-63 Chang P Z, Chiu M, Moon F C, Jin S and Tiefel T H 1992 Grain size dependence of levitation forces in bulk YBa2Cu3O7 superconductors Cornell University Report Mechanical and Aerospace Engineering Chen Q Y, Xia Z, Ma K B, McMichael C K, Lamb M, Cooley R S, Fowler P C and Chu W K 1994 Hybrid high-Tc superconducting magnetic bearings for flywheel energy storage systems Appl. Supercond. 2 457–64 Coombs T A and Campbell A M 1995 An active superconducting magnetic bearing Eur. Conf. on Applied Superconductivity Proc. ( Institute of Physics Conference Series ) vol 148 (Bristol: Institute of Physics) pp 671–4 Coombs T A and Campbell A M 1996 Gap decay in superconducting magnetic bearings under the influence of vibrations Physica C 256 298–302 Coombs T A, Campbell A M and Cardwell D A 1995 Development of an active superconducting bearing IEEE Trans. Appl. Supercond. AS-5 630–3

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Fukuyama H, Seki K, Takizawa T, Aihara S, Murakami M, Takaichi H and Tanaka S 1991 Superconducting magnetic bearings using MPMG-YBaCuO Proc. 4th Int. Symp. on Superconductivity ( Tokyo, 1991 ) (Tokyo: Springer) pp 1093–6 Hayt W H Jr 1981 Engineering Electromagnetics (New York: McGraw-Hill) pp 297–8 Hikihara T and Moon F C 1994 Chaotic levitated motion of a magnet supported by a superconductor Phys. Lett. 191A 279–84 Hikihara T and Moon F C 1995 Levitation drift of a magnet supported by a high-Tc superconductor under vibration Physica C 250 121–7 Homer G J, Randle T C, Walters C R, Wilson M N and Bevir M K 1977 J. Physique 10 879–86 Hull J R 1992 Potential impact of high temperature superconductors on Maglev transportation US DoE Report Hull J R, Mulcahy T M, Uherka K L, Erck R A and Abboud R G 1994 Flywheel energy storage using superconducting magnetic bearings Appl. Supercond. 2 449–55 Jayawant B V 1988 Electromagnetic suspension and levitation techniques Proc. R. Soc. A 416 245–320 Liu J U, Weinstein R, Ren Y, Sawh R P, Foster C and Obot V 1995 Very high field quasi permanent HTS magnet with low creep 1995 Int. Workshop on Superconductivity ( Maui, 1995 ) (New York: ISTEC and MRS) Ma K B, McMichael C K and Chu W K 1992 Applications of high temperature superconductors in hybrid magnetic bearings Proc. 1992 TCSUH Workshop on HTS Materials, Bulk Processing and Bulk Applications (Singapore: World Scientific) pp 425–9 McMichael C K, Ma K B, Lamb M A, Lin M W, Chow L, Meng R L, Hor P H and Chu W K 1992 Practical adaptation in bulk superconducting magnetic bearing applications Appl. Phys. Lett. 60 1893–5 Moon F C and Chang P Z 1994 Superconducting Levitation (New York: Wiley) Nemoshkalenko V V, Brandt E H, Kordyuk A A and Nikitin B G 1990 Dynamics of a permanent magnet levitating above a high-Tc superconductor Physica C 170 481–5 Riise A B, Johansen T H, Bratsberg H and Yang Z J 1992 Logarithmic relaxation in the levitation force in a magnet-high Tc superconductor system Appl. Phys. Lett. 60 2294–6 Takaichi H, Murakami M, Kondoh A, Koshizuka N, Tanaka S, Fukuyama H, Seki K, Tkaizawa T and Aihara S 1992 The application of bulk YBCO for a practical superconducting magnetic bearing Proc. 3rd Int. Symp. on Magnetic Bearings (Technomic Publishing) pp 307–16 Takeda N, Uesaka M and Miya K 1994 Computation and experiments on the static and dynamic effects of high-Tc superconducting levitation Cryogenics 34 745–52 Terent’ev A N and Kuznetsov A A 1990 Rotation of a levitated YBCO superconductor in a low-frequency magnetic field Superconductivity 3 1951–6 Terent’ev A N and Kuznetsov A A 1992 Drift of a levitated superconductor induced by both a variable magnetic field and a vibration Physica C 195 41–6 Wu M K, Ashburn J R, Torng C J, Hu P H, Meng R L, Gao L, Huang Z J, Wang Y Q and Chu C W 1987 Superconductivity at 93 K in a new mixed phase YBaCuO compound system at ambient pressure Phys. Rev. Lett. 58 908–10

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G10 Magnetic shielding

Franco Pavese

Introduction (Electro)magnetic shielding is a pervasive technique in the era of electricity and electronics, more widespread than is usually acknowledged. Compliance with internationally agreed electromagnetic compatibility codes is an integral part of all electronic and of much electrical apparatus in commerce. Compatibility means that apparatus must comply with given limits for electromagnetic radiation, so that the generated interference is below the limits (in amplitude and/or frequency range) stated by the agreed codes. Most domestic appliances, like devices containing electrical motors, magnetic actuators and fluorescent lights, fall in this category. The same applies to radio and television sets, to computers and computer monitors and to telephones. Power lines and stations may be a source of intense electromagnetic noise; radar and broadcasting stations obviously are. One can better feel the extent and importance of electromagnetic interference from the steep increase in the preference given to optical methods for signal transmissions (e.g. fibreoptics for telephone cables). Shielding is therefore necessary in cases where a device must be prevented from radiating (electro)magnetic energy and where it must be protected from picking up (electro)magnetic energy radiated by other devices. In this respect, however, the type and extent of the necessary shielding does not depend much on the former cases, but, due to limitations in the effectiveness of the materials used for shielding, on whether the interfering fields are weak (≈100 times the earth’s magnetic field or less) or strong, or on whether the shielding is intended to reduce a field to within the value allowed by the international codes (e.g. within acceptable health and safety values) or to stabilize a constant local field, or to make the best possible ‘magnetic vacuum’ in an enclosure. After a short introduction to traditional methods, this chapter deals with the use of superconducting materials-both low- and high-Tc -as a means for improved shielding solutions. G10.0.1 Shielding concepts G10.0.1.1 Magnetostatics and attenuation factors for different geometry types Let us consider a perfectly conducting spherical shell of inner radius r and outer radius R, immersed in a static uniform magnetic field. Maxwell’s equations are

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where B is the magnetic flux density vector, H is the magnetic field vector and J is the electrical current density vector. In the spherical homogeneous media of permeability µ = constant, with B = µH , and in the absence of external currents generating magnetic fields ( J = 0, ∇ × H = 0), and expressing the magnetic field in terms of a scalar potential ϕ , H = −∇ϕ , equations (G10.0.1) and (G10.0.2) reduce to the Laplace equation

By imposing the continuity conditions at the spherical shell interfaces, one can deduce the attenuation ratio

where µr is the relative permeability and r and R stand for the inner and outer radius, respectively. Equation (G 10.0.4) results in no shielding ( A = 1) when µr ≡ 1 and in a perfect shielding ( A → ∞ ) for µ → ∞ (infinite permeability) and µ → 0 (perfect diamagnetism). In the first case, equation (G10.0.4) can be simplified (Mager 1970) (see also table G10.0.1)

for thickness t = ( R − r ) « R Similarly, for a cylindrical shell with radii r and R and of infinite length in a transverse field, equation (G 10.0.4) reduces to

For a cubic shell, A⊥ ≈ 0.7µr t/a where a is the cube side length (table G10.0.1). In general, A|| = 4NA⊥ + 1, where N is the demagnetization factor. For a sphere, N = 1/3 . For a cylinder of infinite length in a parallel static field A|| = 1 (N = 0, no shielding). For a finite-length cylinder with both ends closed, A|| .c l . = A||/( 1 + R/L). The demagnetization factor N depends on p = L/2R (L is the cylinder length): N = [1/( p 2 − 1)]{ p/( p 2 − 1)1/2 ln[ p + ( p 2 − 1 )1/2 − 1}. The formula also applies, to a good approximation, for rectangular closed boxes. A disc can be considered as a very short solid cylinder of length (thickness) t « R, the radius. Unlike the former cases, it is an ‘open’ geometry: alternatively, it could be considered as a finite portion of a superconducting layer of thickness t in cylindrical coordinates. This case is important in shielding as it describes a single ‘tile’ of a shield. The closed-form calculations are rather complicated; in general the solution is found by numerical methods (see Frankel (1979) for low-Tc superconductors, Däumling and Larbalestier (1989) for high-Tc superconductors and Mohamed et al (1993) for the granularity effects of the latter types). Basically, at the low fields of interest in shielding, there are three contributions which must be taken into account: ( i ) the dominant intergmin shielding currents flowing through the weak links between grains ( ii ) the infra - grain currents flowing at the surface of the single diamagnetic grains ( iii ) the demagnetization effects, which are dominant near the disc edge. The evidence of the influence of the infra-grain self-field component shows at the disc edges and centre. If the granularity is not taken into account, as is normally the case for commercial low-Tc superconductors Copyright © 1998 IOP Publishing Ltd

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Figure G10.0.1. Field shielding by a disc: (a) computed field lines (a half-disc is represented) ( Jones et al 1992); (b) computed Hz and Hr components: the full and broken lines are for R/L = 10 and 103 (Däumling and Larbalestier 1989); (c) measured field profile above an Nb—Ti disc ≈46 mm in diameter; curves a, b and c are for B < Bl i m : for these cases the current partial penetration in the disc is represented in the lower part. The applied field is that for r 30 mm (Frankel 1979); (d) field calculated taking only the intergrain contribution into account and measured field including the intragrain component ( Mohamed et al 1993 ).

like Nb—Ti and Nb3Sn, the shielding field on the disc axis and the field shape across the centre should have the form of a cusp (figures G10.0.1(a) and (b)): the actual measurements show no cusp and dH/dr = 0. In addition, calculations suggest a field enhancement at the disc border, due to a demagnetizing field (negative H|| in Figure G10.0.1(b): the actual measurements do not show such an enhancement except in (granular) high-Tc superconductors (figures G10.0.1(c) and (d)). The axial shielding field H|| at the disc centre is, for JC( B ) ≠ constant, essentially proportional to Jc( B )t — independent of the disc radius R ( Figure G10.0.1( b)); if JC( B ) = constant it would actually diverge. In addition, a radial field component Hr ∝ Jc( B )t/2 is present, unlike the infinite-length case where it is absent. Therefore the shielding-field vector is not parallel to the driving field except on the disc axis: it becomes fully perpendicular to it (i.e. parallel to the disc surface) at the disc border (Figure G10.0.2(a)). Figure G10.0.2(b) shows the field penetration at the end of a finite-length cylinder. When the shielding factor increases, the cost would rise to the third power, following the increase of the surface and of the thickness. There are, however, some advantages in using multiple shells. In the case of double shells, the shielding factor in transverse fields is ( µr » 1 )

where n = 3 for spheres and n = 2 for cylinders (table G10.0.1). All the previous formulae apply if and only if the material is not saturated. The saturation condiions will be discussed in section G10.0.1.3. In addition, for open-ended finite-length shields, e.g. cylinders, the formulae apply only for applied fields extending beyond the shield length by no more than one radius. If, as is generally the case in technical applications, open-ended shields are used, the field penetration through the openings must be taken into account. For a thin cylindrical shell of inner radius r and R ≈ r and of finite length L and open at both ends in a transverse field, B i n t ⊥ is no longer uniform, but varies as e−k (z /r ). With κ⊥ = 3.832 (first zero of the Bessel function J1 ) the shielding factor reaches its maximum value at the Copyright © 1998 IOP Publishing Ltd

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Figure G10.0.1. (Continued)

centre, Ao p ⊥ = 2/3 e−k ( L/2r ). For L » 2r, typically L/2r » 10, A approaches equation (G10.0.6). For a parallel external field, κ|| = 2.405 (first zero of the Bessel function J0 ) and Ao p|| = 2.6( L/2r )1/2e −k||( L/2r ). The effective shielding factor is in both cases given by 1/Ae f f = 1/Ao p + 1/Ac y l . In the case of semi-infinite cylinders, the internal field at distance z from the opening is given by: H⊥ /He x t ∝ e−3.83z /r and H||/He x t ∝ e−1.84z /r. G10.0.1.2 Shielding factors in alternating fields Unlike in all other magnetic applications, eddy currents—which develop in all conducting materials—are beneficial to shielding. Only when the wall thickness is smaller than the radiation penetration depth δ, the shielding factor nearly retains its static value and this happens only at the higher frequencies. The skin depth δ is calculated as follows

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Figure G10.0.2. (a) Field lines over a disc; (b) field lines at a limit-length cylinder aperture.

where ρ is the electrical resistivity; consequently the lower the frequency f the weaker is the shielding by thickness. Considering µr » 1, t/R « 1, t/δ ≥ 3 and R/µr δ « 1 the following approximations hold

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The effect of an open end in the shield is the same as for static fields for the longitudinal field Be x t || while it is different for the transverse field Be x t ⊥ (Mager 1970). The effective shielding factor must now take the possible phase shift between the two field components into account: 1/Ae f f = (1/A2op + 1/A2 + (2/Ao p A)cos φ )1/2. G10.0.1.3 Shielding with high-permeability materials Traditionally shielding technology has always been based on high-permeability materials. The search for new soft magnetic materials with higher and higher µr values has always been very active. The main problems with these materials are the dependence of permeability on field, frequency and temperature, and its stability in time depending on thermal cycles and on stress. Figure G10.0.3 shows examples of such dependences.

Figure G10.0.3. Typical behaviour of high-permeability materials: (a) two examples of shielding factor of a cylindrical chamber versus applied magnetic field; (b) frequency dependence of µr (top and right axes); the field dependence of µr for the same material is also shown (bottom and left axes); (c) temperature dependence of the shielded field for a material expecially designed to have a Curie point just above room temperature; in normal cases the temperature dependence is much less marked but still nonvanishing V (Vacuumschmeltze 1989).

Table G 10.0.2 shows values of the shielding factors which can be obtained with some one-piece totally closed geometries. Saturation of the magnetic material occurs above a certain field level. A careful check against saturation is required, since the permeability value falls rapidly in the saturation region. Sheets of high permeability concentrate the flux, so that the actual trapped flux density becomes B ≈ 2(R/t)Be x t . G10.0.2 Shielding with diamagnetic materials As shown in equation (G10.0.4), a high shielding efficiency can be obtained with µr values approaching zero, i.e. with good diamagnetic materials. Until the discovery of superconductivity, only a few materials were known to be slightly diamagnetic, so they were of little help. In contrast, the most fundamental property characterizing superconductors is, in principle, perfect diamagnetism.

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This means that, with the material in the superconducting state (T < Tc , H < Hc ), the magnetic flux is totally expelled from its interior. This phenomenon is called the ‘Meissner effect’. Its existence establishes equilibrium thermodynamics for the superconducting phase transition and imposes infinite electrical conductivity: magnetic flux expulsion is achieved by means of electric shielding currents on the surface, which must flow without dissipation. Furthermore, from infinite electrical conductivity σ, it follows that E = J/σ = 0, and therefore from Faraday’s law

we have that B = constant in time. Perfect diamagnetism is expressed by the magnetization M being equal and opposed to the flux density M = −σ0 H. Actually, flux expulsion does not strictly involve the whole material volume, but flux exhibits a small ‘penetration depth’ (37-50 nm in lead), not relevant in shielding technology except at very high frequencies, when the skin effect limits penetration depth to an extent comparable to the penetration depth, ≈10 GHz in the London theory (Hamilton 1970). G10.0.2.1 Type I superconductors The transition from the Meissner state to the normal state is abrupt and takes place at Hc only for the so-called type I superconductors. For these the magnetization versus applied magnetic field increases linearly and is reversible (Figure G10.0.4(a)). Many pure metals like indium, tin and lead belong to this category. A perfect Meissner state, however, is observed only for small dimensions of the superconductor, perpendicular to the impinging magnetic field. In all other cases, the surface has a demagnetization effect N0. Therefore the superconducting material is actually subjected to the field H = He x t / ( 1— N ), and consequently the maximum applicable field for the pure Meissner regime is reduced accordingly to Hc /(1 — N ). For Hc > H > Hc /(1 — N ) the flux density is enhanced in the normal regions and a magnetic- domain structure is generated, called the ‘intermediate state’. In the normal domains the magnetic flux density is µ0 Hc ; they are surrounded by superconducting material, where the shielding currents circulate around the normal domains, also called the ‘pinning sites’. They exist even in pure materials, because grain boundaries, crystal defects and strained regions can act as pinning sites. Calculations on simple superconducting shields with an aperture (Bondarenko et al 1974) are relevant to shielding techniques. Defining a shielding factor κs as the ratio of the applied field to the inner field on the axis at a distance a from an aperture of width 2a, computations were performed for the following geometries, leading to the indicated results: a plane with an aperture and field directed parallel to the surface and transverse to the aperture, κs = 20; a half-space with a groove infinitely deep, κs = 32; a long cylinder with a closed end and with outer and inner radii a and b and field directed along the cylinder axis: κs = 2000; the same with the field transverse to the cylinder axis, κs = 40; a thin sphere of radius

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Figure G10.0.4. The different behaviours of the magnetic field B/µ0 and of magnetization M in a slab versus the external applied field: (a) perfect Meissner state—M increases proportionally with He x t , up to a critical value Hc , where it goes to zero and the internal field B/µ0 jumps from zero to He x t ; (b) type II superconductors with weak pinning—initially and up to a lower critical value Hc 1 < Ht h as case (a), then field gradually penetrates, and M decreases, in the slab until full penetration where B/µ0 increases proportionally with He x t (and M = 0) which occurs at an upper critical value HC 2 > Ht h ; (c) type II superconductors with strong pinning (Bean case)—M gradually increases in the slab up to a value Hl i m where the full slab thickness is crossed, then the internal field starts increasing proportionally with He x t and M remains constant.

a with an aperture υ0 and field directed parallel to the aperture axis: at the centre, parallel component, υ0 = 40°, ks = 100, υ0 = 25°, ks = 104, υ0 = 18°, ks = 106, υ0 = 10°, ks = 108. Another calculation relevant to shielding is that for a ‘porous plate’, i.e. a plane containing an array of holes of radius r and superconducting length (corresponding to the plate thickness) t. Assuming a hole density δ per surface unit, the computation (Matsuba et al 1992) leads to the following shielding factor at the elevation z 0 from the plate for the field component parallel to the plate

G10.0.2.2 Type II superconductors This category is characterized by the fact that the perfect Meissner state exists only up to a field Hc 1 « Hc , called the lower critical field. Then a ‘mixed state’ is formed, characterized by an increasing penetration of the flux into the material up to an upper critical field HC 2 which is higher than the thermodynamic critical field Hc . Figure G10.0.4(b) shows the reversible magnetization versus the applied magnetic field for a type II superconductor with weak pinning. In the presence of a demagnetization factor N0 , the mixed state is found in the field range Hc 2 > H > Hc 1 /( 1 — N ). The flux penetration in the mixed state can be described in the simplest way by the critical-state model (Bean 1962). According to this model, when the density of the supercurrents—equivalent to the eddy currents in normal conducting materials—locally exceeds the critical value Jc , the magnetic flux penetrates the material turned to normal (flux-flow) state. In a steady state, only three local current conditions are then possible: I = 0, Jc , − Jc . From the viewpoint of flux, a gradient establishes itself in the material because of the pinning centres, exerting a force which balances the driving field force B × J , where J is related to H by equation (G10.0.2). The current density may or may not depend on the field. The critical-state equation

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is the force balance equation which determines the distribution of flux, current and magnetization in the material. The form of the solution depends on the geometry of the sample and on the local distribution of Jc ( B ). For a slab in the z − y plane and an applied field in the z direction, dB/dx = ±µ0 Jc( B ); the magnetization gradually increases—linearly for Jc = constant—and then remains constant above the full penetration field Hl i m , as shown in Figure G10.0.4(c). As regards shielding, the field value where the flux crosses the full slab thickness is of interest, as this represents the limiting field shielded by the slab. In table G10.0.1 the value for this limiting magnetization (or flux density B = − M) is given for a number of geometry types and Jc constant. In high-Tc superconductors, the approximation Jc = constant holds with sufficient accuracy only for a small sample thickness, since in general Jc depends on B. The actual deviation from linearity of the flux density inside the superconductor depends on the functional relationship Jc ( B ), which has been found to be different depending on samples and on materials. Using the most common relationship, one obtains (Itoh 1993)

where α ( T ) is the pinning force per unit volume for vortices at temperature T. However, the deviation of equation (G10.0.15) from the values of Bl i m that can be obtained considering Jc = constant does not exceed ≈20% for t ≤ 1 mm (Figure G10.0.5(a)). It is therefore useful to compare results of different authors by defining a ‘shielding capacity’ as Bl i m /t. The shielding factor is reported not to be frequency dependent at least up to frequencies > 107 Hz. Evidence has been reported that a minimum thickness is necessary in a.c. fields, but it is possible that this was an experimental artefact.

Figure G10.0.5. (a) Nonlinearity of the maximum shielded field B versus material thickness t in YBCO according to the behaviour of Jc ( B ) given by Bean’s (Bean 1962) and Kim’s (Kim et al 1962) model (Itoh 1993). The difference between the two models (deviation from linearity) is less than 20% for t ≤ 1 mm. (b) An increase of Bl i m caused by superimposing ferromagnetic shields on an YBCO shield (Ohyama et al 1993).

There are only contradictory indications—most unpublished—about the capability of unconnected superconducting grains (e.g. pressed, unsintered powder) to exhibit shielding: of course, one can take advantage of intra-grain shielding currents only. Several authors report fully negative results, while others report some extent of shielding (e.g. see Eberhardt et al (1988), Johnson et al (1989)). Calculations on arrays of diamagnetic powders lead to shielding factors of ≈2. ‘Superconducting pastes’ have been reported to be effective, but very limited information is available about their preparation. Copyright © 1998 IOP Publishing Ltd

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The simplest way to increase the efficiency of a shield is to increase the superconducting layer thickness. This can be obtained by superimposing several plates. It has been shown that the value Bl i m ,t o t is nearly the sum of Bl i m of each plate, for combinations up to three cylinders of high-Tc superconductors, overcoming the saturation effect of equation (G10.0.15). Superconducting multilayers made of these materials for shielding applications are still under development, while some are already available made of low-Tc materials, such as Nb-Ti/Cu sheets. It has also been shown ( Itoh et al 1996, Ohyama et al 1993 ) that superconducting ceramics can effectively be coupled with high-permeability materials, which easily show much higher Bl i m values, but exhibit a much lower efficiency in shielding small fields. By placing a single low-cost soft-iron cylinder around a ceramic superconducting cylinder, one can easily increase Bl i m to well above 10 mT, and exceed 0.1 T with a double iron shield, while the internal field steadily remains below the sensitivity limit of 10− 8 T ( Figure G10.0.5(b), Ohyama et al 1993). In another test, a semi-infinite superconducting cylinder was externally shielded with a triple iron shield and the open edge was also protected with iron shields of limited length; a transverse field of ≈10−5 T was attenuated to below 10−11 T, starting from a one-diameter length inside the superconducting cylinder. Attenuation factors steadily better than 106 have consistently been observed for single superconducting layers of modest thickness (<1 mm). G10.0.2.3 Shielding with low-Tc superconducting materials The use of superconducting shields using materials which need to be cooled at liquid-helium temperatures is mostly confined to the laboratory scale and is applied to the fabrication of an ‘electromagnetic vacuum’ in enclosures of limited size. Type I superconductors are used for this purpose. Figure G10.0.6(a) shows the Hc-Tc limit curves for several elements. Tinned cans have often been used as the simplest way to make a shield. Another quite common solution is the use of lead cans: a shielding factor of 1-2 × 102 can easily be obtained.

Figure G10.0.6. (a) The superconducting range for several elements at low temperatures (limit Hc - Tc curves shown). (b) The penetration of a 4 × 10−5 T field into different low-Tc materials due to a temperature increase.

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Therefore, having cancelled the earth’s magnetic field, e.g. via Helmholtz coils, one can obtain <10−10 T in a volume of the order of a few litres, equivalent to a few thousand trapped flux quanta (Φ0 = h/2e = 2.068 × 10−15 Wb). This is sufficient for measurements using magnetometers of the flux-gate type for example, whose sensitivity is of the same order of magnitude, but not for the best measurements using SQUIDs, which are much more sensitive (<10−14 T Hz −1/2 ). Recent measurements made with a SQUID magnetometer on low-Tc materials gave a shielding factor A = 1.3 × 106 for Nb and Nb-Ti and A = 5 × 104−5 × 106 for Nb3Sn depending on the metallurgical treatment. A measurable temperature coefficient was observed at 4.2 K, ranging from <1 to ≈50 × 1010 T K−1, depending on the material. Figure G10.0.6(b) shows the penetration of a 4 × 10−5 T field in different low-Tc materials. Type II materials are also used in different shapes. Of course one can use the bulk material, but more economic solutions are available. One method consists of cutting rings from a sheet of high-purity aluminium sputtered with an Nb—Ti single layer 10 µm thick or with a multilayer made by alternating Nb—Ti and Cu, each layer being <1 µm thick. These discs, piled up to obtain the desired total length of a hollow cylinder (typical R/r = 3) and interspersed with identical aluminium discs, are placed inside a copper cylinder coated with an Nb—Ti film. Environment noise of ≈10−7 T Hz−1/2 has been reduced to <10−11 T Hz−1/2 below 200 Hz with only a few frequency components being somewhat higher. Another method uses the superconductor as a wire (Nb-Ti or Nb3Sn) wound in a coil, which is short-circuited on itself. In this case, the quality of the shield critically depends on the resistance of the short-circuit, i.e. on the soldering method. In fact, the higher this resistance is the faster any shielding current will decay. The disc-type shield does not suffer from this problem. Shields made of type II superconductors are effective up to high magnetic fields: ≈0.8 T for the stacked-ring type, ≈3 T for the coil type, which is useful to shield human beings from high magnetic fields produced with magnets working at liquid-helium temperatures; in these cases the residual fields do not need to be much less than the millitesla level. Such an application prompts another possible use of these shields as permanent magnets (see chapter G6 on magnetic separation). G10.0.2.4 Shielding with high-Tc superconducting materials The use of low-Tc superconducting materials provides extremely effective shielding up to high flux densities of the interfering field ( »0.1 T ), but they require liquid helium for refrigeration. Therefore their practical use is extremely restricted, in general, to sophisticated but cost-insensitive applications such the military or space industry. The use of highTc superconducting ceramics combines the same effectiveness with much more easily implemented refrigeration means. However, the peculiar characteristics of these new materials require specific technologies in order to exploit their potential; in some areas these technologies are still under development. The materials generally used are YBCO ( Tc ≈ 92 K ) with or without silver addition—as a powder in sintering or, more effectively, as AgO before reaction—BiSCCO and Bi(Pb)SCCO (Tc ≈ 110 K). The main problems with the material that must be taken into account are: limited Jc , granular nature, brittleness, low thermal expansion coefficient, uniformity and cost. The technology of magnetic shields differs from that used for the production of materials for other applications (electromechanical, levitation) in that:

( i ) Jc cannot be improved by means of methods which take advantage of the material anisotropy if one aims to shield stray fields of random orientation ( ii ) thickness is a fundamental parameter, so that too low values cannot be considered, but too large values simply increase the cost without increasing the effectiveness accordingly, as shown in Figure G10.0.5, and consequently the use of neither thin films nor bulk material should be considered, though the latter has often been used on a ‘better-than-nothing’ basis and for demonstration purposes

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( iii )the integrity of the whole surface is essential, as cracks and pinholes would allow the stray field to leak or to percolate into the shield—as indicated by equation (G10.0.13)—a requirement which is extremely stringent for shields in large-scale applications.

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When considering all these problems together, many of the technological means that are useful for other applications become irrelevant. Table G10.0.3 summarizes the pros and cons of most available technologies. Some of the techniques are aimed at increasing Jc , which in the polycrystalline bulk suffers from the fact that only Josephson links connect the grains of a connected (sintered) material; therefore only the inter-grain Jc plays a role for the Jc ,b u l k and no advantage generally derives from the much higher Jc . g r a i n . In addition the materials are—so far—strongly anisotropic and Jc , b u l k = min( Jc ,υ ) for crystals randomly oriented. Moreover, pinning is not strong enough in common materials to enhance Jc . In shielding, the increase of Jc does not increase the shielding factor but it may be considered desirable because it increases the maximum shielded flux density Bl i m , according to Bean’s theory (see table G10.0.1). With high-Tc superconductors, it must also be taken into account that Jc increases considerably for decreasing temperatures; therefore, in principle, one could take advantage of the higher Bl i m , but at 4.2 K these materials cannot compete with the traditional superconductors from the technological point of view. Of course, one does not need to reach such low temperatures: there is already a gain of a factor of 2–4 in lowering the working temperature from 77.1 K down to the limit of liquid nitrogen ( ≈64 K ), and more in refrigerating down to the limit of the inexpensive one-stage closed-cycle refrigerators (≈50–55 K ) (Figure G10.0.7(a)): Jc(4.2 K)/Jc(78 K) values between 6 and 15 have been measured.

Figure G10.0.7. (a) Typical Jc ( T ) functions for YBCO and BiSCCO. The slope of the curves can change from sample to sample, (b) The most used Jc ( B ) functions ( Jc 0 = 100 A cm−2 ). Jc ( H, T ) = Jc ( T )/[ H/H0( T )]β ( Xu et al 1990 ); H0 = 1. Bean (β = 0) and Kim (β = 1) cases. Jc ( H, T ) = K ( T )/Hα (We and Yamafuji 1967): a—K ( T ) = 10−3, α = 2.6; b—K ( T ) = 10, α = 1.0. Jc ( H, T ) = Jc ( T ) e−H/H0 (Fietz et al 1964, Ravi Kumar and Chaddah 1989). H0 = 1. The shaded area for low Be x t values is intended to warn that Jc ( B ) should not be computed from magnetization measurements below ≈Bl i m .

Most of the methods for increasing the value of Jc are based on the improvement of the grain connectivity along the most favourable crystal orientation. Therefore, the resulting material is strongly anisotropic. This is not a disadvantage for other applications such as wires and tapes; even in levitation, an application that requires Bl i m to be maximized as it is closely related to the trapped permanent field, the bulk sample can be prepared so as to maximize the force in the aimed direction. In shielding, the stray field direction is only seldom fixed and known, so that the material could be aligned with the shielding Copyright © 1998 IOP Publishing Ltd

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Figure G10.0.8. Field penetration in high-Tc superconductors: a comparison of YBCO and BiSCCO types (Miller et al 1993).

supercurrents flowing along the direction of Jc . m a x . In all other cases, isotropic behaviour is necessary and the material must be improved with this condition in mind. High density of the material, larger grains, better pinning and better grain connections are to be pursued. The characteristic Jc( B ) is less important in shielding, provided that the fall is not so strong as to affect Bl i m itself. Typical Jc( B ) functions used in critical-state calculations are reported in Figure G10.0.7(b). With BiSCCO, a higher value of Bl i m is more important, since the shielding factor gradually decreases approaching Bl i m in contrast, with YBCO it stays high up to values much closer to Bl i m (Figure G10.0.8, Miller et al 1993). Therefore, for high shielding factors with BiSCCO one must operate well below Bl i m , losing, at least partially, the advantage of the higher Jc value, with respect to YBCO, that can generally be obtained with this material. Using the bulk material to make plates or one-piece cylindrical tubes or cups with essentially the same techniques as used for traditional ceramic results in objects which are extremely heavy, costly and difficult to cool down without creating cracks which will affect their shielding properties. With this technique applied to small cylinders, limiting fields, Bl i m , up to 20 mT at 77 K (triple-cylinder configuration) have been measured (Itoh 1993) and shielding capacities up to 15 mT mm−1 at 77 K and to 200 mT mm−1 at 4.2 K obtained. Otherwise, the ceramic is deposited or laminated as a thick film on a substrate. A ceramic substrate is therefore the choice that best matches thermal expansion in the whole temperature range— particularly during the high-temperature treatments. A good thermal expansion match is important in many respects: thicker films can be deposited, increasing Bl i m ; pieces can be re-heated up to sintering temperature without cracks for repair in the not-unlikely event that defects develop with time and cool-downs; epitaxial growth is possible, which can improve the overall quality of the crystalline structure. However, cheaper ceramics like alumina or magnesia poison the superconducting ceramic, and resorting to safer ceramics like yttria-stabilized zirconia substantially raises the production costs. In addition, cool-down and refrigeration of all-ceramic parts presents several difficulties in large-scale applications, but not in most small applications, such as those using SQUIDs. Therefore, shielding techniques for large applications have been developed making use of metal substrates. The ‘natural’ metal substrate for YBCO and BiSCCO is silver, but due to the large mismatch in thermal expansion an upper limit for film thickness is <100 µm. This results, however, in Bl i m ≈ 1−3 × 10−4 T at 77 K (2−6 × 10−4 T at 64 K and 20 × 10− 4 T at 4.2 K ), which is sufficient for most purposes (see section G10.0.5). Nickel, and even better stainless steel—which, however, requires a buffer layer against Copyright © 1998 IOP Publishing Ltd

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diffusion of contaminants—allows a better matching and consequently thicker films, up to the limit (few tenths of a millimetre) where a multilayer technique would be more efficient. Shielding capacities up to 11 mT mm−1 at 77 K and 30 mT mm−1 at 4.2 K have been reported (Bykov et al 1996, Hemmes et al 1993, Pavese et al 1992, 1996, Shimbo et al 1993, Vanolo et al 1994). A better configuration to prevent cracks produced by tensile strain is metal cladding. With BiSCCO only silver cladding can be used; with YBCO, nickel or nickel alloys (Gololobov et al 1993, Kozlowski et al 1993) have been used instead. Of course the usual powder-in-tube technique for tape configuration is not suitable for shielding. The equivalent planar technique of lamination has to be used. Multilayers, produced either by multiple lamination (Opie 1993) or by multiple spraying ( Vanolo et al 1994 ), are the natural follow-up technique, but oxygenation problems of the underlying YBCO layers must be solved. Finally, an important parameter for any industrial fabrication is the stability of the shield properties with time. In this respect, and for very large shields only, BiSCCO behaves generally better than YBCO. Not many specific data are available, but degradation of YBCO with time has been observed: it is thought that the reason is exposure to moisture and that protection with a suitable organic or inorganic layer of the surface would prevent this effect. G10.0.2.5 Approaching zero magnetic field The aim of shielding is twofold: ( i ) to limit the (electro)magnetic field within values tolerable to equipment or human beings ( ii ) to suppress (electro)magnetic energy as much as possible—possibly totally. The latter is the aim in experiments of physics and physiology. The minimum attainable field—the ‘electromagnetic vacuum’—is dominated by quantum effects and, ultimately, by Heisenberg’s uncertainty principle. The size of the flux quantum Φ0 = h/2e = 2.068 × 10−7 Wb is well above the detection threshold of SQUIDs. Let us assume we have a perfect shield (A → ∞). If the shield is made of a material of infinite permeability, the residual magnetization Mr of the material should also be zero, or the material should be brought to Mr = 0 with a demagnetization coil carrying a decaying a.c. current and swept over the entire inner surface of the shield. Since nowhere in space is B ≡ 0, the shield will trap in either case some field B0 before it can be sealed off. The problem of eliminating or limiting B0 is an extremely difficult one. In addition, for A → ∞ only, the finite permeability of high-µr materials decreases with frequency and shows a temperature-coefficient characteristic that prompts an increase of the internal magnetization with time. In superconducting shields, supercurrents can develop, generated by thermal electromotive forces (emfs) due to temperature gradients in the shield during cool-down; in addition, there is a ‘magnetic noise’ due to the motion of vortices, not fully prevented because of the limited pinning (see section G10.0.5). The attainment of the lowest fields has been attempted by using superconducting shields. Two techniques have proved to be effective in reducing the trapped field: inflatable bladders (Cabrera 1987) and rotating shields (Vant-Hull and Mercereau (1963) cited by Hamilton (1970)). The first technique starts from a volume Vi n ≈ 0, where the number of flux quanta is much reduced, and increases to a final volume Vf i n > 20Vi n without breaking the shield; this is obtained by using a ductile material, such as lead, below its superconducting transition. The expansion step can be repeated by using concentric bladders: B0 < 10−11 T has been measured. The second technique takes advantage of the asymmetry of ambient stray fields. When the cylindrical shield is rapidly spun during its cool-down through the superconducting transition, an alternating field generates eddy currents in the still resistive material, counterbalancing the stray field so that the shield becomes superconducting in a much lower field and consequently many fewer flux lines are pinned in the shield. As a result, the B0⊥ component is small, while B0|| is the same as for the ambient. Another idea (Hamilton 1970) for reducing trapped field that has been suggested is the so-called

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‘heat flushing’ technique: if the region of shield surface near the superconducting transition is very small (very sharp gradient, as at the liquid-vapour interface of helium), because of flux quantization, it will be impossible for the material to trap any flux. If the shield is then slowly moved into the refrigerant, its entire surface will undergo this process and very few flux quanta will be pinned in. G10.0.3 Shielding by compensation: Helmholtz coils As pointed out in section G10.0.2.5, in order to obtain a low residual field in a superconducting enclosure, the shield should be cooled down in a low ambient magnetic field. It can be put in an enclosure made of high-permeability materials (see section G10.0.5), or, more simply, it can be placed in the centre of a set of compensating coils, called Helmholtz coils. These coils may be of several geometries. The basic geometry consists of a single pair of coils, round or squared. The equations pertaining to the case of round coils of radius R and spacing (distance between coils) d = R are the following

where . At the coil centre Hz 0 = 0.715I/r ’ , where I is the current in the coil. Round coils compensate exactly a field H and its first three derivatives at the central axial point only when the ratio S = d/R = 1 exactly; square coils of side L do the same when S = d/L = 0.5445. Most interesting is the study of the off-axis field map produced by one pair of coils or of more pairs placed orthogonal to each other, in order to obtain an estimate of the compensation accuracy that can be obtained within a given tolerance. Figure G10.0.9 shows two of these maps (Cacak and Craig 1969). It is particularly important to extract the ‘usable volume’ from them, i.e. the volume where the compensation is effective within a given tolerance. Considering that the earth’s magnetic field is of the order of several hundredths of a millitesla and that low-frequency noise in towns is of the order of a few times 10−7 T at best, a full compensation would require a tolerance of a few times 0.1%. Figure G10.0.10 shows that such a degree of compensation is achieved only in a very small fraction of the coil radius or side. More comfortable volumes are provided, of course, by tolerances at the 1% or 5% level. They also better tolerate off-value coil spacing, such as those caused by uncertainties in the evaluation of the winding thickness. G10.0.4 Measuring the shielding factor Only a brief account will be given here, since a book by Boll and Overhott (1989) provides an excellent reference for sensors and the article by Bridges (1968) can be consulted for the measurement techniques of the attenuation ratio in shielded rooms. Table G10.0.4 shows the most common types of magnetometer used for different ranges of the field value. The SQUID sensors (Barone 1992) work only at low temperature, so that the use of the probe must include a cryostat, if the measuring temperature is different from the sensor working temperature (≈4 K or ≈77 K). Models of the other types are available that work from room temperature down to ≈4.2 K. A problem one must be aware of is that the signal-to-noise ratio may limit the sensitivity to values higher than those indicated in table G10.0.4. For example, a flux-gate magnetometer cannot detect field variations better than ≈10−8 T when placed in the earth’s magnetic field ( 0.3-0.5 × 10− 4 T ); a SQUID magnetometer saturates when placed in an excessively large magnetic field. Therefore, the more sensitive the magnetometer used to detect a residual field is, the lower the maximum residual field must be. This means in practice that at present, in order to test the lower limit of the residual Copyright © 1998 IOP Publishing Ltd

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Figure G10.0.9. Field maps for different sets of Helmoltz coils (Cakak and Craig 1969): (a) a single round pair with spacing S = d/R = 0.6; (b) three mutually perpendicular round pairs with spacing S1 = S2 = S3 = d/R = 1.0 (the rightmost crosses indicate the position of one coil of the first pair; the uppermost and lowermost crosses the position of one coil of the second pair; the third pair lies above and below the plane of the paper).

Figure G10.0.10. The maximum usable volume for different tolerances in compensation (Cakak and Craig 1969): (a) a single round pair; (b) a single square pair; (c) three mutually perpendicular round pairs. Copyright © 1998 IOP Publishing Ltd

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field, the experimental set-up must be shielded with traditional high-permeability materials to have a zero-field cooling of the superconducting shield in a sufficiently low trapped field (see section G10.0.2.5 about trapped field problems and section G10.0.5 about the residual field in real superconducting shields). G10.0.5 Shielded enclosures and rooms for vanishing fields: present and future At very high frequencies the main difficulties with large volumes arise in general not from the shielding material itself, but from the very short wavelength of the electromagnetic radiation which makes it easier for the field to leak into the enclosure through the many nonconducting openings (lead feed-throughs, air-conditioning ducts, doors, etc) that a complex shielding structure necessarily needs in order to communicate with the external world. At low frequencies (< 100 Hz), shielded enclosures using high-permeability and conducting materials suffer severe degradation. The intrinsic shielding factors of simple geometry types have been given in

Figure G10.0.11. The shielding factor A for some complex shielded rooms (the interior is approximately cubic with edge length ≈2.5 m). Copyright © 1998 IOP Publishing Ltd

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table G10.0.2: they are generally less than 1000. For the double hollow cylinder, a finite length-to-diameter ratio quickly leads to much reduced values. Complex structures, like shielded rooms, lead to even lower values, as shown in Figure G10.0.11 for five chambers of comparable size: in d.c. fields, a double-layer room has A < 100. Higher values can be obtained with a larger number of layers, but, apart from the cost which is already extremely high for a double-layer room, in situ degaussing becomes impossible for the intermediate layer(s) (Cohen 1970, Kelhä 1981). Degaussing is necessary to eliminate the residual magnetization of the magnetic layers after fabrication, and must be performed in situ with a coil which is swept over the entire surface; the zero so obtained (in general not much better than 10−7 T) is retained only if the temperature at which degaussing has been performed is kept constant within a few tenths of a degree Celsius and, of course, until the material is subjected again to some strong stray field. With either low-Tc or high-Tc superconducting materials shields of comparably simple geometry offer much higher shielding factors at the low-medium frequencies (at high and very high frequencies the advantage of their use is less marked). If the shield dimensions are limited and the shield is used for cryogenic equipment, the use of low-Tc materials is quite straightforward, though generally limited to laboratory use. The types of material described in section G10.0.2.3 can be used: values of A >104 in d.c. fields and >105 in a.c. fields above 3 Hz are reported. Fully shielded rooms using these materials have probably been built in the past for cost-insensitive applications, such as for the military or space, but no data are readily available. In the last few years, high-Tc shields have become an affordable solution. This is the least demanding application, as regards Jc, for the new materials (Pavese 1995), as discussed in section G10.0.2.4. Demonstration shields have already been fabricated using bulk materials. Bi(Pb)SCCO has been used almost exclusively, essentially as a follow-up of its more widespread development for other applications which makes its production technology more advanced with respect to that of YBCO, though the latter would probably be a better choice from the point of view of the material shielding properties. Several demonstration shields have already been tested, most (but not all: e.g. Plechàcek et al 1994) using Japanese technology and mostly made by large companies. All those shields were made in one piece and the characteristics of the most representative of them are listed in table G10.0.5. Some of the designs take advantage of the coupling with ferromagnetic materials to reach lower internal residual fields (Matsuba et al 1992). However, thick films are the only real solution from an economic point of view, and are being made available for supply. They are aimed at extended surfaces for large-scale applications (Pavese et al 1996), or at critical applications, e.g. for use with SQUIDs. In the latter case, where total noise figures of 18 fT Hz−1/2 must be attained (Ludwig et al 1995), a specific problem arises: magnetic noise is generated by the motion of vortices in the material due to imperfect anchoring by pinning centres or to flux creep. The assessment of very low noise figures requires special methods for the measurement of magnetic noise in the shield (Muirhead and Welhöfer 1993) and the use of specially developed materials (Miklich et al 1994, Polushkin et al 1995). Few checks were reported of the effect of joints in the overall effectiveness of the shield, an essential piece of information for the upgrading of the design to large and complex shields. A tile configuration overlapping in three layers was used in one of these tests (Fujimoto et al 1994). In a single layer, full field penetration in the gaps between tiles was obvious, but in the multilayer configuration some attenuation has been observed. However, the attenuation was strongly limited by the structure of the individual tiles, made of large YBCO crystals with gaps (or cracks) between them, through which the flux lines were able to leak. Another test (Kojima et al 1992b) used a cylinder made of four rings, 5 mm thick, separated by a gap equal to the ring height. An axial magnetic field was applied to the assembly: a maximum shielding field Bl i m ranging from 2.5 to 9 mT, depending on the Jc of the material, was observed. These values did not improve when placing superconducting discs as end caps; some improvement was observed when three other rings were added in between the previous four so as to fill the gaps between the discs

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(up to 12 mT). A third study (Kojima et al 1992a) was made by piling up large shields made of small blocks in several chess-board-like (‘grid-type’) configurations. In this way gaps comparable to the block are interspaced in these structures. Cylinders of polygonal cross-section (square, hexagonal, octagonal) and of different length-to-diameter size were tested with a field applied axially. The blocks were joined to each other with a ‘superconducting paste’. Values for A up to 2000 were obtained with a very small grid gap: it decreases almost linearly (in decibels) with the width of the gap. Covering the sides of the square cylinder with four separate sheets of µ-metal produced a small improvement. Of course, all these arrangements are ineffective with respect to transverse field components. They confirm that care in making the joints between the different sheets of the shield for a large-enclosure design is critical to keep A high. This is the only major point that has still to be demonstrated to show the feasibility of using high-Tc superconductors as shields of large dimensions. In most other respects, the present technology is already sufficiently mature to compete with the older types of shield for B 20 mT— or higher with the addition of high-permeability external shields—with an internal residual field below the noise level of SQUIDs. For applications where high-level stray fields must be brought down to the level of the earth’s magnetic field, the development of materials with better characteristics is still needed. References Barone A (ed) 1992 Principles and Applications of Superconducting Quantum Interference Devices (Singapore: World Scientific) Bean C P 1962 Magnetization of hard superconductors Phys. Rev. Lett. 8 250–3 Boll R and Overhott K J (eds) 1989 Magnetic Sensors (Sensors 5) ed W Göpel, J Hesse and J N Zemel (Weinheim: VCH) Bondarenko S I, Vinogradov S S, Gogadze G A, Perepelkin S S and Sheremet V I 1974 Shielding properties of superconducting shields Sov. Phys.-Tech. Phys. 19 824–8 Bridges J E 1968 Proposed recommended practices for the measurement of shielding effectiveness of high-performance shielding enclosures IEEE Trans. Electromagn. Compat. EMC-10 82–94 (this journal and IEEE Trans. Magn. provide a number of useful papers) Bykov Y, Brigor’eva A V and Shkut V A 1996 Synthesis of long YBCO, BSCCO-2212, 2223 HTSC coatings by spraying technology and gas-flame method of crystallization Adv. Cryogen. Eng. 42 at press Cabrera B 1987 Near zero magnetic fields with superconducting shields Near Zero: New Frontiers of Physics ed J D Fairbank et al (New York: Freeman) Cakak R K and Craig J R 1969 Magnetic field uniformity around near-Helmholtz coil configurations Rev. Sci. Instrum. 40 1468–70 Cimberle R, Ferdeghini C, Nicchiotti G L, Putti M, Siri A S, Rizzuto C, Costa C A, Ferretti M, Olcese C L, Matacotta F C and Olzi E 1988 Magnetization measurements on tubular samples of YBCO Supercond. Sci. Tecnol. 1 30–5 Cohen D 1970 Large volume conventional magnetic shields Revue Phys. Appl. 5 53–8 Däumling M and Larbalestier D C 1989 Critical state in disk-shaped superconductors Phys. Rev. B 40 9350–3 Eberhardt F J, Hibbs A D and Campbell A M 1988 Flux trapping and magnetization of hollow superconducting cylinders Cryogenics 28 681–4 Fietz W A, Beasley M R, Silcox J and Webb W W 1964 Phys. Rev. A 335 136–42 Frankel D 1979 Critical-state model for the determination of critical currents in disk-shaped superconductors J. Appl. Phys. 50 5402–7 Fujimoto H, Ban T and Higuchi T 1994 Magnetic shielding of high-Tc superconducting plates Adv. Cryogen. Eng. 40 237–43 Gololobov E M, Prytkova N A, Tomilo Zh M, Turtsevich D M, Shimaskaya N M and Ladut’ko N F 1993 YBCO and BSCCO superconducting coatings on nickel and nickel-based alloy substrates Cryogenic Engineering Conf. (Albuquerque, NM) Hamilton W O 1970 Superconducting shielding Revue Phys. Appl. 5 41–8 Hemmes H, Chaouadi R, Aschern W, Pont M, Rogalla H, Cornelis J, Stöver D and Muñoz J S 1993 Plasma sprayed high-Tc layers for magnetic shielding Adv. Cryogen. Eng. 40 271–80

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Irie and Yamafuji 1967 J. Phys. Soc. Japan 23 255 Itoh M 1993 Influence of wall thickness on magnetic shielding effects of BPSCCO cylinders Adv. Cryogen. Eng. 41 261–70 Itoh M, Pavese F, Mori K, Vanolo M and Minemoto T 1996 Magnetic shielding by superposition of hybrid ferromagnetic cylinder over a YBCO thick-film cylinder J. Magn. Soc. Japan 20 289–92 Johnson J D, Campbell A M, Blunt F J and Alford N McN 1989 Screening of electromagnetic fields by paniculate superconductors Physica C 162–164 1607–8 Jones J N, Dew-Hughes D and Johnson K A 1992 The low field simulation and the study of magnetization in high-Tc superconductors at 77 K Supercond. Sci. Technol. 5 351–4 Kelhä V A 1981 Construction and performance of the Otaniemi magnetically shielded room Biomagnetism ed S N Erné, H D Hahlbohm and H Labbig (Berlin: de Gruyter) Kim Y B, Hempstead C F and Strnad A A 1962 Critical persistent currents in hard superconductors Phys. Rev. Lett. 9 306–11 Kojima H, Ishikawa Y and Yoshizawa S 1992a Magnetic shielding of grid-type high-Tc superconductor unit Adv. Supercond. 4 1073–6 Kojima M, Kohayashi S, Ishikawa Y and Yoshizawa S 1992b High field magnetic shielding using superconducting rings Adv. Supercond. 4 1069–72 Kozlowsi G, Oberly C E, Leese R and Ho J C 1993 Melt-processed YBCO/Ag composite tapes with nickel cladding Adv. Cryogen. Eng. 38 859–66 Ludwig F, Dantsker E, Kleiner R, Koelle D, Clarke J, Knappe S, Drung D, Koch H, Alford N McN and Button T W 1995 Integrated high-Tc multiloop magnetometer Appl. Phys. Lett. 66 1418–20 Mager A J 1970 Magnetic shields IEEE Trans. Magn. MAG-6 67–75 Matsuba H, Yahara A and Irisawa D 1992 Magnetic shielding properties of HTC superconductors Supercond. Sci. Technol. 5 S432–9 Miklich A H, Koelle D, Shaw T J, Ludwig F, Nemeth D T, Dantsker E, Alford N McN, Button T and Colclough M S 1994 Low-frequency excess noise in YBCO dc superconducting quantum interference devices cooled in static magnetic fields Appl. Phys. Lett. 64 3494–6 Miller M M, Carroll T, Soulen R Jr, Toth L, Rayne R, Alford N McN and Saunders C S 1993 Magnetic shielding and noise spectrum measurements of YBCO, BiSCCO and Bi(Pb)SCCO superconducting tubes Cryogenics 33 180–3 Mohamed M A K, Friedrich L and Jung J 1993 Studies of the critical state and demagnetization effects in ceramic disc of YBCO superconductors Cryogenics 33 247–50 Muirhead C M and Welhöfer F 1993 Assessment of thick film YBCO for flux transformer and magnetic screening applications IEEE Trans. Appl. Supercond. AS-3 1695–7 Ohyama T, Minemoto T, Itoh M and Hoshino K 1993 Improvement of magnetic shielding effects: the superposition of a double magnetic cylinder over a high-Tc superconducting cylinder Appl. Supercond. 1 254–60 Opie D B 1993 Physical Science Inc., Alexandria, USA, private communication Pavese F 1995 Magnetic shielding using high-Tc superconductors: a large-scale application with peculiar requirements High Temperature Superconductivity: Models and Measurements ed M Acquarone (Singapore: World Scientific) pp 637–47 Pavese F, Bergadano E, Bianco M, Ferri D, Giraudi D and Vanolo M 1996 Progress in fabrication of large magnetic shields by using extended YBCO thick films sprayed on stainless steel with the HVOF technique Adv. Cryogen. Eng. 42 at press Pavese F, Bianco M, Andreone D, Cresta R and Rellecati P 1992 Magnetic shielding properties of YBCO thick films deposited on silver cylinders with the continuous detonation spray technique Physica C 204 1–7 Plechàcek V, Pollert E, Hejtmànek J, Semdmidubsky D and Knizek K 1994 Improvement of the magnetic shielding and trapping properties of BPSCCO superconducting tubes by the use of multiple thermomechanical processing Physica C 225 361–8 Polushkin V, Buev A and Koch H 1995 Magnetic shielding at liquid nitrogen temperature Appl. Supercond. 2 1–15 Ravi Kumar G and Chaddah P 1989 Extension of Bean’s model for high-Tc superconductors Phys. Rev. B 39 4704–7 Shimbo Y, Niki K, Kabasawa M and Tachikawa K 1993 High-Tc magnetic shields prepared by a low pressure plasma spray Adv. Cryogen. Eng. 40 253–60

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Vacuumschmeltze GmbH 1989 Magnetic Shielding Datasheet FS-M9 Hanau, Germany Vanolo M, Pavese F, Giraudi D and Bianco D 1994 Preliminary results for YBCO thick films sprayed on stainless steel with the HVOF technique, and for multilayers using Ag or CuO Nuovo Cimento D 16 2119–26 Vant-Hull L L and Mercereau J E 1963 Rev. Sci. Instrum. 34 1238 Vlakhov E S, Kovachev V T, Polak M, Majoros M, Dimitriev Y B, Jambarov S, Kashchieva E and Staneva A 1991 The influence of Ag and Te additions on the magnetization of hollow cylinder shaped YBCO ceramics Physica C 175 335–41 Xu M, Shi D G and Fox R F 1990 Generalised critical-state model for hard superconductors Phys. Rev. B 42 10773–6

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PART H POWER APPLICATIONS OF SUPERCONDUCTIVITY

Copyright © 1998 IOP Publishing Ltd

H1 An introduction to the power applications of superconductivity

A D Appleton and D H Prothero

H1.0.1 Foreword This chapter is divided into two parts. The first part (sections H1.0.2-H1.0.7) presents an overview of the potential power engineering applications of superconductivity. This is followed (in section H1.0.8) by an introduction to superconductivity and its associated technologies and also to the economics which ultimately determine the prospects for any superconducting application. Most of the second part is presented in terms of liquid-helium-cooled superconductors, on which most of the design and development work has been carried out to date; however, this part concludes with a discussion of the present state of the art on high-temperature (liquid-nitrogen-cooled) superconductors. These subjects are presented in a form which gives a general background on the the subjects covered and serves as an introduction to those who are considering taking a professional interest in superconductivity. As an introduction, a brief sketch is given of the applications which are discussed in this volume; it may be noted that most of them have been under development since the early 1960s using the liquid-helium-cooled metallic superconductors. Where appropriate, an assessment is also made of the impact of the recently discovered high-temperature superconductors. H1.0.2 Synchronous machines Electrical energy does not cost more than it does at the present time because there have been improvements, particularly over the last 50 years, in the materials for making electrical equipment, better methods of cooling and improved design. Another important area is efficiency and for most types of plant this has been steadily improving over the years. A present-day 500 MW a.c. generator typically has an efficiency of about 98.6%, i.e. the losses are about 7 MW. If these losses could be sold along with the other 98.6%, the increased revenue over the life of a single (base load) generator would be worth about ECU 15M. In practice, however, the heat produced by these losses costs money due to the measures required to remove them from the machine. As the trend towards larger ratings continued, up to the 1980s at least, design engineers of conventional synchronous machines found that almost the only degree of freedom available was the length of the generator. It was possible to squeeze a little more current density out of the windings (often at the expense Copyright © 1998 IOP Publishing Ltd

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of lower efficiency) and perhaps a little more by clever design but, without a radical design change, the only method of significantly increasing machine output was to increase the length. The radical changes considered were: ( i ) to build a four-pole rather than a two-pole machine, but this increases the capital cost and this is not usually what the customer wants, ( ii ) to take the armature winding out of the stator slots and put it in the air gap (development work on this concept started in the 1970s but it did nothing for the limitations imposed by the rotor winding), ( iii ) to water-cool the rotor winding as the next step following hydrogen cooling which was introduced in the 1970s (progress in this direction has been slow). It was against this background that, in the late 1960s, work was started on the development of a.c. generators using liquid-helium-cooled metallic superconductors in the rotor winding. Under nonfault conditions, the rotor winding would be subjected to a constant magnetic field and superconducting a.c. losses would therefore be zero. By contrast, the stator winding would see significant current and magnetic field variations and hence would be subject to large a.c. losses if the winding were superconducting. For this reason, most studies of superconducting synchronous machines have assumed a superconducting rotor and a stator wound from copper. However, in the early 1980s, multifilamentary Nb—Ti composites were developed with extremely fine (i.e. less than 1 µm) superconducting filaments. These composites had extremely low losses and hence made an all-superconducting machine (i.e. both rotor and stator windings superconducting) possible, at least in principle. Detailed design studies were carried out of an a.c. generator with a liquid-helium-cooled ( Nb—Ti ) superconductor in the rotor winding, the stator being wound from copper and at room temperature. Figure H1.0.1 shows a detailed layout of a 1300 MW, two-pole a.c. generator of this type. The benefits of this design, compared with a conventional generator operating at room temperature, include reductions of 50% in the losses, weight and volume; it is possible that the higher-temperature superconductors will increase these benefits and provide a lower capital cost for machines of 1000 MW and above.

Figure H1.0.1. Layout of a 1300 MW superconducting a.c. generator. Copyright © 1998 IOP Publishing Ltd

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H1.0.3 Direct current motors and generators Historically, d.c. machines have a rather limited market because they do not have the simplicity of induction motors or the elegance of synchronous motors and generators. Large d.c. motors are used where good speed control is required and examples of this are found in steel rolling mills, paper mills, colliery winders and ship propulsion. In all of these cases, reliable speed control is critically important. However, the advance of solid-state power electronics has made available large variable-frequency power supplies which, of course, allow a.c. motors to be used. These power supplies are expensive and rather bulky but they do provide an alternative to d.c. motors for the plant engineer. One of the fundamental limitations of d.c. machines, using conventional technology, is that it is impracticable to build a d.c. motor (or generator) of more than 10 MW; furthermore higher powers approaching 10 MW are possible only over a very limited speed range. The reasons for this include practical limits on physical size and weight and problems associated with commutation. However, by the use of superconductors the power range can be extended to meet all probable needs. During the 1960s and 1970s, a considerable amount of work was undertaken on these machines which included the design, building and testing of a number of machines of megawatt rating. This work is described in some detail in chapter H3 of this handbook. Virtually all the technical issues relating to these machines were resolved, although the reliability of the ancillary equipment (refrigeration plant, etc) caused problems. Interest in these machines declined in the 1980s and so far has not revived. H1.0.4 Transformers The development of superconducting transformers is a long way behind that of most of the other power applications because initially it was not possible for metallic superconductors to carry an alternating current without unacceptable power losses. This situation, however, changed in the early 1980s with the development of composites incorporating submicrometre-diameter filaments of Nb—Ti. This approach does not eliminate the mechanisms which cause the losses but it significantly reduces the magnitude of the losses. With superconducting transformers (unlike rotating machines) it is not possible to eliminate the use of magnetic iron; because of transportation problems, this may make the difference in being able to build a three-phase transformer rather than three single-phase units. The winding losses are significantly reduced compared with conventional transformers but other significant problems (such as overload protection) remain which may mean that superconducting transformers will require a lengthy development period. Interest in superconducting transformers has revived to some extent with the development of liquid-nitrogen-cooled superconductors and some detailed studies have been carried out on the effect of these new materials. These confirm the significant benefits that can be achieved with high-temperature superconductors but technical issues still remain relating to overload protection. H1.0.5 Power transmission The power that can be transmitted by an underground cable is limited by the amount of heat which can be removed by the forced cooling of the cable. For some conventional cables, water pipes are laid in the trench beside the power cables to remove the power losses generated within the cable. Underground power cables need to be very reliable and some have been in service for over 60 years; also they have the environmental advantage that they are out of sight. However, overhead power transmission lines have to be used to keep down costs and, in spite of environmental objections, it will be a long time before these are replaced by underground cables. However, overhead lines are increasingly disappearing in urban areas to be replaced with low-voltage underground cables. The possibility of using superconductors for underground cables to reduce losses

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and thereby alleviate the cable-cooling problem has been under investigation since the 1960s. This work went through a peak in activity in the 1970s and subsided in the 1980s prior to the discovery of high-temperature superconductors. One of the reasons for this decline was the fact that a low-temperature superconducting cable would only be economic (compared with a conventional cable) at rather high ratings; different manufacturers put this at between 1000 MVA and 3000 MVA. However, the discovery of high-temperature superconductors has led to a revival of interest and work is at present proceeding in a number of countries on high-temperature superconducting cables.

H1.0.6 Fault current limiters A single electrical power network often covers vast areas of a country with many power stations feeding in and many load points, at different voltage levels, feeding out. The control and protection of this network requires circuit breakers that must be capable of operating safely under all system conditions including fault conditions. Every time a new power station is added to the system, or every time an additional interconnection is made to increase the system flexibility, the level of the fault current tends to increase. To a limited extent, this problem may be contained by increasing the inductance of the system by the installation of fixed (i.e. constant value) devices called ‘reactors’. However, these have the disadvantage of increasing the regulation of the system (i.e. variation of voltage with load) and beyond a certain point they may drive a power system unstable. All the circuit breakers of the system can be upgraded, but this is very expensive and may not be possible in the short term. What is really required is a device which switches from a low impedance under ‘healthy’ system conditions to a high impedance under fault conditions. Such a device is known as a fault current limiter (FCL) and a number of designs have been proposed which use the superconducting/resistive transition of a superconductor to provide this switching action. These designs generally are of the resistive type (which uses the change in resistance between the superconducting and normal state to provide the switching action) or the inductive type (which uses the change in inductance caused by flux exclusion (Meissner effect) in a magnetic circuit during the superconducting transition). Laboratory ‘bench-top’ models of some of these designs have been built and tested, using both liquid-helium-cooled and liquid-nitrogen-cooled superconductors. In most cases, the technical performance of the models has been satisfactory. In addition, in 1978, a large-scale FCL prototype was built and tested by NEI Peebles in Edinburgh, and International Research and Development Ltd in Newcastle-upon-Tyne; this incorporated a liquid-helium-cooled superconducting winding but used a different principle—in this case, the switching action was provided by driving iron cores in or out of saturation. This also performed satisfactorily but failed to reach the market place on economic grounds. Since the discovery of high-temperature superconductors, there has been a significant increase in interest in the FCL application (see ‘Further reading’ at the end of this chapter).

H1.0.7 Energy storage A power system is required to supply energy on demand, not in five minutes time or even five seconds, but immediately. This makes it necessary for spare generator capacity to be available at all times. A TV timetable is an important document in the control room of a power network because it allows some anticipation of power demand changes. The control engineer can become an expert in this anticipation and can make accurate assessments on when to bring in more power stations or when to stand them down. How nice it would be if the system could store some energy to be used on demand and thus make sure of meeting these sudden demands. Ideally, he would like to charge up the store at night, when the demand is low, and use it during the peak load periods.

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This problem has been recognized since the earliest days of the electrical power industry but satisfactory solutions are elusive. If it can be achieved, the savings to be made in the utilization of generating plant are substantial. One solution is the pumped storage of water; this is accomplished by pumping water uphill to a reservoir at night and letting it run down to generate power during periods of high demand. The stored energy of these systems can be very high and a good example is Dinorwig in the UK which stores 8000 MW h. However, suitable sites for such schemes are scarce and the construction costs are very high; hence alternatives are continually being sought. A possibility which has been studied for many years, particularly in Japan and the USA, is superconducting magnetic energy storage (SMES) in which the stored energy is in the form of a persistent current carried in a superconducting coil. In order to store useful amounts of energy by this method, however, it would be necessary for the coil to be very large; the associated electromagnetic stresses would also be very large and a massive structure would be required to support the coil against these stresses. Many studies have in fact assumed that the SMES unit would be located underground in order to provide the necessary reinforcement. In spite of these drawbacks there is continued interest in the possibility of SMES for military and utility applications. Applications for which SMES may be particularly suitable are those which require a modest storage energy and for which power would be delivered for only a short time (typically a few seconds). Examples of these are load levelling, correcting unbalanced voltages and improving the stability of power systems. Depending upon the power system, reasonably small SMES coils would be adequate for these applications.

H1.0.8 Superconductivity, associated technologies and some economics We have discussed in very general terms how superconductors can be used for various applications in the electrical power industry so let us now consider the phenomenon of superconductivity and some associated technologies and also review some economic considerations before we consider these applications in greater detail. The potential impact of superconductivity upon the electrical power industry was recognized by Professor Kammerlingh Onnes when he made the Nobel Prize winning discovery of superconductivity in 1911. This was made during experiments to measure the conductivity of mercury at low temperatures and Onnes attempted to build magnets using mercury superconductors. He soon realized, however, that mercury was easily driven back into the nonsuperconducting state by even a small current density or a small magnetic field. This severely limited the potential applications of superconductivity. Hence, in the years that followed, the technology had little relevance to the electrical power industry; however, advances were made in the understanding of the physics of superconductors and it was a subject of continuing interest in the academic world. This situation changed, as far as the power engineer was concerned, in the late 1950s and early 1960s with the discovery of the so-called type II superconductors which had the capability of operating at high current densities and in the presence of high magnetic fields. One of the first materials of this type to become commercially available was the alloy of niobium and zirconium ( Nb—Zr ) and the world’s first superconducting d.c. motor, described in chapter H3, employed a seven-strand wire of this superconductor. However, the design engineer discovered that nature is not too generous with her bounty because the superconductor would only work if it was immersed in liquid helium which, at atmospheric pressure, has a boiling point of 4.2 K (-269°C). These were completely uncharted waters for power engineers and progress depended upon an assembly of multidisciplined teams to cover all the supporting technologies (vacuum and refrigeration) which were needed. A major part of the development programmes which followed, for a whole range of applications, had to be devoted to cryogenics to ensure that the necessary environment was provided for the liquid-helium systems with very careful attention required to limit ingress of heat by conduction, convection and radiation. A major concern was the availability of a reliable supply

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of liquid helium at a reasonable cost and the associated liquid-helium liquefiers themselves became the subject of development programmes. At the very low temperatures which are required for liquid-helium-cooled superconductors, the specific heats of most materials are very low so that only a small amount of heat would be sufficient to raise the temperature of the superconductor above its transition temperature. Thus, even small amounts of heat leaking into the low-temperature region can be very significant. Furthermore, the cost of removing this heat is high because of the very low efficiency of liquid-helium refrigeration systems. For every watt of heat which leaks into the low-temperature region, it requires more than 1 kW to get it out again. It was against this background that the development of superconducting machines took place. The most active period was between the late 1960s and early 1980s. During this time, there was a remarkable improvement in the quality and performance of the superconductors that became available to engineers. This conductor development work was undertaken jointly between industry, universities and national laboratories. Nb—Zr was rapidly superseded by Nb—Ti which became the principal technical superconductor, and also niobium—tin ( Nb3Sn ). The latter has the advantage that it has a higher critical temperature (about 18 K, compared with about 9 K for Nb—Ti) but it has the disadvantage that it is extremely brittle. The objective of this development work was the achievement of a stable performance from superconducting windings because the substantial financial investment in a large superconducting system would not be justified if reliable performance could not be assured. All superconductors exhibit zero resistivity and zero losses only if d.c. is used; even the necessary action of increasing the current from zero to the design value causes losses in the superconductor and for applications involving a.c. (such as transformers) the losses can be significant. The stability of the superconductor can be increased by encasing it in copper so that the losses in the conductor are readily conducted away by the high-thermal-conductivity copper matrix (this is known in the literature as ‘dynamic stabilization’). A rather simple method of achieving this, shown in chapter H3, figure H3.0.9, was applied in 1965 to the Fawley motor. This conductor consisted of a number of filaments of Nb—Ti in a matrix of copper. In addition to the dynamic stabilization, the copper provides an alternative path for the current to flow along when a transient occurs which causes some of the superconductor to revert to the nonsuperconducting state (a phenomenon known as a ‘quench’). If the copper were not present, the superconductor would rapidly heat up when driven into the nonsuperconducting state and burn-out could occur. With copper present, the current is largely transferred from the superconductor to the copper following a quench. Indeed, if sufficient copper is present it is possible to design the winding that the heat generated in the copper is passed to the liquid helium with a sufficiently low temperature rise that the superconductor can return to the superconducting state (this is known in the literature as ‘cryostatic stabilization’). During periods when the current or magnetic field in the superconductor varies with time, the motion of magnetic flux through the superconductor, and also the copper, will generate losses in both the superconductor and the copper. Two approaches were adopted to minimize this problem. Firstly, it was found that by reducing the size of the superconducting filaments (ultimately to less than 1 µm), the losses per unit volume in the superconductor were correspondingly reduced (this is known in the literature as ‘adiabatic stabilization’). Figure H3.0.11 in chapter H3 shows a commercially available superconductor based on these principles. It consists of 61 filaments of Nb—Ti, of diameter 50 µm in a matrix of copper; the overall diameter of the conductor is 0.6 mm. Secondly, the losses in the copper were reduced by using a three-component Nb—Ti/Cu—Ni/Cu conductor in which each superconducting filament was encased in a Cu-Ni sheath which was itself in a matrix of copper. Figure H1.0.2 is useful in summarizing the basic properties (current density and operating magnetic field) required from the superconductor for many of the applications discussed in part H. It will be seen that some applications are considerably less demanding than others in terms of superconductor properties; however, it is fair to say that Nb—Ti superconductor, as presently available, is capable of meeting virtually all requirements. This, however, is not the full story regarding the commercial prospects for these

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Figure H1.0.2. Superconductor requirements for various applications.

applications. It is also necessary that the cost of the complete device (including auxiliary technologies such as refrigeration) is competitive with a conventional equivalent, if it is available. Secondly, the reliability of the complete device (again including ancillary technologies) must be at least as good as that of equipment at present available. For most power engineering applications, the cost of any ‘down time’, when the equipment is unavailable due to malfunction, tends to be extremely large; hence any reduced reliability of superconducting equipment is almost certain to wipe out any gains in capital or operating costs. Nb—Ti is now an excellent material widely available on a commercial basis and able to meet the requirements for most items of power equipment. Significant advances have also been made with Nb3Sn and it is also available in filamentary form. However, because of its brittle nature, it is more difficult to use than Nb—Ti (which can easily be processed into multifilamentary form) and tends to be used mainly in research magnets. On the question of helium liquefiers, the systems now available are more reliable than they were in the 1960s but are still relatively expensive. To cool helium gas it is necessary to remove energy from it by some mechanical means until it is below the inversion temperature when it can be liquefied by expansion through a Joule-Thomson valve. In one system which is in common use, the gas is compressed to about 10 bar (10 × 105 Pa) and then passed through a series of counterflow heat exchangers. In the steady state there is a temperature gradient from ambient at the top of the heat exchangers to 4.2 K at the bottom. Energy is extracted by small turbines loaded by a paddle wheel in a gas chamber in which the pressure and therefore the load cam be varied. In other designs the turbines are loaded by small electrical generators. The turbine wheels have a diameter of about 10 mm and are driven at high speeds (over 300 000 rev min−1 ) on gas bearings. Usually, there are two turbines, one at a temperature of about 80 K and the other at about 20 K. Refrigerators based on the Stirling cycle (patented by the Rev Robert Stirling in 1816) and also the Gifford-McMahon cycle, have been under development since the late 1940s, initially by Philips in Eindhoven. Coolers are available capable of extracting 100 W at 80 K; lower temperatures may be obtained by having a number of stages in series. It is obviously important to reduce the load on the helium refrigerator and this is achieved by isolating the low-temperature region as much as possible from the ambient temperature surroundings. The superconducting windings are placed in a ‘cryostat’ within which they are thermally insulated by means Copyright © 1998 IOP Publishing Ltd

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of a vacuum and radiation shields; also the mechanical supports for the windings are carefully designed to minimize the heat in-leak and are often counterflow-cooled using the helium gas that is boiled off by the heat which does reach the windings. These techniques are described in greater detail for each of the applications covered by this handbook. One of the main sources of heat in-leak is the current leads which are required to take the current to the windings from a power supply. A great deal of development work has been devoted to current leads and this work has continued with the development of high-temperature superconductors (see chapter D10). In some cases, it is possible to short-circuit the superconducting winding after the required current level has been reached and then remove the current leads. This is called the ‘persistent current mode’ and clearly the current lead losses may be eliminated. This technique is used for some research magnets and for magnetic resonance imaging (MRI) body scanners but in general it is not suitable for electrical power applications because there is usually a requirement, at least in principle, to change the current at any time. Let us leave the basic technologies, which will re-assessed in later chapters, and return to the situation in respect of power engineering applications. It is reasonable to ask: if so much progress has been made with superconductors and refrigeration systems, why is superconducting power equipment not in production? Many of the technical issues associated with the numerous power engineering applications have been solved. Also, in some cases, demonstration equipment has been produced to provide practical evidence of performance. However, when cost projections are made, it is usually found that the superconducting plant cannot compete with conventional equipment unless the ratings are very high. Consider, as an example, a.c. generators. It has been found, on the basis of first costs, that superconducting machines are not competitive unless the rating is at least 500 MW (some manufacturers say 1000 MW). Although it had been expected that the traditional trend to larger unit ratings would continue, this has not been the case in recent years and interest now centres on smaller ratings. It may be noted that the efficiency of a large superconducting generator is estimated to be 99.3% which, for a 500 MW machine, shows a saving of 3500 kW over a conventional generator. If the generator is on base load with a high load factor, the cost of the losses over the life of the machine is about ECU 2500 kW−1, which, in this case amounts to ECU 8.75M. However, it is usually argued that there is an unreliability factor which negates this hypothetical saving and that it is necessary for the superconducting machine to compete successfully on the basis of first costs in order to win acceptance. A somewhat similar situation has been experienced with power cables except that in this case the break-even power rating between conventional and superconducting cables has been variously stated as being between 1000 MW and 3000 MW. In the case of d.c. motors and generators for ship propulsion the situation was somewhat different because the ratings required are much greater than is possible using conventional methods of manufacture. Thus a superconducting machine was the only solution and large machines were built to demonstrate this fact. However, other problems were raised which have almost brought development work to a standstill. These problems are cost and the perceived unreliability of liquid-helium refrigeration systems, particularly for an ocean-going ship. The consequences of these facts were that by the early 1980s most programmes on the power engineering applications of superconductors had fallen to a very low level or had ceased altogether. The development of submicrometre Nb—Ti wire in France and Japan brought about an interest in the possibilities for transformers and totally superconducting a.c. generators (i.e. with the stator as well as the rotor winding superconducting) but the earlier momentum was not regained. An exception was in Japan where development work has continued on almost every application. It is easy to see why the discovery in 1986-87 that some ceramic materials were superconducting at temperatures greater than that of liquid nitrogen (77 K) caused so much excitement. Nitrogen is a common gas, readily available from the atmosphere and at 77 K the specific heats of materials are much higher than

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at 4.2 K (that of copper, for example, is nearly 2000 times higher) so that liquid-nitrogen refrigeration is much more user-friendly than that of liquid helium. Other important points are that industry is familiar with the use of liquid nitrogen and refrigeration efficiency at 77 K for practical systems is 100 times greater than it is at 4.2 K. It follows that the engineering designs of power equipment become much less complex and this impacts on the cost to bring about significant reductions to the break-even ratings for all the applications which have been investigated. However, once again, nature is not too generous and in the cold light of day it was found (in 1987) that the performance of the new superconductors left much to be desired. They were extremely brittle and the current densities and flux densities which they could support were very low; nevertheless, the scientific community throughout the world responded to the challenge. The state of the art in 1995 could be summarized as follows. ( i ) Transition temperature: The highest Tc is 134 K (increases to 164 K at a pressure of 250 kbar (2.5 × 1010 Pa)) for (Hg, Pb)BaCaCuO; this material, however, is chemically unstable. ( Tl, Pb)BaCaCuO, which has good potential as a practical conductor (despite the toxicity of Tl), has only a slightly lower Tc of 125 K. ( ii ) Current density: At a temperature of 77 K and in the absence of a magnetic field, the best Jc values are 107 A m−2, 109 A m-2 and 5 × 1010 A m-2 for polycrystalline materials, melt-textured samples and thin films respectively. At a temperature of 4 K, these values are increased by a factor of ten; hence the sustained interest in using high-temperature superconductors at temperatures less than 77 K (see also ‘Further reading’ at the end of this chapter). The quoted values apply also to short-length (typically 1 cm) samples. They are reduced, typically by a factor of three for 300 m length samples. ( iii ) Magnetic field tolerance: Somewhat material dependent. YBa2Cu3O7 can sustain a magnetic field of about 2 T at 77 K, (Tl, Pb)BaCaCuO should have a somewhat similar value but (Bi, Pb)SrCaCuO is much less on account of poorer flux pinning. Demonstration magnets wound from this last material have been made generating magnetic fields of the order of 2 T but these have all been operated at reduced temperatures (27 K or less). The above has been presented to give a ‘snapshot picture’ of the state of superconductor technology in 1995. It is to be expected that improvements will continue to be made in these materials in future years. Further reading IEEE Trans. Appl. Supercond. AS-5, nos 1, 2 & 3 ( June 1995 ). These three issues give the papers presented at the Applied Superconductivity Conference in Boston ( USA ) in October 1994. This is a recent conference which fully covered the areas relating to superconductors and their applications which are of greatest interest at the present time. Regarding the latter, the relatively large number of papers on fault current limiters should be noted. ( There is also much interest in electronic devices which are outside the scope of this chapter. ) The use of high-temperature superconductors at temperatures below 77 K continues to attract interest ( see last paragraph of this chapter ); such applications would benefit by using current leads made in part from high-temperature superconductors and there are a number of papers on these. It should also be noted that the high level of interest in high-temperature superconductors does not mean that low-temperature ( liquid-helium-cooled ) superconductors have been ignored.

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H2.1 Generators with superconducting field windings H Köfler

H2.1.1 Preface The pursuit of the improvement of electric machinery is a challenging task because the principles of its operational performance have not changed since the invention of the machines in the 19th century. A.c. rotating machinery of conventional design and using conventional materials has attained a high level of performance and high output rates over a period of many years; this has been mainly in the field of generators for central power stations. As we noted in chapter H1, improvements are possible by the use of better materials, better design and by better cooling but, of course, these techniques can only be applied to parts of the complete machine and when the availability of technology permits. It was to be expected therefore that, soon after 1960 when type II superconductors became a feasible product for large magnets, consideration should be given to their application to electric machines. Studies revealed that the application of superconductors operating at high current density in modestly high magnetic fields will result in a reduction of space and weight and an improvement in the efficiency of large a.c. generators. Early development gave confidence that vacuum-insulated rotating vessels can be a safe operating environment for the superconducting magnets and a course was set for a long period of development in many parts of the world. The superconductor which dominated the work programmes was the alloy niobium-titanium (Nb—Ti) and over a period of two decades the performance of this material was enormously improved. Of particular note was the development of submicrometre filaments of Nb—Ti in the early 1980s which, for the first time, allowed consideration to be given to their operation with a.c. An alternative superconductor which allows operation at higher magnetic fields is the compound niobium-tin (Nb3Sn) but problems associated with its mechanical properties prevented it taking the place of Nb—Ti conductors. The discovery of the higher temperature ceramic superconductors caused considerable excitement in the scientific community because of the potential which a higher operating temperature promised for a reduction in both capital and operating costs of the generators. Studies showed that the efficiency could exceed 99% which represented very large cost savings. As we might expect these discoveries stimulated much research worldwide both for generators and for most of the other engineering applications of superconductors (see figure H2.1.1). H2.1.2 Introduction Although in principle many, or even most, of the technical problems identified in connection with synchronous generators with superconducting field windings have been solved (fully superconducting synchronous machines are covered in section H2.3) there is still a need for further refinement, optimization and experience. In the following pages we present a resume in a condensed form of the options, Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.1. Loss reduction from the past to the future.

requirements, technical problems and their solutions for such generators. The statements made and the solutions shown are on the basis of present knowledge and may stimulate further development of superconducting a.c. machinery. A short presentation on the principles which apply to electric machines leads to the different topics which are of main concern in superconducting machinery. The simplified sketch of a synchronous generator in figure H2.1.2 shows the essential components of this machine. Since interest is focused on machines with high ratings as the more likely candidates for economic success, the field winding has to be located in the rotor, otherwise the armature would rotate with the need to transfer the full power via sliprings.

Figure H2.1.2. Sketch of torque production.

Although it is anticipated that the reader is familiar with a basic understanding of the theory of synchronous generators, the principles are presented here for completeness. Consider a conductor with length L metres in a stator winding carrying a current I amperes. This current is exposed to the magnetic field B tesla of the exciter winding which is the rotating member of the machine. As current flows perpendicular to the magnetic field a force F newtons is exerted on the conductor.

This force causes a torque to be exerted on the shaft and the sum of the torques of all of the conductors gives the rated torque of the machine. To give a unified treatment, the currents in the Copyright © 1998 IOP Publishing Ltd

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conductors are expressed as current per unit length of the circumference. The so-called ampere-turn loading, A ampere/metre, of electric machinery is a starting point when designing an electrical machine. Consideration of the design of the longitudinal section of a generator, see figure H2.1.3, with a superconducting rotating field winding shows the differences between the principal components compared with a conventional generator.

Figure H2.1.3. A schematic longitudinal section of a superconducting generator. 1—stator core—provides a path for the flux and shields the environment against excessive magnetic fields; 2—stator winding—location of power conversion from mechanical to electrical and vice versa; 3—field winding—d.c. current winding made from superconductor and cooled with liquid helium; 4—central rotor—helium-tight containment for liquid helium in which the liquid is held on the outer surface by centrifugal forces; 5—torque tubes—transfer members for the reaction torque built between stator and rotor winding, cooled by return flow from the central rotor; 6—drive shaft; 7—supply shaft; 8—central bore—containing current leads, helium supply and exhaust line.

H2.1.3 Superconductor wires for use in a.c. rotating machinery Windings in a.c. rotating machinery are of complicated geometrical form; they are required to shape magnetic fields to prescribed spatial distributions, which are necessary for the production of voltages which vary sinusoidally with time. This task requires a conductor material which can be wound, more or less, in a traditional manner. For copper conductors high standards in correctly shaping windings have been achieved in the past and it is necessary to retain this standard. Metallic superconductors at first sight seem to be the only choice for such windings. However, changes in fabrication technology may give other superconducting materials the chance to be candidate material. H2.1.3.1 D.c. applications in a.c. rotating machinery Development of superconductors in a series of evolutionary steps brought to the market conductors which are able to produce high magnetic fields over large volumes under pure d.c. conditions. In the early magnets it was necessary to pay special attention to the rate of change of current which could be tolerated in reaching the required value. By further development of superconductors the sensitivity of such conductors against rapid change of current was reduced and as a consequence the ability of the conductor to cope with small a.c. currents and a.c. magnetic fields was increased. This performance is precisely that required in the field winding of a synchronous machine. Nominally the field current is d.c. but during regulation of the generator current changes with different time scales are necessary; under fault conditions the superconducting field winding is exposed to a.c. components of the magnetic field and is loaded also with high values of a.c. Copyright © 1998 IOP Publishing Ltd

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currents. These time-varying currents will be addressed in detail in the section dealing with performance. A proper choice of superconducting wires will guarantee reliable operation of a power station generator. (a) Technical standards for proper application At this p oint we will discuss the standards which the superconductors in a rotating field winding must fulfil. In the superconducting rotor of the synchronous machine there are two other magnetic field components which influence the performance of the superconductor. There is the so-called armature reaction field and the reaction field of the damper winding. Therefore a standard situation of operation has to be chosen for which the magnetic flux density values to be met by the superconductor can be calculated. It is found that at the start of design work little information about the windings and their magnetic fields in the machine is available; thus defining the conductor standards is an iterative task. Starting with an estimated flux density value B and specific current turn loading A one makes a trial calculation of the generator dimensions (see section H2.1.4) and also of the field winding. With the availability of this design more reliable values of flux density can be deduced and step by step optimization can proceed. However, the conductor chosen must have electrical insulation suitable for cryogenic use and the standards used for varnish-insulated copper wires may also apply to the superconductor. In the next step, transients in operation have to be considered. After this the number, diameter and twist pitch of the filaments inside the superconducting wire can be defined; also consideration for resistive barriers and for the maximum useful amount of stabilizing copper (or aluminium) can be given. Any specification for the superconductors for a superconducting rotor winding should include the items listed in table H2.1.1 for a monolithic conductor.

In addition the specification may specify, for example, the residual resistance ratio of the stabilizing material and the contact resistance at joints between individual lengths of superconductor. In the case of a cabled conductor table H2.1.1 characterizes the strands but additional criteria have to be added as shown in table H2.1.2. (b) Candidate materials and their performance Superconductor materials which may be used in the superconducting field windings of synchronous generators must be available in sufficient quantity and must be suitable for being formed into windings for rotating machines. These requirements restrict the useful materials (at the present time at least) to the class of metallic superconductors. In this class commercially available superconductors like Nb-Ti are a natural premium choice; an alternative material, Nb3Sn, provides even better electrical properties. However, due to its mechanical properties, at the time of writing Nb—Ti is the most suitable material. When high-temperature superconductors appeared, theoretical assessment work of the potential of this material with regard to superconducting generators was undertaken and, in brief, the result of this work is that high-Tc superconductors must be as easy to apply as the metallic superconductors and show at Copyright © 1998 IOP Publishing Ltd

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least the same current densities at the same magnetic field as the latter. This all has to be fulfilled at the temperature of liquid nitrogen or even higher. Development up to now has not revealed this capability and therefore Nb-Ti and Nb3Sn are unchallenged. For Nb—Ti, which is at present the premium choice for magnet windings in synchronous machines, we can scale the performance of typical conductor critical current in an analytical expression.

Current densities Jc are given in A m−2, magnetic flux density B in T and temperature T in K. The latest developments have increased conductor critical currents, so the factor of 1.4 × 109 may be increased to 2 × 109 A m−2. In the field winding superconductors are subjected to several transient operational states. In these operational states hysteretic losses and eddy current losses are produced in the superconductors and these losses must be taken into account if temperatures in the winding package are to be calculated when such transients occur. During a ramp of the current the losses in the superconductor filament with diameter 2a add up to a density of energy expressed by the following expression:

In this expression Jc can be calculated as explained above. The radius a of the superconductor filaments and the number of filaments have to be taken into account as well. The peak field produced by the ramped current is Bm . Eddy currents are also present in the conductor during ramping. Losses connected with this effect can be taken into account by an expression which includes the properties of the matrix superconductor material

where µe f f is the relative effective permeability of the conductor and µe f f = µ0[(1 − λ)/(1 +λ)] with λ the fraction of the superconductor in the strand; τ is the product τ0 µe f f and τ0 is the time constant of the specific superconductor (see also section B4.3)

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The effective transversal resistivity of the conductor composite is expressed by ρe t and the twist pitch of superconductor filaments by L. i2 is the change rate of the alternating flux density component. All these losses cause heating of the superconductor which in the case of stable operation is kept under control by the cooling fluid. Too low a heat transfer capacity of the fluid causes an increase of the conductor temperature which affects the current-carrvina capacity as well as the rate of losses produced. Introducing these formulae into computer codes allows performance of the superconducting field windings under transient operation to be calculated. An example of such a calculation is shown in figures H2.1.4–H2.1.8. The calculated performance has been tested experimentally and the predicted results compare well with the experimental results. Thus, the use of such calculation codes can improve considerably the prediction of operational performance of superconducting windings in generators. From the figures we can see that the losses in the excitation period raise the temperature to almost 7 K; fortunately this temperature occurs in the very first time period of excitation when the current level is low and is therefore acceptable. Thus the winding discussed here will allow the current changes in the given range without loss of superconductivity. Similar calculations have been carried out for other kinds of transient operation like sudden short circuit, faulty synchronization or power swings after cleared faults elsewhere (Fevrier and Renard 1980).

Figure H2.1.4. A cross-section of a pancake module: the dimensions of interest are marked by measuring arrows, e.g. dimensions of conductor thickness, of conductor insulation and thickness of coil insulation.

Figure H2.1.5. The heat conduction coefficient of coil packages.

Figure H2.1.6. The specific heat of coil materials and packages.

Figure H2.1.7. The temperature in a winding module during excitation (80 A s−1 ).

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Figure H2.1.8. Losses in awinding module during excitation(80 A s−1 ).

H2.1.4 Basic electrical design of a.c. rotating machinery The basic electrical design of a.c. rotating machines entails the definition and design of the windings; furthermore it deals with the magnetic flux linking these windings together. The environmental screen which keeps magnetic field inside the machine is part of the magnetic circuit of the machine and is usually called the stator core. The basic starting expressions employed in the course of design are the apparent power Ps , the voltage U at the terminals, the nominal frequency f of the grid to which the machine will deliver its power and the number of poles p. Of highest interest for the application of superconductors are machines with two poles which give 3000 rev min−1 in 50 Hz grids and 3600 rev min−1 in 60 Hz grids. Such machines are usually called turbogenerators. H2.1.4.1 Stator winding and stator core When designing the stator winding of a superconducting generator one looks for a practical level of flux density and in a first rush of enthusiasm it appears that superconductivity offers a level of two or three tesla; however, the magnetic teeth in the stator cannot carry such high flux densities. If these teeth are eliminated the stator winding can be allowed to occupy all the circumference of the bore. It appears that the barrier to high flux densities is also removed by this change, but this is not the case, as unacceptable eddy current losses in the stator bars are the consequences of high flux densities. The eddy currents can be reduced by expensive methods of stator bar construction or by lowering the first optimistic choice of 1Bm a x to the more realistic level of 1 T or perhaps slightly more. Voltages induced in the stator winding are usually sinusoidal in time and this requires a magnetic field varying sinusoidally in space. In superconducting generators variation in space means variation along the mean circumference of the stator winding assembly, which is of a cylindrical shape with end connections to form the windings. Flux density stands therefore for the maximum of the magnetic flux density wave varying sinusoidally in space (in one pole pitch). The winding assembly is joined together so that best use of this fundamental flux density wave is made and little use is made of higher harmonics introduced from imperfect winding distribution (this imperfect winding distribution is dictated by technical restrictions). This fact is expressed with the winding factor ξ (superscript 1 in the formula below indicates the fundamental component, subscript 1 indicates stator winding, subscript 2 indicates rotor winding)

The numerical factor in the formula is equivalent to Copyright © 1998 IOP Publishing Ltd

and results from sinusoidal variation of

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quantities with time and space. w1 p h stands for the number of turns in one stator phase of a three-phase synchronous generator (connection of phases in a star or delta has to be obeyed in the phase voltage used in the formula). There is still some information missing from this formula. Pole pitch τp and equivalent length L1.2 of the induced winding are not yet known. To get information on these subjects we have to consider a theoretical model of the machine. The power of an electric machine can be expressed with the aid of a formula deduced from the torque developed inside the machine. The torque equation itself simply makes use of the well known expression for force exerted on a current-carrying conductor in a magnetic field. Expressions specific to electrical machines like sine distributed ampere-turn loading A (with its effective value of the fundamental A1.e f f ) appear in this formula. The coefficients used are the result of rough approximation. From the model we find P = 7.0 1ξ1 cos ϕ A1,e f

1 f

Bm a x D 2L 1, 2 f m e c h.

With this expression we can estimate, as a first step, the dimensions of the machine to be built. We are given the apparent power, Ps (equivalent to P/cos ϕ ) of the machine to be designed, the number of revolutions per second fm e c h the nominal frequency fe l of the grid in which it is intended that the machine will operate, the terminal voltage U of the machine and the power factor cos ϕ . Values for the current sheet A1.e f f range from some 104 to some 105 A m−1. Flux density level1 Bm a x has been discussed already ( 1–1.2 T ). The mechanical rotation frequency for two-pole generators is either 50 or 60 Hz. For reasons of good exploitation of magnetic field of the rotor, the winding length, L1 . 2 , is three to four times larger than the diameter. With this information we are able to calculate a diameter D for a first trial design of the stator winding. This diameter D in a superconducting excited machine is the mean diameter of the stator winding because there is no magnetic tooth region present in such machines. In this first trial design several different types of stator winding can be used. Results of the trial give the dimensions of the winding especially the radial extension of the winding. With the availability of the dimensions of the stator winding an analysis of winding factors, stray inductance, losses and cooling of the winding can commence. The unusual design of the coils challenges the imagination of the design engineer because the assumption that the calculation can use the formulae which are valid for ordinary synchronous machines with windings embedded in ferromagnetic slots is not correct. Superconducting machines have eliminated ferromagnetic parts in many regions of the machine, therefore even windings that look like conventional ones have to be calculated by revised formulae. Many variations of style are proposed for the stator windings of superconducting synchronous generators. The fact that ferromagnetic teeth vanish in such machines stimulates the creativity of researchers to think in ultra-high voltages for machine windings and to create unconventional solutions for stator windings. One example of this variety is shown in figure H2.1.9 where the winding consists of six spirally rolled pancake coils. In more conventional solutions the winding can be an ordinary two-layer chorded winding or the diamond or helical type. The possibilities for alternative winding constructions include concentric pancake windings and toroidal windings. In any case the winding must have an equivalent length adequate for the production of the voltage, torque and power. This equivalent length depends on the distribution of the inducing field as well as on the spatial distribution of the windings and their end turns and is not to be confused with the length of the iron yoke. The geometrical correlation of the windings has to be redefined in a new manner in contrast to conventional generators. With some simplifications with respect to shape of the windings and to the variation of flux density along the z axis in the mid-plane of such stator windings, we can find expressions for equivalent lengths of stator windings. It must be remembered that there are different equivalent lengths for the self- and mutual inductances. Figure H2.1.10 shows the basic flux density distributions, effective lengths for voltages induced in the straight part of a stator winding and voltages induced in the curved part of the coil overhang of the same winding. If the real winding is replaced by a very simple window-frame-shaped winding one can calculate Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.9. A spiral pancake stator winding.

the effective length. The following shows the results for this effective length in the case of the depicted typical traces of magnetic flux density along the stator bore. The replacement of the actual windings by the simplified ones seems to be a proper way for obtaining effective length values. Induced voltage is calculated by adding up all of the contributions induced in a loop; the ideal loop is the circumference of an area in which the flux trespasses. The best use of flux is achieved when this area is matched by the physical dimensions of the machine; thus the circumference of that area is a rectangular area with the width of a pole pitch and the length of the stator core or of the exciter winding. The real spread of the winding bars at the circumference of the stator bore always covers zones of 60° of circumference. This is due to the three-phase character of the machines; thus for a one-phase winding the simple model shown is adequate. Seven different profiles of flux density along the z axis of the machine bore are shown; in the circumferential direction sinusoidal distribution of flux densitv is assumed. This, although a rough approximation, meets reality rather closely due to the fact that windings with an appropriate chord use only the fundamental wave of the circumferential flux density distribution. In table H2.1.3, results are given for a 5/6 chorded winding shown on an arbitrary scale but with correct relations of length, width and coil overhang. We can see that approximately one third of the coil overhang is an adequate measure for the voltage contribution of the coil overhang, if flux is declining linearly in the region of the coil overhang.

Flux density along the z axis of the machine depends upon the actual shape of the rotor winding. The field winding is built up of straight and curved parts which contribute to different extents to the flux density profile. The stator winding uses this flux density profile for voltage production to different effect. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.10. Simplified flux density profiles along the z axis of the generator.

This effectiveness of voltage production is best when the flux density is constant over the total area of the stator winding. This can be achieved with a very ‘long’ field winding. Optimal choice of the length of the field winding is important as it effects the slenderness of the rotor. An example of this optimization can be seen in figure H2.1.11 where the indicated experimental machine field winding length is slightly too short (Zerobin 1985). The stator yoke is located at the outer circumference of the stator winding. Its gross dimensions can be found from the data already used for the calculation of the stator winding. The flux driven from Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.11. How to distinguish short and long field windings (Zerobin 1985). E1ph, total induced voltage in stator winding; Es t , induced voltage in overhang of stator winding; Eg , induced voltage in straight part of stator winding; Bm a x , peak flux density in field winding; B0 , centre flux density in field winding; B1 , flux density in centre of stator winding; Bj , flux density on inner stator bore surface.

the rotor winding depends on the winding distribution in the rotor and on stator yoke dimensions with respect to the dimensions of the rotor winding. The rotating flux density is distributed sinusoidally along the circumference of the stator bore and, in the main section of the machine, its maximum value will be almost constant; however, towards the end of the rotor winding as well as to the end of the stator core the maximum value will be smaller. Therefore the stator core will experience an uneven flux density distribution and, in addition, this will be affected by saturation effects in the magnetic core. The yoke is a cylindrical assembly of punched electrical steel sheets. There are no major slots in the inner surface of this cylinder because the stator winding is not buried in slots in such machines. The cylinder is subdivided in the axial direction into several packages and between these packages cooling gas can be circulated to remove heat caused by the iron losses in the yoke. The losses can be calculated in the same manner as employed in conventional machines provided that the different flux density levels are taken into account; these can be found from the procedure which is now described. The magnetic flux in the synchronous machine is produced by the rotor and stator windings and the flux density distributions which they produce interact in all operational modes of the machine. When describing the flux density and flux distribution in the stator yoke the following description refers to an experimental machine examined extensively with respect to these problems. As already described, the yoke is cylindrical as is the spiral pancake winding of the stator. The latter is completely embedded in the stator yoke. The rotor winding can also be seen as a cylindrical member of the cross-section of the machine. Figures H2.1.12 and H2.1.13 show an axiometric sketch of the stator yoke and a cross-sectional view of the location of the windings in this yoke. Ihe five packages of the yoke in figure H2.1.12 are clamped between two end-plates. The gaps between the packages are for the purpose of cooling the stator yoke. The shown construction of the stator yoke in large machines will be altered to yoke packages bolted together with long electrically isolated bolts. In a first step to calculate the fluxes inside the machine one is interested to see the flux generated by the field winding. The superconducting winding which produces the voltage-inducing flux is sensitive to the level of flux density which must not exceed a permitted value. Due to the location of the winding, the central flux density is far greater than that towards the end of the stator yoke (figures H2.1.14 and H2.1.15). This fact can be understood in a very simple manner from figure H2.1.16 which shows a longitudinal Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.12. Axiometric view of a staler yoke.

Figure H2.1.13. A schematic view of the simplified cross-section of a superconducting generator.

Figure H2.1.14. The magnetic flux density over Z: field-excited. Copyright © 1998 IOP Publishing Ltd

Figure H2.1.15. The magnetic flux density over Z: stator-excited.

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Figure H2.1.16. The direction of the flux at the end of the field winding.

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Figure H2.1.17. A comparison of flux density: no-load and three-phase short circuit.

section in schematic form. In the figure the current in the end of the coil winding, which is running in the circumferential direction, is considered to be concentrated in one filament. From the direction of the flux lines (from basic laws) one sees the flux reversal as shown in the previous graph. In no-load operation only the field winding is active. In normal operation of the synchronous machine, however, both windings carry currents and produce magnetic fields. A very special condition which the machine may experience is the steady three-phase stator short circuit and from the simplified theory for this condition the fields almost compensate each other. Figure H2.1.17 compares the flux density distribution at this special operating mode to the flux density of the no-load operation of the machine. It is clear that, as expected from the opposing magnetic fields, the general level of flux density is reduced; however, at the ends of the machine the flux density is higher than under no-load operation. Thus, although it may seem to be reasonable to reduce the cross-sectional area of the stator yoke at the end regions because of the no-load condition, this cannot be done. Reduced cross-sections at the end of the machine will cause the flux at high stator currents to follow paths for which no provision has been made in the design. There are other problems resulting from a short-circuit condition. The flux densities at different locations along the axis of the generator are out of phase with each other and, for clarity, therefore the flux density distribution is more easily shown in a three-dimensional model. The three dimensions are the flux density, the z coordinate and the time; figure H2.1.18 shows such a representation. Numerical integration of this flux plot provides the flux at short circuit. We know from normal phasor diagrams that flux density distributions at any operational state of the superconducting synchronous machine can be found by superimposing the field of the exciter winding and the stator winding. It is, of course, necessary to take into account the fact that there is a phase shift between the fundamental components of the flux density waves as expressed in the phasor diagram. The resulting flux density distribution which is constructed is not a true representation because the harmonics in the distributions are not taken into account by a method based on the fundamental components of currents, ampere-turn distributions and flux density distributions. Nevertheless the method allows the designer to obtain a good appreciation of the different fluxes in the stator at various operating states of the machine. From this information dimensions can be calculated; for all conditions flux density levels in the yoke should not exceed 1.5 T. The machine will then remain in an almost linear magnetic status which is the condition necessary for the superimposing of fields from the different windings. In this analysis, caution is necessary because the ohmic resistance of the stator winding may also cause some phase shift in the superimposing procedure due to the lower synchronous reactance which is a characteristic of superconducting synchronous machines. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.18. Flux densities along the axis under a three-phase short circuit.

H2.1.4.2 Rotor windings Rotor windings of turbogenerators are usually built in dipole form. The difference between this dipole and the ordinary dipole known from particle accelerators is the fact that it spins with high rotational speed and that the useful magnetic field is outside the winding. Therefore the rotor magnet windings have to be embedded in special retaining structures. These structures must keep the winding in position while rotating and transmit the torque which is exerted on the conductors of the winding to the torque tubes connecting the central rotor with the shaft ends at ambient temperature. Different retaining structures are possible. With respect to reliable operation of the superconducting rotor winding a structure which holds the conductors securely in position is most desirable. The most obvious solution is to mount the winding into slots machined into a massive central rotor in the manner adopted for conventional synchronous generators. The superconducting winding is wound and fixed firmly in these slots. However, there are also alternative approaches which use stacked winding structures with flat or slightly bent end windings in a racetrack form. The windings in this case are supported by pole pieces firmly attached to a central pole shaft and by beams which are attached to a cylinder which is shrunk on the rotor. The amount of compression of the racetrack coils produced by the beams when the cylinder is shrunk on is difficult to control and therefore for the very large machines, or prototypes for large machines, which are under construction most of the designs use the massive central rotor with machined slots. The design must give consideration to the cooling of the superconducting winding and the distribution of helium in parallel channels forming individual loops of flow under the rotational forces. The pancake construction with respect to the helium distribution is handled in much the same way as bath-cooled magnets and behaves similarly with respect to cooling performance in a rotating frame (see figure H2.1.19). Superconducting exciter windings in synchronous machines at normal-rated current are operated well below the critical current density of the superconductor. This fact is necessary because of the high direct currents which are added to the normal-rated current during faults. The following expression is a multiplication factor which, to a first approximation, provides the maximum current the winding has to sustain; the inductances shown here are discussed later in more detail. Their values can be introduced in per-unit form (relating all reactances to the nominal reactance of the machine given by the division of rated phase voltage by rated phase current). These per-unit values are very similar in all machines if equivalent constructions are chosen

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Figure H2.1.19. A schematic diagram showing the field winding inside a rotor.

where r1 is the mean radius of the stator winding, r2 the mean radius of the rotor winding and ry the inner radius of the stator yoke. Values of this multiplier range from 1.5 to 2 and depend upon the magnetic coupling between the windings in the machine. The exciter winding additionally will be stressed by a.c. currents of fundamental and twice the fundamental frequency in unbalanced faults. Amplitudes of these currents depend on the damper winding and its shielding performance and will be discussed in more detail in the section on damper windings. The time variation of the current in the superconducting winding can be sketched in a condensed form as shown in figure H2.1.20; this shows the short duration ramp to the peak current and the slow decay to the initial value of the field current. Time constants T2 of the rotor winding and T3 of the damper winding as well as stray coefficient σ12 will be described in the section on the calculation of reactances of the synchronous machine. Currents in lower case letters indicate per-unit values. We will now discuss the overlay a.c. current scheme which is imposed on the d.c. current scheme as shown in figure H2.1.21. The maximum of the a.c. current of fundamental frequency can be calculated with stray factors σ12 , σ13 and σ23 and shielding factor µD as a fraction of the initial exciter current. The duration of this a.c. burst is approximately three times the time constant of the armature. All coefficients

Figure H2.1.20. A simplified d.c. current load scheme. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.21. A simplified a.c. current load scheme.

mentioned here very briefly will be explained in detail in the section dealing with reactances of the superconducting machine. The a.c. currents are superimposed upon the d.c. current shown in figure H2.1.20 and cause additional problems for the stable operation of the superconducting winding. Both ramping and a.c. components cause losses in the superconducting winding, which have to be taken into account. The time variation ot the field current forces the designer to adopt rather complex methods tor the construction of the superconducting wires which make up the superconducting winding. The final conductor for the field winding is built up like a Roebel bar in a stator winding and can use resistive barriers in the matrix in which the superconducting filaments are embedded in order to reduce losses. However, the total exclusion of all possible detrimental effects on the performance of the superconductor will lead to very expensive conductors and to difficult and elaborate fabrication. (a) Damper winding The damper winding of a synchronous generator with superconducting field windings has three different tasks. Firstly, it must prohibit dangerous rotational oscillations of the rotor when sudden changes in torque or electric load of the generator occur; it is mainly for this function that the winding is known in conventional generators. Secondly, it is necessary to compensate for the unbalanced load of the three stator phases and to screen the superconducting rotor winding against a.c. magnetic fields to the extent that the chosen superconductor can remain in the superconducting state even during unbalanced operation of the generator; a similar function is also required in conventional generators during asynchronous operation. Thirdly, the winding will act as a thermal screen which intercepts heat radiating from the warm walls of the rotor to the cold superconducting winding containment. This task is present only in superconducting machines and will be addressed in the section on thermal design. The performance of a damper winding is characterized in an excellent manner by its time constant. However, due to the fact that most of these windings in superconducting machines are shell-like in construction, it is difficult to calculate their resistance and inductance (see figure H2.1.22). The attenuation curves for different impinging frequencies, as shown in figure H2.1.23, inform the designer of the extent to which flux, present at the outer circumference of the damper shell, will penetrate the shell and disturb the superconducting field winding. This attenuation effect is the main reason for using double damper svstems. By this means it is possible to optimize the designs of one shell for damping torque oscillations and one shell to screen a.c. flux density from the superconductor. The losses which are caused from these a.c. flux density waves are shown in figure H2.1.24. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.22. A sketch of damper shells.

Figure H2.1.23. Attenuation versus time constant of the damper shell.

Figure H2.1.24. Losses in the damper winding.

Figure H2.1.25. Storable energy in mass units of the damper shells.

When damper windings are stressed by the unbalanced electrical load of the stator winding, losses heat the damper winding. So only a limited imbalance is possible and also a limited time for unbalanced operation is allowed. Figure H2.1.25 shows some useful data with respect to unbalanced operation. Faults at the terminals of the machine will cause compressive forces to appear on the damper shells. We may consider that, in such events, the magnetic flux in the machine without fault is trapped by the windings. The closer linkage of the damper winding with the stator winding forces the damper winding to keep flux at the level which it had prior to the fault thus causing it to become the most stressed winding in the machine. A shell-like damper will show deformations as indicated in figure H2.1.26. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.26. A deformed damper shell (displacement shown enlarged).

H2.1.4.3 Reactances A forecast of the electrical performance of synchronous machines is accomplished with the aid of lumped parameter equivalent networks. The most critical of these parameters are the reactances of the windings. Analytical magnetic field calculations and some factors of adjustment for the reactances and the resistances are discussed in this section. All are used for the calculation of the time constants of the windings. No discussion of the foundations of d-q (direct and quadrature) axis theory is given here and the reader is recommended to consult textbooks dealing with the derivation of Parks equations. In the elementary approach adopted here only one reactance for each winding in the d axis or in the q axis is used for the equivalent network of the superconducting synchronous machine. It is necessary to mention that the machine is depicted at its basic level by this network. The shell-like structures of the rotor as well as the distributed stator windings can be represented more accurately by a number of T-branched equivalent networks put into a chain of such elements. It may be noted that even the basic level will provide most of the information required for the calculation of the transient electrical performance of the machine. So three distinct time regimes can be identified; steady state, transient and subtransient operation. The respective reactances together with the appropriate formulae for these reactances will now be discussed. The fact that, magnetically, superconducting machines are very linear gives less variation to the values of the reactances in different operational states. Therefore one set of reactances may be used for different conditions. The later discussion of transient performance will be based upon this convenient feature of the superconducting machine. In making a comparison of calculated and measured values a good agreement will be found; this will be shown in the section on performance. Reactances in the superconducting machine are built up from the turns of distributed windings as are the three phases of the stator winding and the dipole windings of the rotor. Between these windings is placed a shell (or more shells) of a conducting material and this acts as the damper winding. All reactances are influenced by the presence of the iron stator core which may be regarded as being infinitely permeable. The route for the calculation will start at the cross-section of an infinitely long machine. By this and the assumption of infinite iron permeability reactances related to the unit length of such a machine may be defined. Adding appropriate factors of length to these expressions finally gives the reactances. Most of the factors of length can be obtained by discussing the magnetic field produced by the winding itself or the field of another winding penetrating the mid-plane of the winding under investigation. This has been shown already in the discussion of the stator winding. The plot of flux lines shown in figure H2.1.27 gives an impression of the linkage of the windings in a machine. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.27. An outline of flux linkages.

The flux originating in either the stator winding or the rotor field winding links together all the windings with different amounts of flux. Lumped parameter equivalent networks express this fact; a very simple but satisfying approach is shown in figure H2.1.28. Self- and mutual reactances in the superconducting machine can be expressed as derivatives of the synchronous reactance in the machine. The discussion is based on per-unit values, which means that actual reactances are related to the main reactance given by the ratio of the nominal phase voltage and the nominal phase current. The central position in our calculation is occupied by the synchronous reactance which we can calculate very easily by using specific values of ampere-turn loading and flux density.

In this formula we find the effective value of ampere-turn loading A1 at the circumference of the stator bore and the peak flux density of the fundamental flux density wave 1B1 m a x in the stator bore. The factor in the brackets takes account of the ferromagnetic iron core. This core enhances the reactance by the factor shown if the permeability of the core is assumed to be infinite. Radius r1 is the mean radius of the stator winding and radius ry is the inner bore radius of the stator yoke. The length l11 is an equivalent length of the stator winding and l12 is an equivalent length expressing linkage of the stator winding and rotor field winding. We can now proceed in a straightforward manner to express other reactances of the machine by the following formulae

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Figure H2.1.28. An equivalent network for a superconducting generator.

As can be seen in the equivalent network, we have different coupling of the stator winding to the field winding and to the damper winding. This is expressed in the set of formulae by additional reactances and factors indicated with asterisks

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With these reactances we can proceed and express reactances transformed to the stator frame

The reactances satisfy the following equations

The subscripts in the different expressions indicate by numbers the location at which the expressions are adjoined, thus the subscript ‘σ ’ indicates stray-flux-based and the subscript ‘h ’ indicates the main-flux-based reactances. A reactance with two equal numbers in the subscript is the complete inductance of the respective winding multiplied by the angular frequency. Two different numbers indicate coupling reactances. In all these formulae we can find geometrical factors expressing the quality of coupling. These stray factors can be calculated by the following formulae

The K factors in our expressions are enhancement factors taking into account the presence of the stator iron core and the respective position of the winding to this yoke. Therefore they bear the subscript ‘ y ’; these factors may also be called imaging factors. Flux and flux density enhancement due to the stator core may also be derived by an imaginary winding in the stator core; the bore surface of the stator yoke is Copyright © 1998 IOP Publishing Ltd

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acting like a mirror. This simple view of the action of the stator core gives reasonably good results with respect to the calculation of reactances but they are only a rough guide to the flux density measured in real machines. If more detailed information on the flux distribution is required a more complicated picture of winding images expressed by magnetic surface polarities sitting at the bore of the stator yoke is necessary. Planar distributions of flux density can be found with commercial finite-element method (FEM) codes. For analytical formulae we use the K factors as an easy and appropriate tool

We also use a prime to indicate transformation of the respective reactance to the stator frame. As in transformers, this transformation is connected with a transformation ratio k1 expressed below. Transformation does not affect power whether it is calculated with original resistances or reactances and the adjoined current or with primed values and the stator winding current

Here m expresses the number of phases (usually m = 3, but in the case of field winding m = 2), ξ is the winding factor with the superscript in front indicating the harmonic number (we use here only the fundamental component, superscript 1) and the subscript indicating the winding. The number of winding bars is expressed by z. The transformation ratio is kt with a subscript indicating whether it is valid for currents I or resistances R. H2.1.5 Cryoengineering of a.c. rotating machinery, mechanical design of key components and optimization of mechanical and thermo-technical design The thermal and mechanical design of superconducting rotors for synchronous machines is different from the design of rotors of ordinary synchronous generators. The fact that within the rotor a temperature gradient must be established from ambient down to the temperature of liquid helium and at the same time the total torque of the machine even in fault conditions must be transmitted from the same cold level to ambient shows clearly the great challenge for cryoengineers. A rotor is built up of basic components which are commom to all of the various designs which have been produced. First we have a driven shaft end and a supply shaft end; these two parts remain at ambient temperature at all times. These shaft ends are linked together by a tube-like outer rotor at ambient temperature; this outer rotor can carry a damper winding but its main purpose is to be the backbone of the rotor and to be the vacuum containment vessel. Inside this vessel we find torque tubes connecting the cold central helium-filled inner part of the rotor with the ambient temperature ends of the machine. These torque tubes act as support members and as a heat exchanger. At least one of the torque tubes must be designed to accommodate the differences in shrinking between the outer (warm) and inner (cold) rotors. Copyright © 1998 IOP Publishing Ltd

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The inner rotor holds the liquid helium and the superconducting exciter winding; the helium is compressed by centrifugal forces. The temperature increase due to this compression is diminished by heat conduction to the massive parts of the rotor. The essential behaviour of the helium in this rotating frame is shown in the enthalpy-entropy diagrams of figure H2.1.29. In the diagram four different isentropic compression routes are indicated; the first route labelled ‘a’ shows the compression of helium supplied with heat at critical temperature. The enthalpy added by compression is related to 3000 rev min−1 and a radius of 0.5 m. The temperature at the periphery will be approximately 7.5 K in this first case. If liquid helium at one bar (105 Pa) is supplied, the final temperature is 5.5 K (trace ‘b’). Supply of subcooled fluid is one possible way of reducing the temperature at the periphery and trace ‘c ’ shows supply of liquid helium at 3 K resulting in 4 K at the periphery. The last example, trace ‘d’, shows the case of a pressurized supply of subcooled helium. The fluid enters the rotor with 5 bar (5 × 105 Pa) and 4 K and reaches at the periphery less than 5 K at a pressure well above 20 bar (20 × 105 Pa). From these four examples one can deduce that the control of the temperature of the cooling helium in the rotor is of prime concern for safe operation.

Figure H2.1.29. Isentropic compression of helium in a rotating frame.

In the enthalpy-entropy diagram of figure H2.1.30 helium enters the rotor in the centre at state 1, say fully liquid at 1 bar (105 Pa) and 4.2 K. On its way to the periphery, located at a radius of 0.5 m, compression energy is added and helium is in state 2 at a higher temperature as well as at a higher pressure. Now an isobaric supply of heat is imaging the heating of helium inside the rotor by thermal losses due to radiation, thermal conduction or due to conductor losses. A further increase in temperature is observed when point 3 is reached; from here helium, on its way back to the centre, is decompressed and ends at state 4. Here the helium is again at the same temperature as it was supplied at but the quality has changed; a considerable amount of liquid has been vaporized and this vapour is now in the vapour-liquid mix of the helium. This cold vapour is further used in the heat exchangers of the torque tubes and in the current leads. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.30. Isentropic compression of helium in a rotating frame with heating.

This process of cooling in the torque tubes takes place at constant radius and therefore the process is isobaric (if the pressure drop in the heat exchanger piping is neglected) and is shown in the enthalpy-entropy diagram. Thus in a good approximation to reality the process proceeds along the constant-pressure line, crosses the phase boundary when all of the liquid is vaporized and ends at the exhaust port of the generator with gaseous helium at a temperature well above 4 K. Under regular operating conditions the supply of helium to a rotor is controlled in such a way that only gaseous helium enters the torque tubes and current leads. The exhaust helium in most cases will be recycled in the refrigerator for further use. Figures H2.1.31-H2.1.33 illustrate the above discussion.

Figure H2.1.31. The temperature course of helium starting with varying state of liquid at the centre of a rotor. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.32. The temperature course for two rotor supply temperatures of helium.

Figure H2.1.33. Pressure in a rotating helium bath with a vapour core.

When the fluid level in the rotor is at a radius larger than zero, compression due to the smaller specific mass of gaseous helium in the centre of the machine is lower and the change in temperature along the radius is different from that shown in figure H2.1.31. Consideration of the figures shows that the supply of cooling fluid with temperatures below 4.2 K will also give a lower operational temperature for the superconducting field winding. In figures H2.1.32 and H2.1.33 the numbers beside the traces in the diagrams indicate the location (radius in metres) of the vapour—liquid interface. A supply of gaseous helium will give the worst conditions of operation for the superconducting winding; this was shown in the enthalpy-entropy diagram of figure H2.1.29 with trace ‘a’. Avoiding gaseous helium in the rotor calls for phase-separating equipment at the entry to the rotor; by suitable bypass constructions using the density difference of vapour and liquid helium, only liquid will enter the central rotor. The vapour fraction in the supply is diverted and used in less difficult cooling areas. This equipment will add some more complexity to the cooling circuit of the rotating rotor. A similar change from less to more complicated structure will occur if self-pumping of the rotor is used. The difference in density of the incoming and outgoing helium on its way through the rotor can be used for lowering the pressure in the central space of the rotor. By this means a lower operational temperature of the winding can be established. Figure H2.1.33 also contains an example of a sub-atmospheric centre pressure. Since the supply Copyright © 1998 IOP Publishing Ltd

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system should be kept under a slight over-pressure otherwise there is a danger of air being sucked into the system, this type of cooling system requires some means of pressure reduction inside the rotor. The different tasks that the rotor has to master may be summarized as the concepts of transmission of torques and forces at minimum risk, reduction of heat in-flow to a minimum and accommodating thermal shrinkage differences in warm and cold components. Mechanically a rotor in a superconducting machine must sustain the same forces and torques as the rotor of a conventional machine. The reduction of heat flow influences the first task as much as the thermal shrinking problem. There are also some differences compared with conventional rotors with copper windings with respect to the electromagnetic forces and torques. The high currents in the superconducting winding cause high forces and the ironless construction of the centre of the machine causes different levels of currents and torques under fault conditions compared with the conventional machine. Figure H2.1.34 gives a simplified synoptic view of the problems and figure H2.1.35 shows solutions, parameters and dimensions to be calculated.

Figure H2.1.34. Problems to be tackled.

It is clear that the torque tube, which acts as a support member between the winding and the driven end of the machine, must also accommodate fault torques, bending stresses caused by the weight of rotor components and stresses due to thermal shrinkage. The latter in the case shown here is to be endured by the torque disk, a membrane which accommodates the shrinkage by regular deflections. The inner rotor shell on the other side will experience less torque but will be subjected to bending stress and centrifugal force stress and current forces. From an idealized point of view it is possible to find, for instance, a maximum of radial pressure which such a shell has to sustain. The worst case is attributed to a three-phase short Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.35. Parameters to be calculated.

circuit at the terminals. In this case we find

where hc is the thickness of the rotor winding; σc the mean mass density of the rotor winding; σt the tolerated stress in the shell and dm i n the minimum thickness of the shell. In these formulae we use the geometrical dimensions of the rotor winding and the shell as well as the mean mass density of the rotor winding and tolerated stress levels of the material which are employed. Stressing of the shell by its own mass forces might require correction of the minimum dimensions. Figure H2.1.36 shows the combined forces as developed by magnetic forces and by centrifugal forces. The discussion of the electrical performance of the damper shell has depicted the shell as being

Figure H2.1.36. Forces on field windings caused by centrifugal acceleration and magnetic field. Copyright © 1998 IOP Publishing Ltd

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deformed due to forces occurring in the fault conditions of the generator. The origin of the forces causing these severe deflections of the damper shell are shown in figure H2.1.36 where radial and tangential pressures at the time instant ω t = π after the short circuit at the terminals of the generator are shown. From torques and other forces and idealized assumptions on load distribution we can find formulae to determine the minimum thickness of such a damper shell

If usual values for the design of the superconducting machine are chosen we may express an approximation formula which takes into account most of the factors in the technically feasible range. We find

The formulae presented above are first guidelines for the mechanical dimensioning of components and these are very different from those for the design of a conventional synchronous machine. However, in other design adaptations the differential shrinkage of the rotor components is accomplished by a warm diaphragm which is part of the outer rotor (see figure H2.1.37).

Figure H2.1.37. A 120/400 MW KWU test rotor.

In the rotor shown in figure H2.1.38 the same task of shrinkage differences is accomplished by a sliding member in the rotor itself. Different approaches are possible also with regard to the cooling in the superconducting generator. Figure H2.1.39 presents some of these approaches with different designs for the key component ‘torque tube’. In version A the torque tube is at the outer diameter of the rotor and is cooled by a separate cooling Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.38. A 250 MW Alsthom prototype rotor.

Figure H2.1.39. (A) A torque tube cooled with a pipe heat exchanger. (B) A torque tube cooled with a distributed heat exchanger. (C) A torque-disc-cooled heat exchanger. Copyright © 1998 IOP Publishing Ltd

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stream flowing through pipes attached to the torque tube. Exhaust helium is returned on a number of manifolds. The design variants B and C have the advantage that the machine can be supplied with one cooling stream; this makes the transfer of cooling fluid to the stationary frame easier. Version C shows a stepped torque tube in which the step is used for a torque disc which accommodates thermal shrinkage. In cases A and B diaphragms have to be added at the locations where the torque tubes end at the shafts. Another possibility in these designs is a split bearing; in this approach the cold rotor at the nondriven end is extended into a separate bearing nested in the main bearing of the rotor. A sliding member (as already shown) and constructions with inclined spokes connecting a hub with a rim can also accommodate the shrinkage. Bellow-like systems may be another approach. The latter seems to be more complicated in construction and calculation than the other proposed solutions. Design C also shows a distinct heat interception at the cooling of the torque disc. Designs A and B have distributed cooling along the torque tube. The position of this cooling is critical with respect to the minimum of heat input introduced to the cold rotor. We now turn to a discussion of some design data on the cooling of the torque tubes. Two different materials are chosen for reference, the more likely candidate, stainless steel, and a possibility for the future, fibre-reinforced plastic. The cooling is supplied either by an ideal Carnot refrigerator or a real Carnot refrigerator which is worse by a variable factor of 10 to 1 in the temperature range from 4.2 K to 300 K. An uncooled torque tube, which means that ambient temperature is connected directly to a component at the temperature of liquid helium will show losses of 212 kW m−1 based upon an ideal refrigerator. The dimensions of this specific loss come from the fact that heat of power P is conducted over a cross-section A along a torque tube of length L giving the expression PA/L to give watts per metre for the specific loss. With a nonideal refrigerator the power necessary at the terminals of the refrigerator naturally is raised to a value of 2130 kW m−1. The corresponding figures for fibre-reinforced plastic are 19.1 kW m−1 and 191 kW m−1. We immediately see that stainless steel from the cooling point of view, at least, is not the best choice. Nevertheless we have to bear in mind that for rotating machines steel is the premium choice; the requirement for vacuum-tight structures which are reliable for long-term operation in a mechanically vibrating application can be achieved only, at the present time, by using stainless steel. Intercepting heat at a location positioned approximately halfway along the length of the torque tube reduces the losses to a great extent; for the ideal and nonideal cases for stainless steel the specific losses are reduced to 47 kW m−1 and 303.5 kW m−1. If heat is intercepted at three locations situated on the torque tube in the range from 0.14 to 0.4 of the torque tube length (calculation starts at cold end) the respective losses are 30.3 kW m−1 and 161.1 kW m−1. A homogeneous heat removal over the total length of the torque tube would result in the lowest limit attainable of 24 kW m−1 and 60.8 kW m−1. For fibre-reinforced torque tubes all values are approximately a factor of ten lower. So for the distributed removal of heat we find 1.5 kW m−1 and 4.8 kW m−1 ( Bodner 1987 ). Dimensions of the torque tube are fixed by the ultimate torque to be transmitted and by the bending characteristics of the rotor. Within the framework of the mechanically necessary cross-sections and length it is possible to determine the optimum design for the synchronous machine under construction. The complexity of the design of the rotor may be elucidated by figure H2.1.40 which shows the modelling of an experimental rotor with respect to torsion and bending oscillations due to imbalance or due to impulse of electromagnetic torques at sudden load changes or faults at the terminals of the generator. H2.1.5.1 Thermo-technical design of key components In the preceding discussion some information has been given on the design of heat intercept at the torque tubes. Therefore we will concentrate here on the task of how to avoid excessive heat influx to the rotor. As usual in cryo devices, heat is best kept from the deep low-temperature parts of the machine by a high vacuum. When all parts are at ambient temperature the reasonable vacuum conditions which can be achieved are lower than 10−3 mbar (0.1 Pa). The experimental rotor used as a reference in this chapter

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Figure H2.1.40. Calculation model for oscillations of a rotor.

had a starting vacuum of 10−4 mbar (0.01 Pa). The thermally insulating vacuum can be either sealed off or continuously pumped during operation. The first approach makes the rotor easier to construct because the pumping port does not need to be accessible during rotation. On the other hand a sealed-off vacuum requires very careful manufacturing and extended leak testing during construction. However, it has been shown that both stationary and rotating cryogenic equipment can be constructed to the standards required for having sealed vacuum systems. During operation the vacuum drops to values in the range of 10−6 to 10−8 mbar (100–1 µPa) due to the cryo-pumping of the cold helium parts. If a pump is included which can also trap any helium which may leak into the sealed-off vacuum even better values may be obtained. Such pumps can use physisorption on zeoliths. The pump can be regenerated in the usual overhaul work. For pumped rotors one has to construct vacuum-tight rotating seals; this task can be solved with ferro-fluidic seals. In any case the supply coupling of such a rotor will become very complicated. Additionally to the high vacuum, heat influx in superconducting rotors is reduced further by thermal screens. One of these screens has been introduced already in the concept of the damper winding. The shell-like member outside the cold inner rotor sitting in the vacuum space of the rotor is, for this reason, connected at both ends to the cooling station of the torque tube. Thus heat flowing by radiation from the warm outer rotor to the damper is conducted to these cooling stations and will not affect the central helium cold rotor. It is necessary for the shield to have a polished surface and by this means most of the incoming heat can be reflected. Adding thermally floating thermal shields such as superinsulation foils will further reduce heat transmission to the cold core of the superconducting machine. Thermal insulation and heat interception can be carried out in such an effective way that the cooling power of a superconducting generator is very low and makes it possible for the efficiency of the generator to exceed 99% (for superconducting excited machines). The principal arrangement of the thermal insulation, the damper screens, the vacuum barriers and the mechanically stiff members carrying forces and torques is Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.41. A cross-section of superconducting rotor insulation layers.

shown in the cross-section of figure H2.1.41. Supennsulation in this arrangement will be compressed by the centrifugal acceleration forces which will make its performance less effective than for uncompressed superinsulation. To ease this problem the multilayer arrangement in most cases consists of very lightweight aluminized foils and, with this safeguard, the performance is acceptable. Multilayer insulation is also an inherent safety factor for the machine in case of accidental loss of vacuum because it helps to keep down the heat influx following a loss of vacuum and helps to control the increase of return helium gas to the refrigerator due to the increased heat influx to the central rotor. The thermally well insulated rotor consists of a considerable number of different materials which must be cooled down to the temperature of liquid helium. This cooling down depends upon the size of the rotor, the amount of coolant which can be passed through the rotor and the efficiency of the rotating transfer system. For convenience, cooling down will be done at low rotational speed. An estimate of such cooling down can be found in figure H2.1.42. The graph shows the temperature versus the product of coolant mass flow and time divided by the mass to be cooled. The product of coolant mass flow and time of cooling expresses the cooling work done on the cooled equipment. Relating it to the unity of mass to be cooled makes the curve applicable to the estimation of any cool-down. The cooled components of the rotor consist of copper, superconductor, stainless steel and electrical insulation. For reasons of over-pressure in the rotor during the very unlikely event of a quench, the cross-section for helium return in the rotor of the experimental machine discussed was made quite large. The pressure drop is low and the utilization of enthalpy of the helium gas inside the rotor is poor and therefore it will return to the refrigerator at a rather low temperature. Therefore cool-down cannot be done with the same efficiency as cool-down of stationary equipment. Inside the rotor ‘cold’ is distributed mainly by heat conduction in poor heat conductors; nevertheless the masses involved are large and therefore thermal Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.42. A normalized cool-down curve for a superconducting rotor material mix.

equilibrium of the rotor will not be reached at the instant when the rotor can first be filled with liquid helium. At an early stage of operation the machine therefore will show a need for an enhanced supply of helium. It follows that the cool-down time for machines with masses in the range of several tonnes will be very long. On the positive side these obviously huge thermal time constants will facilitate operation during a period with a varying supply of cold. A superconducting generator will operate with a substantial safety margin and therefore short-term lack of helium will not influence its performance. Even a lack of helium for hours should not influence the temperatures inside the generator substantially. In this situation reducing load and field current for instance would reduce losses, leading to conditions which open a time window for reestablishing a supply of liquid helium by switching to some other source of cooling. An example of temperature distribution along the torque tube in an experimental machine is shown in figure H2.1.43. Different mass flow rates for heat interception change the heat withdrawn at the cooling stations which affects the temperature distribution. The mass flow rates in the cooling station are indicated in figure H2.1.43 too. In this figure we can see again the effective reduction of heat influx to the low-

Figure H2.1.43. Temperatures from ambient down to liquid helium along the stretched path: shaft flange, torque tube and the central rotor. Copyright © 1998 IOP Publishing Ltd

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temperature region by one cooling intercept at the location of the torque disc. The slope of the temperature curve is a measure of the heat flowing towards the cold end. Two distinct changes in this slope can be seen; the most remarkable change being found at the location of the torque-disc heat exchanger. H2.1.6 Electrical characteristics and regular operation including a comparison with conventional a.c. rotating machinery The main advantage of superconducting generators in comparison with conventional generators is improved efficiency. This topic will be discussed in the section dealing with the economic benefit expected from superconducting generators. Among other advantages are some points arising from the sole operation of such a generator in a grid. In the superconducting machine the excitation is large and therefore armature reaction due to stator current is smaller than in conventional machines; this arises from the value of the synchronous reactance. Figure H2.1.44 shows the variation of the ratio of the pull-out torque of any generator with regard to its synchronous reactance. We see that superconducting machines with their reactance well below 1 p.u. show increased ratios of pull-out torque to rated torque.

Figure H2.1.44. Ratio of pull-out torque to rated torque.

This can also be expressed in terms of the rotor angle which is smaller in superconducting generators than in conventional machines for the same load and therefore there is higher stability against load changes in power networks with superconducting generators. Emphasis is often placed on the so-called critical clearing time after a fault when making a comparison between different machines. In this time definition the inertia H, synchronous reactance xd and transient reactance x ’d are linked. As an example, decreasing xd results in an increased clearing time and therefore for superconducting machines this should give an additional operational advantage. However, the inertia of the superconducting generator is reduced to a smaller value than the inertia of the conventional machine and this leads some authors to the conclusion that the advantage of a reduction of reactance is balanced by a reduction of inertia in superconducting machines. This may be true when only the generator is taken into account but the effect on the clearing time must take account of the inertia of the total rotating string of the prime mover and generator. In this case the reduction of inertia due to the smaller frame size of the superconducting generator is relatively small and the advantage in the critical clearing time remains an advantage of superconducting generators. The current circle diagram at leading and lagging currents in the machine clearly shows the difference in operation between the superconducting generator and conventional generators. Figure H2.1.45 shows such a circle diagram and we see that operation with a leading power factor is improved in superconducting Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.45. Current circle diagram.

generators. We can see two different static stability thresholds. One is valid for conventional generators and sits inside the current circle diagram; it reduces the possible currents in pure leading conditions to less than half the rated current. The static stability threshold for superconducting generators is outside the current circle diagram so that any point in leading operating conditions is available to the operator of such a machine. The remaining excitation at full nominal current in pure leading conditions is still large enough to maintain stable machine operation. An advantage one can see from the current circle diagram is that the restriction of load current under lagging conditions due to excessive losses in the rotor winding of conventional machines can be abandoned. Superconducting field windings are dimensioned such that all points of the current circle diagram can be reached without endangering the operation of the generator. This statement is correct for stator currents higher than the rated values. Thus we have a generator able to deliver rated power at any active and reactive power at rated current and, with respect to over-heating of the rotor winding, at higher than rated values. Excitation of the superconducting machine needs only a small power for holding the field current at the preset level. Fast changes in the field current will require higher voltages and this voltage multiplied by the currents usually used for excitation of superconducting field windings gives a high rating of the excitation source. The costs for this can be avoided if the machine is used in an operational mode where only slow changes in excitation level are necessary; this can be achieved without difficulty because the machine is not sensitive against under-excitation or when the active load changes suddenly. H2.1.6.1 Faults and transient operation including comparison with ordinary a.c. rotating machinery Under fault conditions the superconducting machine behaves slightly differently compared with the conventional machine. The large air gap reduces the linkage of the windings; this has been discussed in the section dealing with reactances. At the same time it was shown that the equivalent network of the machine remains very close to that of the conventional machine. From this it follows that the formulae for currents in the stator and in the field winding can be taken from the literature for conventional machines. This is especially true for the three-phase short-circuit condition but for a two-phase short circuit some modification may be necessary. In this context, unlike the conventional generator in which the interest of the designer and user is focused upon the stator current, in the superconducting generator we are more interested in the field current which stresses the superconducting winding. Therefore only the formula for the stator current in a three-phase short circuit is written down here. We can see that the components of the current are the same as can be found in conventional machines. As a result of the low synchronous reactance, the steady-state short-circuit current is bigger by a factor of three to six than in conventional Copyright © 1998 IOP Publishing Ltd

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machines. This should not be a problem because the machine will be taken out of operation as soon as such a fault occurs. In transient and subtransient conditions currents are almost equal to the currents which might be expected in conventional machines. This is linked to the fact that transient and subtransient reactances differ only slightly from the values found for conventional machines. We can calculate the current with the aid of the following formula

Phases b and c can be calculated with the appropriate time shift in the angular functions. The load angle δ, which indicates from what load the machine was short circuited, can be set to zero when looking at the traces of the currents. A typical result for a superconducting machine generator (SMG) is shown in figure H2.1.46. The stationary component related to 1/xd is approximately half the value of the maximum current. In the field winding current, figure H2.1.47, we can see the effect of the damper winding inhibiting the sudden increase of the field current and the long decay time caused by the large time constant of the field winding. The relative current trace can be calculated as follows

Figure H2.1.46. The stator current at short circuit at no-load condition δ = 0 in the SMG.

The reading from an oscilloscope screen shows traces of the current from an SMG with xd = 0.32 p.u. at short circuit from no load. The respective field winding current is shown in figure H2.1.47. Figure H2.1.48 shows a short circuit under the same conditions as before. The field winding was changed to a winding with tighter coupling to the stator winding. The synchronous reactance remained constant but the parameters x ’d , x’’d and µD changed their values. The principal shape of the currents is comparable but the ultimate values are different. Figure H2.1.49 shows typical traces of both the stator and field currents in the case of a two-phase short circuit. The curious shape of the field winding current is due to the double-frequency component present in the field winding. The current consists of components not decreasing exponentially (constant Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.47. The rotor current at short circuit from a no-load condition.

Figure H2.1.48. Currents at short circuit in an SMG.

amplitude of a.c. currents and constant value of d.c. current) and components decreasing exponentially with three different time constants. Two of these time constants are only present in two-phase short circuits, namely Td I I and Td I I ; the other one is Ta , already used in the three-phase short-circuit calculation. Calculation of the field winding current is possible only with a long equation. The formula given here is only valid for the case of symmetry of the direct and quadrature axis subtransient reactance. The coefficient A in this formula takes into account the fact that, in superconducting synchronous generators, no common main reactance for all the windings is valid. The lack of magnetic material links damper winding and field winding in a different way to the stator winding. The same is valid for the coupling of the field and damper windings. The coefficient is as follows

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Figure H2.1.49. The relative stator and field winding current for a two-phase short circuit under no-load conditions.

Figure H2.1.50 shows the traces of stator current and of rotor current from a very rough experiment from which we can see that mechanical oscillation frequencies of the rotor are in the same range as can be found in conventional machines when sudden load changes or faulty synchronizing is applied to the generator. As an additional example for insight into the machine’s behaviour calculated and measured values of flux density of the stator winding are compared. The calculation is elaborate because the calculation code must cover three-dimensional flux density distributions. However, a code based on Ampère’s law Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.50. Resynchronization of the idle running synchronous generator (correct speed, correct voltage but out of phase).

Figure H2.1.51. Plot of lines with equal flux density in a longitudinal cut of an SMG.

can deliver these data if the presence of iron is taken into account by mirror windings or image magnets at the iron surfaces. Results of such calculations can be seen in figures H2.1.51 and H2.1.52. The results have been compared with the results taken from the experimental SMG and show good agreement. Figure H2.1.51 shows a section through the ends of the field coil winding. The line of symmetry is the z axis which in this case is identical to the axis of rotation of the rotor. The distance between the coil Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.52. Plot of lines with equal flux density in a cross-section of an SMG.

ends contains the long side of the field winding which cannot be seen in this sectional view. In figure H2.1.52 lines of constant flux density in a cross-section of the same field winding are shown. The cross-section is located at the centre of the machine so that the winding covers equal distances from the cut plane to the ends of the winding (see decay of flux density along z coordinate of figure H2.1.53). In figures H2.1.51 and H2.1.52 it can be seen that at the nominal field winding current the highest value of flux density is 3.5 T. From the figures we can conclude that the highest flux density in the field winding is reached somewhere in a corner at the inner end of the field winding. With due consideration we may expect the maximum to be and this has been confirmed experimentally. The configuration and length of the damper winding shell make a considerable contribution to the level of magnetic field to which the field winding is exposed and this must be carefully evaluated during the design process. An example of a comparison of such a field in two experimental machines is shown in figure H2.1.54. In view of the rather complicated calculation it was decided to build a scale model. On this model the flux density was measured and compared with the distribution found in the full-size machine and the conclusion is reached that building models for these purposes might be a less expensive way of getting reasonably accurate results than undertaking complex computations. Scaling of the dimensions and electrical conductivity of the damper shell in the models has to follow distinct laws which take care of the theory governing eddy current screening. A final example of the transient behaviour of a superconducting machine covers the very unlikely Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.53. Plot of lines with flux density in a length cut of an SMG (calculated—full curve; measured—dotted curve).

Figure H2.1.54. A comparison of flux densities along the z axis of a superconducting machine with a damper shell.

event of a quench of the superconductor winding in the machine. Figure H2.1.55 shows the changes of the current in the field winding and the voltage at the terminals of the field winding during a quench. The voltage is used by the safety device (the quench detector) to indicate whether or not there is a quench and this is seen to be a very noisy signal. However, by integrating over a preset period of time the noise may, to a large extent, be eliminated and the true d.c. offset caused by resistivity somewhere in the superconducting winding can be detected. The removal of noise takes some time which we can see in the trace of the current. The results shown in figure H2.1.55 come from a machine where the current source for the field winding had a voltage threshold set to a value only slightly higher than the voltage necessary to drive the current through the superconducting winding. As soon as a resistive zone in the winding appears, the current begins to drop until the detector initiates the switch which commutates the current to a discharge resistor. Immediately after this commutation the highest voltage appears at the terminals of the field winding. The current and voltage after this commutation decrease exponentially to zero at a time constant governed by the inductance of the field winding and the value of the discharge resistor; the latter is optimized at the design stage. By this optimization procedure the dielectric stress of the field winding and the heating of the superconducting coil have to be balanced and usually rather low voltages for discharge are used. Moreover, the damper winding acts like a quench tube known from ordinary superconducting coils. Part of the magnetic energy stored in the field winding is transferred to Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.55. Traces of current in and voltage at the field winding prior to and during a quench.

the damper shell during the rather rapid discharge and is dissipated there in the resistance of the shell. In most field windings, due to the effects described here, quench causes only a modest increase in the temperature of the coil; thus cooling to a safe operating temperature following a quench should not be a major problem when running such a machine in the public grid. H2.1.6.2 A review of operating experience Experience with superconducting generators has been obtained with different operating states of such machines. Most of the experience was gained with experimental machines running for rather short periods and the conclusions from most of the reports are that, in principle, the operation is as expected but, due to the experimental nature of either the machine or the test, not all of the objectives of the tests could be fulfilled. What are the problem areas in operating superconducting generators? First of all we find reports dealing with problems in cool-down and in maintaining low temperatures. These difficulties in most cases can be traced back to problems with the vacuum and it is often the case that systems with sliding vacuum barriers in the rotating cryostats are not as reliable and tight as expected from preliminary tests. Another reason for poor vacuum performance is inadequate quality of the welding of the vacuum vessel. With high-quality welding, avoidance of porous parts of welding seams by careful selection of materials and the proper use of purge gas and technology, it is possible to build sealed-off structures which hold their vacuum for one or two years. For instance in the experimental SMG the sealed-off vacuum never degraded to a level worse than 10-3 mbar (0.1 Pa) which is sufficient to commence a cool-down of the machine. Nevertheless having a vacuum lower than this at the start of cool-down reduces the consumption of cryogens during the cool-down but it will not shorten the cool-down time because cryopumping of cold surfaces will improve the worst starting vacuum drastically and the machine ends up at the same conditions as the machine with superb starting vacuum values. A typical cool-down of an SMG is shown in figure H2.1.56. The cool-down is split into three time phases. The first covers the cool-down from ambient to the temperature of liquid nitrogen, often performed with liquid nitrogen as cooling fluid. The second covers the cool-down from 80 K to the temperature of liquid helium and the third covers the phase where the rotor is filled with liquid to the necessary level and is maintained at this level during operation. In the example below, the cool-down with liquid nitrogen is not shown but this takes approximately one hour. The cool-down to helium temperature is performed with helium taken from small transport dewars (60 1 capacity). The trace of the cool-down temperatures therefore is interrupted at the time of change of supply. The short interruption in supply causes the rotor to rise in temperature which is shown by the trace of temperatures starting at approximately 40 K in the added part of the graph shown in figure H2.1.56. (For steady-state operation in superconducting conditions helium was delivered from a 1000 1 dewar connected Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.56. Cool-down of an SMG.

to a helium liquefier.) The cool-down of a generator in a power station will be done in a closed circuit with a dedicated liquefier and in this case one can expect a smooth change in temperature in the rotor. The rate of temperature change will be controlled by the amount of cooling fluid transferred to the rotor. The different parts in the rotor have different thermal time constants and will, therefore, cool at different rates. The cool-down rate must not allow too much time lag because the temperature distribution will be uneven and considerable thermo-mechanical stress will be introduced. A very common value for differences in the temperature permitted in cryogenic installations is 40 K. In the example shown this difference is never exceeded but there have been shock cool tests on this equipment which have passed without any damage. The area of disturbed operation of the superconducting machine has been described already with regard to various fault conditions. The numerous questions which arise in connection with this transient operation have been examined in many reports both theoretically and experimentally. The conclusions from these reports state that no unexpected performance of the superconducting machine will be observed; however, it must be said that up to now no superconducting generator was exposed to full short-circuit conditions. Calculation codes checked with short circuits at less than full short circuits show that the effects to be expected can be tolerated by the components of the superconducting generator. Steady-state operation of superconducting generators is reported under no-load conditions. From tests at low power performed with the SMG one can conclude that such tests are very unspectacular. The superconducting machine runs very smoothly on the grid and one can find no deviation from stable running conditions. Loaded tests are reported with small power machines and on back-to-back tests of a 20 MVA superconducting generator. These reports do not show any spectacular deviations from stable operation either (Jones et al 1984). The situation regarding measuring equipment is rather difficult to evaluate; from some reports one may conclude that measuring sensors in the rotating frame are far from being tested for industrial use. However, most of the equipment in the experimental machines was chosen with short-term operation in mind. Also in the SMG several sensors failed, for instance due to the large compressive forces in the rotating frame. However, in the SMG a central core of instrumentation was functional during all the different tests; these were very rugged sensors and heavy gauge measuring wires were used. Transfer of signals via sliprings and with rotating electronic multiplexing was a very useful method of acquiring data from the rotating frame. In most cases the transfer of cooling liquids to the rotor was done with bayonet structures and the thermal losses of such transfer units seemed to be approximately in the range of 10 W. The long-term performance of such transfer units was checked in separate test stands and reports described their operation Copyright © 1998 IOP Publishing Ltd

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as being satisfactory. No data on the performance of transfer units are available from actual machines. From tests on the SMG one can report that the main problems in this area come from cold return gas and from excessive return gas in the case of quench of the field winding. In the SMG Teflon-coated rubber ring seals are used for the return gas path to separate it from ambient; these operate with sufficient performance when lubricated with some oil outside the helium space but do not seal at enhanced pressure in the helium return following a quench. This leads to some loss of helium gas but it is such a rare event in the operating history of a generator that it is not a serious problem. There were no problems in the SMG due to abnormal operation of the superconducting rotor. Quench of the field winding produces a large amount of return gas which can be handled by the recovery system through the use of intermediate gas storage. During quench no disturbance in the running performance of the machine could be observed and even in the catastrophic case of a quench circuit failure when the field current arced inside the rotor and burned a hole into the vacuum chamber of the rotor no disturbances in the running could be observed. The sudden breakdown of the vacuum did not cause more gas flow than an ordinary quench at the critical current level would produce. Balance of the running parts of the machine did not change from the condition of ambient temperature throughout the rotor to cryogenic temperature in the central rotor and gradually increasing temperature in the components connecting this central rotor to ambient. After this arc-burning, the machine could be brought to a standstill without any problems, showing that there was no imbalance. Due to the modular design of the rotor it was possible to repair the hole which the arc had produced. After the repair, the overhauled rotor reached the same values of current and thermal performance as prior to the failure and was in balance. This leads to the conclusion that the repair of such equipment is possible. H2.1.7 Economic aspects What, if any, are the economic consequences of superconducting generators? Can the positive benefits balance the greater complexity of the product? These questions have been discussed with diverging conclusions by scientists, engineers and economists. Beyond any doubt the superconducting generator will show greater efficiency and will have a smaller frame size than the conventional generator. Some of the improvements in efficiency originate in the reduction of stator core mass and associated losses in this part of the machine; however, the improvement is due mainly to the drastic reduction of losses in the field winding. Superconducting windings do not have ohmic losses but they need refrigeration to remain in the necessary state of superconductivity. In the field of refrigeration considerable improvements have been made in the 1980s as may be seen from the power input which is necessary for a refrigerator to cool 1 W at liquid-helium temperature. This figure, which is representative of the performance of the refrigerator, has been reduced from 500 W/W to 200 W/W for large refrigerators. There is no doubt that these improvements can be achieved in smaller refrigerators. Figure H2.1.57, showing the efficiency of generators versus different load states, refers to the conservative figure of 500 W/W. The loss represented by the power required to cool the cryogenic part of the rotor is essential data regarding the efficiency of the generator. There are few data available regarding the helium consumption of the machines which have been constructed but figure H2.1.58 shows calculated data for machines as a function of the rating of the machine. The most important source of losses is the heat conduction along the structural parts of the rotor; another source of loss is that due to the current leads. These losses may be reduced by the optimization of the design of these components but this has not been fully carried out for the existing experimental machines (Ramsauer 1991). For this reason the data of figure H2.1.58 may be somewhat misleading because they indicate a requirement of 40 1 h−1 for machines with very small ratings whereas the experimental SMG (rated at 2 MVA, see literature) did not require so much. For the test case in which the rotor of the SMG is filled

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Figure H2.1.57. Change of efficiency under different loads.

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Figure H2.1.58. Liquid-helium consumption.

with liquid helium and no liquid is replenished the consumption of the rotor alone was approximately 10 1 h−1; under the best operating conditions of the transfer equipment in the same machine 15 1 h−1 was observed. This value can easily rise to 20 or 25 1 h−1 if the bayonet is not adjusted properly and if helium in the complete return flow path downstream of the rotor exit finds a large pressure drop. From this it is clear that not only the rotor of the superconducting machine but also the return and supply lines outside the rotor have to be included in the optimization of the design. Conventional synchronous generators for power stations are highly sophisticated products and when making a comparison with superconducting generators the main difference is the wider experience in the construction and handling of conventional generators. The cost per unit of electrical energy, capital costs, time for design and construction, prognosis of safe operation and other important information are already at hand for conventional machines. Superconducting machines have been built only on a small scale in experimental construction shops, and are designed as unique pieces of equipment with the main objective being not to fail when demonstrating the excellent features of these machines. Thus the lack of reliable data makes the tasks of estimating the cost of the electrical energy, capital costs, time for design and construction, prognosis of safe operation and reliability of operation highly tentative. Therefore there is a wide range of diverging statements with respect to these important data; it is possible to find the price of a superconducting generator below that of a conventional generator. This can be understood from figure H2.1.59 because of the huge power of the machine under investigation. An equivalent result can be found for the same range of generated power, as shown in figure H2.1.60. From this one may conclude that lower ratings of the superconducting generator end up with the same, or higher, construction costs than the conventional generators. Taking into account the higher efficiency of the superconducting generators (see figure H2.1.61) and the value of the consequent savings which are made over the life of the machine, it is found that the superconducting generator is less expensive than a conventional generator down to a power rating of below 1000 MVA. Futhermore the crossover point, for example at 300 MVA, might be reduced to 200 MVA given a better knowledge of the technical problems of the superconducting machine; we know for instance that, in Japan, 70 MVA is considered to be a representative rating for large power generators. Even for a small SMG the calculated efficiency could be higher than for a conventional machine if the excellent figure of performance of large helium refrigerators can be achieved in small-scale refrigerators. Finally we conclude that, with reliable experience in design, construction and operation of superconducting generators, their potential economic benefit will be assessed with confidence and the product will find its way onto the market, thus reducing the waste of primary energy for the prosperity of man. Copyright © 1998 IOP Publishing Ltd

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Figure H2.1.59. Material utilization (kVA kg-1).

Figure H2.1.60. A comparison of capital costs.

Figure H2.1.61. A cost comparison taking efficiency into account.

References Bodner B 1987 Erwäarmung und Wärmeabfuhr im Rotor eines supraleitenden Synchrongenerators Dissertation University of Technology Graz, Austria Bodner B and Käfler H 1988 Cryogenic helium cooling in rotating parallel cooling ducts with common supply channels Proc. ICEC 12 (London: Butterworth) Fevrier A and Renard J C 1980 Thermal, electrical and magnetic behaviour of a superconducting winding 1980 ASC IEEE Trans. Magn. MAG-17 224–7 Copyright © 1998 IOP Publishing Ltd

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Intichar L et al 1989 Development of superconducting generators for power plant VDI Bericht, Fachtagung Supraleitung in der Energietechnik ( : VDI) pp 121–35 Jones W D et al 1984 Performance testing of a 20 MVA superconducting generator Cigre 984 Session (Paris, 1984) doc. 11–08 Köfler H et al 1985 Investigation on fault current limits in superconducting exciterwindings of turbogenerators Proc. MT 9 (Züurich, 1985) Krainz G 1991 Rechnergestützte Betriebsüberwachung einer supraleitenden Synchronmaschine Diploma Thesis University of Technology Graz, Austria Liebmann W 1980 Die Verteilung des magnetischen Flusses im Ständerjoch einer supraleitenden Synchronmaschine Diploma Thesis University of Technology Graz, Austria Ramsauer F 1991 Strom- und Kühlmittelzuführung für Rotoren von supraleitend erregten Synchrongeneratoren Dissertation University of Technology Graz, Austria Schnapper C 1978 Waärmetransport und Wärmeübergang mit Helium in rotierenden Kanäalen Dissertation TU Karlsruhe Schnapper C et al 1977 Thermodynamics of a self pump cooling cycle for superconducting generator application Cryogenics 429 Zerobin F 1985 Rotoren mit supraleitender Erregerwicklung—Entwurfsgrundlagen Dissertation University of Technology Graz, Austria

Further reading D . c . a p p l i c a t i o n s i n a . c . r o t a t i n g m a c h i n e r y : the papers in this section discuss the basics of the generator with superconducting field winding Bratoljic T 1973 Turbogeneratoren mit supraleitender Erregerwicklung Bull. SEV 64 1040 Bumby R 1983 Superconducting Rotating Electrical Machines (Oxford: Clarendon) Fürsich H 1975 Probleme bei der Steigerung der Leistung von Synchrongeneratoren, insbesondere bei der Anwendung einer supraleitenden Erregerwicklung Bull. SEV 66 318–23 Lehuen C 1974 Principaux problémes constructif et tendances des characteristique d’exploitation des alternateurs à inducteur supraconducteur Rev. Gén. Elect. 83 707–15 S u p e r c o n d u c t o r w i r e s f o r u s e i n a . c . r o t a t i n g m a c h i n e r y : the selection of papers in this section will elucidate the superconductors used in generators with superconducting field windings Köfler H and Telser E 1987 Wire insulation applicable in cryogenic service Proc. CIGRE Symp. on Modern Materials (Vienna, 1987) Köfler H and Ramsauer F 1988 Operating performance and losses of coil joints in a superconducting rotor winding Proc. ICEC 12 (Southampton, 1988) (London: Butterworth) Kwasnitza K et al 1975 AC losses of superconducting composites with 8 µm NbTi filaments in a DC magnetic field with a superimposed AC component at 1 < f < 50 Hz Cryogenics 723 Walker M S et al 1973 Alternating field losses in the superconductor for a large high speed AC generator Adv. Cryogen. Eng. 19 59–65 S t a t o r w i n d i n g a n d s t a t o r c o r e : the selection of papers in this section will discuss special concepts of stator windings and the environmental iron screen in generators with superconducting field windings Aichholzer G 1975 Neue Lösungswege zum Entwurf grober Turbogeneratoren bis 2 GVA, 60 kV E u M 92 249 Aichholzer G et al Generatorwicklungen für extrem hohe Spannungen ETZ 2 1160–3 Anderson A F et al 1980 Analysis of helical armature windings with particular reference to superconducting AC generators IEE Proc. C 127 129–43 Casinovi G et al 1982 Analysis of the outer electromagnetic shield of a superconducting turbo-generator IEEE Trans. Magn. MAG-18 665 Flick C 1979 New armature winding concepts for EHV and high CFCT applications of superconducting turbine generators IEEE Trans. Power Appl. Supercond. PAS-98 2190–200 Ishigohka T 1982 The optimum environmental shielding systems of superconducting generators Proc. ICEC 9 (Kobe, 1982)

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Kirtley J L 1977 Armature of the MIT-EPRI superconducting machine IEEE Trans. Power Appl. Supercond. PAS-96 88–96 Kirtley J L et al 1985 Switching surge voltage distribution in superconducting generators IEEE Trans. Power Appl. Supercond. PAS-104 2190–7 Manowarda M 1986 Theoretische und experimentelle Untersuchung von koaxialen Wicklungssystemen Dissertation TU Graz Morini A et al 1985 Electromagnetic force analysis on stator and rotor windings of a superconducting generator IEEE Trans. Magn. MAG-21 672–5 Patel M R et al 1982 Viscoelastic interface design for airgap armature in large superconducting generators Proc. ICEM (Budapest, 1982) Petrenko N et al 1978 Ankerwicklungsbefestigung und Grenzleistung nutenloser Synchronmaschinen Elektrie 32 595–7 Takahashi M et al 1982 A helical airgap winding for superconducting generators Proc. CEC 9 (Kobe, 1982) R o t o r w i n d i n g : the selection of papers in this section is devoted to deepening the information on rotor windings in generators with superconducting field windings Ashkin M et al 1983 Superconducting generator field winding design for high fault tolerance IEEE Trans. Magn. MAG-19 1035 Bratoljic T et al 1978 Three dimensional calculation of fields and losses in the rotor of a superconducting synchronous generator IEEE/PES Summer Meeting (Los Angeles, 1978) Dagalakis N et al 1975 Protection of superconducting field windings for electrical machines by the use of an inertia free electromechanical shield IEEE Trans. Magn. MAG-11 650–2 Köfler H, Ramsauer F and Sammer J 1990 Quench signals from a superconducting rotor Cryogenics 30 Suppl. Smith J L et al Superconducting field winding for a 10 MVA generator Adv. Cryogen. Eng. 25 137 Ueda A et al An experimental study concerning field windings for a 70 Megawatt class superconducting generator with low response excitation Cryogen. Eng. Japan 26 362–9 (in Japanese) D a m p e r w i n d i n g : the selection of papers in this section will give some additional information on dampers in generators with superconducting field windings Allinger G 1977 Das Betriebsverhalten von Synchrongeneratoren mit supraleitender Erregerwicklung und getrennt gelagertem Aubendämpfer Dissertation TU München Ashkin M 1980 A theoretical analysis of finite length electromagnetic shields of superconducting generators IEEE/PES Summer Meeting (Minneapolis, MN, 1980) Bratoljic T 1977 Negative sequence losses and eddy current losses in the rotor of a superconducting generator Proc. MT 6 (Bratislava, 1977) pp 206–11 Casinovi G et al 1982 Analysis of the electromagnetic shield of a superconducting turbine generator IEEE Trans. Magn. MAG-18 665–9 Echtler K et al 1977 Feld-, Schirm- und Schutzfragen bei einem Turbogenerator mit supraleitender Erregerwicklung ETZ A 98 270 Er-Jian F 1985 Three dimensional analysis and experimental studies of fields in superconducting generator with multishield screening systems IEEE Trans. Magn. MAG-21 679–82 Han S et al 1985 The investigation of the multi shielding system of the superconducting generator IEEE Trans. Magn. MAG-21 676–8 Lawrenson P J et al 1976 Damping and screening in the synchronous superconducting generator Proc. IEE 123 Markus H et al 1977 Die Abschirmung magnetischer Felder bei zylindrischen Anordnungen insbesondere bei einem Turbogenerator mit supraleitender Erregerwicklung Arch. Elektrotech. 59 270–89 Monti C et al 1975 Steady analysis of the magnetic fields and eddy currents in the rotating screen of a superconducting alternator Arch. Elektrotech. 57 319–28 Muta I 1984 Characteristics of electromagnetic shield control for superconducting synchronous generator Proc. ICEM (Lausanne, 1984) 1114 Scurlock R G et al 1981 50 Hz rotating superconducting magnet for screening studies IEEE Trans. Magn. MAG-17 2198–200 Stoll R L 1984 Damping tests on a small superconducting synchronous generator Proc. ICEM (Lausanne, 1984) p 1126 Copyright © 1998 IOP Publishing Ltd

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Tegopoulos J A 1982 Eddy current distribution in cylindrical structures caused by rotating magnetic fields Proc. IEE B 129 64–74 R e a c t a n c e s : the selection of papers in this section will improve the ability to deal with reactances and equivalent networks in the generator with superconducting field winding Bratoljic T et al 1988 Identification of parameters for a superconducting generator having free or fixed outer rotor, based on tests and calculations IEEE Trans. Energy Conver. EC-3 164–71 Echtler K W 1976 Dreidimensionale analytische Berechnung des Erregerfeldes eines Turbogenerators mit supraleitender Erregerwicklung Arch. Elektrotech. 58 259–71 Echtler K W 1977 Dreidimensionale analytische Berechnung des Statorfeldes eines Turbogenerators mit supraleitender Erregerwicklung Arch. Elektrotech. 59 87–105 Lambrecht D et al 1984 Investigation of the transient performance of superconducting generators with an advanced network-torsion-machine program IEEE Trans. Power Appl. Supercond. PAS-103 1764–72 Miller I 1977 Analysis of fields and inductances in air-cored and iron-cored synchronous machines Proc. IEE 124 121 Prior D et al 1984 Air cored alternators: a numerical analytical transient field model EEE/PAS 1984 Winter Meeting (Dallas, TX, 1984) Sergl J 1974 Berechnung der magnetischen Felder und Wicklungsinduktivitäten bei einem Turbogenerator mit supraleitender Erregerwicklung Arch. Elektrotech. 56 180 Ula A H M S et al 1988 The effect of design parameters on the dynamic behaviour of the superconducting alternators IEEE Trans. Energy Conver. EC-3 179–86 Cryoengineering of a.c. rotating machinery, mechanical design of key components, o p t i m i z a t i o n o f m e c h a n i c a l a n d t h e r m o - t e c h n i c a l d e s i g n : the rather long selection of papers in this part deals with the behaviour of the cryogenic fluid helium and the material selection in generators with superconducting field windings Barkov A V et al 1981 Investigation of a rotating liquid helium transfer coupling Cryogenics 21 654–6 Bejan A 1974 Material selection for the torque tubes of large superconducting rotating machines Cryogenics 14 313–5 Bodner B, Köfler H and Ramsauer F 1988 Transient thermal behaviour of current leads for SC generators when operated with fault current up to 150% nominal duty Proc. ICEC 12 (Southampton, 1988) (London: Butterworth) Brynskii E A et al 1980 Investigation of thermal bridge assembly for rotor of superconducting turbogenerator Sov. Elect. Eng. 51 83–7 Camporese R et al 1977 Thermal problems in the rotor of a superconducting alternator Proc. MT 6 (Bratislava, 1977) pp 194–205 Collins E W et al 1977 Magnetic and thermal properties of stainless steel and Inconel at cryogenic temperatures Adv. Cryogen. Eng. 22 159–73 Eckels P W et al 1983 A rotating liquid helium transfer system Adv. Cryogen. Eng. 29 Formin B I et al 1987 Main stages of manufacturing a 300 MW superconducting generator Cryogenics 27 243–8 Haseler L et al 1976 Thermodynamic considerations for the refrigeration of rotating superconducting machinery Cryogenics 331 Haseler L E et al 1985 Rotating cryostats for heat transfer and fluid flow studies on the helium cooling of superconducting generator rotors Cryogenics 25 355–65 Hillig W B et al 1977 Application of fiber reinforced polymers to rotating superconducting machinery Adv. Cryogen. Eng. 22 193–204 Hofmann A 1980 Properties and behaviour of liquid helium in a rotating machinery Proc. ICEC 8 (London, 1980) p 554 Hofmann A et al 1977 A method for calculating temperatures in a superconducting rotor cooled by 2-phase helium Adv. Cryogen. Eng. 23 146–50 Igra R et al 1988 Reverse convection in helium and other fluids in the high speed rotating frame: negative and positive buoyancy effects Adv. Cryogen. Eng. 31 447–54 Intichar L et al 1985 Experimental simulation of helium cooling system for a superconducting generator Proc. MT. 8 Intichar L et al 1985 Construction and testing of a rotating helium coupling for a superconducting generator Adv. Cryogen. Eng. 31 Keim T A et al 1984 Design and manufacture of a 20 MVA superconducting generator IEEE/PAS 1984 Winter Meeting (Dallas, TX, 1984) Copyright © 1998 IOP Publishing Ltd

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Kim Y I et al 1981 Axial transfer tube heat link of rotating superconducting machines IEEE Trans. Magn. MAG-17 130–3 Kirichenko Yu A et al 1979 Heat transfer to helium I during different boiling regions at high centrifugal accelerations Cryogenics 19 49–51 Kirichenko Yu A et al 1983 Calculation of the thermodynamic state in rotating cryostats Cryogenics 23 137–8 Kirtley J L et al 1991 Cryogenic isolating torque tubes for a superconducting generator. Detailed model and performance analysis IEEE Trans. Energy Corner. EC-6 267–73 Laskaris T E 1977 A cooling concept for improved field winding performance in large superconducting AC generators Cryogenics 17 201–8 Laskaris T E et al 1984 Liquid helium rotational reservoir managment in a 20 MVA superconducting generator Adv. Cryogen. Eng. 29 401–9 Litz D C et al 1981 High speed test rig to study natural convection in liquid helium Adv. Cryogen. Eng. 27 799 Nathenson R D 1986 Thermal stress analysis and design of a 300 MVA superconducting generator IEEE Trans. Energy Conver. EC-1 141–7 Ogata H et al 1978 Thermal design of a cryogenic rotor for a 50 MVA superconducting generator Proc. ICEC 8 (London, 1978) p 276 Petukhov B S et al 1986 Cryogenic liquids forced boiling heat transfer in a rotating channel Cryogenics 26 226–33 Pfotenhauer J M et al 1984 Stability and heat transfer of rotating cryogens, Part 2: effects of rotation on heat transfer properties of convection in liquid 4He J. Fluid Mech 76 239–52 Ries G 1988 Numerical experiments on helium flow in superconducting generators, MT 10 IEEE Trans. Magn. MAG-24 1485–8 Schnapper C et al 1977 Thermodynamics of a self pump cooling cycle for superconducting generator application Cryogenics 429 Sergrev S I et al 1984 Liquid boiling process in rotating vessels and channels J. Eng. Phys. (Engl. Transi.) 46 84–90 Tye R P et al 1977 Thermal conductivity of selected alloys at low temperatures Adv. Cryogen. Eng. 22 136–44 Yamaguchi K et al 1984 Superconducting rotor development for a 50 MVA generator IEEE Trans. Power Appl. Supercond. PAS-103 1795–800 Yamaguchi K et al 1989 Rotor design of a 1000 MW superconducting generator IEEE Trans. Energy Conver. EC-4 244–9 Zhukov V M et al 1986 Study of the boiling crisis in the flow of a subheated liquid in a rotating radial channel High Temp. (Engl. Transi.) 23 869–75 Electrical characteristics and regular operation including comparison with conventional a . c . r o t a t i n g m a c h i n e r y : the selection of papers in this section will facilitate the calculations behind the stationary operation of generators with superconducting field windings Alyan M A A S et al 1987 The role of governor control in transient stability of superconducting turbo generators IEEE Trans. Energy Conver. EC-2 38–46 Bischof H et al 1989 Practical experience on the operation of a 320 kVA synchronous generator, with a superconducting field winding IEEE Trans. Magn. MAG-25 1791–4 Bratoljic T et al 1984 Tests on a 320 kVA superconducting generator IEEE Trans. Power Appl. Supercond. PAS-103 771–81 Edmonds J S et al 1982 Stability of superconducting generators—power system and cryogenic system effects Proc. 44th Annu. Meeting Am. Power Conf. vol 44 pp 723–8 Einstein T H 1975 System performance characteristics of superconducting alternators for electric utility power generation IEEE Trans. Power Appl. Supercond. PAS-94 310–8 Engl W 1989 Ein digitales Überwachungs- und Schutzsystem für den transienten Betrieb supraleitender Wicklungen—Darstellung am Beispiel eines 320 kVA-Modell-Synchrongenerators Fachtagung Supraleitung in der Energietechnik, VDI Bericht ( : VDI) pp 431–6 Ilgushin KV et al 1981 Regulation features of superconducting synchronous generator Power Eng. (Engl. Transi.) 19 60–4 Köfler H, Ramsauer F and Fillunger H 1989 First results from the start up phase of the 2 MVA superconducting generator SMG Proc. MT 11 (Tsukuba, 1989) (Amsterdam: Elsevier) pp 519–24

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Muta I et al 1980 Comparison of dynamic stability of superconducting and conventional generators Elect. Eng. Japan 100 98–106 Muta I et al 1983 The basic test on the 20 kVA superconducting generator IEEE Trans. Magn. MAG-19 1043–6 Nitta T et al 1985 On-load test of the 20 kVA superconducting generator IEEE Trans. Magn. MAG-21 668–71 Faults and transient operation including comparison to ordinary a.c. rotating machinery : the selection of papers in this section will give additional insight into the transients in generators with superconducting field windings Ashkin M et al 1981 Superconducting generator transient interaction analysis using three dimensional models IEEE Trans. Power Appl. Supercond. PAS-100 2880–8 Ivanov A et al 1979 Electrodynamic forces appearing under abrupt short circuits in a AC machine with toothless stator and nonmagnetic rotor Elektrotechnika 50 703 Koronides A G 1980 Transient simulation of superconducting synchronous machines PhD Thesis University of Pittsburgh Kowarski M E et al 1981 Forces and stresses in cryoturbogenerator rotor in presence of short circuit Power Eng. (Engl. Transi.) 19 51–9 Mironov O M 1981 Investigation of transients in a superconducting synchronous generator Power Eng. (Engl. Transi.) 19 30–6 Nitta T et al 1989 Transient characteristics of parallel running of the 20 kVA superconducting synchronous generator and a conventional one IEEE Trans. Magn. MAG-25 1775–8 Titko A I 1980 Electromagnetic field in a cryoturbogenerator in the presence of an abrupt three phase short circuit Power Eng. (Engl. Transi.) 18 16–23 E c o n o m i c a s p e c t s : the selection of papers in this section will elucidate the economic calculations behind generators with superconducting field windings Edmonds J S et al 1980 Large superconducting generators for electric utility applications—the prospects Proc. Am. Power Conf. 1980 vol 42 pp 629–38 Fujino H 1983 Technical overview of Japanese superconducting generator development program IEEE Trans. Magn. MAG-19 533–5 Glatthaar R 1982 Refrigeration systems of superconducting generators for large power plants BMFT-FB-T82–071, June 1982, Germany Glebov I A et al 1983a High efficiency and low consumption material electrical generators IEEE Trans. Magn. MAG-19 541–4 Glebov I A et al 1983b Large superconducting devices: cooling techniques and reliability Power Eng. 21 20–5 Intichar L et al 1983 Technical overview of the German programme to develop superconducting generators IEEE Trans. Magn. MAG-19 536–40 Köfler H et al 1985 SUSI und SMG—ein Entwicklungsprogramm für Synchrongeneratoren mit supraleitender Erregerwicklung Elin Zeitschrift 34–43 Lambrecht D et al 1981 Status of development of superconducting AC generators, MT 7 IEEE Trans. Magn. MAG-17 1551–9 Laumond Y 1981 Electricité de France—Alsthom Atlantique Superconducting turbogenerator development program IEEE Trans. Magn. MAG-17 890–3 Maki N et al 1988 Development of superconducting AC generator, MT 10 IEEE Trans. Magn. MAG-24 792–5 Mole C J 1989 Future development of large superconducting generators IEEE Trans. Magn. MAG-25 1783–6 Smith J 1983 Overview of the development of superconducting synchronous generators IEEE Trans. Magn. MAG-19 522–8 Uyeda K et al 1988 Research on superconducting generators and materials in Japan Proc. Am. Power Conf. 1988 vol 50 pp 454–9 Because of the large amount of written material on superconducting power generators the selection of given references is restricted. However, it should be possible for the reader to expand the number of titles with the aid of the reference sections of the given publications.

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H2.2 Motors with superconducting field windings H Köfler

H2.2.1 Introduction Electric machinery as described in H2.1 will operate as a motor as well. However, machines in the range of power capacities of turbogenerators are seldom operated as motors. Synchronous machines with smaller capacities are more often used as motors in various applications. The cost assessments for generators - indicate that, as in the case of motors with superconducting field windings, small power engines have to compete against conventional motors. So it is quite natural that the superconducting route was investigated only for special purpose machines and that only a few special drive motors with superconducting windings have been built. From the principle of operation, a.c. synchronous and d.c. machines are well suited for superconductor applications. The a.c. induction machine is not a convincing candidate because first of all no d.c. excitation winding is present and secondly the rotor winding with the low-frequency current has to have resistance for the purpose of torque production. This says that a superconducting rotor winding in the induction machine must be connected to an external ohmic resistance. A.c. synchronous and d.c. machines are more or less from the same family, which can be seen if the a.c. synchronous machine is fed by an inverter with variable frequency. In this case it will show performance near to the operational performance of the d.c. machine. In this chapter only synchronous motors are addressed. The parts dealing with superconductors and some cryogenic features of the superconducting generator will apply to motors as well. Therefore the following will describe a few examples of superconducting motors, either studied or built. H2.2.2 Superconducting drive motor In special high-power drive systems with low speed conventional synchronous machines are already used. If we want to substitute these drives by superconducting machines we should be able to offer some advantages. Study work showed that we can expect the same benefits as found with superconducting generators, namely size and weight reduction and increase in efficiency. The greater complexity in the case of a motor is a more severe burden than it is in the generator business and therefore no machine has been built up to now. The material presented in this chapter is taken from an unpublished study. We shall refer to a motor power of 50 MVA at a nominal speed of 80 rev min−1. The synchronous motor is fed from an inverter with variable frequency. This machine will operate at the rather high level of the first harmonic peak value of flux density of 1.93 T. This high level is possible because of the low frequency at which the machine is operated. Eddy current problems in the stator winding are linked to frequency and flux density squared and therefore reduction of frequency by a factor of five would permit an increase of flux density by the same number if the stator bar construction developed for superconducting generators were used. In fact flux density is increased only by a factor of two, which indicates that we can expect a Copyright © 1998 IOP Publishing Ltd

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simpler design of stator bars. The stator winding will operate at a voltage of 10 kV and the pole number of the machine will be 2p = 16. With respect to this pole number another quality assessment for the field winding has to be adopted. In the generators described in section H2.1 we have investigated exclusively two-pole rotors. The flux lines in this two-pole case are very special as we find homogenous flux density inside the field winding inner bore and an almost sinusoidal flux density distribution outside the field winding. The machine with 16 poles differs drastically from the two-pole case. First of all the magnetic flux density is no longer homogenous inside the field winding bore. We find the field squeezed into an angular sector of 22.5°. The pole pitch τp is less wide with respect to the distance from the outer diameter of the field winding and the mean diameter of the stator winding, which may be said to be the air gap δ of such a machine. This gap is large because thermal insulation of the superconducting field winding containment has to be done in this space. In the two-pole machine we find values of ten and more for this ratio. In the case of the high-pole machine we arrive at three or even less. The ratio expresses the fact that in the high-pole machine a smaller portion of the flux produced from the field winding can be used by the stator winding than is possible in the two-pole machine. This lower rate of flux exploitation and the fact that neighbouring poles influence the magnetic field can be expressed by usage of the coupling coefficient k. As can be seen in figure H2.2.1 the coupling depends strongly on the ratio of δ/τp .

Figure H2.2.1. The coupling coefficient between stator winding (1) and field winding (2): 1—ratio r1 /ry o k e = 1; 2—ratio r1 /ry o k e = 0.97; 3—ratio r1 /ry o k e = 0.9.

The graph is based on two-dimensional calculations but can be used for three-dimensional cases too. The coupling of the machine under discussion is indicated by an arrow on the graph and shows that the rather short machines which are constructed for low-speed applications have reduced coupling compared with infinitely long machines due to the influence of the end-winding effects. From the figure we can easily check our previous statement on coupling. Two-pole machines will exploit flux produced from the field winding by at least twice as much as the high-pole machine described here. The poor coupling forces the designer to provide excitation ampere turn loading to be distributed sinusoidally on the circumference of the rotor. By this the higher harmonics in the field density distribution can be controlled. This was not so necessary in the two-pole generator where trapezoidal distribution of the ampere turn loading produced satisfactory flux density distributions. A tentative cross-section of such Copyright © 1998 IOP Publishing Ltd

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Figure H2.2.2. Flux lines and location of windings in a 2p = 16 superconducting motor.

a machine (only part of it is shown) and the flux lines due to field winding excitation are shown in figure H2.2.2. A few features of a suggested construction of such a motor with water-cooled stator winding can be seen in figure H2.2.3. The benefits of such a motor can be summarized as follows. We can expect that this motor will show 60% of the mass of a conventional motor with water cooling in all windings and that it will have an efficiency of 98% over the load regime ( 1/4 to 5/4 load). From the point of superconductor application we will find a high peak flux density level in the field winding (6.3 T). It is supposed that the field winding will not withstand a sudden short circuit as also the p.u. reactances of the machine are low (x1 d ≈ 0.065, xd ≈ 0.18). The electrical performance with the intended supply will be studied carefully to obtain more information about the benefits of such a drive motor. Copyright © 1998 IOP Publishing Ltd

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Figure H2.2.3. Construction layout of a superconducting motor.

H2.2.3 Superconducting torquer The next example of a superconducting motor is related to the fully superconducting generator because the stator winding is built from superconductors. Special information on the features of the fully superconducting generator can be found in chapter H2.3. There is no field winding in this small machine but magnetic field is supplied by permanent magnets with a high energy density product. The nominal operation speed of the torquer is 300 rev min−1; however, it may operate at up to 600 rev min-1 and therefore the maximum supply frequency is 50 Hz. From this it follows that we have 2p = 10. The reduction of nominal frequency to 25 Hz and the rather low level of magnetic flux density produced from the samarium-cobalt magnets in the air-cored machine made it possible to manufacture the three-phase stator winding from Nb—Ti superconductor with very fine filaments and a Cu—Ni matrix. Some specifications of the torquer are shown in table H2.2.1. The motor just discussed has the superconducting winding at rest and the permanent magnets rotating. This feature facilitates the construction and the helium management. With respect to the performance of an electric motor the example is not to be used as a basis for comparisons. This torquer has a very special application in space research and is not to be confused with applications of ordinary electric motors. Finally we want to mention that many superconducting motors have been studied in the area of Copyright © 1998 IOP Publishing Ltd

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Figure H2.2.4. A sketch of a cross-section of a superconducting torquer.

d.c. machines (see chapter H3) and also some theoretical work can be found on air-cored asynchronous machines in which the superconducting field winding supplies additional excitation to improve the poor power factor of these machines. Except for the d.c. proposals, none of the theoretically studied machines has been built. The introduction of the high-temperature superconductors has stimulated research again into the motor application of superconductors. There exist some reports on demonstration models with these high-temperature conductor materials which show that, as already known, superconductors will be of good use in electrical machines. Calculations regarding such motors in any case have to follow the route shown in the generator section. In addition more attention has to be paid to abnormal motor operation (including in fact the electrical faults studied in the generator section) like starting and blocking and also to the supply by inverters and the adjoined switching procedures. Further reading Sabrie J L, Grunblatt G, Carminati J, Gorisse M, Testard O, Bonnal J F, Blondel C, Sirou M, Champenois G and Rognon J P 1986 Superconducting torquer for isocam Proc. 11th Int. Cryogenic Engineering Conf. ed G Klipping and I Klipping (London: Butterworths) Marshall R 1983 3000 horsepower super conductive field acyclic motor IEEE Trans. Magn. MAG-19 876–9 Erikson J-T, Mikkonen R, Paasi J and Söderlund L 1995 A 1.5 kW HT superconducting synchronous machine Applied Superconductivity 1995 (lnst. Phys. Conf. Ser. 148) ed D Dew-Hughes (Bristol: Institute of Physics)

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H3 Direct current machines

A D Appleton and D H Prothero

H3.0.1 Introduction In this chapter we are going to discuss the design and development of superconducting d.c. motors and generators since the early 1960s. Of course, as we have seen in chapter H1, over these three decades there have been very significant improvements in the quality and performance of the superconductors themselves which have created the need for an on-going review of the impact which these materials have had upon the state of the art of these machines. Historically, the bulk of the development of superconducting d.c. machines has been directed towards homopolar machines which will, therefore, be the main topic in this chapter. Almost all of this work has targeted the requirements of marine propulsion but there are numerous other land-based applications for which these machines have potential. All of the work on motors has been for large, slow speed machines usually with ratings that cannot be achieved using the conventional technology that has been developed for heteropolar motors. Originally (1964) the choice of the homopolar motor was made because it was less demanding on the performance required from the superconductor but, in fact, for these very high-torque machines the same choice would be made today even though the Nb-Ti superconductors are enormously improved compared with those early days. There is, of course, a lot of common ground between large synchronous motors and synchronous generators. Although we shall be mainly dealing with d.c. motors in this chapter, virtually all the topics covered will apply equally to d.c. generators. Features which are peculiar to d.c. generators are considered separately in section H3.0.13. Turning now to the main topic of this chapter we are going to trace the development of homopolar machines because this is the best way of appreciating the problem areas and where further development may be required in the future. We shall see that the key technology for these machines is not superconductivity but current collection at the sliprings and the need to advance this technology has resulted in an enormous investment in solid brushes and liquid-metal current collection systems. The improvements in the metallic superconductors, particularly the alloy Nb—Ti, have been quite dramatic; there have been worldwide contributions to the advancement of this material which may now be used reliably with a guaranteed performance. The other important technology required for these machines is that of helium refrigeration; in the mid-1960s, the reliability of liquid-helium refrigeration plant was a major problem but significant improvements have since been made. It is probably the case that engineers whose responsibility it is to ensure the trouble-free operation of plant have acquired a dislike for d.c. machines because of the need to employ electrical brushes which Copyright © 1998 IOP Publishing Ltd

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require maintenance and which, surprisingly, still defy a complete understanding of how they work. Most d.c. machines are not homopolar but heteropolar which requires commutators and high-resistance carbon/graphite brushes. Homopolar machines need to have sliprings and very low-resistance brushes or current collection systems. A rotating electrical machine may be built in a few minutes using some copper wire, a battery and a few odds and ends from a workshop; this was demonstrated by Michael Faraday in 1831 when, having discovered the laws of electromagnetic induction, he built the first homopolar motor. This comprised a disc which was capable of rotating about its axis and electrical connections to the shaft and to the perimeter of the disc. When this arrangement was placed in a magnetic field and a current was passed through the disc it began to rotate. In this chapter we will examine in some detail the extension of this simple concept to large high-power motors and generators. In practice, one of the disadvantages of homopolar machines before the advent of superconductivity was the fact that they were fundamentally low-voltage devices; typically it was difficult to achieve more than a few tens of volts which meant, of course, that for any significant power output a high current was required from the supply system. It was for this reason that there were very few applications of homopolar machines and from Faraday onwards a major industry was built up on heteropolar motors and to a lesser extent generators. As industry developed there was a need for larger and larger motors with variable speed capabilities and the design of d.c. motors was pushed to their practical limits. The limitation at lower speeds is simply the size and weight of the motors; attempts to exceed these limits rapidly result in machines weighing hundreds of tonnes. The limitations at higher speeds are mechanical strength considerations and the very important question of commutation. It is not appropriate here to provide a long exposition on commutation in electrical machines; suffice to say that if the reactance voltage at the point of commutation is too high there will be sparking at the brushes which will create damage and a lot of problems. However, at both low and high speeds, it is possible by using superconducting windings to generate higher flux densities than are possible in machines with conventional conductors which dissipate ohmic losses. As a result, the use of superconducting windings enables the ratings of d.c. machines to be increased. In the 1960s when metallic superconductors began to become available commercially and when a need appeared for much more powerful d.c. motors than could be produced by conventional means, it did not take long to reach the conclusion that the homopolar route was the best approach. As we shall see there were a number of reasons for this, not least our ignorance, at that time, of the behaviour of superconductors. It was known, for example, that the superconductors were sensitive to time-varying magnetic fields such as would be experienced by the excitation winding of a heteropolar motor; it was also known that movement of the superconductor in a magnetic field could cause problems and consequently it was desirable to limit the mechanical forces on the superconductor. The consideration of the homopolar machine showed that both of these problem areas could be avoided and this became the starting point for development work on these machines. Engineers designing electrical machines were faced with numerous other problems which were alien to them including the need to immerse the field winding in liquid helium, to refrigerate the boil-off gases and to prevent excessive amounts of heat reaching the cold or cryogenic parts of the machine. Thus refrigerators and cryostats were added to the list of components for the motor and each required much development in their own right. It is sometimes forgotten that the prototype for the first commercial helium refrigerator was not available until after World War II and that reliable and efficient performance took a long time and much ingenuity to achieve. In retrospect it is clear that one of the reasons for the failure of these machines to reach the market-place was the fear of the unknown as far as the use of liquid helium was concerned. A more logical reason was the fact that the cost of the engineering required to meet the conditions necessary for the economic use of liquid helium was very high. Nevertheless, since the 1960s, very large development programmes have been undertaken and considerable progress has been made.

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The action which started the activity in the UK in 1964 was a statement by the Royal Navy that they had a requirement for a d.c. motor to produce 20 MW (26000 hp) at 200 rev min−1 with a weight of not more than 25 t and a low external magnetic field. This specification was far beyond anything that could be produced by conventional means but in the following sections of this chapter it will be shown to be feasible if superconductors are used. H3.0.2 The basic homopolar machine In 1831 Michael Faraday arranged a circular metal disc on a spindle so that it could rotate and placed it in a magnetic field as shown in figure H3.0.1. He then demonstrated that if a direct current is made to flow in the disc, either radially inwards or radially outwards, then the disc will rotate. In this machine the magnetic field always cuts the current-carrying conductor in the same direction and hence the name homopolar. It is clear that if a current is to flow in a rotating disc there must be electrical connections to the centre of the disc and to its perimeter and these connections are made, usually, by electrical brushes on sliprings. Some of the early experiments of Faraday were performed with the perimeter of the disc dipping into a pool of mercury as one of the points of contact and one of the methods of current collection, as an alternative to solid (carbon-based) brushes, is a liquid-metal system.

Figure H3.0.1. The principle of the Faraday homopolar disc.

Let us use figure H3.0.1 to investigate why it is that this disc rotates. A brush is placed on the outer rim and on the hub and a current of i amperes flows in the disc. The disc is now placed in a magnetic field, B tesla, which is parallel to the axis of rotation, and we can see from figure H3.0.1 that this magnetic field acting on the current, i, produces a force on the disc in a direction at right angles to the directions of both the current and the magnetic flux; this is the force which causes the disc to rotate. The force on a radial element of the disc, of length dr, is the product of the magnetic field, B, the current, i amperes, and the length, dr. By putting a number of brushes around the periphery of the disc, a number of radial currents, i, are generated in the disc. The force on a small radial element, dr, of the disc (figure H3.0.1) is equal to iBdr and therefore the torque is equal to iBrdr, thus the torque on the disc is given by T * = i B ∫0r r dr + i B r02/2 N m (where r0 is the outer radius of the disc). 0

The torque on the disc may be calculated by a different method; let us also assume that the spindle on which the disc rotates has a negligible diameter so that the whole of the area of the disc is cut by the Copyright © 1998 IOP Publishing Ltd

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magnetic field. A current of I amperes is now passed radially in the disc via brushes on the spindle and on the perimeter. As we have seen, the disc will rotate and at a speed of n revolutions per second, there is a voltage, E volts, generated in the disc where E = n πr02 B. This follows from Faraday’s law of electromagnetic induction which states that voltage is equal to the rate of cutting magnetic flux and it is clear that the total flux cut by the disc in one revolution is πr 02 B and the time of one revolution is (1/n) seconds. The power, P * watts, developed by the disc as it rotates is equal to the product of E and i, i.e. P * = Ei = nπr02Bi. The power P * is also equal to 2πnT and from this we get the same result as above. The common name for the voltage, E, which is produced in a motor is ‘back electromotive force ( emf )’ ( it is usually given the symbol Eb , but for the present we will keep E ). We can now progress to investigate the efficiency of this machine. Let us assume that the disc is made of copper, it will have a resistance and there will also be a resistance in the brushes; let us call the total resistance R ohms so that when the current, I, is passed there will be a voltage drop of IR volts. Thus the voltage of the power supply to the disc, or motor, which we will call V volts, must be sufficient to overcome the voltage E (which opposes the applied voltage which is why it is called back emf) and also IR. Thus V = E + I R; if we now examine the power requirement, P watts, from the supply we see that P = VI = EI + I 2 R where I 2 R represents the losses which do not contribute to the output of the motor but merely heat up the disc and the brushes. The efficiency, η, of the motor is the ratio of the output to the input of the motor and this is seen to be.

η = E/( E + I R ). The simple relationships developed above define completely the Faraday disc machine provided that the magnetic flux density is uniform. Let us now examine why it is that the homopolar motor (or generator), with the technology that was available to Michael Faraday, and for the next 100 years or so, was never able to reach the market-place to any significant extent. To achieve a reasonable magnetic flux density, rotating electrical machines use magnetic iron; in other words, an iron core is placed around the Faraday disc shown in figure H3.0.1. For a reasonable design, the maximum flux density that can be persuaded to cross the air gaps, on either side of the disc, is about 1 T. If we take a disc diameter of 1 m (and already the motor is getting rather heavy) and a speed of 200 rev min−1, the above derivations show that E = n π r2B = (200/60) × π × (0.5)2 = 2.62 V. This low voltage is a characteristic of homopolar machines and, apart from a few very special applications such as a low-voltage, high-current generator, Faraday’s homopolar machine did not find much demand. On the basis of this simple example, the 20 MW, 200 rev min−1 motor specification by the Navy would require a power supply current of over 9 × 106 A which, of course, is impracticable. Increasing the diameter to 2 m will reduce the current to about 2 × 106 A but this is still not very satisfactory and in any case the weight is rapidly increasing to hundreds of tonnes. It may be noted that a generator with a 1 m diameter disc in a field of 1 T at 3000 rev min-1, which might just be feasible, would produce a voltage of less than 40 V. There are at least two configurations for the homopolar machine and these are the disc-type and the drum-type as shown in figure H3.0.2. In each case the principle is the same. In the disc-type there is one excitation winding and the rotor is a disc with one slipring at the outer perimeter and one slipring close to the axis of the machine. In the drum-type there are two excitation windings with current in the opposite

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Figure H3.0.2. Disc- and drum-type homopolar machines.

directions as shown in figure H3.0.2; in this case the rotor is a cylinder, or drum, and the current flows axially between sliprings at each end of the drum. The rotating disc or the rotating drum are normally referred to as the armature of the machine. The excitation, or field windings can either be of normal conductors (copper or aluminium) or superconductors. We will now examine the impact which the availability of superconductors made upon the concept of the homopolar machine. H3.0.3 The basic superconducting homopolar machine We have seen that the flux density of a homopolar machine may, by the use of iron, be increased to about 1 T but this does not greatly improve the prospects of this machine ever being widely used in industry. By the use of superconductors we can achieve magnetic flux densities in excess of 5 or 6 T, and possibly even higher values. The first observation to be made is that in most cases the iron core is largely redundant. The magnetic iron does provide an enhancement of the magnetic field, even with the use of superconductors, but in most cases this does not justify the additional weight and cost of using iron. So, in this chapter, we will consider only air-cored homopolar machines and, in the first instance, only the disc-type. Figure H3.0.3 shows a superconducting coil and a disc-type rotor comprising a shaft, a disc and two sliprings; one of the latter is near to the shaft and the other is at the outside diameter of the disc. The reader will notice that the brushes on the outer diameter are arranged on the underside of the slipring and this is because it is important to ensure that as much as possible of the excitation flux cuts the disc. Unlike the iron-cored machine, where most of the flux stays in the iron except where it has to cross the air gap, in the air-cored machine the maximum flux density is at the field winding. Thus, because we wish to maximize the total flux cut by the rotating disc, the brushes are arranged as shown. Ideally the brushes at the inner diameter should be as far away from the field winding as possible and it is advantageous to move this slipring in the axial direction away from the machine. A little consideration will show that by doing this it is possible to increase the diameter of the inner slipring (to give more accommodation for the brushes) without losing any of the valuable magnetic flux. Figure H3.0.3 shows a stationary conductor that carries the armature current from the brushes at the outer diameter down towards the shaft where it then moves axially away to form part of a co-axial circuit with the conductor that connects to the brushes at the inner diameter. This stationary conductor may take Copyright © 1998 IOP Publishing Ltd

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Figure H3.0.3. The basic superconducting disc machine.

the form of a disc and it is clear that it is carrying the same current and is in the same magnetic field as the rotating armature. Thus, this conductor will experience the same forces as the armature and this is, of course, the torque reaction of the machine. We note therefore that the torque reaction does not appear on the superconducting field winding. Let us now see what happens to the magnetic field caused by the armature current, because as we might be beginning to suspect, (and will see later), the armature current is very high. Consideration of figure H3.0.3 will show that this magnetic field is confined to the space between the rotating armature and the stationary disc that takes the current from the outer brushes, and also in the co-axial conductors that bring the armature current to and from the power supply for the motor. To a very good approximation the flux due to the armature current is everywhere orthogonal to the excitation flux of the field winding. The conclusion from this situation is that unlike the heteropolar motor the homopolar motor has no armature reaction effects. Armature reaction is a term which refers to a modification of the magnetic field of the excitation winding caused by the magnetic field of the current in the armature winding. The absence of armature reaction in homopolar machines is important for two reasons. Firstly, it simplifies the speed control of the motor because the excitation is unaffected by the load imposed upon the motor; secondly, the superconductor will not be exposed to any time-varying magnetic fields which may cause some losses in the low-temperature regions. The lack of armature reaction and the manner in which the torque reaction is accommodated had a Copyright © 1998 IOP Publishing Ltd

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major influence on the choice of homopolar machine when the specification for a marine drive was first presented in the 1960s. However, having said this, the homopolar motor would, in any event, remain the first choice because of its robustness and very high torque capability even at zero speed. Let us now consider the design of a superconducting, disc-type, homopolar machine, some of the features of which have been discussed above. Without the iron core the magnetic field of the superconducting field winding, which from now on we will call the excitation winding, is free to choose its own path outside the machine, although all of it must pass through the bore of the winding. This external field represents a problem for whatever environment the machine may find itself in and is discussed at length in section H3.0.11. It rapidly becomes apparent to the design engineer that current collection at the sliprings is a subject that is going to require a lot of attention. It is evident that a large current has to be transferred to and from the sliprings and that the options appear to be to use fairly conventional electrical brushes (possibly of the type used on slipring induction motors) or some form of liquid-metal system. It can be stated now that, in general, solutions to this problem have been identified but it is a subject that has dominated the three decades of the development of these machines. We will examine this topic in some detail in section H3.0.7. H3.0.4 Machine equations for a practical machine (a) Calculation of torque Unlike the iron-cored machine, the flux density which cuts the disc in a superconducting machine is not uniform and hence the calculation of the machine parameters is rather more complex than that presented in section H3.0.2. The magnetic field in the centre plane of the coil is in the direction of the coil axis. The total armature current is I amperes and the total current in the excitation winding is i ampere turns. Consider a small distance dr at a radius r; the flux density is B and is a function of r as indicated in figure H3.0.1. The force on the element dr is I Bdr and the torque, dT, at that point is given by dT = I B( r )r dr. Thus total torque on the disc is given by T = I ∫ B( r )r dr N m. (b) Calculation of back emf The magnetic flux dΦ through an annular ring at radius, r, and thickness, dr, is dΦ = 2π B( r )r dr. Thus the total flux cutting the disc is given by Φ = ∫ 2π B( r )r dr Wb. If the speed of rotation is N rev min−1, the voltage generated, or back emf, is given by E = N Φ/60 = (2 N π/60) ∫ B( r )r dr.

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(c) Calculation of power developed by the motor The power developed by the disc is the product of the back emf and the current which flows in the disc; this is given by

(d) Discussion of the integral ∫ B (r )r dr The most significant parameter in the machine design is the integral ∫ B(r )r d r which appears in all of the machine equations. The limits of the integral are the radii of the inner and outer slipring and clearly its magnitude must be maximized. The flux density in the bore of the excitation winding is created by the ampere turns in this winding and any flux that is not included in this integral represents a waste of the investment in the superconductor. Figure H3.0.4 shows the variation of axial field with radius for a solenoid coil (such as would be used for the excitation winding of a disc-type homopolar machine).

Figure H3.0.4. Axial magnetic field due to a solenoid coil.

In considering figure H3.0.4, it is important to realize that the design objective is to make as much as possible of the flux cut the disc between the inner and outer sliprings. The most important region is that between the superconducting winding and the outer slipring where the flux density is greatest. Within this space it is necessary to accommodate: ( i ) the steel vessel which supports the winding and contains the liquid helium for cooling the winding; ( ii ) a thermal radiation shield to reduce the amount of heat reaching the low-temperature regions; ( iii )a vacuum space for thermal insulation; Copyright © 1998 IOP Publishing Ltd

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( iv )the steel vessel which contains the vacuum and which encloses and supports the vessel containing the winding, (note that the above items form part of what is called the cryostat ); ( v ) a clearance space between the rotor and the cryostat; ( vi )the thickness of the outer slipring. Figure H3.0.5 shows the relative location of these regions for a superconducting disc-type homopolar machine.

Figure H3.0.5. The arrangement of a disc-type homopolar machine.

Very careful design is required to minimize this radial distance because approximately 20% of the total flux is lost in 10% of the distance between the axis of the winding and the superconducting winding. We will return to the details of construction of these machines after giving further consideration to the machine equations. If the number of ampere turns in the excitation winding is i and the mean radius of this winding is A metres, it is shown in appendix A at the end of this chapter that the value of the integral is given by

∫ B( r )r dr = C µ0 i A where C is a constant, provided that the limits of integration are a fixed percentage of the mean radius of the excitation winding. Obviously it is necessary to undertake a precise calculation for a specific detailed design but this approximation is helpful when trying to optimize the parameters to meet a particular specification. (e) Design optimization The specification is likely to state the output power and speed; it will probably also impose restrictions on the height of the shaft of the motor. Using the above expression for the integral, B(r )r dr , the equations Copyright © 1998 IOP Publishing Ltd

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given in paragraphs (a), (b) and (c) above become T = C µ0i IA N m Φ = 2π C µ 0 i A Wb E = (2π/60)C µ 0 i A N V P = (2π/60)C µ 0 i A N I W. It is stressed that the use of the constant C is restricted to an optimization exercise after a precise calculation has been completed to determine the value of C. In the following sections, we will proceed to consider the design of practical superconducting homopolar machines commencing with a discussion of the main components and those aspects which have required a considerable amount of development work, particularly current collection. All of the design and development to date has been in the context of the use of liquid-helium-cooled metallic superconductors and this is where we will commence our discussion; the impact of the new higher-temperature ceramic superconductors will be reviewed in section H3.0.14 at the end of this chapter. The discussion of homopolar machines will draw on the experience of the following three machines built under the direction of one of the authors (ADA) at the International Research and Development Company (IRD) at Newcastle upon Tyne, UK:

Figure H3.0.6. The superconducting Model Motor and the Fawley Motor. Copyright © 1998 IOP Publishing Ltd

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( i ) the ‘Model Motor’, rated at 37.3 kW, built in 1966 which was the world’s first superconducting motor; ( ii ) the ‘Fawley Motor’, rated at 2.4 MW, which followed the Model Motor in 1969; figure H3.0.6 shows the Fawley Motor and the Model Motor together and figure H3.0.7 shows the Fawley Motor under test at Fawley power station in 1969 (note that the Fawley Motor itself is on the right-hand side of figure H3.0.7; the structure on the left-hand side is the Fawley cooling water pump); ( iii )the ‘CMS concepts’ which were developed during the 1970s. This work covered the construction of a motor and generator set, rated at 1 MW. These are shown in figure H3.0.8. The generator (at the rear of the photograph) operated at 1500 rev min−1 and provided the armature current for the motor (at the centre of the photograph) which ran at 375 rev min−1. The output from the motor was fed to a water brake (at the front of the photograph) which simulated a propeller load. The CMS Generator was designed for rapid field changing which necessitated a special type of superconductor (see section H3.0.5). The CMS work also included extensive studies on a number of aspects, such as coil stresses and stray magnetic field, of a fully rated (20 MW) motor. We shall consider the technical implications of these design aspects in the following sections.

Figure H3.0.7. The superconducting Fawley Motor.

H3.0.5 The superconducting excitation winding The superconductor employed in the Model Motor is now only of historical interest; it was a seven-strand niobium-zirconium (Nb-Zr) cable, purchased in 1965 from the company ‘Supercon’ in the USA. ( The material Nb—Ti, which was to become, and remains, the most widely used superconductor, was not readily available at that time). Each Nb—Zr strand was 0.25 mm in diameter and copper plated to a thickness of 0.025 mm. At that time ideas on improving the stability of superconducting windings were beginning to emerge (Wilson et al 1970) and it was decided to wrap copper wires around the superconductor to increase its diameter from about 0.91 mm to about 1.32 mm and bond it all together with indium. The Copyright © 1998 IOP Publishing Ltd

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Figure H3.0.8. The CMS Motor.

superconductor was insulated with a weave of glass-fibre threads and wound onto a stainless steel coil former; after test the winding was installed into the cryostat. The total number of turns was 2140 and during testing the coil quenched five times at currents between 235 and 238 A. This was somewhat below the short-sample performance of the superconductor but, nevertheless, it was adequate for the purposes of the machine under construction. By way of contrast the excitation winding of the Fawley Motor contained over 5 t of Nb—Ti superconductor—copper composite. Five wires of Nb—Ti were co-drawn in a copper billet to form a 10 × 1.8 mm2 composite conductor; the final diameter of the Nb—Ti wires was 0.5 mm. Figure H3.0.9 shows a short length of the conductor with some of the copper removed to exhibit the superconducting filaments. The maximum current that can be carried by a superconductor is given by its short-sample characteristic. This is obtained, as the name suggests, by taking a short length of the superconductor in the form of a U-bend, placing it in a magnetic field and passing an increasing d.c. current until it quenches (see also B7.3). This is repeated for a number of values of magnetic field to produce the curve shown in figure H3.0.10. The load line of a coil shows the maximum magnetic field produced by the coil (for a solenoid coil, in the absence of iron, this occurs at the mid-plane of the coil at its inner Copyright © 1998 IOP Publishing Ltd

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Figure H3.0.9. The superconductor used for the Fawley Motor.

Figure H3.0.10. Coil load line and superconductor characteristics.

bore). The coil load line is a straight line as shown in figure H3.0.10. The maximum performance of the coil is where the load line intercepts the short-sample characteristic and this is called the short-sample performance. However, particularly for the early superconducting coils, the maximum current achieved by the coil tended to be less than the short-sample performance. This was due to heating effects within the coil which are discussed later in this section. As an example, the short-sample performance of the excitation winding of the Model Motor was 400 A, but the coil quenched at about 238 A, i.e. about 59% of the short-sample value. The difference between these two numbers is a measure of the ‘goodness’ of the winding and this is determined by a number of factors including the quality of the superconductor and the degree to which the winding has been stabilized. The difference between the superconductors for the Model Motor and the Fawley Motor is that the latter has a much larger ratio of copper to superconductor. Also the winding of the Copyright © 1998 IOP Publishing Ltd

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Fawley motor was arranged in a number of pancakes with a cooling channel for the liquid helium between them. As a result, the Fawley Motor achieved about 85% of the short-sample value (compared with 59% for the Model Motor). A detailed exposition of the stabilization of superconducting windings is beyond the scope of this chapter but the subject is extremely well documented (Wilson 1983). However, to give an indication of some of the factors which have to be considered in this work it is necessary to calculate the heat transfer required from the copper around the superconductor to the liquid helium to ensure that stable conditions are achieved. For the superconductor used in the Fawley Motor, this varied from about 0.204 W cm−2 to 0.325 W cm−2; this was sufficiently low to ensure that nucleate boiling was maintained in the liquid-helium coolant. It must be noted that the transition temperature of the Nb—Ti under the field conditions in the Fawley Motor is less than 8 K and the temperature of the superconductor when the heat transfer rate is 0.325 W cm−2 is 4.65 K. One of the factors in these calculations is the conductivity of copper which, from the point of view of minimizing the heat generated in the copper, should be as high as possible. However, it is necessary to subject the copper to some work hardening to increase its mechanical strength but this also reduces the conductivity. In practice, therefore, a compromise is necessary and a conductor with a work hardening of a few per cent was chosen. Cable conductors, made up of copper and superconducting composite, were used for the superconductor for the CMS Motor and Generator. The superconducting composite utilized technical advances in superconductor construction achieved at the end of the 1960s, principally by IMI Ltd in Birmingham, UK and the Rutherford High Energy Laboratory (RHEL) near Oxford, UK. The conductor itself was manufactured by IMI. The superconductor used was Nb—Ti, drawn into fine filaments of diameter about 50 µ m and embedded in a matrix of high-conductivity copper to form a ‘strand’ of diameter 0.5 mm and a copper to superconductor ratio of 1.35. Figure H3.0.11 shows, on an enlarged scale, a section through one strand of the superconducting composite conductor. This strand was twisted along its axis with a pitch of about 25 mm. The complete conductor consists of six of these strands along with 13 strands of high-conductivity copper with the same diameter (0.5 mm); they are co-processed in a hexagonal pattern into a soldered cable. The finished cable was twisted along its axis with a pitch of about 40 mm. Because the superconductor was in the form of fine filaments, the ‘hysteresis losses’ generated in the conductor were very small and the losses which did occur were readily dissipated into the copper. Because of the twisted nature of the strand and the finished conductor, any coupling current losses between the superconducting filaments were reduced to an acceptable level. Finally, fabricating the conductor in a soldered cable form meant that it could be produced in the long single lengths (of the order of 1 km) necessary for winding the coils of the CMS Motor. This conductor therefore represented a significant technical advance on its predecessors and is in fact essentially similar to the Ni-Ti conductors which are available today. The overall cable conductor, used for the superconducting coils of the CMS Motor, had a diameter of 3 mm and was capable of carrying a current of about 850 A in a magnetic field of 4 T. The excitation coils of the CMS coil system consisted of a number of ‘pancakes’ wound using this conductor. Each pancake had a diameter of about 930 mm, a radial depth of 68 mm and an axial width of about 30 mm. Figure H3.0.12 shows a group of these pancakes being lowered into the cryostat during assembly of the motor. In order to provide mechanical rigidity, each wound pancake was impregnated with an epoxy resin. The epoxy resin used was a standard commercially available product supplied by Ciba—Geigy, the coils being ‘impregnated’ at a slightly elevated temperature (60°C) and ‘cured’ by maintaining them for about 24 h at a higher temperature (typically 100°C). In addition to the excitation coils, some degaussing coils were also wound. These were identical to the excitation coils in cross-section, type of conductor and method of manufacture but were larger in diameter (1700 mm instead of 930 mm). The theory behind the degaussing coils is described in some detail later in this chapter (section H3.0.11).

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Figure H3.0.11. A section through the superconducting composite used in the CMS Motor.

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Figure H3.0.12. Superconducting coils of the CMS Motor being assembled into the cryostat.

In addition to providing mechanical reinforcement for the coils, the use of epoxy resin impregnation prevents movement of the coil conductor within the winding. This is important because, in liquid-helium-cooled systems, material specific heats are very low and hence the frictional heat generated by conductor movement could cause the superconductor to go normal. The following numerical examples should help to illustrate this point. ( i ) Consider a liquid-helium-cooled Nb—Ti superconductor at 4.2 K and suppose that this superconductor would quench at 6.5 K (a typical operating value). Then, using published data on the specific heat of Nb—Ti, it is found that the energy input required to drive the superconductor normal (by heating it from 4.2 K to 6.5 K) is about 2 × 104 J m−3. ( ii ) Suppose next that the conductor is carrying a current density, J, of 3 × 108 A m−2 in a magnetic field, B, of 4 T. Suppose further that the conductor moves a distance, ∆x, of 30 µ m under the action of this force. Then, the associated energy density release, given by JB∆x, is 3.6 × 104 J m−3. From (i) above, this is sufficient to drive the superconductor normal. Thus the function of the resin impregnation in preventing conductor movement is a crucial one. This benefit does not come entirely free, however. When a superconducting coil is energized, electromagnetic stresses are generated (we will consider these later in section H3.0.10); hence when the coil is energized for the first few occasions, localized cracking can occur within the resin in regions where it is stressed to its yield point. This leads to a localized release of energy density of magnitude σy2/2E where σy is the yield stress of the resin and E its elastic modulus. Typical values for these parameters are 3 × 107 N m−2 Copyright © 1998 IOP Publishing Ltd

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and 3 × 109 N m−2 respectively; these give a value of 15 × 104 J m−3 for the energy density release. From ( i ) above this is certainly large enough to cause quenching of the superconductor. However, it has been found that such cracking only occurs, if at all, on the first few occasions when the coil is energized. On subsequent occasions, for circular potted coils at least, the coil can be reliably energized to the same quenching current. For the excitation winding of the CMS Generator coils, the same basic approach was used (i.e. epoxy-impregnated main and degaussing coils with the conductor being in the form of a soldered cable made up of strands of copper and superconducting composite). The nature of the superconducting composite conductor was, however, different to accommodate the fact that the generator coils, unlike the motor coils, were subject to a relatively rapid change of field (typically from zero to a rated field in about 10 s). Such a relatively rapid change would cause very large ‘coupling current’ losses between the superconducting filaments if the generator superconductor had been of the same form as that of the motor; hence, in order to reduce the losses a special three-component (TC) superconductor composite was used in which each filament of Nb-Ti (see figure H3.0.11) was enclosed in a sheath of copper-nickel. The high-resistance copper-nickel increased the electrical resistance between the superconducting filaments and hence reduced the coupling losses; however, the presence of copper in the composite ensured that it was still adequately stabilized and protected. H3.0.6 The cryostat and liquefaction system The cryostat of the Model Motor, shown in figure H3.0.6, is a stainless steel vessel which contains the Nb-Zr superconducting magnet. It is necessary for this vessel to be welded up to prevent the loss of helium which, apart from other considerations, would ruin the vacuum needed to prevent heat getting to the excitation winding. The top of this annular vessel is connected to a stainless steel tube, of 10 cm diameter, which is the liquid-helium reservoir; this tube is connected to a further tube of 7.5 cm diameter which supports the weight of the superconducting coil from the flange at the top. To reduce the heat in-leak to the low-temperature region by conduction, over a distance of about 15 cm, the wall thickness of this tube is reduced to about 0.25 mm. Within the vacuum space which surrounds these tubes is a co-axial annular vessel which contains liquid nitrogen to protect the liquid-helium vessel from the thermal radiation from the room-temperature walls of the cryostat. At its lower end this liquid nitrogen vessel connects to a series of pipes which cool a stainless steel thermal radiation shield to protect the vessel containing the superconducting winding. Figure H3.0.6 shows the assembled cryostat complete with its diffusion pump (located behind the motor). It is necessary to obtain a vacuum of better than about 10-6 Torr (about 10-6 mbar) to reduce conduction to negligible levels. The helium vessel and the thermal radiation shield were wrapped with ‘superinsulation’; this is Mylar of about 0.025 mm thickness coated with aluminium of about 0.0025 mm thickness. Up to about 20 layers may be required and it is effective in reducing thermal radiation. The cryostat for the Fawley Motor (also shown in figure H3.0.6) is very similar in concept except that the vessel containing the coil is supported by stainless steel cables. For both of these machines the current leads from the excitation winding to the power supply are taken up through the cryostat to a flanged connection at the top. In the case of the Fawley Motor the leads are cooled with a stream of helium gas from the liquid helium which boils off in the cryostat. By this means the heat conduction down the leads is reduced; this is most important because the lead losses are about 50% of the total losses. In the case of the Model Motor the liquid helium was transferred into the motor by means of a vacuum-insulated transfer line. To reduce the quantity of liquid helium required for cool-down the liquid-helium space was first filled with liquid nitrogen and the residue pumped out. In the case of the Fawley Motor there were permanent connections from the top of the cryostat to a helium liquefier. The compressor

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for this liquefier was a modified reciprocating air compressor which gave a great deal of trouble, resulting in impurities in the helium which deposited in the heat exchangers in the cold box and also the very high speed turbines used to extract energy from the helium. It must be remembered, however, that these were rather early days (1968) for large helium liquefiers and the problems were all part of the learning process. The cryostat of the Fawley Motor, and of subsequent machines, was directly connected to a dedicated helium liquefier operating as a closed system. The operation of the liquefier was based on a Claude cycle with two very high-speed gas-bearing turbine expanders as a means of abstracting work from the helium gas. One turbine had a steady-state inlet temperature of about 80 K, the other of about 20 K. Because liquefier and cryostat formed a closed circuit, it was essential to ensure that this circuit was leak-tight to a high degree before commencing cool-down and that it remained so during operation. This is particularly important with regard to the low-pressure parts of the circuit, particularly those close to the suction side of the compressor. This proved to be a daunting task for early superconducting machines such as the Fawley Motor. Leak-testing prior to starting the cool-down was essential. It was usually carried out by pressurizing the complete circuit—cryostat and liquefier—to a relatively high nitrogen pressure and checking for large leaks by monitoring any loss of pressure and by means of soap solution applied to all connections and joints. When it was found to be leak tight to that coarse degree, the whole circuit was evacuated and back filled and pressurized with helium gas so that a thermal conductivity ‘sniffer’-type leak detector could be used to probe all external joints and connections. When the complete circuit was considered to be leak tight, it was evacuated and backfilled with helium two or three times; the resulting gas purity was checked with a katharometer to ensure that the helium content was greater than 99%. It was important to achieve this high degree of initial purity since residual impurities and those entering the circuit during operation (from suction leaks and even make-up gas) could overload the capacity of the absorber incorporated into the liquefier or condense onto the heat exchanger surfaces or in the expansion turbines. For successful long-term operation, it is essential to prevent the ingress of water vapour and carbon dioxide from the atmosphere. The liquefier compressor was then started up and the helium gas circulated through the cryostat/liquefier closed circuit. During this time, the moisture content of the helium was checked and the absorber bed in the liquefier was used to trap the water vapour. When the moisture content was reduced to less than 100 ppm (parts per million by volume) the absorber was temporarily taken out of the circuit and regenerated by heating it electrically and evacuating it. If the water level in the circulating helium remained low, the plant was ready for cool-down. Cool-down commenced by allowing the gas-bearing turbine expanders to run at full speed (several thousand revolutions per second) instead of idling. To ensure a reasonably quick cool-down, a liquid-nitrogen-helium heat exchanger, incorporated into the high-pressure stream, was supplied with liquid nitrogen to expedite cool-down of the liquefier and the cryostat to 77 K. At temperatures below 77 K, the cool-down was achieved solely by the use of the turbine expanders; below 20 K, the Joule-Thomson effect began to lake over, becoming the dominant coolant mechanism at temperatures below 10 K. The liquefier circuit was equipped with a number of bypass arrangements to ensure that cold gas was fed directly to the cryostat to facilitate cool-down. With the cryostats of the Fawley Motor and later machines, it was possible to achieve cooling rates up to 2 K h−1 without risk of structural damage from thermal stresses. The Fawley Motor had a separate vacuum-insulated stainless steel dewar vessel interposed between the liquefier and the cryostat. Once liquid helium had started to form in the cryostat, a further five days was required before the cryostat (700 1 in volume) and the dewar (1000 1) were completely filled. Once the dewar was full, the superconducting coils were energized and the machine put into operation. The dewar provided a reserve capacity of liquid helium to enable the liquefier to be shut down for a few hours in an emergency while the cryostat was held full of liquid helium. In that event, the boil-off helium gas would be vented to the atmosphere.

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H3.0.7 Current collection We have seen previously (section H3.0.4(d)) how important it is for the effective diameter of the armature disc to occupy as much as possible of the bore of the excitation winding. Also we have seen, in the basic disc-type machine, the convenience of having two discs in the bore of the winding, each with their own sets of sliprings, connected electrically in series. We have touched upon the question of current collection and seen that we may use conventional solid brushes or a liquid-metal system. In the case of the work in the UK on the Model Motor and all subsequent machines it was decided that the development programmes would be based upon the use of conventional brushes; the reason for this was the Royal Navy did not wish to have liquid metals, such as mercury or sodium-potassium (Na-K), in its ships. The programme in the USA, which started at about the end of the 1960s, opted for the alternative route and almost all of their work on superconducting homopolar machines has been based upon the use of Na-K liquid-metal slipring systems. We will commence our discussion of the armature with an examination of the current collection problem. The relatively low voltage of homopolar machines makes it necessary for large currents to be taken to and from the sliprings of the armature and the question arises as to what is the most appropriate current collection system. On heteropolar machines it is necessary to use high-resistance carbon brushes on the commutator but for sliprings it is common practice to use brushes with a high content of copper, i.e. a very low resistance. Most of the available experience of the latter is with slipring induction motors but these, usually, have only a few brushes. Large superconducting homopolar machines are required to have a large number of brushes and, in 1964 when this work started on the Model Motor, there were virtually no data on the performance of such current collection systems. It was particularly important to know the best type of brush to use in order to obtain the maximum current density that could be carried without excessive wear, the voltage drop and the most appropriate materials with which to make the sliprings. Let us first of all examine the magnitude of the current collection problem. A motor with an output of 11.2 MW at 200 rev min−1 could be designed to have two Faraday discs, arranged back to back; assuming that it is practical for these discs to have a diameter of 3 m and that an average flux density of 5 T could be obtained using a superconducting excitation winding, we may note that a possible set of design parameters would be as follows: ( i ) back emf, E = 335 V approximately; ( ii ) armature current, I = 33400 A approximately; ( iii )outer slipring velocity = 32 m s−1. Thus the current loading on the outer sliprings of each disc is about 3500 A per metre of the slipring circumference but, assuming that it is not possible for more than 60% of the circumference to be loaded with brushes (because of the need to have brush supports, etc), the current loading becomes about 5800 A m−1. If we assume that the width of the brush is 2.5 cm, the brush current density is about 23 A cm−2. The slipring velocity is reasonable for the type of brushes under consideration. For a practical machine, it is necessary: ( i ) to minimize the contact voltage drop between the brush and the slipring in order to reduce the losses; this is most directly achieved by increasing the pressure on the brush; ( ii ) to reduce the friction losses between the brush and the slipring; this is most directly achieved by reducing the brush pressure; ( iii )to obtain a ‘reasonable’ wear rate on the brushes; this implies testing a number of different types of brush; ( iv )to obtain a ‘reasonable’ wear rate on the slipring which again implies testing a number of materials.

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A major contribution to the achievement of good performance can be made by using a brush capable of a high contact current density to minimize the brush area. Copper-carbon brushes are not only capable of high current density but also exhibit a low voltage drop and, for these reasons, such brushes were used on the large Canberra homopolar generator (Rashleigh and Marshall 1978). After consultations with brush manufacturers it was decided to use the Morganite grades CMIS and CM2. The detailed composition of the brushes is proprietary information; however, both brush types consist of about 80% copper, the remainder being made up mainly of carbon with some trace elements (zinc, tin and lead) also being present. With all types of brush, the friction and electrical contact losses show a marked dependence upon brush pressure; friction increasing and contact loss decreasing as the pressure increases. It follows from this that there is an optimum brush pressure (for a given value of current) but since the optimum depends also on current density and slipring speed it can be determined only by experiment. Unsatisfactory brush behaviour can have a large number of causes, but the effects are the same; high and uneven wear rates of both brushes and slipring and high power losses at the brush contacts, accompanied sometimes by sparking. It is well established that the satisfactory operation of solid brushes on metal sliprings requires the formation on the sliprings of a stable film of metal oxide. In the case of the homopolar machine, it was uncertain whether the satisfactory formation and operation of this film would be affected by the high percentage cover of the slipring surface. Brush, behaviour can be profoundly affected by atmospheric conditions, and quite small quantities of some substances, notably silicones, can have a highly damaging effect upon performance. It is also known that operation in an inert atmosphere, e.g. in nitrogen gas, and with careful control of humidity, can improve the performance of brushes. Brush performance becomes problematic at high slipring speeds because, apart from increased wear rates, there is a tendency for the brushes to lift by a wedge of air drawn under them. As a general guide, for slipring speeds up to about 30 or 40 m s−1, solid brushes may be designed to work in a satisfactory manner; if the speed is higher than this it is necessary to introduce special measures, such as putting grooves in the sliprings under the brushes. For continuous operation at speeds in excess of 100 m s−1 it is almost certain that a liquid-metal system must be employed. Yet another problem is caused by the uneven sharing of current between a large number of brushes on a slipring; this can result in severe overheating and excessive wear of the brushes. Fortunately this problem is not as severe with the high copper content brushes as it is with the more resistive brushes. Prior to the completion of the design of the Model Motor it was decided to undertake an experimental programme on current collection which would simulate as closely as possible the conditions in the full-size machine, i.e. a slipring speed of 32 m s−1 and a brush current density of about 23 A cm−2, as defined earlier in this section. Also the tests would have regard to the geometry of the slipring at the outer diameter, i.e. with the brush on the underside of a slipring, as shown in figure H3.0.3. The first slipring to be used was of sand-cast copper-nickel and had a helical groove on each brush face to prevent aerodynamic lifting of the brushes. Later sliprings were of centrifugally cast copper-nickel. The rig was driven by a d.c. motor capable of speeds up to 2880 rev min−1 (corresponding to a slipring speed of 52 m s−1 ). The slipring was loaded with brushes on both sides, the total brush area on each ring being 130 cm2 (i.e. 20 brushes, each of area 6.5 cm2 ). The d.c. power supply was capable of 3500 A which allowed a brush current density of up to about 27 A cm−2. After the brushes had been bedded in at reduced speed and current (brushes should not be operated at zero current) for 50 h the tests were commenced. The initial brush pressure was set at 1.3 N cm−2 and steady-state conditions established over a period of about 8 h, with continuous monitoring of brush voltage drop, slipring temperature and brush temperature, with regular readings of individual brush currents and motor torque. These tests were repeated for a range of brush pressures up to 4 N cm−2. It was found that the CM2 brushes had the lower losses under these conditions. The wear rates of the brushes are given in table H3.0.1 from which it is seen that the CM2 brushes have the lower wear rate by a significant factor. In fact the wear rate for CM2 brushes was found to be about 1700 h cm−1 which would be acceptable for most applications, bearing in mind that the machine

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is unlikely to run continuously for months on end. It was also noted that the slipring wear with CM1S brushes was quite high with severe grooving of the ring after about 300 h. In contrast to this the CM2 brushes gave very little wear of the slipring. It should be noted that in respect of slipring wear, the rate decreases sharply once a ‘steady state’ groove has been set up. Another important parameter in assessing the performance of a brush is the degree to which the current is uniform across the cross-section of a brush. It was found that this was closely similar for the two brush grades, both showing a marked improvement in current sharing with increase in brush pressure. However, the optimum brush pressure for CM2 (determined on the basis of voltage drop and friction losses—see (i) and (ii) earlier in this section) was found to be higher than CM1S and hence, under its optimum conditions, the current distribution was better in CM2 than in CM1S. Thus it is clear that on all three counts—losses, wear and current sharing—the CM2 brush emerges as superior to the CM1S. The above results have been given to acquaint the reader with some of the complexities of the current collection problem using conventional electrical brushes. This discussion allows us to proceed with the design of the armature.

H3.0.8 The armature The rotor of the Model Motor was very similar in detail to the schematic diagram shown in figure H3.0.3 except that two Faraday discs were present instead of the single one shown in figure H3.0.3. One Faraday disc was located on each side of a central stainless steel support disc, each Faraday disc having outer and inner sliprings fully laden with brushes. The series connections between the two Faraday discs were made through the support disc; this allowed the power supply to be brought to one side of the motor. The inner sliprings of the motor (one on each side of course) were copper cylinders located on, and insulated from, the stainless steel shaft. It was noted at the beginning of this discussion on superconducting homopolar machines that the output voltage from practical machines tends to be very low. This was not a problem for the Model Motor, which was essentially a device to demonstrate the principle; however, for the larger machines (the Fawley Motor and the CMS Motor and Generator) this voltage limitation was a significant technical issue, particularly in respect of the power supplies which were required. A number of methods are, however, possible for increasing this voltage and these are discussed in the remainder of this section. Some of these were applied in the Fawley and CMS machines. For the Fawley Motor, the useful magnetic flux, i.e. that which cuts the rotor between the sliprings, was 6.45 Wb which for a speed of 200 rev min−1 would give a back emf of 21.5 V per disc; for two discs, arranged back to back, the voltage would be 43 V which would require an armature current of 58 000 A. It was decided that this was unsatisfactory and an alternative design was sought. The possibility of using a large number of discs in series (i.e. more than two which is quite convenient) was rejected because of the severe mechanical complications and maintenance problems in gaining access to all of the brushes. Copyright © 1998 IOP Publishing Ltd

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It was required to keep the working conductors as close as possible to the mid-plane of the excitation winding for two reasons; simplicity of design and to employ the magnetic field to the maximum advantage. A new concept was developed for the rotor of the Fawley Motor which is shown during assembly in figure H3.0.13. The inner and outer sliprings on each of the discs were divided into a number of segments with each segment insulated from its neighbours. Each of the outer slipring segments was connected to the corresponding inner slipring segments by a radially disposed copper conductor. It is clear that the emf generated in each of these radial conductors is in the same direction either radially inwards or radially outwards. Electrical brushes, arranged on every other slipring segment, ensured that the emf will add so that if there are 2n segments the back emf on each disc is nE volts and the full armature current is flowing in each conductor bar joining one pair of segments. With a slight movement of the rotor, the brushes bridge two sets of segments; the voltage between the brushes is unaltered but the current is now shared between two segments. When the rotor has moved on further the brushes are once again on one pair of segments. It is clear that the voltage between the brushes is unchanged but the current in each of the conductors joining one pair of segments varies from the full value to half value and back to full value again.

Figure H3.0.13. The armature of the Fawley Motor.

It will be observed that this switching of the armature current is quite different from the commutation in a heteropolar motor or generator where part of the armature winding is short-circuited as commutation proceeds. In the case of the segmented homopolar machine the current is not trying to reverse but is Copyright © 1998 IOP Publishing Ltd

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merely switched from bar to bar. There are a number of points to be observed; firstly, the inductance of the conductor bars must be as low as possible in order to reduce the possibility of sparking at the brushes and, secondly, there is a limit to the voltage that can be tolerated across the insulation between two segments. The first of these points can be dealt with by design but the second is not quite so easily accommodated. With the scheme which has been described, the voltage progressively builds up as one proceeds around the rotor, so that, at one location, the voltage between adjacent segments will be nE volts, where n is the number of segments on a disc. This would give an unacceptably large voltage across this pair of segments. It is known that for a commutator on a heteropolar motor the voltage between segments should not exceed about 20-30 V or severe sparking will occur and although, as we have seen, the segmented homopolar machine does not involve commutation, this was regarded as a good yardstick for design purposes. The solution to this problem is achieved by arranging the brushes so that the segments on one half of the disc are in parallel with the segments on the other half of the disc and the voltage between adjacent segments never exceeds E, the back emf generated in one rotor conductor. It was necessary to establish the maximum voltage that could be tolerated between segments and this was achieved experimentally by a modification to the Model Motor; it was concluded that 30 V was a safe design value and that this could probably be exceeded as experience was gained with these machines. The disc which supports the stationary armature conductors also takes the current from the brushes on the outer slipring to the power supply. As we have seen earlier, it is these conductors which carry the torque reaction; this is transmitted from the torque reaction disc by a large stainless steel cylinder surrounding the disc. This is fixed to the frame of the motor and thence to the foundations. We have seen already that the voltage of the homopolar machine may be increased by having one Faraday disc on each side of a support disc; when these are connected in series the terminal voltage of the machine is doubled. In principle it is possible to add more Faraday discs, each with its own pair of sliprings, and this was achieved a few years after the Fawley Motor was built. Another machine was built with four sets of discs on each side of the support disc. Furthermore each of the discs was segmented to increase further the terminal voltage. However, although it has been shown to be possible, multiple discs in a disc-type machine do lead to a complicated design. With a drum-type homopolar machine (figure H3.0.2(ii)) other configurations are possible. As we have seen, with the drum-type geometry, there is no inner slipring because the armature drum has a slipring at each end and these must be as close as possible to the superconducting winding. The voltage output of a drum-type machine can be increased by having a number of rotor bars mounted on the rotor instead of the single rotor bar shown in the schematic diagram in figure H3.0.2(ii). Each rotor bar would be connected, via brushes and sliprings, to a stator bar and all rotor bar/stator bar pairs would be connected in series. It is, of course, important that the rotor bars are electrically insulated from the rotor and from each other and figure H3.0.14 shows schematically an arrangement for achieving this. For simplicity, only two bars are shown in the illustration. The CMS Motor and Generator were drum-type machines with multiple rotor and stator bars of this type. However, with this arrangement the location and alignment of the sliprings for the brushes requires attention. The simplest arrangement is an axial slipring arrangement (figure H3.0.15); however, if this slipring were cut by radial magnetic flux, a voltage difference would develop across adjacent sliprings. Extended over an axial arrangement of many sliprings, this would lead to a large imbalance in the currents carried by different stages; this in turn would lead to unequal wear of the brushes at different stages. Because of this effect, an axial arrangement of sliprings can only be located over a limited axial length where the radial component of magnetic field is very small. This severely limits the number of stages which could be used in practice. In principle, this problem could be circumvented by using tilted sliprings so that the surface of all the sliprings lay on the same flux line. In practice, however, it can be seen from the profile of the flux lines shown in figure H3.0.2 that the construction of the brush and

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Figure H3.0.14. Sliprings, rotor conductors and an insulating bar on the rotor drum.

Figure H3.0.15. The axial slipring arrangement and voltage generated between sliprings.

slipring arrangement of such a machine would not be straightforward. In practice, therefore, it is best to retain the arrangement shown in figure H3.0.15 and to limit the number of stages to not more than about ten. Other machine geometries are possible in addition to those which have been discussed in this section. For example, the rotating armature could be placed outside the stationary excitation winding. This has the important advantage of providing easier access to the sliprings although access to the cryogenic regions is compromised. Designers therefore have the capability of selecting the machine configuration most suitable to their requirements. Copyright © 1998 IOP Publishing Ltd

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H3.0.9 The output of homopolar machines As a result of the modifications described in this section, we need to revise the power output equation (equation (H3.0.1)) which we derived earlier. In its revised form, this equation becomes:

where N, I and B(r ) have the same definitions as before but the new term p is the number of stages, i.e. the number of times the voltage is multiplied by using segmented or multiple sliprings. Equation (H3.0.2) suggests that in principle any value of power, P, could be achieved by suitably increasing p, I or ∫ B(r )r dr. In practice, there are limits to the magnitude of each of these variables. These are as follows. ( i ) Number of stages, p—increasing the number of stages increases the brushgear in the motor which increases the maintenance requirements and ultimately may lead to problems regarding access to the brushgear. Furthermore, in the case of drum machines, there is an intrinsic limitation to the number of stages, as mentioned in section H3.0.8. ( ii ) Armature current, I—the armature current must be transferred via the brushes. Hence the allowable armature current is limited by the current density which can be carried by the brushes. For a given axial width of brush, the current can be increased by increasing the diameter of the motor. ( iii )Flux, ∫ B(r)rdr—the flux is generated by the current in the superconducting coils. For both disc-and drum-type motors, the flux can be increased by increasing the coil current but this also causes the magnetic field in the coil windings itself to increase. The maximum flux which can be generated by the coils is therefore limited by the maximum magnetic field which the superconductor can tolerate. The flux output of the motor can be increased by increasing the diameter of the coils. Thus, a required output can always be achieved, in principle, by increasing the diameter of the superconducting coils to the required size. For most applications, however, there is frequently a need to limit the diameter of the coils. Furthermore, even if a size limitation is not explicitly present, increasing the coil size can lead to mechanical problems (coil forces and stresses) as we shall see in the next section. To focus the discussion, table H3.0.2 below gives details of a possible coil configuration which has been considered for a 22 MW disc-type homopolar motor. This table gives the dimensions and basic parameters of the system and also gives values for a number of key parameters which are discussed in this and subsequent sections. The voltage output from the coil described in table H3.0.2 is 30.5 V. It was envisaged that there would be two pairs of sliprings and each slipring would contain 32 voltage stages. The generated voltage for the machine would therefore be 30.2 × 32 × 2 = 1950 V. With an armature current of 12500 A, this would give a power output of 22.4 MW. H3.0.10 Electromagnetic stresses Magnetic stresses are set up in the superconducting coils of both the disc- and drum-type machines. For both these machines, these stresses can be quite significant for large coils. In practice, these stresses are most conveniently calculated using computer codes; one of these (described by Mulhall and Prothero 1973) has been used to derive the values in table H3.0.2. In order to understand the parameters which determine these stresses, we shall use a simple model which is described below. Two components of electromagnetic stress are most important in the excitation coils: (i) hoop stresses, which are carried primarily by the conductor, and (ii) axial stresses, which are transmitted transversely across both the conductor and coil insulation.

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The mean hoop stress in a current loop is given by the expression σ h = J Bz r where r is the radius of the loop, J is the current density and Bz the axial flux density at the loop. As a simple approximation, a coil winding can be considered as a stack of such loops of increasing radius. In practice (see figure H3.0.4) the value of Bz varies markedly within the windings; at the inner bore of the coil it has the same direction as the central flux density but is larger in magnitude. As we travel through the winding, however, its value decreases and passes through zero so that at the outer radius of the coil it is reversed in sign and somewhat smaller in magnitude. Hence, the large hoop forces on the innermost ‘loop’ are partly supported by the outer ‘loops’. As a result, the peak hoop stress in the winding occurs, as one might expect, at the inner bore, but the variation of hoop stress through the winding is much less than one might expect from the variation of axial magnetic field. This is illustrated in figure H3.0.16. The hoop force on the coil is carried primarily by the conductor itself since the coil insulation has a much lower elastic modulus than the conductor and hence would carry very little of the hoop load. The values for hoop stress given in table H3.0.2 are overall values, i.e. averaged over the conductor and the insulation. Since the hoop load would in practice be carried largely by the conductor and since the conductor (in a typical coil) would occupy about 50% of the winding cross-section, it follows that the hoop stress values given in table H3.0.2 should be multiplied by a factor of two in order to obtain a realistic estimate of the hoop stress in the windings, In addition to the hoop stress in the coil, which is carried by the conductor, axial stresses, σ2 , are also present due to the electromagnetic attractive forces between the conductor turns which are carrying current in the same direction. For the disc-type machine these axial stresses are zero at the axial edges of the coil but rise to a maximum at the coil mid-plane. These axial stresses are also generally calculated using computer codes but a rough estimate can be obtained by the following very simple analysis (see also figure H3.0.17): ∂ σz /∂z = − J B r . For an isolated coil, Br = 0 at the coil mid-plane (z = 0) and for other values of z is given to a reasonable approximation by Br = K µ 0 Jz Copyright © 1998 IOP Publishing Ltd

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Figure H3.0.16. Variation of axial magnetic field and hoop stress within the coil windings.

where K is a constant which in practice is somewhat less than 1. Hence, if z0 is the axial length of the coil, then the above expression can be integrated to give the maximum axial stress at the mid-plane of the coil. ( σz ) m a x = µ 0 J 2z 02/8. Unlike the hoop stress, the axial stress is transmitted through the coil insulation as well as the coil conductor. For a coil with the parameters given in table H3.0.2 ( J = overall current density over conductor and insulation = 23 A mm2 and the axial length, z0, of the coil is 0.96 m ), this gives a peak axial stress of the order of 10 N mm−2. This is not a negligible value when it is remembered that this stress is sustained by the coil insulation. Hence the axial stresses are important for large coils carrying significant current densities. A further restriction applies to drum-type machines. It will be seen from figure H3.0.2 that the two coils in a drum-type machine are carrying current in opposite senses. As a result, an electromagnetic repulsive force exists between these coils. The magnitude of this force is calculated in practice using computer codes; however, it will come as no surprise that the force increases as the square of the ampere turns in each coil and also tends to increase as the coils come closer together. Detailed calculations on a typical coil system required for a 25 MW, 200 rev min−1 machine show that the magnitude of the electromagnetic axial force for such systems is large—of the order of a few MN (i.e. a few hundred tonnes). To accommodate these forces it is necessary to have a suitable force restraint structure (such as that shown schematically in figure H3.0.18) consisting of a cylinder or an assembly on both the inside and outside of the coils. This force restraint structure has implications for the cryogenic design which we will consider later in this chapter. If each coil of the drum machine is enclosed in a separate cryostat, then the structure will Copyright © 1998 IOP Publishing Ltd

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Figure H3.0.17. Axial forces on a coil element.

Figure H3.0.18. A force restraint structure for drum-type machines.

need to pass through a liquid helium-ambient temperature boundary. The heat in-leak at low temperature would therefore be increased which would increase the size of the refrigerator—clearly an undesirable feature. An alternative arrangement is to enclose both coils, together with the support structure, within a single vacuum vessel. This reduces the overall heat in-leak and is therefore the preferred arrangement. H3.0.11 Stray magnetic field Another consideration which requires attention is the stray magnetic field generated by the superconducting excitation coils. This is of course particularly important for any defence-related application but it is of significance also for nonmilitary applications because of the effect of stray magnetic field on equipment and personnel in the vicinity of the machines. In practice, again, stray magnetic field calculations for a particular system are carried out using appropriate computer codes; however, to understand the principles involved, knowledge of underlying theory is essential. A simplified version of this theory is presented here; for a more detailed and rigorous analysis, the reader is referred to the appropriate texts (for example, Montgomery 1969). Consider a sphere just large enough so that it encloses all current-carrying coils. At any point, outside this sphere, with spherical polar co-ordinates ( r, θ, φ ), the magnetic scalar potential can be written as a series

where the summation is from n = 2 to ∞ and the coefficients An are related to the Legendre polynomials Pn(cosθ ). The radial and circumferential magnetic field components, Hr and Hθ , are given by the usual relations Hr = −∂ V/∂ r

Hθ = − ∂ V/∂θ

(not surprisingly, the magnetic field is independent of φ for an axially symmetric system). For a disc-type machine, with the coil configuration as shown in figure H3.0.2(i), it is found that the odd-powered coefficients, A3 , A5 , A7 , etc , in equation (H3.0.3) are all zero. The expansion for the magnetic scalar potential therefore has the form V( r ) = A 2 /r 2 + A 4 /r 4 + A 6 /r 6 + A 8 /r 8 + A 10 /r 10 + … . Copyright © 1998 IOP Publishing Ltd

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At reasonably large distances from the centre of the machine, the magnetic scalar potential varies as the square of the distance and hence the magnetic field varies as the cube of this distance. The drum-type machine, shown in figure H3.0.2(ii), has two coils with equal and opposite excitations. A detailed application of the above theory shows that for this system the magnetic scalar potential term in (1/r )2 vanishes in equation (H3.0.3), leaving the (1/r )3 term as the dominant one. At large distances, the magnitude of the stray magnetic field therefore falls off as (1/r )4 The-drum type machine therefore has the advantage over the disc-type machine that its stray magnetic field is lower and falls off more rapidly with increasing distance. A more detailed inspection of the series for the magnetic scalar potential for a drum-type machine shows that it consists only of odd-powered terms, namely V( r ) = A 3 /r 3 + A 5 /r 5 + A 7 /r 7 + A 9 /r 9 + A 11 /r 11 + … . It is possible to reduce the stray magnetic field drastically by the use of degaussing coils, arranged as shown for disc- and drum-type machines in figures H3.0.19 and H3.0.20. A detailed exposition of the subject of degaussing is beyond the scope of this chapter; however, a simple application of degaussing for the disc-type machine (figure H3.0.19) can readily be seen. If the excitation of the degaussing coil is designed so that its A2 coefficient in equation (H3.0.3) is the same as that of the main coil, then the magnetic scalar potential of the complete system (main coil + degaussing coil) will fall off as r 4 and hence the stray magnetic field will fall off as r 5, resulting in lower stray field values. Table H3.0.2, which gives stray field values for a 22 MW coil system with and without degaussing coils, shows this effect. With drum-type machines, even greater reductions in stray magnetic field are possible. Two further points should be noted before we finally leave the subject of stray magnetic field. ( i ) The closer the degaussing coils are located to the main coils, the better the overall degaussing (because the higher power terms also diminish); however, this gain is achieved at the penalty of reducing the flux

Figure H3.0.19. A coil configuration incorporating degaussing coils for a disc-type machine. Copyright © 1998 IOP Publishing Ltd

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Figure H3.0.20. A coil configuration incorporating degaussing coils for a drum-type machine.

output from the main coils and also introducing large electromagnetic forces between the main coils and the degaussing coils. These disadvantages are reduced by increasing the size of the degaussing coils but this increases the size of the overall machine which is not an attractive feature for most applications. ( ii ) An alternative method of degaussing the superconducting excitation winding, which may have occurred to the reader, is to use a screen of high-permeability material. This option was examined, as an alternative to the use of degaussing coils, but it was found that the mass of screening material required would be extremely large (i.e. several tonnes). Hence, the use of a high-permeability screen is not attractive for large d.c. machines. H3.0.12 Cryogenic considerations The superconducting coil system must be maintained at low temperatures, using a refrigeration system, and it is therefore desirable generally to reduce as far as possible the thermal burden on the refrigeration system. This is particularly important for liquid-helium-cooled systems on account of the high cost associated with liquid-helium refrigeration. Since most of the work to date on superconducting machines has used liquid-helium-cooled systems, the discussion in this section is confined to the use of these systems. The consequences of the use of liquid-nitrogen-cooled systems are described in a later section. At this stage, it is useful to review the main contributors to the thermal load on the refrigeration system of a liquid-helium-cooled d.c. machine. (a) Heat in-leak down the current leads The coils of a d.c. machine would be connected to a power supply at room temperature by means of two current-carrying leads of copper. One end of the lead would be at ambient temperature (typically 20-40°C) and the other end at liquid-helium temperatures (i.e. at or close to 4.2 K). Heat is therefore conducted down these leads even when they are carrying no current; when they are carrying current, the heat in-leak is increased. The heat in-leak down a lead can be reduced, however, by counterflow-cooling the lead with boil-off helium gas provided that the surface area (for heat transfer between the boil-off gas and the lead) is sufficiently large. In practice, this can be achieved by fabricating the current lead from an assembly of copper foils or by using a finned conductor for the current lead. Figure H3.0.21 shows a typical characteristic describing how the heat in-leak down the lead varies with the mass flow rate of boil-off gas flowing up the lead. Copyright © 1998 IOP Publishing Ltd

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Figure H3.0.21. Heat in-leak at the cold end of a current lead against the coolant mass flow rate.

Direct current machines

Figure H3.0.22. A protection circuit for a superconducting coil.

Detailed analytical work on current leads shows, not surprisingly, that for a given boil-off gas flow rate, the heat in-leak down the lead increases as the current to be carried by the lead increases. There is thus an incentive to reduce to a minimum the operating current. At the same there is a need to protect the superconducting coils adequately in the event of a ‘quench’, i.e. an unscheduled reversion of the coil to the normally conducting state. A convenient way of achieving this would be by using the coil circuit in figure H3.0.22. A quench would be detected by an appropriate detection device (figure H3.0.22 shows one such device—a search coil wound in a bifilar manner to the main winding; when a quench occurred, a nonzero voltage would be registered by the voltmeter, V). When a quench was detected, a relay would cause the switch to open and the coil to discharge into a dump resistor. In practice, the dump resistor could be a nonlinear one (i.e. voltage largely independent of current); however, for simplicity, we will consider the case of an ohmic dump resistor of resistance R. Following the opening of the circuit breaker, the current density in the coil would vary with time, t, as J0 exp(—Rt/L) where L is the inductance of the coil system; hence the adiabatic rise in temperature of the coil following a quench would be given by

where C(θ ) is the volumetric specific heat and ρ(θ ) is the electrical resistivity of the conductor, both expressed as a function of the temperature, θ; E is the stored energy of the coil system. Θ is the temperature of the winding after the coil current has decayed to zero. Hence

where V = IR is the voltage across the coil and . In practice, X(Θ) is evaluated numerically from published data of the resistivity, ρ, and volumetric specific heat, C, of the overall conductor as functions of temperature, θ. Hence, if Θm is the maximum allowable conductor temperature, the minimum allowable value for the coil current, I, is given by

In practice, Θm is usually limited to about 100 K since thermal expansion would occur at higher temperatures which would lead to local but high stresses in the winding. The value of X(Θm ) will Copyright © 1998 IOP Publishing Ltd

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vary with the composition of the conductor; however, for most practical conductors which are likely to contain a significant proportion of copper, it can be shown that the value of X at a temperature of 100 K would be about 0.7 × 1017 (A m−2)2 s. The maximum design voltage across the winding would in practice be limited to about 2 kV in order to avoid voltage breakdown. To assess quantitatively the effect of all this, consider the coil system described in table H3.0.2. The values for the conductor current density, J0 , and stored energy, E, for this coil system are as follows: J0 = 4.6 × 107 A m−2 (assuming that the winding occupies half the coil section) and E = 7.5 MJ. Hence, from equation (H3.0.4), the minimum current which will enable the coil to be adequately protected is 113 A. Using larger coils, with a larger stored energy, the minimum allowable operating current would be larger. (b) Heat in-leak down the support structure In order to support the coils, some structure is necessary which must be fixed at some point to ambient temperature. Heat therefore is conducted down this structure into the liquid-helium space. As with the current leads, the heat in-leak can be reduced by cooling the support structure with boil-off helium gas; the curve of heat in-leak at 4 K versus counterflow gas mass flow rate is qualitatively similar to that for the current leads (figure H3.0.21). (c) Heat in-leak through superinsulation The superconducting coils are immersed in liquid helium in a cryostat which is surrounded by a vacuum layer filled with superinsulation. This has a very low, but nonzero, effective thermal conductivity (of the order of 10−4 W m−1 K−1); because the surface area of the cryostat is very large (it would be several square metres for a 20 MW motor), there is a significant heat in-leak through the superinsulation. Unlike the heat in-leaks through the current leads and the support structure, the in-leak through the superinsulation cannot readily be modified by the use of boil-off gas. An important part of the cryogenic design is clearly to minimize the overall thermal demand on the refrigeration system, i.e. to minimize the liquefaction rate of the refrigerator, since this has a bearing on its cost. A graphical method by which the optimum operating point can be identified is shown in figure H3.0.23. The heat in-leak/coolant mass flow rate characteristic for two current leads carrying their operating current is shown at the top of the construction and a similar curve for the support structure is shown at the bottom; in between is a base load representing the heat in-leak through the superinsulation. Linking the curves are a series of straight lines, AA’, BB’, etc, with the same gradient of 20 J g−1 which is the latent heat of liquid helium. Each of these lines represents one mode of operation of the system. Thus, when the system is operated according to the line AA’, the boil-off gas mass flow rate up the two current leads together is m1 (i.e. the boil-off gas up each current lead is m1/2) and the boil-off gas mass flow rate up the support structure is m2. The sum (m1 + m2 ) represents the total liquefaction rate required from the refrigeration plant. It can also be seen that there is one particular line (actually BB’ in figure H3.0.23) for which the sum (m1 + m2 ) is a minimum and this is clearly the preferred operating point from the point of view of minimizing the overall refrigeration requirement. H3.0.13 Direct current generators The discussion so far in this chapter has been entirely in terms of homopolar machines used as d.c. motors; however, the same machine configuration can also be used as a generator although only one of the machines which have been described (the CMS Generator) operated in this mode. The principal operational difference between motors and generators is that the latter typically run at much higher speeds. For the CMS machine, for example, the motor speed was 375 rev min−1 whereas the generator speed was 1500 rev min−1. By referring to equation (H3.0.2), it can be seen that by increasing N (the number of revolutions per minute), it is possible to achieve the same output, P, with a reduced

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Direct current machines

Figure H3.0.23. Construction to determine the optimum operating point for a superconducting system.

number of stages, p, and/or a reduced flux output, ∫ B(r )r dr, which can be achieved with a reduced size of machine. In practice, with homopolar machines, reducing the size of the machine is not an efficient option since the radial distance over which flux is lost (identified as items (i) to (vi) in section H3.0.4(d)) cannot be correspondingly reduced. Hence the preferred option for the CMS Generator was to retain the same diameter as the motor but reduce the number of stages by a factor of four. This had the advantage of simplifying the slipring and brush arrangement. However, it also incurred a severe penalty in that the rubbing speeds of the brushes was increased by a factor of four which considerably aggravated brush-wear problems. To cope with these more demanding conditions, carbon-fibre brushes were used rubbing onto silver-plated segmented sliprings. The performance of these was far from satisfactory however (erosion problems were particularly severe); as a result later conceptual studies envisaged the use of an a.c. rather than a d.c. generator. It was also envisaged that rapid field changing would be required for the CMS Generator and hence the excitation coils of the generator incorporated a three-component superconductor composite to limit the conductor; this aspect has already been fully discussed (section H3.0.5). However, apart from the current collection aspects and the nature of the superconducting composite used in the excitation winding, nearly all the design aspects discussed between sections H3.0.5 and H3.0.12 for d.c. motors apply equally to d.c. generators. H3.0.14 The use of high-temperature superconductors The discussion so far on d.c. machines has been presented entirely in terms of low-temperature (liquidhelium-cooled) superconductors because these were the only materials available up to 1986 when the bulk Copyright © 1998 IOP Publishing Ltd

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of the work on d.c. machines was carried out. Relatively little work has been carried out on d.c. machines since 1986 when high-temperature superconductors were first discovered. It is appropriate at this point to consider the difference which high-temperature superconductors would make to the technical and economic prospects for these machines. The use of high-temperature superconductors would mean the replacement of liquid-helium refrigerator technology with that of liquid nitrogen. This would undoubtedly lead to a big reduction in cost (both capital cost and running cost) and an improvement in reliability. This would clearly be a big benefit. Regarding the superconductor itself, however, the benefits are not so obvious. In particular, assuming that the superconducting coil designs were similar to those developed for the liquid-helium-cooled machines and described earlier in this section: ( i ) the peak magnetic field in the vicinity of the winding would be several tesla with a conductor current density of the order of 50 A mm−2 ; ( ii ) the conductor would need to be in wire or tape form with a length of several kilometres. Both (i) and (ii) would represent severe problems for high-temperature superconductors which are available at the present time. Surveying the progress made on high-temperature superconductors over the last nine years, it seems likely that issue (ii) will be resolved satisfactorily in the near future but issue (i) may well prove more intractable. Furthermore, it should be clear from the discussion of the previous sections that a number of the technical issues of d.c. machines do not relate specifically to the superconductor. These issues include the use of multistage current collection systems and (particularly for military applications) the degaussing of a system with large-diameter electrical coils. These technical issues would remain significant in d.c. machines using high-temperature superconductor materials. Finally, the large stresses and electromagnetic forces acting on the coil system should not be forgotten. The cable conductors, incorporating multifilamentary superconductors used at liquid-helium temperatures, are ductile materials with an effective tensile strength of the order of 100 N mm−2; by contrast, the high-temperature superconductors which have been developed to date are brittle materials with much lower tensile strength values. The large electromagnetic forces are therefore likely to be a significant technical issue for any d.c. machine using high-temperature superconductors.

Appendix A The integral ∫B(r)rdr Consider the excitation windings of the disc-type homopolar generator shown in figure H3.0.A1. To a reasonable approximation, this winding can be represented by a single loop of conductor of radius A and carrying a current i. It is shown in standard texts that the axial magnetic field at the centre of this loop is given by B ( 0 ) = µ 0 i/2 A. The average magnetic field B(r) over the area of the loop will be somewhat larger than this but can be written in the form B ( r ) = K µ 0 i/2 A where K is a dimensionless constant whose magnitude would be greater than unity. In practice, the precise magnitude of K would depend on the dimensions of the excitation winding. The integral ∫ B(r )r dr is therefore given by

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Direct current machines

Figure H3.0.A1. The magnetic field due to a circular current loop.

i.e.

where the constant C = K/4. This simplified expression is used in section H3.0.4(d) of the main text. It should be noted that the same expression can also be used for a drum-type homopolar machine although the magnitude of the constants K and C would then be different. References Montgomery D B 1969 Solenoid Magnet Design (New York: Wiley-Interscience) Mulhall B E and Prothero D H 1973 Magnetic stresses in solenoid coils J. Phys. D: Appl. Phys. 6 1973–7 Rashleigh C S and Marshall R A 1978 Electromagnetic acceleration of macroparticles to high velocities J. Appl. Phys. 49 2540–2 Wilson M N 1983 Superconducting Magnets (Oxford: Clarendon) ch 6 and 7 Wilson M N, Walters C R, Lewin J D, Smith P F and Spurway A H 1970 Experimental and theoretical studies of filamentary superconducting composites J. Phys. D: Appl. Phys. 3 1517

Further reading Wilson M N 1983 Superconducting Magnets (Oxford: Clarendon) (This deals with most aspects of liquid-helium-cooled superconducting systems. The various applications are described only briefly but there are detailed treatments on magnetic field configurations, magnetic stresses, current lead design and quench protection. There is also a detailed discussion on low-temperature superconducting materials, stabilization and losses.)

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H4 Transformers

Y Laumond

H4.0.1 Introduction Ever since the discovery of superconductivity, the idea of replacing copper windings by superconducting windings in all electrical machines has been in the minds of people specializing in advanced electrical technology (Harrowell 1970, McFee 1961, Riemersma et al 1981, Westinghouse Electric Corporation 1982). Whereas in a superconducting transformer there are no Joule-effect losses in the case of a d.c. supply, losses caused by electromagnetic forces can be observed during a.c. operation (see B8.2). When considering the use of superconductors, an initial condition has to be observed. The cooling energy cost must be sufficiently low so that the efficiency of the machine is at least as satisfactory as that of a conventional machine fulfilling the same function. Progress made in recent years in the field of a.c. conductors allows this objective to be reached. This means that the use of superconductors in a transformer can be envisaged (Brunet and Tixador 1987, Fevrier and Laumond 1986, Van Overbeeke 1986, Van Overbeeke et al 1985). The purpose of this chapter is to show the specific aspects, advantages and inconveniences of superconducting transformers as well as their interest for some applications. H4.0.2 Superconducting transformers: specific aspects H4.0.2.1 Reduction of losses Conventional transformers with copper windings are reliable machines of very satisfactory efficiency levels (95-99%, depending on the case). The first positive aspect of superconductors is the absence of losses due to Joule-effect heating. Unfortunately, in the case of an a.c. supply, losses of hysteretic or induced-current type appear and only highly sophisticated superconducting strands allow these losses to be maintained at an acceptable level. The advantage, in terms of energy, of superconducting transformers can be expressed in the following way. Let there be two machines, a conventional machine (C) and a superconducting machine (S) fulfilling the same function. For machine (S) to have an advantage with respect to machine (C), the following inequality must be checked.

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Transformers

where COP is the coefficent of performance of the refrigerator. In recent years, losses during a.c. operation have been decreased (submicrometric multifilamentary strands) whereas the coefficients of performance of cryogenic coolers have been increased. The progress made in these fields allows us to fulfil the above condition. The efficiency of transformers of the superconducting-winding type is higher than that of transformers of the resistive-winding type. H4.0.2.2 Ampere turns—iron core reduction The Boucherot formula (see textbooks on transformers) is used in the theory and calculation of transformers. In this formula the voltage of the primary winding is expressed as follows

where Bm a x is the peak value of flux density (∼1.7 T in the iron core), n1 the number of turns at the primary, Sf the magnetic core iron section and f the frequency (50 Hz in general). If the nominal current circulating in the turns is In one has

where J is the current density in the conductor (∼3 A mm−2 in Cu) and sc the section of the turn (n1sc = Sc , conductor section). The rated output can thus be written or

where nI is the number of ampere turns

The use of a superconductor allows current densities which are much higher than those of copper. In this case, the number of ampere turns is more important, which, for the same output, allows the magnetic core section to be reduced. The output of the transformer is of the type

where L is the characteristic length. Its mass is of the type

Thus, we note that the rated power per volume unit (MVA m−3) is of the following type

The use of a superconductor allows the number of ampere turns to be increased and the size and weight of the transformer to be decreased. Copyright © 1998 IOP Publishing Ltd

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H4.0.2.3 Short-circuit current limitation In the case of a short circuit, the current increases and exceeds the superconductor’s critical current. This results in a quench of the superconductor which becomes resistive. This sudden appearance of the resistance has the effect of limiting the current (see H6). Under these conditions the windings can be dimensioned in such a way that in the case of a short circuit a certain value, for instance 2In , is not exceeded.

H4.0.2.4 Specific constraints The advantages described above are counterbalanced by a certain number of constraints. ( i ) Cryogenic problems can be solved and the COPs obtained allow very good efficiencies. The major inconveniences will thus be the weight, the size, the reliability and the cost. The advantage represented by a weight and volume reduction of the transformer must not be ruined by the weight and the volume of the cooler. Depending on the configuration selected, cryogenic enclosures made of insulating materials (glass-fibre-reinforced resin) can also be used to avoid eddy current losses in metal cryostats resulting in disadvantageous, or even unacceptable, cryogenic losses. ( ii ) Dielectric problems: when using high-voltage windings in a cryogenic fluid it must be known that dielectric strength values are often very bad. The disruptive voltage for 1 mm is 15 kV in He and N2 gas, 25 kV in liquid He, 45 kV in helium at 9 bar (9 × 105 Pa) and 5.5 K and 4 kV in air (see also D2). ( iii )Current leads (see also D10): current transfer from the ambient temperature to the temperature of the cryogenic bath (4.2 K in helium) is achieved via resistive current leads, which results in thermal losses of the cryogenic bath and thus decreases the efficiency of the machine. At present, in the case of current leads in copper (between 300 K and 4.2 K) a loss value of approximately 1.2 W kA−1 can be admitted. The use of Cu-BSCCO composite current leads should make it possible to divide these losses by five. ( iv )Sophistication of conductors: the use of superconductors in transformers requires the manufacture of multifilamentary strands with ultra-fine and twisted filaments embedded in a highly resistive matrix. These strands are then assembled so as to obtain conductors which will allow a high current to be conveyed (see B8.2). ( v ) Recovery of the superconducting state: as explained above, exceeding the critical current results in a transition to the resistive state which allows a current limitation and an increase in the temperature of the conductor at the same time. At a given moment the current must be switched off. The maximum temperature reached depends on the time elapsed between quenching and interruption Td . The maximum temperature is Tm a x and can be calculated with

This relation supposes that all the energy is dissipated by the Joule effect in the windings. In addition it is supposed that the velocity of quench propagation is very fast and there is no hot spot. This is actually true for superconducting wires designed for 50-60 Hz applications (see also B8.2). Of course the time necessary for the superconductive state to be recovered depends on the maximum temperature reached and the coefficient of thermal diffusivity in the winding. In the case of a network, this time should be smaller than 3 s.

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H4.0.3 Different types of transformer When considering conventional transformers, they are generally classified in terms of power range (kVA, MVA), current type (single-phase or three-phase) or type of use. In this chapter superconducting transformers will be divided into two categories, i.e. ‘with iron core’ and ‘without iron core’. We shall only consider single-phase transformers and distinguish between the primary and secondary superconducting case and the superconducting/copper hybrid-type case.

H4.0.3.1 Transformer with an iron core (a) Fully superconducting transformer For all transformers, there is an optimum iron section with respect to losses (Bekhaled and Huve 1985). When increasing this section, the iron core loss development is quicker than the winding and cryostat loss reduction. In the same way, when reducing this section, the winding loss development is more rapid than the iron core loss reduction. This means that transformers without an iron core will only be interesting for questions of weight or simplicity or when the reduction of winding, cryostat or current lead losses can be even more important. The use of currently available superconducting materials decreases the optimum iron section by a factor of four approximately when the iron is at ambient temperature. Moving the optimum towards an iron section reduction is very interesting in terms of weight and iron core losses. This reduction of the optimum iron section is all the more interesting as the total losses are at a lower level. So, in the case of a comparative study carried out on a 220 kVA single-phase transformer with a warm magnetic core, the weight and the losses of the magnetic core can be decreased by a factor of 15 and the total losses by a factor of 24. Another aspect of interest is the possibility of lapping the primary and the secondary in spite of the insulation difficulties this may represent. The lapping reduces the magnetic field applied to the conductors; this increases their transmission capacity and reduces their losses. In our example the lapping is double. This double lapping seems to be sufficient with respect to the loss levels and the critical current density of superconducting materials currently in use. An additional reduction of the iron section (or even its removal) requires an increase of the ampere turns, which leads to a higher magnetic flux density on the conductor or to a higher degree of low-voltage and high-voltage lapping. The volume of superconducting material will be greater and thus more expensive, which will only be acceptable if the present losses are decreased. When considering the efficiency of the machine, we note that with lower losses in the superconductor it will be possible to reduce the iron section; with losses equal to zero in the superconducting winding there would be no need for a magnetic core. To illustrate this example, a 220 kVA single-phase superconducting transformer without an iron core and made out of 14 low- and high-voltage lappings would have approximate winding dimensions of 140 mm outside diameter and a height of 170 mm. Its losses would be 4 kW approximately, i.e. four times more than in the case of the transformer selected; its weight would be 10 kg approximately (hypothesis concerning losses: P (W m−3) = 1.2 × 105 B m a x ). In all cases the superconducting transformer with an iron core seems to be a good solution, but weight and size will often have to be reduced (whilst taking account of the cooling); this is the case with all train, air, ship or satellite-borne equipment. A comparative study in the concrete case of a 4 MVA, 25 kV/1100 V transformer gave the results specified in table H4.0.1 (Poittevin 1992). Note that when taking account of the weight of the cryogenic machine, the total mass must be divided by two in comparison with classical transformers.

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(b) Hybrid-type transformer As specified in section H4.0.2 it is always difficult to use high voltages at cryogenic temperatures. It is, however, possible to construct a new transformer where the low-voltage winding is superconducting and the high-voltage winding conventional. A quick calculation shows that the design of such a transformer is very similar to that of a conventional one. On one hand it is not possible to change the dimensions of the high-voltage winding radically; on the other hand the presence of the cryostat between the high voltage and the low voltage ruins the gain obtained by the superconducting winding. This hybrid-type transformer, which at first does not seem to be very attractive, becomes more interesting when considering its property of ensuring a galvanic insulation between the windings. Such a transformer allows the energy to be transferred from a low-temperature area to an ambient-temperature area without thermal bridges. Thus, the hybrid-type power transformer becomes of interest if it is connected, for instance, to a completely cryogenic generator (see H2.2). Even though completely cryogenic generators are not the subject of this chapter, it must be said that the problem should be considered as a whole. H4.0.3.2 Transformer without an iron core (a) Analysis of a transformer without an iron core When studying electrical circuits connected electromagnetically, variables referred to as inductances (L1, L2, M, I1, I2,…) are used. These variables are always of the type n 2/R where n represents a number of turns and R the reluctance of the ‘magnetic core’ crossed by the magnetic flux. The latter depends on the magnetic permeability of the medium and finally R is of the type I/µS. Ferromagnetic media (transformer laminations) have permeabilities which are approximately 1000 times greater than those of air. Therefore, if the iron core is ‘replaced’ by air, the reluctances become very important (Ra i r Ri r o n ) and consequently the inductances, and also possibly the mutual inductances, become very low. The first consequence of this is a very high magnetization current

With respect to the rated current I1, this current is negligible in a transformer with an iron core but not when the iron core is removed. Although the use of superconductors allows the ampere turns to be increased, I1m is not negligible. In order to make a simple and quick analysis of the transformer without an iron core, we shall use the ‘method of total inductances’. The voltages and currents of the primary and secondary circuit are connected electromagnetically by the following equations (independent of the iron core)

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Transformers

where r1 and r2 are the ohmic resistances of the primary and secondary, L1, L2 are the primary and secondary self-inductances, M is the mutual inductance and

is the coupling coefficient,

is the dispersion coefficient, l1 , l2 are the primary and secondary leakage self-inductances and

is the transformer reactance. In the case of a sinusoidal operation of angular velocity ω and by assuming superconducting circuits (r1 = r2 = 0), the equations become

These equations are general and applicable to all single-phase superconducting transformers with or without an iron core. In general, when defining a transformer, the voltage of the primary Vn 1 and the frequency (50 Hz) as well as the voltage of the secondary V2 n are known. The operation of the transformer can then be calculated from three parameters, σ the dispersion coefficient (Kc ) which is dependent on the geometry of the windings and magnetic cores, the power factor cos ϕ 2 n of the load at the secondary and the expected reactance Xt 2 . The presence or absence of a magnetic core affects the dispersion coefficient σ. Using these equations and as a function of the three parameters indicated above, it is possible to calculate the ratio of apparent powers S1 n /S2 n

In the case of conventional transformers with an iron core the coupling coefficients are close to unity, σ is approximately 10% and the ratio S1 n /S2 n is close to unity. The study of the function above shows that when the dispersion coefficient increases (which is the case for transformers without an iron core) the ratio S1 n /S2 n increases and, as a result, the primary is over-dimensioned with respect to the secondary. This result is well known and is why transformers without a magnetic core are seldom discussed. However, if the losses of the superconductor were nearly equal to zero, despite an over-dimensioning of the primary, it would be interesting to remove the magnetic core. If the magnetic core is to be removed, it is necessary to define a new geometry in which the coupling must be an optimum and, consequently, the leakage fluxes a minimum (see figure H4.0.1 with Xt 2 = 13%). As it is necessary to have very strong coupling and very low leakage fluxes, despite the absence of a magnetic core, coupled toroidal windings should be selected. (b) A toroidal transformer without an iron core (TTSF) As has already been specified above, the choice of a toroidal configuration has the advantage of reducing the leakage flux; it also allows the use of metal cryostats which are more reliable than insulating cryostats Copyright © 1998 IOP Publishing Ltd

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Figure H4.0.1. Transformer without iron core. Sn /S 2 n = f (σ , cos ϕ2 n ) for xt 2 = 0.13.

of glass-fibre resin. It is, however, necessary to keep a minimum distance between the primary and secondary windings for dielectric reasons; therefore, the magnetized volume increases, which penalizes σ. Studies carried out on this type of transformer show that it is recommendable to have a large radius R and a high ratio R2 /R (R2 is the mean outer radius of the winding and R the mean radius of the torus). Table H4.0.2 allows us to compare several types of transformer.

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1620 Transformers

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Transformers

Within the framework of a study concerning a 4 MVA transformer, we demonstrated that there was no weight reduction when comparing a machine with an iron core with a machine without an iron core The only advantage in the case of a machine without an iron core is the metal cryostat; however, the tact that it absorbs much of the primary current is a great disadvantage and, in the case of an on-board transformer, it would be necessary to compensate for this inconvenience on site. For some very special applications where pulse operation is sought, the transformer without an iron core can be of interest, e.g. in the pulse discharge of a superconducting magnetic energy storage system via a transformer without an iron core on a copper secondary or in a high-current supply for cable testing (40 kA).

Figure H4.0.2. 220 kVA transformer Alcatel Alsthom Recherche-GEC Alsthom (from Fevrier et al 1986).

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Examples of manufacturated prototypes

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H4.0.4 Examples of manufacturated prototypes A certain number of experiments have been carried out on prototypes. Generally, the prototypes tested up to now have had Nb-Ti-based superconducting windings: a distinction can be made between transformers with iron cores and inductive transformers used for high d.c. tests on cables. In table H4.0.3 the major characteristics of the following single-phase transformers with an iron core are given chronologically: • • • • •

220 kVA transformer, Alcatel Alsthom Recherche-GEC Alsthom (Fevrier et al 1996) (see figure H4.0.2) 3.3 kVA transformer, CNRS CRTBT Grenoble (Brunet and Tixador 1987) 2 kVA transformer, Kyushu University (Funaki et al 1988) 100 kVA hybrid-type transformer, Nagoya University-Takaoka Electric MFG.G (Kito et al 1991) 330 kVA, the most recent transformer manufactured by ABB (Hornfeldt et al 1993, 1994).

Table H4.0.4 was published by Twente University (Mulder et al 1990). We added the data for the GEC Alsthom transformer which was manufactured for CERN in 1992.

H4.0.5 Interest in high-critical-temperature superconductors A certain number of technical and economic studies started to be published in 1987 when high-critical-temperature superconductors were discovered. Copyright © 1998 IOP Publishing Ltd

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A first study carried out by Kernforschungszentrum Karlsruhe in 1988 (Yamamoto et al 1988) compared a conventional transformer with a 1000 MVA three-phase high-Tc superconductor transformer. In this study the critical currents and a.c. losses were assumed to be equivalent to those of low-temperature superconductors (Nb—Ti with submicrometric filaments). Table H4.0.5 shows the major results.

A second study (Yamamoto et al 1988) made in Japan concerned a 100 MVA 50 Hz 66/22 kV three-phase transformer. Considering a high-Tc superconductor with the following properties, Jc = 5 × 104 A cm−2 and an a.c. loss = 200 kW m−3 at 0.3 T peak field with a cooling efficiency of 10%, the authors showed that the losses are divided by three with respect to the conventional case (115 kW instead of 415 kW). A third study (Giese et al 1992, Wolsky et al 1988) made in the USA by the Argonne National Laboratory in 1989 evaluated the relative costs and performances of three 1000 MVA transformers (conventional, low-Tc and high-Tc ) of service life equal to 30 years. The results are given in table H4.0.6 ( Jc = 105 A cm−2 at 0.2 T and 77 K).

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Conclusion

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Finally the most recent study taking account of the latest progress made in the field of high-Tc superconductors was carried out by Mumford in 1994. This technical and economic study concerns power transformers with high-critical-temperature superconductors within the 30 to 1500 MVA range. It compares the conventional, the low-Tc superconductor and the high-Tc superconductor cases (figure H4.0.3). The losses were evaluated at 10 kW m−3 for low- and high-temperature superconductors. For different values of Jc , the author showed that the overall cost is 0.3 to 0.6 times that of the conventional case for all ratings between 30 and 1500 MVA. In addition, the loss reduction with respect to the conventional case is approximately 40%.

Figure H4.0.3. Relative life cycle cost (from Mumford 1994).

The interest in using high-Tc superconductors is obvious in situations where it is necessary to manufacture conductors of long lengths with sufficiently high critical currents in the presence of a magnetic field and with a.c. losses at an acceptable level, and which can be easily wound. The initial interest is of an economic nature. Indeed, cooling at liquid-nitrogen temperature (77 K) is 10 to 20 times less expensive than at 4.2 K, which is advantageous in terms of efficiency (Mumford 1994). This advantage is very interesting when considering ‘embarked applications’ (train, plane, ship, satellite, etc). When considering operation in the superconducting state, one will note that the main advantage concerns stability. This is due to the fact that at temperatures such as 77 K, the specific heat is much higher than at low temperatures (4.2 K). On the other hand, in the case of a quench, the propagation velocities of the normal zone are very low. Therefore it is necessary to detect the quench and to switch off the current rapidly, which is easy if the current densities are not too high. H4.0.6 Conclusion The various experiments carried out throughout the world have shown that it should be possible to manufacture transformers of higher efficiency by using Nb—Ti conductors with ultrafine multifilamentary strands. In addition to this higher efficiency, weight reduction and self-protection would represent two further advantages. This last property allows us to consider the use of superconductors when weight is considered an important element, which is the case in railway traction situations (several MVA under 25 kV). Provisional studies made on transformers using high-Tc superconductors show that the other advantages to be gained could be even more important, the simplification of the cryogenics being one supplementary asset. Copyright © 1998 IOP Publishing Ltd

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References Bekhaled M and Huve C 1985 Dimensionnement d’un transformateur triphasé à enroulements supraconducteurs Note Interne GEC Alsthom Brunet Y and Tixador P 1987 Small scale experiments on static devices using a.c. superconductors Electr. Power System Res. 12 149 Fevrier A, Bottini G, Kermarrec J-C, Tavergnier J-P, Huve C and Bekhaled M 1986 220 kVA superconducting transformers Proc. ICEC 11 ( Berlin, 1986 ) (Guildford: Butterworth) pp 474–8 Fevrier A and Laumond Y 1986 Prospective use of superconductors for 50/60 Hertz applications Proc. ICEC 11 ( Berlin, 1986 ) (Guildford: Butterworth) pp 139–52 Funaki K, Iwakuma M, Takeo M and Yamafuji K 1988 Preliminary test and quench analysis of a 72 kVA superconducting four-winding transformer Proc. ICEC 12 ( 1988 ) vol 6 (Guildford: Butterworth) pp 729–33 Giese R F, Sheanen T P, Wolsky A M and Sharma D K 1992 High temperature superconductors: their potential for utility application IEEE Trans. Energy Conversion EC-7 589–97 Harrowell R V 1970 Feasibility of a power transformer with superconducting windings Proc. IEE 117 131–40 Hornfeldt S, Albertsson O, Bonmann D and Kônig F 1993 Power transformer with superconducting windings IEEE Trans. Magn. MAG-29 3556–8 Hornfeldt S, Albertsson O, Kônig F and Bonmann D 1994 Transformateurs supraconducteurs Revue A. B. B. 1/1994–13 Kito Y, Okubo H, Hakayama N, Mita Y and Yamamoto M 1991 Development of 6600 V/210 V 100 kVA hybrid-type superconducting transformer IEEE Trans. Power Delivery PD-6 816–23 McFee R 1961 Superconducting power transformers—a feasibility study Electr. Eng. 80 754–60 Mulder G B J, Ten Kate H H J, Krooshoop H J G and Van de Klundert L J M 1990 On the inductive method for maximum current testing of superconducting cables 11th Int. Conf. on Magnet Technology ( MT-11 ) vol 1, pp 479–84 Mumford F J 1994 A techno-economie study of high-Tc superconducting power transformers Int. Conf. on Electrical Machines ( 1994 ) pp 630–5 Poittevin J 1992 Avenir du transformateur supraconducteur, Association Suisse des Electriciens: La supraconductivité dans les techniques de l’énergie Report Berne Riemersma H, Barton M L, Litz D C, Echels P W, Murphy J H and Roach J F 1981 Application of superconducting technology to power transformers IEEE Trans. Power Appl. Systems PAS-100 3398–105 Schauer F, Jûngst K P, Komarek P and Mawrer W 1987 Assessment of potential advantages of high-Tc superconductors for technical applications of superconductivity Kernforschungszentrum Karlsruhe Report KfK 4308 Van Overbeeke F 1986 On the application of superconductors in power transformers PhD Thesis Twente University Van Overbeeke F, Oordt K and Van de Klundert L J M 1985 Design and operation of a protection system for transformers with superconducting windings Cryogenics 25 687–94 Westinghouse Electric Corporation 1982 Application of low temperature technology to power transformers US Department of Energy Report DOE/ET/29324–1 Wolsky A M, Daniels E J, Giese R F, Harkness J B L, Johnson D R and Zwick S A 1988 Advances in applied superconductivity: a preliminary evaluation of goal and impacts Internal Report Argonne National Laboratory (January 1988) Yamamoto M, Ishi Gohka T, Kaiho K and Kimura Y 1988 Conceptual design of a power transformer with high Tc superconductor Proc. ICEC 12 (1988) (Guildford: Butterworth) pp 734–8

Further reading Aichholzer G and Schauer F 1984 Coaxial turn transformer Proc. ICEM 84 vol 1, pp 58–61 Fevrier A 1988 Preliminary tests on a superconducting power transformer IEEE Trans. Magn. MAG-24 1477–80 Fevrier A 1989 Applied Superconductivity Technology — Now and Future Transformer ch 5 Schauer F 1990 Superconducting coaxial turn transformer Cryogenics 30 866–88 Tixador P and Laumond Y 1991 Projets et développements en cryoélectrotechnique J. Physique 1 237

Copyright © 1998 IOP Publishing Ltd

H5 Power transmission

J Gerhold

H5.0.1 Introduction Electrical power transmission dates back to the late 19th century when the first remote hydroelectric power stations were erected. The benefits of high-voltage over high-current long-distance transmission were soon recognized because, of course, high-voltage transmission incurs a minimum of power loss. Voltage transformation up and down is easy in alternating current (a.c.) systems but costly with direct current (d.c.) and, for this reason, most of the power grids around the world employ a.c. transmission; the three-phase system is dominant for its inherent balancing. D.c. power transmission is of relatively minor importance but it is used occasionally for very long-distance-very high-voltage transmission, or for short interconnects between large a.c. systems. Utilities prefer overhead systems because the main insulator, i.e. the ambient air, is freely available and the aging of the dielectric is of little concern. It is necessary, however, to use underground power cables in urban and suburban areas, or for submarine transmission. Enormous progress has been made between installation in 1890 of the first 10 kV a.c. Ferranti cable and the present 400 kV/1 GVA systems. Generally three-phase a.c. cables are employed; however, d.c. must be used for long submarine cables to avoid unacceptable capacitative loading. Urban areas are characterized by a very high power density and limits on the cable capacity became evident in the early 1960s. The unavoidable ohmic and dielectric losses produce heat, the removal of which, by natural soil sinking, is restricted as illustrated in figure H5.0.1 for a 110 kV cable. It is seen that without cross bonding there are high jacket losses. Note also that natural soil sinking is limited to less than 70 W per unit cable length. It follows from this that a further increase of the conductor cross-section cannot increase the rated power. Higher voltages allow the rated power to be increased but it is still limited to below 300 MVA and, in the early 1960s, this was considered to be too low a level in view of future demands. Forced cooling, e.g. by water, was generally recognized as an adequate remedy to allow the rated power to rise to 1 GVA (Kiwit et al 1985). At about the same time that forced cooling was deemed to be necessary, enormous progress was being made in cryogenics and superconductivity; the latter offered two very uncommon and exciting features for power transmission: ( i ) no ohmic losses; ( ii ) extremely high real current densities. Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.1. Losses in a 110 kV paper/oil cable. Reproduced from Kiwit et al (1985) by permission.

The first reaction to these new features was to suggest a power transmission with high current/low voltage because no limitations for current rating were anticipated. For example, a 60 kA, 20 kV d.c. line was mentioned by McFee in 1962, and Klaudy proposed in 1965 a fully flexible a.c. cable with 10 kV and 8.7 kA. This would have allowed a direct coupling of the generators to a consumer grid, and would have eliminated the need for transformers in the power station; as a further step direct consumer voltage transmission was considered even to the extent of using direct current. However, existing a.c. systems were an established fact and this prevailed to bring about more realistic designs. In the medium-distance transmission, of the order of tens of kilometres, superconducting a.c. cables of 1 GVA ratings at 100–150 kV were found to be the most acceptable and were developed up to the prototype stage and then field tested by utilities in the late 1970s. However, due to the cost of converters, long-distance transmission was seen as the domain of d.c. and very high-rated power systems, for instance, 600 kA at 200 kV yielding 120 GW, were claimed. In a more detailed study a 1000 km long, 200 kV d.c. line for 100 GW was proposed (Garwin and Matisoo 1967). These figures may circumscribe the initial prospective broad regime of transmission lines with classical superconductors. The discovery of new superconductors with high critical temperature, i.e.>100 K, in the late 1980s, has shifted the interest to lower a.c. power ratings, i.e. <1 GVA. Superconducting cables with moderate rated power are deemed to be of serious interest for retrofitting existing ducts in urban areas (Ashworth et al 1994). Table H5.0.1 provides an overview of the widespread efforts around the world to develop cables with classical superconductors; some cryoresistive cables cooled by liquid nitrogen are also included and the acquired knowledge may be very useful for further work with the new high-critical-temperature superconductors. The table also gives information about commonly used abbreviations of companies and research centres. H5.0.2 Main constraints and benefits It is necessary for superconducting cables to use complex cryogenic technology and this means that the benefits of having heavy currents in a nonresistive mode are offset by the cost which this brings. In practice superconductivity in metals can be used up to temperatures of the order of 10 K. This allows an appropriate Copyright © 1998 IOP Publishing Ltd

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margin from the highest critical temperature of about 20 K. Therefore the only available cooling medium with classical superconductors is helium. The new high-Tc superconductors still need cryogenic cooling, e.g. by liquid hydrogen, liquid neon or preferably by liquid nitrogen. In either case an effective heat shielding system, i.e. a cryogenic envelope, must be employed to enclose the superconducting cable core. The envelope uses the well-proven means of cryostats, i.e. vacuum, radiation shields and supports with low thermal conduction, to ensure a most efficient thermal insulation. H5.0.2.1 Constraints Heat shielding can never be perfect so there are always some small heat leaks from outside into the coolant. Thus, even a d.c. cable without any electrical losses results in entropy being given up to the ambient by the refrigerator compressors as waste heat, irrespective of the actual cable load. For a.c. transmission cables it is necessary to add some dielectric and superconductor losses, the latter being dependent on the actual load. Early claims mentioned an efficiency of better than 99.5% for a 200 km d.c. line but a more realistic estimation stated almost 99%, including the terminations, for a 10 km a.c. cable for instance. These figures are based on classical superconductors. High-Tc superconductors show distinct hysteresis losses but the heat pump factor is much lower than for classical superconductors. Thus, the efficiency of an a.c. high-Tc superconducting cable may be of similar order. The efficiency figure is not very different from competing cables, e.g. water-cooled cables. Loss saving is therefore not a very strong argument in favour of superconducting cables, especially not in favour of a.c. cables. H5.0.2.2 Benefits Electrical system engineers are fascinated by the inherent compactness of superconducting cables in trenches; there is no thermal interaction and the heavy current outward and return paths can be arranged very closely. The level of magnetic stray fields can easily be reduced to a point that virtually eliminates electromagnetic pollution. It has been shown that subdivision into parallel circuits can already limit the magnetic field levels very effectively (McFee 1962, Kafka 1969, Klaudy 1966). A complete stray field elimination can be guaranteed when using a superconducting shield conductor. No cross bonding will be required for shielded three-phase systems.

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A second advantage is the cooling by refrigerators at distinct points, i.e. by preference at the cable ends. This eliminates the problem of heat removal along the cable route because there is practically no heat exchange between the cable and its surroundings. The elimination of interfering electromagnetic stray fields and heating of its surroundings allows much more licence for route design than would otherwise be the case. Right-of-way costs can be minimized and emotional objections to heavy power transmission may be overcome more easily than with overhead lines because, additionally, superconducting cables cause no corona noise and do not affect the landscape at all. There is practically no thermal cycling of a superconducting cable during its operation. Only the conductor losses depend on the actual load, the dielectric losses as well as the heat in-leak from ambient being constant. No severe variations to the temperature of the coolant may occur. Thus aging of the cable due to thermal cycling is neglible. Finally, because the current density in classical superconductors may be as high as > 1010 A m−2, the quantity of expensive superconductor could be kept at a minimum and the classical cable core has been claimed to be made lightweight. Industrially manufactured high-Tc superconductors, however, have a lower current density at present, i.e. of the order of 108 A m−2, so the quantity of high-Tc superconductor will become somewhat more important. The weight of the cryogenic envelope on the other hand can be kept at a low level because it contains mainly a vacuum space with superinsulation. The coolant fluid is an inert gas, i.e. helium for classical superconductors, or preferably liquid nitrogen for the high-Tc superconductors, which is found naturally in the atmosphere and does not present a hazard. The disposal of such a cable at the end of its operational life is straightforward because there is no irreversible impregnation of the components, nor any harmful material, and a far-reaching recycling programme may be anticipated. Thus superconducting cables may fit very well into a future scenario that requires materials to be recycled. H5.0.2.3 Superconducting cable components A superconducting cable system has four main components. ( i ) The electrical cores embodying the live conductor, the annular dielectric insulation and the grounded shield. Cables for d.c. transmission may have the outward and return lines at high voltage and symmetrical to ground, i.e. in the form of two monopole cores. This is indicated in figure H5.0.2. Alternatively coaxial conductors, where the grounded outer conductor carries a current in the opposite direction to that in the inner live conductor, have often been proposed, see figure H5.0.3; for this design there are no electrical or magnetic fields outside the cable, which has also been commonly accepted for three-phase a.c. cables. To ensure superconductivity the core has to be cooled by a suitable refrigerant and for the metallic materials, e.g. Nb, Nb—Ti or Nb3Sn which have low critical temperatures, helium was the sole refrigerant which could be used in practice. Boiling liquid helium has been found not to be suitable because bubbles may impair the dielectric strength and the two-phase conditions may impede a safe coolant flow. Supercritical helium on the other hand is in fact a very suitable cooling and insulating fluid. The live conductor may be cooled from the inside whereas the grounded shield conductor is cooled from the outside. Cooling of high-Tc superconductor cable conductors may be provided by pressurized liquid nitrogen instead of supercritical helium for instance; again, bubble formation has to be prevented. Of course, the overall core dimensions should be minimized to limit cost and the Poynting vector in the annular dielectric space must be optimized to allow the maximum power density. This leads to the choice of circular conductors, similarly to conventional cables. ( ii ) The cryogenic envelope which has to minimize the heat in-leak into the cooled core. Since heat in-leak rises with increasing radial envelope dimensions this is another reason for achieving the maximum power density.

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Figure H5.0.2. A superconducting d.c. cable with two monopole cores: 1—cooling helium; 2—conductor; 3—tape dielectric; 4—cooling helium; 5—core enclosure; 6—spacer; 7—suspension system; 8—radiation shield; 9—intermediate temperature liquid-nitrogen duct; 10—spacer; 11—superinsulation; 12—vacuum enclosure.

Figure H5.0.3. A superconducting cable with concentric live/ground conductors: 1—cooling helium go; 2—live conductor; 3—tape dielectric; 4—ground conductor; 5—cooling helium return; 6—helium duct enclosure; 7—radiation shield; 8—intermediate temperature liquid-nitrogen duct; 9—vacuum enclosure.

( iii )The terminations which have to link the cold core with the outside power system at the cable ends. This problem, which is not met with conventional cables, requires a transition from a highly stressed cable dielectric into ambient air and a temperature transition from cryogenic to ambient temperature. ( iv )The cooling circuit which requires special refrigerators to remove the heat in-leak through the cryogenic envelope and the heat due to core losses. In addition, the cooling system must provide for the coolant flow along the cable and for this adequate cooling ducts must be provided. There are severe distinctions between the cores of d.c. and a.c. cables which are due to different physical factors. Also, the cable terminations are different for a.c. and d.c. cables. However, the cryogenic envelope as well as the requirements for the cooling circuit are similar for both types, at least in their general design and performance although the numerical data may be different. A lot of work has been performed on classical superconductors and a broad spectrum of data and ingenious technological solutions have been collected around the world. This is discussed quite comprehensively in the following sections in order to make clear the inherent problems. This work stopped in the early 1980s due to the effect of the energy crisis. Many of these earlier ideas may be transferred, however, into the high-Tc superconductor domain with minor adaptions. The new high-Tc Copyright © 1998 IOP Publishing Ltd

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superconductors with different physical factors give rise to additional challenges and prospects of course, which will be suggested in comparison with the classical standard solutions. The reader is asked to assess whether a standard solution may still be adequate or should be better replaced by an improved technology. To ease the reading of the subsequent sections, all equations are written in SI units and the units may not always be indicated. However, diagrams and numerical examples use the more comfortable multiples of the basal units which are in common use in practice. H5.0.3 Cable cores The cores are the active electrical part of any cable. In a superconducting cable the current flows in the live conductor which has no resistance and which is insulated from ground by a dielectric space. In the case of d.c., the current density is restricted to below the critical value which may be up to 109-1010 A m−2 in type II superconductors such as Nb—Ti, Nb3Sn or Nb3Ge. Bismuth-based high-Tc superconductors of the 2223 type cooled by liquid nitrogen are currently being developed for cable applications, with a current density considerably below 109 A m−2 at worst. High-Tc superconductors of the 2222 type may have a higher critical current density but will need a lower cooling temperature, e.g. cooling by liquid hydrogen or liquid neon, the latter resulting in a more expensive refrigeration circuit. For a.c. there are losses which limit the allowable current density to a lower value. Pure Nb or Nb3Sn were the top candidates for use in classical a.c. cables, and 2223 high-Tc superconductors seem to have become the top candidates for high-Tc superconductor a.c. cables. Because of its effect upon current density it is necessary to reduce the magnetic field to the lowest possible level and this requirement favours the use of live conductors with circular cross-section surrounded by an annular dielectric space. The electrical field then is purely radial and uniformly distributed when providing a grounded Hochstädter screen. In the case of d.c. transmission, it has been proposed to use two separate monopole cores as shown in figure H5.0.2; the live conductors which carry the outward and return current, respectively, are operated symmetrically to ground with a voltage of ±U. The magnetic fields are moderate so operation is close to the self-field condition. The concentric arrangement as shown in figure H5.0.3 is also of interest in the case of d.c.; the grounded shield conductor now carries the return current and no magnetic field is present outside the core. The concentric arrangement is most interesting for a.c. transmission because stray magnetic fields would otherwise cause eddy current losses within cold surrounding metallic parts, e.g. in the heat-shielding system. H5.0.3.1 Cores for a.c. cables To ensure the elimination of stray fields in a concentric arrangement, as shown in figure H5.0.3, the shield conductor has to carry precisely the instantaneous but opposite current to the live conductor and short circuiting the shield conductors of a three-phase system is very advantageous (Maddock and Male 1976). An ingenious circuit proposed by Siemens is indicated in figure H5.0.4; there is no net neutral current in the system. Neither an electrical field, nor any magnetic field exists outside the shields, even under load imbalance conditions, and consequently no voltage drop appears along the superconducting shields. The sophisticated techniques such as jacket cross-bonding which are common with conventional cables are dispensed with for superconducting cables. (a) Conductors for a.c. cables Superconductors carry d.c. currents without ohmic losses; however, when carrying an a.c. current, hysteretic losses occur. These are small at weak magnetic fields but rise sharply above some tens of millitesla, i.e. considerably below the critical field for the superconductor. Current and field penetration is still restricted to a very thin surface region in the classical superconductors; thus, current flows on the outside of the

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live conductor but on the inner side of the shield conductors, as shown in figure H5.0.4. The bulk of the superconductor is not involved and therefore thin superconducting surface layers on an appropriate substrate with annular cross-section were claimed to be sufficient and cost effective. Current flow within the bulk conductor must be accepted for the high-Tc superconductors on the other hand, due to the considerably lower real critical current density. These new materials will require a somewhat larger cross-section but this will still form a thin surface layer on a core with circular cross-section. Silver cladding over the high-Tc superconductor layer may be a promising technology in order to ensure chemical stability.

Figure H5.0.4. A three-phase superconducting a.c. cable: 1—transformer with remote common ground; 2—live conductor; 3—transformer secondary; 4—shield conductor; 5—neutral point; 6—current flowing on outer surface of live conductor; 7—current flow on inner surface of shield conductor; 8—circumferential magnetic flux lines; 9—radial electrical field lines.

Figure H5.0.5. Typical a.c. losses of Nb3Sn tape and of Nb-Cu strip conductor at 4.2 K/50 Hz (from Maddock and Male 1976).

Losses Figure H5.0.5 shows the measured loss increase for the relevant classical superconductors, i.e. pure Nb or Nb3Sn, which have been discussed as being preferred for a.c. cables (Maddock and Male 1976). Actual surface conditions such as roughness, surface layers and inclusions may have a strong influence on the performance: the real field penetration depth then is of the order of 0.1 µ m only. The problem is to maintain extremely good surface conditions during an industrial production process; relying on usual metallurgical Nb surfaces may shift the Nb curve to the left so the available field may significantly be reduced. The increase in surface loss becomes more and more important above 300 mW m−2, and 100 mW m−2 has Copyright © 1998 IOP Publishing Ltd

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often been claimed as an appropriate upper level for normal operation. This provided some margin for overload operation. The magnetic field along the surface must be as uniform as possible to avoid strong field regions and this condition again strongly favoured circular concentric conductors, where the current is distributed uniformly over the surface. The related magnetic field is circumferential with a minimum possible value at the live conductor surface for any actual current I. No applied field is built up from the shield conductor, so the live conductor sees only its self-field. It has become common practice to classify cable conductors of diameter d by their surface current density Ss rather than by the magnetic field H

Note that PI and Ss have exactly the same numerical value. The classification correlates very clearly the rated current with the live conductor diameter. However, the surface current density in a type II superconductor can be directly correlated with the penetration depth δs via

provided the Bean model is applicable with constant critical current density Jc . The actual superconducting surface layer must have a somewhat larger thickness ds of course. The inner cross-section of a thin-walled annular conductor can be used, for example, as a cooling fluid duct. High-Tc superconductors will need a considerably larger layer thickness than Nb3Sn for instance, due to the much lower critical current density of the former. Note that the critical current density in an high-Tc superconductor depends strongly on the applied magnetic field strength. Figure H5.0.4 indicates neatly that Ss is highest around the live conductor. The shield conductor produces negligible losses in theory, and, therefore, shielding currents were claimed to be carried without practical loss penalty in superconducting a.c. cables. Unfortunately, a long cable cannot be operated under completely isothermal conditions and, when cooling with supercritical helium or pressurized liquid nitrogen, respectively, the temperature will vary along the cable route of total length l, according to heat in-leak and cooling circuit parameters. According to the critical field decrease of any superconductor when approaching the critical temperature Tc, a corresponding lowering of the applicable surface current density must be anticipated in an a.c. cable conductor. This has been clearly demonstrated by the CERL group as well as by Siemens researchers; figure H5.0.6 illustrates this effect for tubular Nb conductors. It is evident that a high current rating needs the operating temperature to be as low as possible and Ss is determined by the maximum temperature in the conductor, not by the average value. Nb provides only a narrow temperature margin, since its Tc is only about 9 K. Example H5.0.1 An Nb live conductor is required to carry 5.2 kA root mean square (rms) at a loss level 100 mW m−2, and the maximum core temperature, Tm a x , is set at 6.5 K. A relative surface current density of 0.7 is found from figure H5.0.6 and a current density of 7 × 104 A m−1 at 4.2 K from figure H5.0.5. Therefore, a surface current density of 4.85 × 104 A m−1 can be claimed at 6.5 K. The diameter is found to be 0.034 m from equation (H5.0.1). The superconductor loss per unit core length is then ≈0.01 W m−1, neglecting the shield losses. It is seen that Nb was in fact one promising candidate for superconducting a.c. cables. However, relying on the more realistic loss characteristics of industrial surface quality only 3 × 104 A m−1 may be permissible; the corresponding diameter being 0.055 m. An alternative and promising type II superconductor, i.e. Nb3Sn, has been investigated very successfully by the BNL group (Forsyth 1987). The theoretical hysteresis losses at frequency f which are

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Figure H5.0.6. Relative surface current density of Nb versus temperature.

Figure H5.0.7. Relative surface current density of Nb3Sn versus temperature.

dissipated in an ideal surface layer of thickness ds around a circular substrate can be calculated from

where I is the actual rms current carried within the superconducting layer. Figure H5.0.7 shows the relative surface current density versus operating temperature. Ss at 4.2 K is similar to that in Nb, but the temperatures can be considerably higher with Nb3Sn due to its higher critical temperature (Tc = 18 K). The real critical current density in Nb3Sn is > 1010 A m−2, and the magnetic field may only penetrate several micrometres into the surface layer which must have an adequate thickness of course. A valid comparison between Nb and Nb3Sn must include the different heat pump factors of the particular helium refrigerators, i.e. the required power to be fed into the refrigerator for every watt of heat which is put into the low-temperature region. Cables with Nb conductors may require up to 500 WAV in Copyright © 1998 IOP Publishing Ltd

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a temperature range 4.5–6.5 K but the group at BNL has found 250 WAV for 6.7–8 K operation, which seems appropriate for Nb3Sn superconductors. Example H5.0.2 We assume the current in an Nb3Sn conductor to be 4.1 kA. According to the lower heat pump factor, the permissible loss level is set at 200 mW m−2. The maximum operating temperature may be allowed to come up to 8.0 K. The corresponding 4.2 K surface current density is 6.5 × 104 A m−1, see figure H5.0.5. Figure H5.0.7 suggests a relative surface current density of 0.89, i.e. an operating value near 5.5 × 104 A m−1. Hence the conductor diameter is found to be 0.024 m. This illustrates the theoretical advantage of using Nb3Sn instead of pure Nb. However, there are additional parasitic losses so the final diameter had to be larger and the final loss level was considerably higher; this will be discussed later on. A high-Tc superconductor will yield higher a.c. losses. A much larger penetration depth of the order of 0.1 mm must be anticipated, according to the rather modest critical current density. It is seen from equation (H5.0.3) that enforcing as low as possible a diameter, which may be of interest for retrofitting with very compact cables, yields high losses. Materials with high critical current densities must then be used to prevent excessively high a.c. losses (Hara et al 1992). However, a higher loss level than with classical superconductors can be accommodated since the heat pump factor with liquid-nitrogen cooling may be of the order of ten only. Example H5.0.3 A thin 2223 high-Tc superconductor layer on a circular substrate with diameter 31 mm is assumed to carry a rated current of 4.1 kA. The circumferential self-field amplitude will amount to 6.2 × 104 A m−1; this is again synomynous with the surface current density Ss. The cooling entrance temperature Te n = 70 K at x = 0 is assumed for simplicity to increase linearly with χ up to Tex = 85 K at the coolant exit where χ= l. The critical current density Jc may decrease from 5 × 108 A m−2 down to 3 × 108 A m−2. The amplitude of the field penetration is found from equation (H5.0.2) to increase from 0.12 mm at the coolant entrance up to>0.2 mm near the coolant exit; a high-Tc superconductor layer thickness ds = 0.4 mm may be appropriate. The per unit length loss in the conductor will vary from 1.7 W m−1 at the coolant entrance up to almost 3 W m−1 and is proportional to 1/( 5 × 108– 2 × 108x/l ), according to equation (H5.0.3). The mean losses are found from integrating the actual loss per unit length along the cable route with respect to Jc and then dividing by the total length l. This results in an average of 2.2 W m−1, which would already be an unacceptably high level, even with a heat pump factor of 10 WAV, since additional parastic losses have to be borne in mind. Thus, improved materials with higher critical current density have to be developed. In addition to the minimization of losses by means of the concentric conductor design there is an additional important argument for the latter. The high currents, i.e. several kiloamps, cause strong electrodynamic forces; however, in a concentric arrangement these forces cancel out. Thus the need for heavy structures in the dielectric space is eliminated as are the mechanical oscillations of the conductors. Such oscillations would cause additional losses, and also might give rise to premature mechanical aging. This may be very dangerous for high-Te superconductors. Mechanical restrictions Cable conductors must allow for thermal expansion, contraction and bending. Conventional cables often use helically wound segmented conductors, and discrete expansion elements between rigid conductor sections are common with long gas insulated lines. Thermal contraction control was a very severe problem in early superconducting cable designs, which were based on rigid tubes. Discrete bellows elements were proposed to compensate for the contraction which is about 0.3% when cooling down from ambient to the operating temperature. For a 10 km cable this results in a shrinkage of 30 m which cannot be compensated for by simple elastic strain in a rigid

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tube; the forces are unacceptably high and would exceed the elastic limit of the materials. All the rigid tube designs reached a dead end because a large number of bellows had to be incorporated between the rigid conductor elements of limited length and these had to be superconducting. The costs were found to be prohibitive particularly for the installation of the cables in the field. Two concepts have shown promise for overcoming the thermal contraction problem and allow manufacturing in long unit lengths: ( i ) segmented conductors consisting of helical strips; this technique has been developed for Nb- as well as for Nb3Sn-based conductors. It seems to be still adequate for the new high-Tc superconductor cable conductors; ( ii ) tube conductors with helical corrugation; this technique has been shown to be successful with Nb. Attempts to extend it into the domain of Nb3Sn have been made but never concluded. There are no known development efforts in the high-Tc superconductor domain. Segmented conductors A segmented conductor with helical strips is shown in figure H5.0.8. If we assume an unchanged conductor length on cool-down it follows that the pitch, p, is also unchanged. The strip length, l = ( p 2 + (π)2 )1/2, shrinks according to the unit thermal contraction εt h by an amount ∆l = εt h l. The radial shrinkage is found from π∆d/πd. Thus, we find

This shrinkage is higher than that of an equivalent rigid cylinder by an amount p 2/π2d 2. To allow the helix to shrink needs adequate butt gaps, b: Σ b should roughly equal π∆d so the gaps disappear at the operational temperature. This is favourable in view of a smooth conductor surface.

Figure H5.0.8. A cable conductor with helical strips.

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Example H5.0.4 A conductor helix of 0.03 m diameter is required to shrink radially by 1%; this figure is set in view of the dielectric. Thus, ∆d = 0.3 mm and Σ b must come up to almost 1 mm. A unit thermal contraction of 0.3% is assumed; this is typical for niobium and copper or for stainless steel. The appropriate pitch p is then found to be 0.14 m and the pitch angle γ is 56°. Care must be taken to allow for core bending. This may not be a severe problem when bending is restricted by the ambient temperature going up and down during cable installation, i.e. drawing an Nb-based core into a rigid cryogenic envelope, because the butt gaps which have been provided will allow for a sufficient strip displacement. However, Nb3Sn is a very brittle superconductor and strain must never exceed 0.2% and, as a consequence, bending a Nb3Sn core causes some problems. Bending may become still more critical with high-Tc superconductors where irreversible damage can even occur when strain exceeding 0.1–0.2% is applied. There is another problem caused by conductor helices in a.c. cables; in a simple helix, the current is subdivided into the single strands with no current flow between the strands. This current produces a considerable axial field, which increases the a.c. losses. The problem may be overcome by employing a double helix with the result that the axial field is eliminated (Forsyth 1987); figure H5.0.9 illustrates this for the live conductor and for the shield conductor. A condition for a zero axial flux is a pitch angle of ±45° but this was not fully acceptable in practice. Measurements carried out by the BNL group yielded for instance residual axial fields of roughly 0.1 mT in a 13 kA core. Such a low uncompensated flux was considered to be tolerable in view of additional losses in the strips.

Figure H5.0.9. Current flow in a double helix a.c. cable core: 1, 2—current along the live conductor; 3, 4—current along the shield conductor. Reproduced from Brookhaven National Laboratory Informal Report PTP#69 (27 December, 1976) by permission.

It must be mentioned here that, with classical superconductors, strips were claimed to be only clad by the expensive superconductor, the bulk material being a normal metal like copper. A complete cladding over the whole strip surface would have been appropriate. Otherwise, so-called edge losses may arise, the nature of which has not yet become very clear. In the case of complete cladding with pure Nb at 5 K, the measured helix conductor losses corresponded well to figures H5.0.5 and H5.0.6. Nb-clad strips have been used by CERL and Siemens for example, but BNL could not cover the strip edges completely with Nb3Sn. As a consequence losses were found to be much higher, i.e. 200 mW m−1 at 7 K in a 4.1 kA conductor whilst Ss = 4.4 × 104 A m−1. High-Tc superconductor strips and wires, respectively, are manufactured for instance by the powder-in-tube process. The superconductor is completely covered by hardened silver. Eddy current losses must be faced in addition to hysteresis losses in the high-Tc superconductor. Some prototype cable conductors have been built with multilayer helices, for instance six layers, which does not seem to be a very appropriate final configuration. Investigation of the additional parasitic loss mechanism is in progress (Fujikami et al 1994). Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.10(a) shows a complete Nb-clad 1 mm thick copper strip, as used in the CERL conductor. The corresponding Nb3Sn strip used by BNL and shown in Fig. H5.0.10(b) is more complicated, mainly because it was necessary to eliminate excessive strain in the brittle Nb3Sn layer. The Nb3Sn is located close to the neutral zone, which is achieved between a copper/stainless steel composite. The parasitic eddy current losses in the stainless steel and solder have been found to be tolerable. A monocore silver-clad high-Tc superconductor strip is illustrated schematically in figure H5.0.10(c) for comparison. The high-Tc superconductor region cannot be located close to the neutral zone and microcracks are an inherently serious problem; multifilamentary strips which are under development at various laboratories seem to be a more promising configuration.

Figure H5.0.10. Tapes for a.c. cable cores: (a) niobium clad copper strip (from Maddock and Male 1976); (b) BNL conductor tape (reproduced from Forsyth (1988) by permission); (c) silver-clad high-Tc superconductor strip; top—before bending, bottom—after bending (reproduced from Patel et al (1995) by permission).

Of course, any helical conductor needs appropriate struts inside. Several formers have been proposed and tested. Figure H5.0.11 illustrates a technique used by CERL. A competitive strut technique has been used by Siemens. BNL relied on helical bronze struts; of course, strut shrinking had to fit into the radial contraction of the conductor. Corrugated conductors Helical corrugation is an old established technique in the cable industry. It has been advocated by Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.11. The flexible a.c. cable core proposed by CERL. Reproduced from Edwards (1988) by permission.

Kabelmetal and consequently used in the ATF a.c. cable. In theory helically corrugated tubes can be manufactured in endless lengths; however, transportation sets a limit for practical individual lengths. Conductors as made by Kablemetal are longitudinally welded, the bulk material being high-resistance-ratio copper. The copper thickness δ of such a corrugated tube could not be chosen much below 1 mm for mechanical reasons. The adjacent Nb cladding of 10 µ m thickness on the outside of the live conductor and inside of the shield conductor was therefore interrupted in the axial direction by a small slot, see figure H5.0.12. This prevented helical current flow without increasing the a.c. losses; the welding was vacuum tight. However, the superconductor surface was larger than that of a straight tube of equivalent mean diameter and, of course, there was a field enhancement at the wave peaks. It has been found appropriate to base the a.c. loss estimation on a fictitious surface current density near the mean diameter = (do u + di )/2, and to use the total area of the corrugated surface. Surface quality of the Nb was as is usual after careful rolling, a 100 mW m−2 level for a.c. losses being related to Ss ≈ 4 × 104 A m−1. The strain during cool-down could be roughly calculated assuming circular arcs for the corrugations (see figure H5.0.13), the core length being maintained by stretching the arcs within the elasticity limit.

Figure H5.0.12. The manufacturing scheme for a helically corrugated niobium/copper cable conductor according to Kabelmetal: stage 1—cold welding of copper particles onto a niobium sheet; stage 2—electrolytic copper reinforcement and rolling; stage 3—tube forming and longitudiual welding, corrugation. Reproduced from Klaudy (1980) by permission. Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.13. The wall of a helically corrugated conductor.

The use of normal conductors, e.g. a copper substrate clad by the superconductor, or silver sheathing of a high-Tc superconductor core strip, reveals a very important operational advantage because the superconductor is inherently stabilized by a low-resistive shunt. This is a technique widely used for most applications of superconductors. The most important aspect in the design of the stabilization of cable conductors is in the handling of fault currents. Instantaneous short-circuit currents may easily be more than ten times the rated current and, under these conditions, superconductivity cannot be maintained in an economic design. As a consequence of the high resistivity of a superconductor after quenching, a highly conductive parallel path has to be provided and pure copper or silver can be used to make an appropriate shunt. The total amount of shunt material has to be chosen according to the acceptable heat production during the unprotected fault as a worst case scenario. This heat causes a pressure rise of coolant which must be accommodated in the cooling ducts. Example H5.0.5 An Nb-clad helical corrugated copper conductor with a mean diameter d = 0.055 m is stressed at a temperature of 6 K by a fault current the thermal mean equivalent of which is It h = 30 kA rms. The circuit breaking time At is 0.1 s. The high copper residual resistivity ratio (RRR) of 150 results in a low skin depth of only 0.3 mm at 50 Hz so the effective a.c. conductor resistance R after quenching amounts to 8µΩ per unit length. Heat production is then It2h R ∆t = 0.72 kJ m−1. This heat is primarily stored in the copper (M = 1.45 kg m−1) by raising its temperature Te to ≈35 K, according to It2h R ∆t = Mc ( Te — 6 K ) with c for the averaged specific heat. Such a temperature would increase the copper resistance but raises the skin depth, the final resistance being of the order of 20µΩ. Using the latter value as a worst case limit yields a heat production of 2 kJ m−1 and a final temperature near 50 K. However, there is supercritical helium inside the conductor and assuming its initial state with 0.5 MPa at 6 K, for instance, yields a helium mass of 0.25 kg per unit length. The helium cannot expand during the fault so it takes approximately the full amount of heat produced and is warmed up under constant volume conditions, i.e. to about 7.5 K at a pressure of 1.1 MPa; this may be found easily from the internal energy difference along the isochore. For the worst case limit the result is 9 K and 1.4 MPa. This is a very conservative approach. The instantaneous heat transfer from the conductor into the helium is difficult to estimate, but would not contribute much to conductor cooling during the short breaking time. Therefore the pressure rise may be less than estimated above. There is no doubt that fault current control could be effective in classical superconducting cables without damage. Figure H5.0.14 sketches the general course of the temperature balancing which can extend to several seconds. High-Tc superconductor cable cores may be more seriously affected by short circuits. There is an inherent danger of liquid-nitrogen vaporization, and restoration of normal operation must then overcome two-phase flow, which may be troublesome. This point needs careful attention when designing a high-Tc superconductor cable (Forsyth 1993). To shorten the recovery time after a fault requires a limit to excessive Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.14. The temperature course of a cooling fluid-filled cable conductor in the case of a short circuit: 1, 2—conductor and helium temperature respectively, very weak heat exchange; 3—perfect heat exchange; 4, 5—conductor and helium temperature respectively, true heat exchange (from Gerhold and Schachinger 1980).

currents, or a reduction in the breaking time may be advisable. Superconducting fault current limiters can be very helpful. (b) Electrical insulation of a.c. cores The phase-to-ground insulation is provided within the annular space between the line conductor and the shield, see figure H5.0.4. The insulation must show very low dissipation during normal operation and it must withstand all the transients according to the particular Basic Insulation Level (BIL) regulations. Early rigid tube conductor designs relied on discrete spacers embedded in cold helium; the even earlier design of spacers in vacuum have been eliminated because of their unreliability. However, spacers in helium have also been found to be inadequate, notwithstanding the good experience gained using such techniques in common gas insulated lines. Inadequate spacer styling may have been one reason. In any case, the CERL group for instance measured impulse strengths up to only 15 MV m−1, which was claimed to be insufficient in a cable where the operational electrical stress should be near 10 MV m−1 (rms). The investigations into helium-impregnated lapped-tape insulations were much more successful. A lot of work was carried out around the world, e.g. by CERL (Swift 1975), Siemens (Bogner et al 1979), EdF/CGE (Fallou and Breteau 1975), BLN (Forsyth 1991) and ATF (Klaudy and Gerhold 1983). Some important features are summarized in the following. ( i ) Partial discharge (PD) onset depends on the local butt-gap stress and the filling helium strength. A typical lapped-tape dielectric is indicated in figure H5.0.15. The assessment has often been made by using the CERL approach, see figure H5.0.16: the insulation PD inception voltage Vi is found from the helium strength versus the density course, according to

with Eg = Vg /t′ (Eg is the helium breakdown field at spacing t′ , which is the void depth; t the total insulation thickness; εr the relative permittivity of the tapes). Many appropriate tape candidates have εr 2–3, whereas εr of helium is near that of empty space. However, even if no PDs can be detected there may be some aging. In fact the CERL picture is too simple, as may be seen from the butt-gap equipotential contours as communicated by BNL. There are Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.15. A representation of a taped insulation, (a) The structure of the insulation package: 1—inner conductor; 2—shield conductor; 3—dielectric tape; 4—metallized screen tapes; 5—metallized surface; 6—butt gap between dielectric tapes; 7—gap between screen tapes; 8—double depth butt gap. (b) Individual butt gap with equipotential contours. Reproduced from Forsyth (1991) by permission.

Figure H5.0.16. Butt-gap inception sparking voltages versus helium density (from Meats 1974). Copyright © 1998 IOP Publishing Ltd

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obviously field components parallel to the solid surfaces, which may give rise to sliding discharges. Nevertheless, the BNL accelerated life tests revealed a lifetime >30 years at a 10 kV mm−1 working stress, which is reasonable. Of course, the tape material must be chosen appropriately; polypropylene or polyethylene as well as cellulose paper have been found to be suitable from the no-discharge and lifetime points of view. An important point is the safe impregnation of an insulation package. Kraft paper is in fact easier to impregnate than plastic tapes, but supercritical helium exhibits no surface tension which is very comfortable in either case. It is deemed to be appropriate to have the imprégnant helium density as high as possible; however, pressurizing was limited in practice for mechanical reasons. A safe design would rely on the plateau-like region, as seen in figure H5.0.16. The helium strength is then only moderately affected by thinning the helium, for instance down to ≈50 kg m−3. Example H5.0.6 A lapped paper insulation with 0.08 mm thick butt gaps is impregnated with supercritical helium of density 80 kg m−3. Figure H5.0.16 indicates a gas breakdown strength Eg = Vg /t ′ = 40 kV mm−1. A 10 mm thick paper package yields an inception voltage of 230 kV peak. This corresponds to an rms inception strength of 16 kV mm−1, which seems adequate for an anticipated 10 kV mm−1 working stress level. Taped insulations for high-Tc superconductor cables will be impregnated by pressurized liquid nitrogen for instance, instead of helium. PD onset strength will be high provided gas bubbles can be safely excluded. Many earlier experimental data can be used for guiding future development (Weedy and Swingler 1987). ( ii ) The dielectric loss factor tan δ of a helium-impregnated, or a liquid-nitrogen-impregnated taped insulation, respectively, is determined by the tape losses, not by the impregnant. Of course, no partial discharges must be involved. The average package permittivity comes out from the solid filling factor. Only a few materials were known for classical superconducting cables, e.g. PTFE, PPP, polyethylene, all of which show at 6 K tan δ ≈ 10−5 at 50 Hz. Kraft paper, however, has a higher loss factor of tan δ ≈ 3.5 × 10−4. This conflicted with a broad a.c. application of this cheap and well known material. A much broader variety of materials is now available for future high-Tc superconductor cables. The tan δ will be higher by one to two orders of magnitude; this being offset by the much lower heat pump factor. Figure H5.0.17 illustrates the dielectric losses in some selected materials. Kraft paper can be used without too high a loss penalty. The dielectric heat production in an insulation package of length l is given by

It varies across the annular space, according to the electrical field strength profile

where r is the actual radius; U* the phase-to-ground voltage and D the shield conductor diameter). The dielectric loss number, ∈r tan δ, is not affected by the field strength below partial discharge onset, and is only slightly dependent on temperature along the cable route. The total dielectric core losses per unit length are found by integrating equation (H5.0.6) over r, and increase with the stressing voltage U*2 , as usual.

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Figure H5.0.17. Dielectric losses in liquid-nitrogen-impregnated insulation packages at 50 Hz: a—Kraft paper; b—Melinex; c—polycarbonate; d—polyethylene (from Weedy and Swingler 1987).

Example H5.0.7 For the already mentioned 0.055 m diameter conductor with 5 mm corrugation depth and a 10 mm helium-impregnated lapped paper insulation, tan δ ≈ 3.5 × 10−4 and ∈r ≈ 1.5. This yields a 50 Hz loss per unit core length of about 0.17 W m−1 for a rated rms voltage the dielectric losses are much higher than the a.c. losses of the Nb conductor. In fact, dielectric losses can severely affect the cable efficiency. Polypropylene for instance would drop the loss down by roughly one order of magnitude. Kraft paper may only be acceptable for moderate voltage levels in classical superconducting cables with a heat pump factor > 300 WAV. A corresponding liquid-nitrogen-impregnated Kraft paper package on the other hand can be assumed with tan δ ≈ 3 × 10−3 and ∈r ≈ 2.5, yielding a loss per unit core length of almost 2.5 W m−1. This level is of the order of hysteresis losses in the high-Tc superconductor and seems to be acceptable for a heat pump factor 10 W/W. ( iii )The breakdown strength is high. Figure H5.0.18 may give an idea of the crest breakdown values found in various laboratories with helium-impregnated samples. It is interesting to note that the a.c. breakdown strength of liquid-nitrogen-impregnated packages can be even higher, up to 50%. However, the impulse strength is very high in general, i.e. >100 kV mm−1. Short-term breakdown itself is not a main problem in the taped dielectric of superconducting cables. The excellent performance is attributed to the multibarrier effect of the overlapping tapes, which cancel out local weak points. Wide tapes are recommended to prevent sliding discharges, but bending may restrain the width to <35 mm. ( iv )The insulation package must be matched mechanically to the live conductor and to the shield conductor. Unfortunately, all of the low-tan S tapes are brittle at low temperature and contract by an order of magnitude more than metals. Plastic tapes are compatible only with segmented helical conductors when fitting the radial shrinking. Equation (H5.0.4) can again be used, but εt h is now 1.5% for instance, or even higher. Large pitch angles are needed to maintain the required constant pitch length. However, there may be some need to vary the pitch angle during winding, according to the radial build-up of the insulation. Sophisticated techniques have to be used in practice. Figure H5.0.11 showed the different pitch angles of conductor and insulation tapes. Elastic strain can be used for compensating shrinking in part so a residual tape strain of ≈ 1% Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.18. The dielectric strength of solid insulating materials in the form of single- and multilayered tapes versus insulation thickness; results from various European laboratories. Reproduced from Bogner (1975) by permission.

may be assumed in practice. Some compressive stress due to elastic tape tension is very advisable; it prevents package loosening with inherent void formation. A too high tape stress on the other hand may yield cracks, which also creates voids. Example H5.0.8 A tape width of 30 mm plus a butt gap of 2 mm yields a pitch length of 32 mm. The corresponding radial shrinkage during cool-down is found from equation (H5.0.2) to be 1.1%; this fits well with the 30 mm diameter helical conductor radial shrinking of 1% mentioned earlier. The appropriate pitch angle is ≈70°. The situation was less favourable with a corrugated tube conductor. Radial conductor shrinking is almost negligible and strain compensation requires the elastic tape elongation to be higher than the thermal contraction of the tapes. Only very few materials were known in the 1970s so Kraft paper was investigated with special emphasis on these factors in spite of its inherent high tan δ. The thermal contraction of paper is low, i.e. near that of metals, and elastic elongation at low temperatures is considerable. Paper gave no mechanical problems at all, and winding techniques were common in the cable industry. Moreover, paper was very cheap compared to plastic tapes; paper insulation has been used in the ATF a.c. cable for instance. Apart from the electrical and mechanical performance of cable insulation there are additional restraints; the first of these is to avoid any large voids because the field distortion effect is drastic (Kiwit et al 1985). Large voids in lapped-tape insulations occur preferentially near the conductor surface and an interlaying smoothing screen is mandatory. Semiconducting screens, e.g. carbon black paper, are commonly used in conventional cables to smooth the surface of segmented conductors. An additional bedding layer may be advisable in superconducting cables to give a margin for incomplete contraction matching between conductor and insulation package. However, semiconducting screens and layers, respectively, cause parasitic losses: an additional series resistance is introduced between the metallic conductor surface and the dielectric. A semiconducting layer near the shield conductor adds a loss of similar order. Figure H5.0.19 shows the CERL core with screening and bedding layers. Selecting the tolerable loss level which is caused by the layer yields the permissible resisitivity for any wanted layer thickness. Carbon black paper may not be acceptable for a low-loss insulation in a classical superconducting cable, yielding a tan δ increase of the Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.19. Cross-section of a superconducting a.c. cable core for 6 kA and 230 kV line-to-line. Reproduced from Swift (1975) by permission.

order of >10−6. The BNL cable therefore was equipped with metallized screen tapes, see figure H5.0.15. Of course, an equivalent electrostatic screen has also been provided on the exterior package surface, and an appropriate compression had to be maintained to eliminate voids. This could be guaranteed by having a proper pitch angle of the adjacent helical strip shield conductor, an additional compression layer being advantageous. For the corrugated tube conductor cable, however, aluminized paper sheets have been found to be more favourable with the aluminium being punctured to allow easy impregnation. There was no contraction difference against the paper package and additional compression was not a prerequisite. Parasitic losses of this origin may be of much less concern in future liquid-nitrogen-cooled high-Tc superconductor cables. The insulation package loss factor is near 10−3 and the resisitivity of carbon black paper, for instance, will be significantly reduced. A parasitic loss increase ≈ 10−6 is completely negligible. One of the most severe problems is industrial manufacturing and bending of the insulated cores. For the CERL cable a bending radius of >2 m has been claimed. This figure sets the maximum tape width as well as the butt-gap width. Note that the innermost side of the core is shortened according to

when Rb is the mean bending radius. The outermost core side is lengthened simultaneously. The tapes must be able to slide reversibly; crinkling would occur otherwise, which is not acceptable since large voids are created within the insulation. Example H5.0.9 A core D = 100 mm has to be reeled up onto a cable drum with Rb = 2 m. Then εb using lapped tapes with 30 mm width yields a minimum butt-gap width of 0.75 mm; there must be additional space for manufacturing inaccuracies: 1 to 2 mm may be appropriate in practice. All these conditions must be borne in mind when designing and manufacturing the core. An appropriate winding tension must be maintained and CERL recommended <10 MPa for polyethylene tapes. A computer-controlled taping machine was used by BNL to wind its special propylene tapes. Kraft paper tapes, however, could be handled as usual in the cable industry. Great care must be taken when applying an insulation package to a high-Tc superconductor to avoid irreversible damage. Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.20. The BNL superconducting a.c. cable core for 4.1 kA and 138 kV line-to-line; diameter over jacket 58.4 mm. Reproduced from Forsyth (1987) by permission.

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Figure H5.0.21. Design of a corrugated conductor superconducting a.c. cable core for 5.25 kA and 110 kV line-to-line; outermost diameter 99 mm; 1, 6— cooling helium; 2—live conductor; 3—carbon black paper screen; 4—helium-impregnated paper tape dielectric; 5—aluminized paper tape screen; 7—shield conductor with niobium layer inside; 8—core enclosure.

(c) Design and performance of a.c. cores As a result of the various constraints which have been discussed, an a.c. cable core looks complicated when compared with the former simple basic arrangement of figure H5.0.4. The main components are the live conductor, the annular dielectric space with adjacent screens and the shield conductor. Cooling ducts have to be provided in the live conductor and outside the shield conductor to remove the losses. The CERL design has been shown already in figures H5.0.11 and H5.0.19, and other designs as related to the most advanced projects relying on classical superconductors are shown in figures H5.0.20 and H5.0.21 respectively. The BNL core has been built and tested with full voltage/current in a synthetic circuit (Forsyth 1987). The ATF/Kabelmetal core was extrapolated from field trials of a 60 kV cable in a utility power station (Klaudy and Gerhold 1983). Some of the new high-Tc superconductor cable designs are similar to the BNL core (Engelhardt et al 1992). Alternative designs, e.g. with an extruded all-solid insulation or even a warm dielectric, will be discussed later on. However, no field experience at all exists with high-Tc superconductor cables. Copyright © 1998 IOP Publishing Ltd

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Two aspects are given preference when searching for an optimum design of cables with classical superconductors. ( i ) The average of the Poynting vector should be maximized. This allows a minimum outer core diameter which saves costs. It also reduces the heat in-leak since the cryogenic envelope is minimized in its size. ( ii ) The core data must fit into an overall system; this predetermines power, voltage and impedances to some extent. However, the classical superconductor domain was only seen in bulk power transmission of the gigavolt amps order. Power density As mentioned already, the Poynting vector [EH ] assesses the local power density in the annular dielectric space. Of course, the rms values are of interest. E and H are limited by the conductor and dielectric physics respectively. Any further optimization can only rely on geometry and should give a maximum power density in the full core cross-section Ac = πD 2/4. From the phase-to-ground voltage (see equation (H5.0.7)), we find U* = 0.5d Em a x In(D/d ) where Em a x is the rms stress near the live conductor. With the rms magnetic field strength Hm a x as found from the synonymous surface current density (see equation (H5.0.1)), the averaged power density Pc is calculated

Variation of D/d yields a power density maximum if D/d = 1/2 → D/d = exp( 1/2 ). Thus, Pc found to be

max

is

Note that the geometrical optimum as given by equation (H5.0.10) does not depend on the material properties at all; this is a special feature of shielded superconducting cables. In conventional cables where H is not limited by conductor physics, D/d may be close to exp(1) in order to make best use of the dielectric. Example H5.0.10 The 60 Hz BNL cable used an Nb3Sn helical strip conductor with d = 2.95 cm and Hm a x = 442 A cm−1. Hence, D should come up to 4.86 cm. The rated voltage when admitting 10 kV mm−1 maximum working stress yields a voltage of 74 kV phase-to-ground, i.e. almost 130 kV line-to-line. The rated current is 4 kA. Thus, Pmax = 16 MVA cm−2 and the core rated power is 0.3 GVA. This is almost the prototype core as shown in figure H5.0.20. The single core losses per unit length have been measured as ≈0.3 W m−1 for the helix conductor and shield conductor, and <0.1 W m−1 for the insulation, including all parasitic losses. This would amount to 6 MW refrigerator driving power for a 20 km long three-phase system or 0.6% of the rated power. Comparing these data with a 50 Hz corrugated system gives a wealth of information. The niobium/copper conductor rms magnetic field is set with Hm a x = 300 A cm−1 for a mean diameter d = 55 mm so the rated current is 5.2 kA. The outer conductor diameter of the live conductor must be enlarged to 60 mm due to the corrugation. Inserting the mean diameter into the optimum condition yields a corrugated shield conductor mean diameter of 89 mm. From the corrugation depth, 79–99 mm may be appropriate. The lapped paper insulation thickness then is 9.5 mm; the maximum stress being limited to 7.7 kV rnm−1. This modest stress gives a near optimum efficiency of the cable, including the cryogenic envelope heat in-leak. The phase-to-ground voltage is found from equation (H5.0.7) to be 64 kV. The actual core rated power comes out to be 0.33 GVA and the power density is 4.3 MVA cm−2 for 99 mm outermost diameter. This is considerably below the theoretical optimum value, i.e. 10 MVA cm−2. The

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core is shown in figure H5.0.21. The conductor losses were claimed to be <0.1 W m−1 and the dielectric losses were estimated at <0.2 W m−1 per unit core length. Hence the refrigerator driving power to pump the electrical losses of a 20 km long three-phase system would come up to 9 MW, i.e ≈0..75% of the rated power. This figure is similar to the BNL core performance. Comparing the actual power density against the theoretical optimum highlights the obvious penalty for using the simple and foolproof corrugated tube conductor system instead of a sophisticated helical strip conductor system. Finally, the performance of a liquid-nitrogen-cooled high-Tc superconductor core is discussed for comparison. The overall 50 Hz losses in a helically wound conductor consisting of six layers of Bi-2223 Ag-sheathed multifilamentary wires with a total high-Tc superconductor cross-section of 16 mm2 and inner/outer diameter 19/23 mm have been measured at 77 K to be of the order of 1 W m−1 at 1000 A, which is below the critical d.c. current limit. The circumferential rms magnetic field is only 140 A cm(tm)1. The losses increase approximately as I 2. This indicates parasitic effects in addition to hysteresis losses in the high-Tc superconductor filaments. Dielectric losses of the order of 1–2 W m−1, and shield conductor losses of <1 W m−1, must be added for the total core losses. A 20 km long 110 kV three-phase system with 200 MVA rated power will then need a refrigerator driving power of <2.5 MW, i.e.>1% of the rated power; this figure may not be fully satisfactory. However, conductor losses may be even higher since the core temperature varies along the cable route. Hence, strong efforts are needed to develop high-Tc cable conductors with a lower loss level. The key to this end has already been made evident in equation (H5.0.3), i.e. a higher critical current density of the high-Tc superconductor material. Equation (H5.0.10) yields a first-order cost optimum for classical superconducting cables since the cryogenic envelope amounts to a significant part of the total cost. This may not be the case with high-Tc superconductor cables where the amount of superconducting material will be very decisive. The strong Jc degradation with increasing adjacent magnetic field strength must be borne in mind (Ashworth et al 1994). Thus, the optimum live conductor diameter may be somewhat larger than the minimum feasible diameter. Utility system integration Power transmission has to meet utility system needs. Overhead lines are favoured not only by their relatively low cost, but also by their high surge impedance ζ (∼300 Ω). The three-phase natural load 3U2* /ζ is below the rated power 3U2*/Zl where Zl is the load impedance. This forces an inductive current which is very suitable with regard to power-system stability; stability is essential for long-distance transmission. Conventional cables have limited current ratings resulting in a high load impedance of the order of 100 Ω or more. The surge impedance is only of the order of 20–50 Ω, This gives rise to a capacitive load, which is undesirable. Therefore, conventional a.c. cable transmission often is restricted to short lengths in the few kilometre range; long submarine cables must be operated with d.c. Superconducting cables offer the unique possibility of a fairly good matching between surge impedance and rated load impedance; neither inductive nor capacitive currents have to be transmitted. The implications of a matched bulk power transmission element have been discussed at length (Forsyth 1987, 1991). In a concentric superconducting cable, the surge impedance follows from ζ = (L′/C′ )1/2 = . The per unit length capacity C′ and inductivity L′ are given by C′ = 2π∈r∈0 /ln(D/d ) and L′ = µ 0 ln(D/d )/2π, respectively, because there only exist thin current sheets. An optimized core with ∈r ∼ 2 yields a theoretical surge impedance of 21 Ω. Comparing ζ with the load impedance Zl = U* /I leads to . This ratio should be near unity so

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Thus, the field strength ratio Em a x /Hm a x should be of the order of 300 Ω. A conventional cable with its inherent thermal limit for the rated current may shift Em a x /Hm a x into the multi-kilo-ohm regime since Hm a x may remain below the 15 A cm−1 limit. A superconducting cable can carry much more current than a copper conductor of the same size. A dielectric working stress of almost 10 kV mm−1 fits well to a surface current density on the live conductor (=Hm a x ) near 400 A cm−1. This figure is very close to the BNL core data. The initially surprisingly low system voltage of 138 kV for a 1 GVA system is justified by the impedance matching condition. There is no compulsion to use very high voltages. The option of natural load matching may also be within reach for high-Tc superconductor cables of an adequate design, i.e. by appropriate selection of Hm a x and Em a x . However, high-Te superconductor cables might not be used in the near future for bulk power transmission but rather in the several hundred MVA domain for retrofitting. Natural load matching then is not of primary concern for limited cable lengths, i.e. <20 km. Most of the high-Tc superconductor cables proposed hitherto were designed with modestly rated voltages of the order of 100 kV, which fit well into existing systems in urban areas. One of the most important design features may be a low overall diameter. This forces the maximation of the power density using higher magnetic fields and dielectric stresses (see equation (H5.0.10)). Operation of a.c. cores Any superconducting cable core is only effective when cooled safely to its working temperature. However, there are heat losses which must be removed by the flowing refrigerant and pumped to the ambient by the refrigerator. A considerable part of the heat input comes in via the cryogenic envelope and is not affected by any electrical working condition. Assuming satisfactory refrigerator and flow conditions, the removal of this heat can be guaranteed. The core is not affected by extraneous conditions. The cooling of classical superconductors was believed not to be difficult. Direct contact with the refrigerant could be easily provided, and heat transfer was quite sufficient to remove the a.c. losses safely during normal operation, thanks to the prevailing turbulent helium flow conditions. A critical heat transfer situation seemed very unlikely, even under overload conditions. High-Tc cable conductor cooling must be scrutinized more seriously. A direct wetting by the turbulent coolant flow can only be secured for the innermost strip layers in a multilayered helix, the outer layers being indirectly cooled via heat conduction across contacts and stationary fluid. Thus, the outermost strip layers which see the highest adjacent magnetic field may exhibit a higher temperature, which will severely affect the total current-carying capacity. Relying on a double-helix configuration as in the BNL cable requires an adequately high critical current density of the high-Te superconductor, but will render conductor cooling much more reliable. However, dielectric losses have to be added. The loss variation over the radius has been indicated already (see equation (H5.0.6)). The produced equivalent heat per volume, q, has to removed by heat conduction across the package; there is little chance for imprégnant convection in the butt gaps. From the integration of the Fourier equation

the radial temperature profile T(r ) is found, the thermal conductivity k as well as the dielectric loss number being assumed to be constant. In the case of cooling flow inside the live conductor and outside the shield conductor, the temperatures TD and Td at the respective diameters D and d are fixed. The maximum temperature then will occur inside the package and is found from

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Otherwise, the maximum temperature will occur at the outer package surface; this temperature will be somewhat higher of course. Example H5.0.11 To get a rough idea we may assume that the live conductor temperature, Ta , is identical to the shield conductor temperature TD . Thus, the maximum temperature is found at a radius rm = 0.5(D/d )1/2 with Tm a x = (πf∈r ∈0 tanδ U 2* /4k) + Td . The temperature will be insignificant in cables with classical superconductors; even with Kraft paper only 0.1 K can be assumed at 110 kV. High-Tc superconductor cables will show typically an order of some kelvin due to the considerably higher dielectric loss number. This temperature rise may still be not very harmful, but should be verified during the design procedure. The maximum package temperature increases with the square of the voltage, as usual. Thus, operational overvoltages can become dangerous, especially with simultaneous overload current in high-Tc superconductor cables. No bubbles must be generated at all within the imprégnant otherwise partial discharges can occur, due to the lowered dielectric strength in vapour-filled butt gaps. This would in fact be a very dangerous condition. Therefore, an adequate high impregnant pressure must be maintained. BNL relied on typically ∼ 1.5 MPa; this figure had to be compatible with the refrigerator since the flowing helium directly impregnated the dielectric. Detailed data for liquid-nitrogen-cooled high-Tc superconductor cables are still missing. In the corrugated niobium/copper tube system as used in the ATF cable there was no exchange between the cooling fluid and the imprégnant so the latter pressure could be adjusted independently. This was claimed to be a great advantage, apart from avoiding any damaging particles which may be present in the fluid. No ideas have been communicated hitherto to utilize this inherent advantage in high-Tc superconductor cables with impregnated lapped-tape insulation. It seems that normal operation of a cable core gives no severe technical problems provided an adequate design has been made. However, there was a rather narrow bandwith for economic operation of classical superconductors. The constant part of the total losses, i.e. heat in-leak via the cryogenic envelope as well as dielectric losses, is independent of the actual power loading, i.e. these losses had to be taken as essentially zero load losses. It was of course never advisable to have a classical superconducting cable running near to the zero load condition. However, fluctuations are rather modest with the average level coming up to almost 90% of the peak power in a high-power grid system (Kiwit et al 1985). Classical superconducting cables would have fitted well into such scenarios. The constant part of the total losses may be less significant in high-Tc superconductor cables, due to the much lower heat pump factor. These cables may much better fit varying load conditions which are usual with moderately rated power levels, without too much loss penalty. There was also only a limited margin when running a classical superconducting cable under overload conditions. Conventional cables exhibit a considerable thermal time constant, due to a high heat capacity and to the moderately increasing loss as the temperature increases. Therefore, some overload may be accommodated over reasonable periods. The cryogenic core of cables with classical superconductors has practically no heat capacity in its conductor and solid dielectric, and has a very limited capacity in the cooling and impregnating helium. It is obvious that any overload current which produces excessive losses in the superconductors raises the core temperature. Overload capacity is therefore limited by the overload performance of the cooling circuit and buffers integrated into the refrigerators were thought to be very advantageous. Looking at the a.c. loss characteristics of superconductors is very instructive. With Nb for example, the losses increase from 100 mW m−2 up to 300 mW m−2 on raising the current, and hence the adjacent surface field, by roughly 30%, provided the cooling temperature can be maintained. This was believed to be a modest margin. BNL could operate its Nb3Sn cable with almost 50% overload on the other hand, which was accepted as an appropriate figure. A high-Tc superconductor core will show a hysteresis

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loss increase with I 3, but total core losses may increase as I2 only as has been already mentioned; 50% overload then means more than twice the normal running loss design level. Overload operation may not be a very serious design problem. H5.0.3.2 Cores for d.c. cables The two-monopole-core circuit with symmetrical +/- voltages to ground and the concentric arrangement with the return current conductor at ground have both been discussed. Magnetic d.c. stray fields in the former arrangement cause no eddy current losses and can be tolerated therefore along unpopulated cable routes. Figures H5.0.2 and H5.0.3, respectively, show these different arrangements. Any d.c. cable has to be connected to an utility system by a.c./d.c. converters which can limit excess fault currents to twice the nominal value. Therefore short-circuit protection of the cable is not needed but an appropriate safety margin against the critical current limit of the conductor is mandatory. Similarly to a.c. cables, the need for flexible cores was recognized very early. The techniques to ensure flexibility are essentially the same as discussed for a.c., i.e. segmented helical strip or wire conductors with an internal cooling duct. The helices are supported by strutting elements and there is no pressing need to provide double helices with opposite pitch angles since an internal longitudinal magnetic field would not seriously affect the conductor performance. However, a two-layered helical strip conductor allows a much higher rated current.

Figure H5.0.22. A superconducting d.c. cable with concentric conductor arrangement, 5 GW at 200 kV, designed at the Los Alamos Scientific Laboratory. Reproduced from Electric Power Research Institute: Cost Components of High Capacity Transmission Options EPRI EL-1065 (vol 2, 1979) by permission.

The conductor had to be surrounded again by a lapped-tape insulation, impregnated with supercritical helium. There is no dielectric loss so a wide variety of materials could be envisaged. Figure H5.0.22 shows a cable core of the concentric type with helical strip conductors. The corrugated-conductor type also has been investigated, but no really satisfactory heavy current corrugated conductor for d.c. applications has yet been built. (a) Conductors for d.c. Classical type II superconductors have been found to be a most economic choice in the past. High current densities can be maintained at a high external field. However, type II superconductors suffer from flux Copyright © 1998 IOP Publishing Ltd

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jumps, and filamentation and cryogenic stabilization are mandatory for heavy current conductors. Safe operation under varying load conditions must be guaranteed. Some early proposals were based on commercially available superconductors such as Nb—Zr and Nb-Ti; however, Nb3Sn was claimed later on as being a favourite candidate due to its higher critical temperature. In the case of the concentric conductor arrangement the applied field is simply found from equation (H5.0.1). In the case of the two-monopole-core circuit, an additional field component has to be superimposed. On the other hand Nb3Ge was investigated as an alternative in the USA; Nb3Sn and Nb3Ge are both very brittle materials. An arrangement where the superconductor is situated near the neutral strain zone of a composite strip has already been shown in figure H5.0.10. Such a strip configuration is adequate for d.c., too. Serious design studies with high-Tc superconductor materials have so far been scarce. (b) Electrical insulation for d.c. cores As in the case of a.c. cores, a lapped-tape insulation fitting on to the conductor helix must again be used to meet the mechanical needs of a flexible core. However, tape selection is much easier due to the lack of dielectric losses. Helium-impregnated plastic tapes as well as Kraft paper have been proposed, the latter having been investigated with special emphasis by AEG (Bochenek et al 1975). Pure d.c. stress would not cause considerable PDs. However, there could be switching surges so a PD-free normal operation seemed advisable. Therefore, the impregnating helium conditions as well as butt-gap sizes were taken over from the a.c. insulation package. A high breakdown strength could be guaranteed at low temperatures only, see figure H5.0.23. It may be noted that an elevated helium pressure, e.g. 1 MPa, considerably increases the strength; the precise amount can be derived from the helium density, according to figure H5.0.16. However the dielectric strength requirement may set an upper temperature level of 12 K.

Figure H5.0.23. The breakdown strength of Kraft paper packages under d.c. stress. The pressure of the impregnating helium is 0.27 MPa (from Wimmershoff 1974).

The high-voltage testing of d.c. cables may produce very high local stresses in general; space charges are built up in ordinary d.c. cables so a test with voltage reversal is often required. These test conditions may also be relevant for superconducting cables, where little is known about space-charge phenomena. There is no question that cold helium-impregnated lapped-tape insulation has proven to be adequate for d.c. high-voltage insulation. The occasionally expressed doubts about cryogenic insulation which have Copyright © 1998 IOP Publishing Ltd

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led to a two-monopole-core circuit scheme with the insulation outside the core, e.g. at ambient temperature, seem not to be justified. (c) Design and operation of d.c. cores The integration of a d.c. cable into a utility grid does not imply any severe stability problems, due to the inherent a.c./d.c. converters; this is illustrated in figure H5.0.24. However, energy losses in converters are serious. Only long d.c. cables are of interest. The optimum voltage/current for a given rated power may again be obtained according to equation (H5.0.10). A d.c. core is considerably smaller than an equivalent a.c. core; this is due mainly to the much higher permissible d.c. surface current density.

Figure H5.0.24. Power transmission circuits with d.c. cables: (a) two monopole cores; (b) concentric conductor arrangement.

Example H5.0.12 An Nb3Sn two-monopole-core d.c. cable is being designed with a circular conductor of diameter 28 mm, consisting of six helical tapes of 12.7 mm × 0.3 mm cross-section. The critical current is 29 kA at 4.2 K, which produces a circumferential magnetic field of 3.3 kA cm−1 (0.41 T). This is, in fact, close to a self-field condition. However, operating the Nb3Sn conductor up to 12 K has been postulated, the critical current being reduced considerably below that at 4.2 K, i.e. down to 17 kA. A rated current of almost 9 kA was claimed to be within reach, which is more than twice the equivalent figure in an a.c. cable. The conductor helix is insulated to ground by an insulation package with D = d exp(1/2) = 5 cm. This fits to an operating voltage U = ±100 kV if an operational stress of about 10 kV mm−1 is acceptable. The rated monopole core power is 0.85 GW, i.e. almost three times higher than in an a.c. core of similar size. Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.25. A cross-sectional view of a high-Tc superconductor d.c. cable prototype. Reproduced from Beales et al (1996) by permission.

A recent proposal for a coaxial high-Tc superconductor d.c. core to be cooled by helium gas (15-65 K) is illustrated in figure H5.0.25 for comparison. The rated current and rated voltage are 10 kA and 40 kV respectively. Long-distance transmission, i.e. 100 km, is claimed (Beales et al 1996). However, using dielectrically weak helium gas up to 65 K for impregnating a lapped-tape insulation may give rise to serious insulation problems. Steady-state operation of d.c. cores may give no severe problems. No a.c. losses arise so cooling is strictly reduced to remove the heat in-leak through the heat shielding system. This heat in-leak is never affected by the actual power load, which should be constant; however, some safety margin to overcome transient currents must be provided. H5.0.4 Cryogenic envelope The cryogenic envelope is an auxiliary component, but is costly. It is required to prevent heat in-leak into the cable core. The temperature span, i.e. 300 K to 6 K with classical superconductors, and 300 K to ≈80 K with high-Tc superconductors, is exorbitant for utility engineers but very common for cryogenic engineers. Any cable cryogenic envelope is in principle an extremely long cryostat and all of the early designs, as seen for instance in figures H5.0.2 and H5.0.3, were based on common insulation techniques: high vacuum, intermediate thermal radiation shields, superinsulation and sophisticated suspension elements. The technology relied mainly on rigid tube systems. The innermost tubes had to contain the cooling fluid and withstand the pressure, even in the case of an emergency such as a fault current. Thermal contraction had to be compensated for by bellows elements, similarly to the early rigid conductor tube systems. The heat in-leak must be kept as low as possible for which an upper limit may be set by a cable efficiency figure. For instance, a 20 km a.c. cable should have an overall efficiency near 99% at 1 GVA rated power. Then, 10 MW can be provided to run the refrigerators, i.e. mainly for the driving power for the compressors. This equals 20 kW heat to be pumped with 500 WAV from 6 K for instance to cool classical superconductors. The per unit length core losses may be assumed to be approximately 0.2 W m−1. Thus, 4 kW of the 20 kW must be attributed to them and a similar figure should also be provided to cool the cable terminals. Assuming that about 10 kW are available for the total envelope heat in-leak means a per unit length in-leak /l < 0.5 W m−1 into the 6 K region for a three-phase system. Example H5.0.13 A two-monopole-core d.c. transmission line with 2 GW rated power is provided with two separate cryogenic envelopes; the trench length is 200 km. Assuming a heat in-leak per unit length into the cooling ducts of Copyright © 1998 IOP Publishing Ltd

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2 × 0.5 W m−1, a total heat of 4 × 105 W has to be removed by the refrigerators. With a heat pump factor of 200 WAV at a maximum superconductor operating temperature near 10 K, a driving power of 80 MW is needed, which amounts to 2.4% of the rated power. This figure seems to be acceptable for a 200 km long transmission line. There is no doubt that heat in-leak ≤ 0.5 W m−1 can be guaranteed in ordinary cryostats and, in fact, sophisticated techniques may yield heat in-leaks lower by an order of magnitude. However, the cost of the latter may be prohibitive so an economic design must find an optimum solution. Two points seem to be very important. ( i ) Operation must be simple. Utilities would not easily accept a scheme with two different cooling fluids, i.e. helium and liquid nitrogen, as has been proposed in all the early designs (see figures H5.0.2 and H5.0.3). Similarly, continuous attended operation of vacuum pumps is not acceptable. ( ii ) Installation must be economic. In the USA, due to the considerable experience with pipelines, rigid tube systems are common practice. Therefore a rigid cryogenic envelope composed of individual elements of 20 to 30 m is readily acceptable, the flexible core being pulled in after. BNL has advanced this technique. However, because cable laying in Europe is commonly performed by reeling flexible cables from drums in long sections, a fully flexible envelope was of greater interest. This technique has been advanced by Kabelmetal/AEG/ATF. H5.0.4.1 Performance of heat shielding Most of the heat in-leak is due to radiation although spacers and a less than perfect vacuum may contribute a significant amount. The per unit area heat transfer rate can be derived theoretically from

where To and Tc are the ambient and cold boundary temperatures respectively; N is the number of reflecting superinsulation foils with emissivity ε; σ = 5.7 × 10−8 W m−2 K−4 (Stefan-Boltzmann constant); k is the summary heat conductivity of spacers, foil contacts and remaining gas with pressure <10−2 Pa during operation and t is the insulation thickness. The respective heat flux per envelope unit length has to be found approximately from the mean insulation diameter de n = 0.5(De n + de n )

However, the improved overall performance of a system with cooled intermediate shields was well known to cryogenic engineers. In the mid-1970s, a heat flux density from ambient into liquid-nitrogen-cooled shields of < 2 W m−2 and from the shield into the helium duct of , < 0.1 W m−2 was claimed to be representative for a rigid tube envelope by Siemens. CERL assumed a per unit length in-leak into the liquid-nitrogen shield of 2.5 W m−1, and of 0.15 W m−1 into the helium duct for its 5 GVA cable. These figures seemed to be optimistic. It later became necessary to eliminate intermediate shields in advanced systems because of utility objections and high installation cost; this decision was made in spite of the inherent efficiency benefits of intermediate cooling. BNL first demonstrated a heat flux density of about 0.5 W m−2 in a rigid tube system of evacuated and sealed individual sections of 30 m length, including the joints between sections to be made in the field. The corresponding per unit length heat flux was measured to be less than 0.6 W m−1. This system, which can house the three cores of a complete three-phase a.c. cable, may also be used without severe modification for future high-Tc superconductor cables. It is shown in figure H5.0.26. There is no doubt that rigid tube envelopes can be built and installed reliably with an appropriate low heat in-leak. However, the technology is sophisticated and costly.

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Figure H5.0.26. BNL rigid cryogenic envelope; the three cores are to be pulled in after installation: (a) assembly with three cores; (b) details of section. Reproduced from Forsyth and Thomas (1986) by permission.

Example H5.0.14 The BNL envelope has a measured heat in-leak of 0.6 W m−1, including joints. Assuming that the claimed heat pump factor of 250 WAV is correct yields a refrigerator driving power of 3 MW for a 20 km cable. This is only 0.3% of the rated power. When this envelope is used in a high-Tc superconductor cable with a heat pump factor of only ten, the refrigerator driving power will drop down to 0.12 MW. The heat in-leak will not be much affected by the higher core temperature. The competing flexible corrugated envelope as shown in figure H5.0.27 is much simpler, but it suffers from poor heat shielding performance. The outermost diameter is limited to 26 cm due to transportation, and since a three-phase system with classical superconductors could not be housed within one single envelope three envelopes had to be provided. An envelope with 26 cm outermost diameter for instance could accomodate a single core as shown in figure H5.0.21. A per envelope unit length heat input into a 6 K core of 0.8 W m−1 for each envelope yielded >2% of 1000 MVA rated three-phase power in a 20 km cable, which was not very exciting. However, little progress has been made since the early 1980s. Superinsulations without need of separate spacers have been developed, but this technology has not yet been integrated into industrial manufacturing (Edwards 1988). However, such an evelope was still attractive for a classical superconducting d.c. cable where the core size is reduced considerably. The corrugated envelope, however, seems to be very interesting from the point of view of housing Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.27. A fully flexible cryogenic envelope: 7—shield conductor; 8—core enclosure; 9—superinsulation; 10—outer envelope tube; 11—corrugation sheath.

future high-Tc superconductor cable cores. A per unit length heat in-leak of a few W m−1 will be permissible compared with the electrical core losses, and bearing in mind the moderate heat pump factor. H5.0.4.2 Operation The steady-state operation of a well designed cryogenic envelope may be achieved without any severe problems; vacuum tightness should be sufficient to guarantee a fairly good thermal insulation without the need for permanent pumping. Both cryopumping and cryosorption were deemed to help to maintain an excellent vacuum over the cable lifetime, and adsorbens were generally provided for classical superconducting cables. Liquid-nitrogen-cooled envelopes on the other hand can be pumped by cryosorption only with a limited period of activity, and elimination of permanent pumping will be a great challenge for future high-Tc superconductor cables (Forsyth 1993). It is, however, necessary to provide adequate ducts for the cooling fluid inward/return flow. Preferably the live conductor should be employed as the coolant inward duct, and the space between shield conductor or jacket and innermost envelope tube for the coolant return duct. This can be seen from figures H5.0.20 and H5.0.26. Alternatively, inward and return flow may be accommodated in the respective conductors in a two-monopole-core d.c. circuit. The per unit length heat input into any cooling duct, , is being taken over by the respective coolant flow according to ∆l = ∆h, where ∆l is the cable section length and ∆h indicates the enthalpy rise of the coolant. The mass flow rate, , forces an appropriate flow velocity υ so = Aρυ in the flow Copyright © 1998 IOP Publishing Ltd

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cross-section A and ρ is the actual fluid mass density which varies along the cable. The pressure drop ∆ρ can be found from ∆ρ = ζυ 2ρ∆l/(2dh ) where dh is the hydraulic diameter of the duct and ζ stands for the friction factor. Flow may be turbulent in general, due to the low fluid viscosity which gives high Reynolds numbers. There is a considerable difference between rigid tube ducts and corrugated tube ducts or helically wound strip conductors with internal struts, in view of the pressure drop. In rigid tubes, ζ may be of the order of only 0.02. In corrugated tubes and helically wound conductors, ζ can rise up to more than 0.1. The relations mentioned above allow an order of magnitude estimation for the appropriate cooling station distances between refrigerators, i.e. the section length

Distances between 5 and 60 km have been reported in the literature for helium cooling of classical superconducting cables. High-Tc superconductor cables with liquid-nitrogen cooling may be built with lengths of several kilometres (Ashworth et al 1994). Internal friction losses must be borne in mind then and must be added to the external heat input. Example H5.0.15 CERL proposed a rigid tube cable envelope with = 0.30 W m−1 total per unit length heat input into a cooling duct at 4.5 K. The helium inlet was specified with 4 K/0.47 MPa and the outlet with 5 K/0.4 MPa, respectively. The averaged helium mass density ρ is close to 130 kg m−3 and ∆h is found to be 4 kJ kg−1. The flow velocity, υ, is about 0.2 s−1 within a cross-section A ≈ 0.05 m2. The hydraulic diameter dh may be near 0.02 m. Hence, a refrigerator distance ∆l = 19 km can be estimated for ζ = 0.02. Friction losses are very low compared with the heat input. A more accurate calculation carried out by the CERL group concluded that ∆l = 16 km. Such a calculation must incorporate the density/enthalpy variation along the cable and can only be performed by using appropriate computer programs. The cool-down of a cable may be much more difficult than cooling a compact system. First of all, achieving a vacuum is very time consuming in the regime below 1 Pa even if factory-sealed and evacuated sections are employed, even when using corrugated tubes of long individual lengths. Cooling reduces the vacuum pressure very effectively by cryopumping and an initial vacuum of 0.1 Pa is adequate for cooling to commence. However, coolant density is low when cooling begins, and the mass flow rate is modest. In general, cool-down time increases with ∆l 3 (Schauer 1991). Order of magnitude estimations for a helium-cooled cable were three to eight weeks for a 15 km refrigerator distance. Estimations for a high-Tc superconductor cable are outstanding. It is obvious that any warm-up during the operational lifetime of the cable may be very troublesome for a utility and it is thus preferable for it to remain in the cooled state for as long as possible. An additional benefit of this is that any aging effects due to thermal cycling are reduced. H5.0.5 Terminations and joints Any cable requires links to the outside world. Superconducting cable terminations must span a temperature range ≈ 6-300 K when employing classical superconductors; with high-Tc superconductors the temperature span range will be reduced, e.g. 80 K to 300 K. This is a very uncommon situation for utility engineers. Joints on the other hand are needed to couple the individual sections of a cable system. Sections can be manufactured and transported in limited length only, e.g. 1000 m in the case of a flexible core.

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H5.0.5.1 Cable terminations A superconducting cable termination has various functions: ( i ) to feed the current into the live conductor—this requires an appropriately rated high current lead; ( ii ) to provide high-voltage insulation of the live conductor lead against ground and the respective transition to the highly stressed lapped cable dielectric; ( iii )to feed the current into the shield conductor—this again requires an adequate current lead but at ground potential and in some situations low voltage insulation; the shield conductor is of relevance only in a concentric conductor arrangement; ( iv )to feed the coolant into the live conductor—this requires a feeding device with a high-voltage break since the refrigerator is grounded; ( v ) to feed the coolant into the annular channel around the grounded shield conductor—this is of relevance for a concentric conductor arrangement; ( vi )to feed the impregnating fluid into the dielectric space—this function being relevant only with helium-tight corrugated conductor systems; ( vii ) to provide all of the required control elements because there is little chance to accommodate distributed sensors along the cable core. Occasionally some simplifications may be possible; for instance, the common grounded neutral point for the shield conductors (see figure H5.0.4) may be established at low temperature; no shield current leads are then needed for the respective terminations.

Figure H5.0.28. A schematic diagram of a.c. cable termination. Copyright © 1998 IOP Publishing Ltd

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Items ( i ) to ( vii ) represent a general scheme: figure H5.0.28 illustrates this with an a.c. cable termination as used for the ATF cable field trials (Gerhold 1984). A corresponding d.c. termination which was used by AEG in its laboratory cable tests is shown in figure H5.0.29.

Figure H5.0.29. Single-pole termination for a two-monopole-core d.c. transmission line for 12.5 kA and 100 kV. (a) Assembly with flexible cable (reproduced from Wimmershoff (1979) by permission), (b) A cross-section: 1—current lead heat exchanger; 2—supporting ring; 3—superconducting tapes; 4—cast resin insulator with metallized potential surface; 5—paper insulation; 6—supercritical helium inlet; 7—liquid-nitrogen inlet; 8—cryogenic envelope; 9—line connection; 10—porcelain sheath; 11—paper insulation (reproduced from Heumann (1972) by permission).

(a) High voltage current lead The current lead problem has been a popular subject for many papers. However, the general design rules are simple. We have a conductor which carries a resistive current, and simultaneously conducts heat from the warm end down to the cold end where the cable superconductor is connected. The total cooled end heat which is being fed into the core coolant is partly due to Joule losses from the current and partly due to heat conduction. A minimum arises when the warm-end temperature gradient vanishes. According to the Wiedemann-Franz-Lorentz law, optimization has to be performed by choosing an appropriate ratio of lead cross-section to lead length. Then, any unit current may cause almost 40 mW total heat input into the cooling fluid for each individual lead; this figure is not too dependent upon the cold-end temperature Tc e . However, 40 mW A−1 is not attractive for heavy currents when Tc e is very low, i.e. in the case of classical superconductors. Fortunately, counterflow cooling of the lead was known as a very effective means of reducing the cold-end heat input, c e , which may be <1 mW A−1. The counterflow helium gas temperature, T, increases from Tc e up to ambient, but the power to recool the counterflow gas must be added. The refrigerator is running in a so-called mixed duty operation; therefore the heat, T , being extracted from the conductor by the counterflow gas stream has to be weighted with the appropriate mixed duty factor ϕ . According to t = c e + ϕ T we find the equivalent total refrigerator load, , at Tc e . t

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Carnot cycles yield ϕ = 0.05 if Tc e ≈ 5 K. Industrial Claude refrigerators show a similar figure when equipped with several expansion stages; however, φ may depend on the actual c e / r ratio. The cold-end heat input of the lead depends on the counterflow gas stream so = c e{ t}; this is often an almost hyperbolic relationship depending on the actual lead heat e exchange conditions. A minimum can be found for any particular lead refrigerator combination by varying T . Example H5.0.16 A well performing copper lead into liquid helium has been measured with c e = 0.8 mW A−1 near the dewar cooling condition, T = 80 c e The vaporized liquid is just fed as the precooling gas. c e decreased with 1/ t around the dewar cooling point. The refrigerator showed a constant mixed duty factor ϕ = 0.05. Hence, the optimum is found very close to the dewar cooling so t = c e + 80ϕ c e = 4 mW A−1, the corresponding per unit current mass flow rate of the counterflow helium gas being of the order of 50 µ g s−1 A−1. To ensure such attractive performance, an adequate heat exchange between counterflow cooling gas and conductor has been identified to be important and large wetted surfaces had to be provided, see for instance figure H5.0.29. High pressure drops could be admitted in cable current leads so turbulent flow was easier to maintain than with usual cryostat leads. For a 4 kA cable at 6 K, 4 mW A−1 means 16 W refrigerator load per phase and terminal, i.e. ≈ 100 W for a full three-phase system with two terminations. Bringing in the heat pump factor 500 WAV yields a compressor power near 0.05 MW, which was claimed to be an acceptable figure of merit. However, there were some additional requirements for classical superconducting cable leads; first of all, the coolant had to be fed in near the cold end with most of this coolant flowing into the live cable conductor and only a small part going into the lead. This part had to be extracted from the warm end and fed back to the grounded refrigerator. Therefore an adequate high-voltage break had to be incorporated at ambient temperature, as shown in figures H5.0.28 and H5.0.29. Current leads for a.c. cables also suffer from eddy current losses and since the skin depth may be minute at very low temperatures the conductor cross-section had to be subdivided into many strands. These strands may oscillate under current forces and cause additional losses, which should be prevented. Any surrounding metallic parts, e.g. vacuum envelopes, may also contribute eddy current losses, since there is no electromagnetic field shielding as in the cable core. These parasitic losses also have to be minimized. The technological expenditure for future high-Tc superconductor cable leads may be significantly reduced. According to the modest heat pump factor with liquid-nitrogen cooling, a cold-end heat input of 40 mW A−1 can be easily accepted and counterflow cooling by cold nitrogen vapour can be dispensed with. Hence, no high-voltage break to feed back any warm gas to the refrigerator will be needed. However, a high-voltage break to feed the liquid nitrogen into the live conductor must still be provided. Eddy current losses may be of less concern. (b) High-voltage insulation Particularly difficult conditions for high-voltage insulation had to be overcome with cables using classical superconductors. Helium gas is a very weak dielectric at ambient temperature and it can be used only in the cold state. To span the temperature range from ambient to the cryogenic region with an appropriate long solid insulator called a bushing has been the commonly accepted practice. This bushing must be carefully screened so that warm helium gas is never stressed with an electric field. A very convenient capacitor bushing for a.c. is seen in figure H5.0.30. The bushing cold end has a reverse stress cone which is covered by cold helium at high pressure. This type of bushing was first used by Siemens in its cable termination (Bogner et al 1979) and later on in the ATF design. The cable core dielectric on the other hand ends in a conventional stress cone made out of similar materials to that used in the core dielectric. The stress cone has to reduce the local field strength, which stresses the helium, and has to provide an appropriate long flashover path. Note that surface discharge inception may occur down to less than 2 kV mm−1 stress parallel to the solid-helium interface.

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Figure H5.0.30. Design of a single-phase a.c. cable termination: 1—He gas outlet (high voltage); 2—Cu busbar; 3—insulating tube; 4—bushing with capacitive layers; 5—oil outlet (heating at high voltage); 6—He gas outlet (ground potential); 7—electric heating (ground potential); 8—current connection; 9—nitrogen inlet; 10—He outlet (supercritical); 11—boiling He (current lead); 12—He inlet (supercritical); 13—nitrogen outlet; 14—distributor valve; 15—He cooled current leads; 16—boiling He bath; 17—He gas exchange tubes; 18—stress cone with capacitive layers; 19—supercritical He for cooling the cable core; 20—outer conductor; 21—inner conductor (cable core); 22—nitrogen shield; 23—He shield (He return flow); 24—He tube. Reproduced from Bogner et al (1979) by permission.

The a.c. bushing design had some constraints. Dielectric losses occur in a solid and must be removed from the cold end, i.e. near the bushing stress cone. It seemed not to be appropriate to employ counterfiow gas cooling since heat transfer from the insulator inside to any helium gas on the insulator surface is very limited. However, an optimization was found to be possible, similar to the so-called adiabatic current leads. By using the equivalent heat flow scheme but replacing the heat sources per unit volume according to equation (H5.0.6), a minimum for the cold-end heat input could be defined (Gerhold 1984). Note that heat conduction incorporates any electrostatic screens, which are integrated into the insulator. The radial insulator size was set appropriately to the electrical stress conditions, especially for the helium covering the stress cone. The discharge inception stress is controlled by the helium density. 5 kV mm−1 (rms) was taken to be an appropriate total stress level for normal operation. Any optimization could then only vary the bushing length for given radial dimensions. For this particular length there is theoretically no temperature gradient at the warm end, which is analogous to an adiabatic current lead; however, the internal insulator temperature can be considerably higher, with some radial temperature gradient, therefore it seemed reasonable to rely on a maximum temperature near 310-320K. Example H5.0.17 A 50 Hz bushing for maximal 65 kV rms line to ground has a tubular cross-section insulator with an outer/inner radius of 0.137 m/0.1055 m. The minimum cold-end heat input at 7 K is found to be ≈25 W Copyright © 1998 IOP Publishing Ltd

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and the optimum length will be approximately 1 m, which is surprisingly short. It is interesting to compare the cold-end heat input with the heat input due to pure heat conduction, i.e. without any stressing voltage. The insulator with integrated aluminium screens may conduct ≈16 W which is ≈2/3 of the total heat input at the operating voltage. It is obvious that a.c. bushings can contribute a similar order of heat input as the live conductor leads, since insulating materials with low dielectric loss number and low heat conductivity were mandatory in classical superconducting cable terminations. However, only a few insulants matched the thermomechanical requirements, especially in view of mechanical stress during the cable cool-down period. Ethylene propylene rubber may be one of the top candidates for future high-Tc superconductor cable bushings where a somewhat higher heat input will be permissible. These bushings may be considerably smaller than existing designs. Bushings for d.c. do not suffer from dielectric losses and the heat in-leak is due only to heat conduction in the solid. A long insulator may suppress the heat in-leak at a low level. However, as there is no capacitance grading, the design of a satisfactory d.c. bushing with safe field control along stress cones may be even more critical than for an a.c. bushing: a successfully tested design example is presented in figure H5.0.29. The cold-end heat removal into the covering fluid must be carefully checked in any design. This was found to be a particularly critical region in the terminations for classical superconducting cables. The heat transfer had to rely safely on free convection. There was no margin for large temperature differences between insulator surface and bulk helium, since the helium layer adjacent to the stress cone would be thinned and impaired in its dielectric strength, which could be very dangerous. Similarly, no vapour layer adjacent to the stress cone at the cold bushing end could be tolerated in future high-Tc superconductor cable terminations as this condition sets a limit for the permissible heat input. A competing a.c. design has been completed by the BNL group, as shown in figure H5.0.31. The live conductor lead is insulated against the ground shield conductor again by means of a capacitor bushing, the 6-300 K transition being arranged horizontally, since the BNL cable could not be bent. An ambient temperature elbow provided the final usual vertical position for the outdoor insulator (Schauer 1991). (c) Shield conductor current lead Shield conductor current leads are required for concentrically arranged conductors, as mentioned already. Counterflow helium cooling was again the proper means to limit the total heat input in the case of classical superconductors; however, electrical insulation was not critical. Any a.c. shield conductor lead should surround the live conductor lead concentrically as can be seen in figures H5.0.30 and H5.0.31, respectively. This eliminates magnetic stray fields thus reducing eddy current losses which may otherwise arise in the heavy walls of the cryogenic vessel. (d) Coolant feeders The live conductor must be fed with the coolant coming from a grounded refrigerator and a satisfactory break in the high-voltage insulation is a critical part of this. The ATF design used a separate coolant breaking device (see figure H5.0.28), which was completely separated from the termination, the helium coming into a cold vessel from the refrigerator by means of a vacuum insulated transfer line and the high-voltage insulation being provided by a capacitor bushing with a matched field control electrode. The design was simple and reliable, but caused additional losses, mainly due to the bushing. BNL and Siemens on the other hand were able to incorporate the cold helium break into the termination directly, as shown in figures H5.0.30 and H5.0.31. This eliminated any additional heat in-leak; however, in the event of sparking the termination could be destroyed completely. The break was also incorporated into the AEG d.c. termination, see figure H5.0.29. The connection of the return coolant at ground had to be achieved in a manner which is common in cryogenic technology. Impregnation of the cable dielectric was provided directly from the inward coolant

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Figure H5.0.31. Termination of the BNL transmission line; the cable end is on the right. Reproduced from Brookhaven National Laboratory Informal Technical Note 119 (14 July, 1981) by permission.

stream, as in the BNL case and the Siemens design, this scheme being adequate for high-Tc superconductor cables, too. Of course, leakage across the dielectric must be not excessive. The ATF cable with its separate dielectric space needed a separate impregnating helium inlet, again at ground potential. (e) Thermal insulation The termination thermal insulation is based on common cryostat techniques. Pressure vessel regulations must be borne in mind. These were very rigorous for the final ATF termination since the cable had to be installed in a public utility station. This forced a very conservative mechanical design. The structure heat input has been estimated with almost 25 W into the 6 K region, which is quite a high figure. Termination structure heat in-leak may be of much less concern with high-Tc superconductor cables. (f) Performance Normal operation of cable terminations for helium-cooled classical superconducting cables has, in general, been found to be adequate. However, quite often the actual heat input has been measured to be higher than the design value. This might be due to some manufacturing degradation, especially for the outdoor test sites. A realistic estimation for an a.c. 4 kA/110 kV cable may be 0.5 kW total heat input into the 6 K region. This figure is equivalent to about 1 km cable of the BNL design. Hence, classical superconducting cables were only attractive for lengths >5 km. AEG reported individual heat input of 120 W at 12.5 kA/200 kV d.c. operation. A full cable route with two separate monopole cores would then Copyright © 1998 IOP Publishing Ltd

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require 0.15 MW refrigerator driving power. Transient performance has been found to be less satisfying and problems with current flow have been reported even though the helium-gas-cooled current leads were, in general, properly designed. However, extensive voltage tests have identified shortcomings in the insulation. Weak points were identified where multiple barriers against breakdown could not be provided, e.g. in the high-voltage cold-helium break in the BNL and Siemens terminations. Similarly, the AEG termination showed first flashovers in the cold-break region at 210 kV tests. The ATF cable termination failed during an elevated a.c. voltage test at almost 140 kV rms, possibly due to thermal runaway of a bushing. The critical transfer of the enhanced dielectric losses into the pressurized helium covering the cold bushing stress cone might have been underestimated. Transient performance also incorporates cooling down of the cable system. Solid bushing insulators may easily crack when the internal temperature gradients are too large and this may limit the admissible cool-down rate. It is not advisable for instance to fill a liquid cryogen directly into a termination, as has been occasionally proposed with the aim of achieving an accelerated cable cool-down. This point needs special attention for the design of future high-Tc superconductor cable terminations, as liquid nitrogen has a high specific heat and the cool-down rate may have to be limited by careful temperature control. No detailed design data, or even operational experience, are known for high-Tc superconductor cable terminations at present. However, due to the less conflicting conditions mentioned already, future terminations may become easier to design in an adequate manner when incorporating the acquired knowledge with classical superconducting cables. H5.0.5.2 Joints Joints have to connect the individual cable sections and no transition to ambient temperature is needed. The technology is less expensive than in the case of the terminations and only a few groups have made detailed designs. Joining classical conductors by welding the stabilizing copper was thought to be quite easy. The low resistances did not cause difficulties but, in the case of corrugated tubes, the welding had to be helium tight. The most severe problems may be caused by the dielectric insulation. Graded stress cones, similar to those in the terminations, seem to be an economic approach, as indicated in figure H5.0.32. Thermal insulation can be provided by an appropriate box which connects the cryogenic envelope sections. This technology may also be adequate for high-Tc superconductor cable joints. However, no joints have ever

Figure H5.0.32. Joint for a superconducting a.c. cable. Reproduced from Bogner et al (1979) by permission.

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been built and therefore no operational experience exists. H5.0.6 Refrigeration The prevailing heat load into the cooling fluid comes in via the heat in-leak to the cryogenic envelope; dielectric losses and a.c. superconductor losses are an additional load with a.c. cables. The mass flow of the fluid has to be chosen according to the fluid heat capacity, as has been already discussed in section H5.0.4.2. As long as the respective ducts are thermally insulated from each other, see for instance figure H5.0.2, there is no interaction between the inward and return stream. However, interaction may have a strong impact on temperature profiles when a heat exchange between a coolant inward/return stream is effective, see for instance figures H5.0.3, H5.0.20 and H5.0.21. The maximum allowable superconductor temperature must never be exceeded at any point along the cable and this requires a careful detailed design. Note that there is a Joule-Thomson expansion along the cable and also a cooling effect whenever a flow-inward stream is thermally coupled with a recooled flow-return stream. Thus, the cable end temperature may be lower than the maximum level with any particular coolant. Two-phase flow must be safely eliminated otherwise vapour locking could occur, which is a very dangerous condition both for the conductor and for dielectric performance. The general temperature profile of the fluid along the cable with incorporated pressure drop is given by

where the subscript Q stands for the per unit length constant heat input i n plus heat exchange rate with a coupled flow stream and h for the isenthalpic expansion due to pressure drop. The heat input ex term may be written as (∂T/∂l )Q = ( i n + e x )/(cp ) where cp is the constant pressure specific heat. The isenthalpic term is simply (∂T/∂l )h = ψ (∂p/∂l ) where ψ is the Joule-Thomson coefficient, which can be positive or negative. The pressure-drop rate ∂p/∂l is found according to section H5.0.4.2.

Figure H5.0.33. The cooling circuit for superconducting cables. Reproduced from Croitoru et al (1974) by permission.

Several helium cooling schemes for classical superconducting cables have been discussed in the past and all serious workers ended with an inward stream/return stream principle using supercritical helium. The elimination of adverse flow oscillations forced the results into a high-pressure range so that the near-critical-point regime with its high specific heat could be avoided. The simplest cooling scheme is indicated in figure H5.0.33. To ensure the necessary flow, either a liquid pump or an appropriate refrigerator can be used, as shown in figure H5.0.34 (Croitoru et al 1974). The pump circuit completely decouples the flow rate through the cable from the refrigerator flow rate, thus allowing some degree of freedom in the design. Advanced multistage Claude-type refrigerators are considered to be most advisable in the case of helium coolant. Unfortunately, at low temperatures pump losses arise and these have been found to be prohibitive. Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.34. Circulation of supercritical helium in a superconducting cable loop: (a) by means of a cold pump; (b) by direct refrigerator pumping. Reproduced from Croitoru et al (1974) by permission.

Using the refrigerator directly for fluid pumping resulted in the incorporation of the whole cable into the refrigeration scheme and the flow rate through the cable had now to match the refrigerator mass flow rate, i.e. the cable had to become part of the refrigerator. Some heat exchange between the inward and return fluid in the concentric conductor design, as shown previously in figure H5.0.26, had also to be incorporated. BNL were able to proceed to a logical conclusion by using the cable deliberately as a heat exchanger. A far-end expansion turbine provided the necessary temperature adjustment, as shown in figure H5.0.35 (Forsyth 1987). This scheme seemed to be the most advanced for classical superconducting cables and resulted in surprisingly low overall heat pump factors (≈250 WAV for a maximum temperature of ≈8 K, including losses due to fluid flow pressure drop) for the whole circuit. However, the exchanged heat had to cross the lapped-tape dielectric and this implied severe restrictions on the heat conductivity; an extremely low conductivity for instance would eliminate any heat exchange and this is not acceptable in view of the dielectric losses. A very high conductivity could be appropriate Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.35. The counterflow cooling circuit for superconducting cables. Reproduced from Forsyth and Thomas (1986) by permission.

Figure H5.0.36. Temperatures and pressure drops in a counterflow-cooled cable. Reproduced from Forsyth (1987) by permission.

for steady-state conditions, but this prevented cable cool-down. BNL used a dielectric heat conductivity below 0.09 W m−1 K−1 otherwise the maximum temperature in the cable was calculated to become excessive. This conductivity figure could be easily maintained with the usual helium-impregnated lapped-tape insulations. Figure H5.0.36 indicates typical calculated temperature and pressure variations along a 30 km cable. Additional cooling capacity for the cable terminations had to be provided. The cooling needs had already been specified to be of the order of 0.3 kW per cable end. The total refrigerator capacity, for instance for the 30 km BNL design, may easily come up to 30 kW, requiring a compressor input power of the order of 5 MW. The equivalent waste heat energy must be dissipated locally to the ambient. The ingenious BNL cooling scheme cannot be transferred to high-Tc superconductor cables (Forsyth 1993). There is no chance of relying on supercritical nitrogen for instance since the critical pressure is much too high. Hence, any pressure drop may lead to a temperature increase since the Joule-Thomson coefficient will always be negative; the pumping losses can be of the order of the external heat input into the coolant. A far-end turbine expansion may also not be feasible. A cooling scheme as shown in figure H5.0.37 where the return stream is recooled in a refrigerator before again being fed into the respective cooling duct seems to be preferable. Figure H5.0.38 illustrates schematically the calculated

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Figure H5.0.37. The cooling circuit for a high-Tc superconductor cable.

Figure H5.0.38. Temperature and pressure along a liquid-nitrogen-cooled high-Tc superconductor cable; go nitrogen flow. Reproduced from Ashworth et al (1994) by permission.

temperature/pressure profiles along a cable section. Again, the maximum conductor temperature does not occur at the inward stream exit (Ashworth et al 1994). There was wide experience in the field of steady-state helium cooling of long ducts with low heat load input which could be used for initial designs of classical superconductor cable cooling. Most of this experience came from the cooling of large magnets when using cable-in-conduits. Flow modelling seemed to be reliable to predict the conduction for long power cables reasonably precisely. However, little operational experience is known for helium-cooled ducts with considerable heat load, and with lengths of some tens of kilometres running over extended periods. Doubts about the long-term reliability of cable cooling could never be dispelled completely. Some long-term operational knowledge of high-Tc superconductor cables may be anticipated from industrial liquid-nitrogen pipeline transfer and distribution systems. Hence, acceptance by utilities may be easier for these new superconducting cables. Cool-down of long power cables may be much more ambiguous than the steady-state operation. The enthalpy of the core solids is large and cooling down by means of previously stored liquid helium has often been proposed for classical superconducting cables, this liquid being blown down the cable. The refrigerator need not then be designed for a specific cool-down capacity. However, although recovery of the evaporated helium at the cable exit was necessary, cool-down might be more effective when relying on a one-way stream instead of a counterflow scheme with inherent heat exchange. Note that temperature gradients may be dangerous, due to the step cold wavefront travelling along the cable. Whatever method is employed, cool-down periods of several weeks had to be anticipated for long Copyright © 1998 IOP Publishing Ltd

Pilot facilities

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helium-cooled cables. The cool-down period of a liquid-nitrogen-cooled high-Tc superconductor cable may be not much shorter, even when blowing down liquid nitrogen without end gas recovery, which is a serious option now; the limitation coming from the permissible temperature gradients. Normal operation cannot begin immediately after cool-down because the safe filling up of the taped dielectric space with the impregnating fluid as well as establishing genuine stable temperature distributions along the cable may add a considerable additional delay time of up to one week. H5.0.7 Pilot facilities Some of the projected cables with classical superconductors have proceeded up to an advanced stage of testing which gave very valuable information about the benefits and drawbacks of the various technologies. The most significant of the built and tested cables are highlighted below. Much of the acquired experience could be very helpful for the development of future high-Tc superconductor cables. Although many laboratory a.c. cable links have been built and tested, there is only one d.c. prototype cable. The Kryzhanovsky Power Engineering Institute in the former USSR was not very successful in its 100 m long three-phase design (15 kV line-to-line; 21 kA), due to inherent mechanical problems with the rigid conductors which were employed. The more promising All Union Scientific Research Institute concept of an Nb3Sn-clad corrugated tube conductor had to be closed down in the early 1980s, before a final assessment could be made. H5.0.7.1 Laboratory test facilities CERL in Leatherhead finished its a.c. cable development in the late 1970s by demonstrating a 5 m long 6.3 kA core, without high-voltage terminations. This core has already been shown in figures H5.0.11 and H5.0.19. Siemens were able to proceed much further with a.c. cable development by installing and testing a 35 m long single-phase prototype with adequate terminations, as shown in figure H5.0.39. The three-phase rated power of the cable was specified with 2 GVA/110 kV line-to-line (Bogner et al 1979). However, development had to be stopped in 1977, partly because of a study commissioned by the German Federal Department of Research and Development. This study claimed that water-cooled cables were economically more competitive than superconducting cables; however, some important views such as natural load matching were not treated adequately so the conclusions of this particular study were seriously called into question later on (Forsyth 1991). AEG in Germany relied on a d.c. two-monopole-core circuit, each core being housed within its own cryogenic envelope, and one core with its envelope was built and tested. The rated current was 12.5 kA (overload current 50 kA) and the rated voltage was 100 kV. The simple helix conductor with 35 mm diameter consisted of 12 Nb3Sn strips, stabilized by copper. The annular helium-impregnated paper insulation package had a thickness of 14 mm; smoothing carbon black paper screens were provided inside and outside. This flexible core was heat shielded by a flexible corrugated envelope built by Kabelmetal. The current rating of the core was demonstrated by a precursory ring test with the shortened core being the secondary of a tr ansformer. The primary coils were slowly excited and 25 kA was reached by this technique. Voltage tests with 200 kV d.c. and polarity reversal were performed for a respective ten-day period using the terminations as shown already in figure H5.0.29. A first flashover along the cold cast resin insulator occurred at 210 kV. The test site is shown in figure H5.0.40. H5.0.7.2 Outdoor test facilities Only two superconducting cable projects reached the status of outdoor testing.

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Figure H5.0.39. The 35 m test facility of a semi-flexible superconducting a.c. cable at Siemens, Erlangen: (a) the three-phase cable model; (b) the test installation. Reproduced from Bogner (1975) by permission.

BNL erected a spectacular a.c. cable test site and, notwithstanding the basic problems, the team carried out very relevant investigations. Of special note is the excellent work on electrical system engineering and on the cooling circuit design with its inherent low heat pump factor. Figure H5.0.41 shows an overall picture of the BNL test site. Two cores of three (see figure H5.0.20) were inserted into the rigid cryogenic envelope and tested artificially with rated current and rated voltage. The ingenious test circuit is shown in figure H5.0.42. Many potential engineering problems such as long-term vacuum reliability of the cryogenic envelope, cooling performance, terminations and BIL compatibility were investigated and largely solved in the programme. However, no true power has ever been transmitted across the BNL cable. The second outdoor a.c. facility was installed by the team of the late Professor Klaudy (ATF) in a power station near Graz, Austria. The cable was built by Kabel metal. Since the cable was of the fully flexible corrugated-type design (see figure H5.0.43) it could be laid underground in a similar manner to any conventional cable. To reduce the costs only one phase of the cable was installed. The electrical operation scheme is shown in figure H5.0.44. The single-phase cable was connected into an existing Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.40. Test assembly of superconducting d.c. cable at AEG. (a) Flexible cable. 1—struts; 2—superconducting tapes; 3—carbon black paper; 4—wrapped paper insulation; 5—carbon black paper; 6—innermost tube of cryogenic envelope; 7—superinsulation and spacers in vacuum; 8, 10—enclosure of liquid-nitrogen duct; 9—spacer; 11—superinsulation and spacers in vacuum; 12—outermost tube of envelope; 13—corrosion sheath, (b) Current rating ring/test, (c) 200 kV voltage test. Reproduced from Bochenek et al (1975) by permission. Photographs courtesy of Daimler Benz. Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.41. The outdoor superconducting power transmission line test site at BNL. Reproduced from Forsyth (1993) by permission. Photograph courtesy of Brookhaven National Laboratory.

Figure H5.0.42. The test loop for the artificial a.c. cable test by resonance tuning of a current loop. Reproduced from Forsyth (1987) by permission.

overhead line. The rated voltage was 60 kV line-to-line, and the rated power at the consumer end was specified at <10 MW, which was in fact much below the cable capacity. However, the grid was equipped with a compensating coil so the BIL had to be set according to a 60 kV line-to-ground level, i.e. 110 kV line-to-line rated voltage (Klaudy et al 1981). The cable performed well under normal operation and gave confidence regarding cryogenic installations and reliable control under real outdoor utility conditions. Moreover, it was the first superconducting cable in the world which had transmitted true power to be paid for by the consumer. Note that the 1 GVA corrugated fully flexible cable as shown in figures H5.0.21 and H5.0.27 has been extrapolated from these trials. However, development work had to be stopped in the mid-1980s, due to lack of funding. Copyright © 1998 IOP Publishing Ltd

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Figure H5.0.43. The outdoor a.c. cable test site at Arnstein. (a) The fully flexible cable: 1—live conductor; 2—helium-impregnated lapped-Kraft-paper insulation; 3—shield conductor; 4—core enclosure; 5—superinsulation + spacer in vacuum; 6, 7, 8—liquid-nitrogen duct; 9—superinsulation + spacer in vacuum; 10—outermost tube with corrosion sheath, (b) The layout of the cable, (c) Terminations. Courtesy of ATF.

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Figure H5.0.44. Test circuit at the Arnstein power station.

H5.0.7.3 Assessment All the cited a.c. cable tests showed clearly that superconducting transmission lines for d.c. as well as for a.c. could be built and operated with techniques already existing in the late 1970s, i.e. by using classical superconductors cooled by supercritical helium, and progress in technology gave rise to an optimistic future view. Shortcomings which were found such as the terminations during elevated-voltage tests indicated weak design points but the significance of these should not be overstated. There is no doubt that superconducting cables were one serious option for future huge power transmission demands. Much improvement could be anticipated from synergetic effects when extracting the particular benefits of each of the developed systems. In particular the most advanced projects, i.e. the BNL cable and the ATF cable, resulted in some important ideas. These can now be used profitably when developing cables which make use of the new high-Tc superconductors. H5.0.8 Economics and acceptance The classical superconducting cable was, and any high-Tc superconductor cable will be, a new product for utilities which employ new technology and, for the latter to be accepted in practice, important advantages must be inherent in the product. The economics, especially investment cost, must be attractive but above all reliability and suitability for integration into an existing system are of crucial importance. H5.0.8.1 Economics The total cost of any transmission system is composed of two main parts: ( i ) investment cost including cable manufacturing and design, right of way expenditure and civil engineering, transportation and installation, auxiliaries such as vacuum equipment, refrigerators, control equipment and the refrigerant inventory; also the need for redundancy may require additional investment; some of these items may be of no concern for high-Tc superconductor cables, for instance when providing these cables for retrofitting; Copyright © 1998 IOP Publishing Ltd

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( ii ) running cost such as loss removal, operation control, coolant losses due to leaks, refrigerator maintenance. It has been commonly accepted from the time of the earliest developments that superconducting cables may never compete economically with overhead lines because there is a cost advantage of between 5:1 and 20:1 in favour of overhead lines. However, there are some factors which may force utilities to adopt cable transmission, for example overhead lines cannot be erected in densely populated areas and they are easily disturbed by atmospheric influences. Superconducting cables may be not affected by these external shortcomings. Superconducting a.c. cables have to compete mainly with other advanced cable techniques, especially with modern XLPE cables which show very low dielectric losses. Also apart from any economics, XLPE cables, in general, use natural cooling and therefore there is no cooling fluid to be pumped and hardly any control requirements. It is claimed that these cables can be installed and then ‘forgotten for their lifetime’. However, some spectacular XLPE cable failures in the past render such a philosophy in a somewhat different perspective. Superconducting d.c. cables on the other hand have to compete with well established extra-high-voltage overhead lines for long-distance transmission and their use in the submarine domain has not been given much attention in the past. Submarine transmission may become an interesting niche application in future. Short d.c. links between large a.c. systems which may be needed to limit excessive short-circuit currents could also be of future interest. Superconducting cables require rather exotic cooling fluids and rather complex refrigerators from the utility point of view. The genuine direct losses are very low but have to be pumped up to ambient temperature and removed as waste heat; the economic value of the latter is modest but nevertheless finite. Heat is removed locally and may be used for moderate temperature and steady heating. Numerous cost estimates for classical superconducting cables have been produced in the past but all of them are

Figure H5.0.45. The relative cost of underground power transmission: (a) a.c. cables; (b) two-monopole-core d.c. cables (from Croitoru et al 1974).

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incomplete in absolute values; nevertheless they indicated one important tendency: the cost per unit power and length was claimed to decrease much more rapidly than with any kind of conventional cables, as shown in figure H5.0.45. These data were communicated by EdF in 1974. A distinct break-even point in the comparison with any competitive technique was claimed. However, the actual break-even-point power rating has often been discussed in an intuitive rather than a rational manner. Weighing the benefits or drawbacks where costs are ambiguous may shift the break-even point within one order of magnitude. Particular group interests also might have been of some influence. For instance, how can the benefits of no electromagnetic pollution, which is restrictively inherent in shielded superconducting a.c. cables, be capitalized on against the need of permanent cooling control? What about the right-of-way cost without any electromagnetic noise which is under discussion at present by the public? It came out that all of these points can be discussed with relevance only when planning a particular transmission system. The list in table H5.0.2 may give a rough impression of break-even points as communicated by various competent workers in the field. The break-even points depend not only on the competitive technique but also to a large extent on the particular superconducting cable technology. All data refer to classical superconductors cooled by helium.

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BNL has not given break-even figures for its advanced a.c. cable; however, the published comparative values for specified bulk transmission lines are much more informative. Table H5.0.3 indicates a summary derived from the so-called ‘Pennsylvania study’. The rated voltage/power was 550 kV/5.4 GVA. Two three-phase circuits with 495 km length have been assumed.

EPRI, on the other hand, relied for a.c. cable evaluation on an extended BNL version (345 kV/4.8 GVA) with 80 km and 322 km length, respectively. Details are indicated in table H5.0.4. The importance of the cryogenic envelope cost is obvious (≈40% of the total cost). The total 1979 cost for this cable type has been claimed to be economically attractive. GEGB came up with a similar relative cost situation for a.c., as may be seen from figure H5.0.46 (Maddock and Male 1976). Klaudy and his team relied on manufacturing figures as communicated by Kabelmetal for the a.c. cable cost estimation. Again, the cryogenic envelope was estimated to amount to

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Figure H5.0.46. The relative cost of underground a.c. power transmission: (a) pro rata cost of superconducting cable items; (b) comparison with competing techniques (from Maddock and Male 1976).

Figure H5.0.47. The relative cost of superconducting cables versus operating temperature. Reproduced from Edwards (1988) by permission.

Figure H5.0.48. Transmission costs CT* versus critical current density Jc for high-Tc superconductor cables. Reproduced from Ashworth et al (1994) by permission.

30% of the total cost. The latter was evaluated in 1982 at 21 × 106 US$ for a 10 km long three-phase cable with 1 GVA rated power. However, the estimation might have been somewhat too optimistic. A considerable reduction in total cost when increasing the operating temperature had already been anticipated long before the discovery of the new high-Tc superconductors: GEGB for instance had shown this in 1967, see figure H5.0.47. Some more detailed recent cost estimates have been published for shielded a.c. high-Tc superconductor cables with liquid-nitrogen cooling. However, superconducting materials costs and core manufacturing costs may still be rather speculative at present. A break-even current density may be seen from figure H5.0.48 for a high-71- superconductor cable with rated power 1000 MVA. There is a distinct optimum live conductor diameter which depends on the rated power, see figure H5.0.49 (Ashworth et al 1994). All these cost estimations are hard to verify in detail. Copyright © 1998 IOP Publishing Ltd

Economics and acceptance

Figure H5.0.49. Transmission costs CT* versus live conductor diameter for high-Tc superconductor cables; Jc = 200 kA cm−2. Reproduced from Ashworth et al (1994) by permission.

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Figure H5.0.50. Transmission costs CT* versus high-Tc superconductor cable rated power P retrofitting into 150 mm duct. Reproduced from Ashworth et al (1994) by permission.

Cost optimization is based on different conditions when retrofitting in already existing ducts is considered. A Japanese study for a 1000 MVA/66 kV high-Tc superconductor cable resulted in an overal expenditure of only 79% compared with a competing conventional cable transmission system so retrofitting was claimed to be very attractive (Hara et al 1992). However, superconductor costs were assumed to be similar to Nb3Sn and a high critical current density of the high-Tc superconductor of almost 1010 A m−2 is anticipated. A more recent evaluation is shown in figure H5.0.50. Retrofitting by a high-Tc superconductor cable seems to become economical provided high current densities of >1109 A m−2 can be achieved in the superconductor. The rated power may be surprisingly low, i.e. only a few hundred mega volt amps. H5.0.8.2 Acceptance Reliability is one of the most important criteria for utilities and means must be provided to compensate for any actual component failure within a reasonable delay so that the consumer can be supplied with energy without any inconvenience. Therefore, it is not very advisable to put all the bulk power into one single cable otherwise reserve capacity of this order would have to be provided. In Europe, bulk power lines of much more than 1 GVA are rare and super conducting a.c. cables may not be acceptable unless cost competition can be guaranteed near, or even considerably below 1 GVA rated power and distances of the order of 10-20 km. In the case of d.c., the acceptance is even more doubtful. Also the need for permanent cryogenic cooling is a serious impediment. Cryogenic technology must be seen distinctly in the context of quality control during manufacturing and assembly: neither the current-carrying capacity, nor the dielectric integrity, can be tested before final laying of and cooling down of the full cable in site, any intermediate quality control being restricted to secondary related properties, e.g. metallurgy of conductor strips, mechanical correctness of helices and wrapped insulation packages, vacuum tightness, etc. However, the more general objections against cryogenics are common with any other super conducting power equipment such as fault current limiters, generators and transformers and super conducting magnetic energy storage. Acceptance of cryogenics has to be discussed in a much broader context; it is not a specific cable problem. Copyright © 1998 IOP Publishing Ltd

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The technical feasibility of cables with classical superconductors has been demonstrated and the inherent features, i.e. no thermal or electromagnetic interference with the environment along the route, as well the lack of any thermal cycling, has again rekindled interest in the subject. Matching high-Tc superconductors are on the way to being developed up to real industrial manufacturing and handling performance. Design work for integrated superconducting power systems is being carried out again. In Japan, for instance, special urban grid configurations are being discussed in view of the profits seen from a.c. cables. These cables may fit into existing narrow trench routes. Siemens on the other hand proposed some years ago an all-superconducting power-transmission system based on the new high-Tc superconductors; this is illustrated in figure H5.0.51. Also some European utilities are beginning to discuss superconducting power equipment in a broader context. The modest power ratings as claimed to be economical for retrofitting in figure H5.0.50 could make high-Tc superconductor cables much more attractive for utilities and limit the risk when introducing a very new technology such as cryogenics into power systems.

Figure H5.0.51. A high-Tc superconductor power transmission system for 1 GVA at an operating temperature of 65-77 K. Reproduced from Bogner (1990) by permission.

H5.0.9 Prospects of novel cable designs The discovery of the high-Tc superconductors in 1986 renewed the interest in superconducting cables dramatically. Many speculative papers appeared around the world concerning a.c. cables, not all of them being in fact very relevant. Critical temperatures of more than 110 K rendered cooling with liquid nitrogen instead of helium feasible. Only a very few papers propagated lower-temperature domains, e.g. cooling by liquid hydrogen, liquid neon or helium gas, notwithstanding the availability of a broader variety of superconductors such as the Bi-based 2222 types. One reason might have been the anticipated negligible inventory cost of filling a cable with liquid nitrogen, but helium costs have not been found to be very prohibitive for the former classical cables, see for instance figure H5.0.46. However, gas storage tanks might be dispensed with when using liquid nitrogen. There are some definite and fascinating facts when considering the use of the higher operating temperatures. First of all, any kind of heat input is less critical, due to the much lower heat pump factors of the refrigerators. Secondly, some points with great potential may be found from state-of-the-art engineering: ( i ) the cryogenic envelope techniques as previously developed for helium cooled cables are well suited for liquid-nitrogen cooling; ( ii ) commercially available refrigerators, pumps and installation techniques to ensure fluid flow control and heat removal need no, or very little, special development;

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( iii )reliable electrical insulation is available; in a similar manner to helium-cooled superconducting cables, lapped-tape insulation packages impregnated by the actual refrigerant may allow the build-up of flexible cable cores; all-solid insulation may also be available in future, provided thermal contraction problems can be overcome, and there are also proposals to rely on an ambient temperature dielectric; ( iv )terminations become simpler.

H5.0.9.1 Performance potential of the cryogenic envelope Either the rigid envelope technique from BNL or the flexible corrugated tube technique from ATF/Kabelmetal seems promising with high-Tc superconductors. The per unit length heat in-leak is not much affected by the operating temperature so 0.6 W m−1 can still be assumed for the BNL envelope; however, some performance degradation when simplifying the design may be advisable and a safe cryosorption must be secured for a 30 year lifetime. A promising competing prototype semi-flexible envelope system, which drastically reduces the number of cold welds to be assembled during field mounting, has been built and tested in the ATF. The per unit length heat input into the inner tube with diameter 100 mm has been measured at 0.6 W m−1, irrespective of the actual filling fluid. This would correspond to 1 W m−1 for a proper size, housing a 1 GVA cable. Significant cost reductions compared with the original BNL design have been claimed (Schauer 1991). The fully flexible corrugated envelope is another competing system. Commercially available corrugated transfer lines for cryogenic fluids with a rather simple and cheap insulation yield 1-2 W m−1 into a 39 mm inner diameter duct. This would amount to 5 W m−1 for a 100 mm inner diameter envelope (see equation (H5.0.13)), which may easily house a 1 GVA single-phase high-Tc superconductor cable core for 1 GVA rated three-phase power. Thus, 15 W m−1 may be taken as an upper total three-phase heat input level using well established industrial techniques. The refrigerator driving power in the case of liquid-nitrogen cooling will amount to only 0.3% of the rated power, which may be a very acceptable figure. H5.0.9.2 Electrical insulation As in the case of helium-cooled superconducting cables, flexible cores using an impregnated lapped-tape dielectric seem to be a reliable solution. The mechanical performance is very similar and the tapes are of comparable brittleness. The dielectric strength of pressurized liquid nitrogen is adequate. The impregnant fluid again controls the respective butt-gap inception voltage and insulation packages may be designed by using the proven rules. A lifetime of 30 years seems to be compatible with 10-20 kV mm−1 rms operating stress. The dielectric losses are acceptable, as has been shown in figure H5.0.17. A liquid-nitrogen-impregnated lapped-tape insulation is of course one serious option for future high-Tc superconductor cable insulation. However, a sufficiently high liquid-nitrogen pressure must be maintained to prevent bubble formation within the package. A competing technique has been developed in the form of an all-solid insulation in Japan. Ethylene propylene rubber (EPR) for instance can be extruded over radially stiff conductor tubes; the cores remain flexible and radial thermal contraction can be controlled safely down to 40 K. Any electrical stress along the cable route is limited to the extruded solid. The coolant is never stressed so bubble formation is not critical from the viewpoint of insulation performance. EPR is very resistive against treeing, and dielectric losses of the order of 10−3 at liquid-nitrogen temperature are very acceptable. This dielectric has sometimes been claimed to be some kind of electrical ‘super-insulation’. Insulation quality control could eventually be carried out at ambient temperature before laying down and cooling the full cable length. However, longitudinal contraction is difficult to overcome; a design which fully satifies this particular point is still outstanding.

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Figure H5.0.52. Extruded XLPE-insulated cryogenic a.c. cable: (a) cable configuration; (b) test installation. Reproduced from Kosaki et al (1992) by permission. Photograph courtesy of M Kosaki.

The all-solid insulated cores are believed to result in an extremely small core size, which is essential for pulling a cable in already existing narrow ducts for retrofitting. Figure H5.0.52 shows a view of a cable test site at Toyohashi University (Kosaki et al 1992). A second competing technique has been proposed in the USA, i.e. an electrical insulation at ambient temperature. Figure H5.0.53 gives an impression of the three-phase design (Engelhardt et al 1992). However, this technique may suffer from some serious drawbacks. ( i ) Eddy currents are induced within the cryogenic enclosure, since no superconducting shield conductor can be provided. The related losses have to be dissipated by natural soil sinking, and the current capacity of these cables will be limited to an order of 2500 A. Magnetic stray fields will cause interactions between the three live conductors, which results in additional a.c. losses in the live conductors. ( ii ) An evacuated annular space has to be provided for thermal insulation of the conductor, this annulus being maintained at high voltage. A similar configuration where liquid nitrogen was provided as a dielectric was proposed for classical d.c. cables in 1964. However, the principal objections concerning evacuation of the thermal insulating duct have never been defeated. Evacuation and maintaining a

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References

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Figure H5.0.53. A schematic diagram of a room temperature dielectric high-Tc superconductor cable. Reproduced from Engelhardt et al (1992) by permission.

high vacuum is a critical process with liquid-nitrogen-cooled cables, as has already been mentioned. Multiple pumping necks may have to cross the high-voltage insulation to this end, at the worst with distances of only a few hundred metres, which is a complicated and costly technology in any case. Any neck will also be a point of enhanced breakdown risk. The main argument in favour of an ambient temperature insulation is the anticipated low development risk which is related essentially to high-Tc superconductors. H5.0.9.3 Assessment A final assessment of the various high-Tc superconductor cable designs being discussed at present is hard to find. Each of the designs has its benefits as well as its drawbacks. Which of the particular techniques will finally be successful will depend on the power ratings of interest for the utilities as well as on progress being made by the individual development groups. Different solutions may be accepted eventually in the various continents, according to particular utility conditions.

References Ashworth S P, Metra P and Slaughter R J 1994 A techno economic design study of high-temperature superconducting power transmission cables Eur. Trans. Elect. Power Eng. 4 293-300 Beales T P, Friend C M, Segir W, Ferrero E, Vivaldi F and Ottonello L 1996 A dc transmission cable prototype using high-temperature superconductors Supercond. Sci. Technol. 9 43-7 Bochenek E, Franke H and Wimmerhoff R 1975 Manufacture and initial technical tests of a high-power d.c. cable with superconductors IEEE Trans. Magn. MAG-11 366-72 Bogner G 1975 Cryopower transmission studies in Europe Cryogenics 15 79-87 Bogner G, Pencynski P and Schmidt F 1979 Development of a superconducting high power ac cable Siemens Forsch. Entwickl. Ber. 8 1-22 Broad R J (project manager) EPRI 1979 Cost components of high-capacity transmission options EPRI EL-1065 vol 2, project 568-1, final report Croitoru Z, Deschamps L and Schwab A M 1974 Transport d’energie electrique par cryocables Entropie 56 47-56 Edwards D R 1988 Supertension or superconducting cables IEE Proc. 115 9-23 Engelhardt J S, von Dollen D and Samm R 1992 Application considerations for HTSC power transmission cables Proc. 5th Annu. Conf. on Superconductivity and Applications (Buffalo, NY, 1991) (New York: Elsevier) pp 692-711 Copyright © 1998 IOP Publishing Ltd

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Fallou B and Breteau J P 1975 Comportement dielectrique sous haute tension des structures rubanees impregnees de fluids cryogeniques Rev. Gen. Electr. 84 748–57 Forsyth E B 1987 Superconducting cables Encyclopedia of Physical Science and Technology 13 465–92 Forsyth E B 1988 Energy loss mechanisms of superconductors used in alternating-current power transmission system Science 242 147–9 Forsyth E B 1991 The dielectric insulation of superconducting power cables Proc. IEEE 79 31–40 Forsyth E B 1993 Superconducting power transmission systems-the past and possibly the future Supercond. Sci. Technol. 6 699–713 Forsyth E B and Thomas R A 1986 Performance summary of the Brookhaven superconducting power transmission system Cryogenics 26 599–614 Fujikami J, Shibuta N, Sato K, Ishii H and Hara T 1994 Characteristics of the flexible high Tc cable conductor Appl. Supercond. 2 181–90 Garwin R L and Matisoo J 1967 Superconducting lines for the transmission of large amounts of electrical power over great distances Proc. IEEE 55 538–48 Gerhold J 1984 Design criteria for high voltage leads for superconducting power systems Cryogenics 24 73–82 Gerhold J and Schachinger E 1980 Thermische Überlastauslegung tieftgekühlter elektrischer Leiter Kälte Klimatech. 33 6–10 Hara T, Okaniwa K, Ichiyanagi N and Tanaka S 1992 Feasibility study of compact high-Tc superconducting cables IEEE Trans. Power Delivery 7 1745–53 Heumann H 1972 Kabeltechnische Probleme bei Projektierung und Bau von Übertragungsstrecken mit Supraleitern Mitteilungen Kabelwerke 4/72 1–7 Kafka W 1969 Entwurf eines SupraSeitungs-Drehstromkabels Elektrotech. Z. A 90 89–92 Kiwit W, Wanser G and Laarmann H 1985 Hochspannungs- und Hochleistungskabel (Frankfurt: Verlags- und Wirtschaftsgesellschaft der Elektrizitätswerke mbH) Klaudy P 1980 Elektrische Hochleistungsübertragung besonders mittels supraleitender Kabel und die Entwicklung und erstmalige Erprobung eines solchen Kabels (Bauart Klaudy) im Kraftwerk Arnstein der STEWEAG in der Steiermark Ost. Z. Elekr. 33 214–27 Klaudy P A 1966 Some remarks on cryogenic cables Advances in Cryogenic Engineering vol 11 (New York: Plenum) pp 684–93 Klaudy P A and Gerhold J 1983 Practical conclusions from field trials of a superconducting cable IEEE Trans. Magn. MAG-19 656–61 Klaudy P A, Gerhold J, Beck A, Rohner P, Scheffler E and Ziemek G 1981 First field trials of a superconducting power cable within the power grid of a public utility IEEE Trans. Magn. MAG- 17 153–6 Kosaki M, Nagao M, Mizuno Y, Shimizu N and Horii K 1992 Development of extruded polymer insulated superconducting cable Cryogenics 32 885–94 Maddock B J and Male J C 1976 Superconducting cables for ac power transmission CEGB Research September 11–32 McFee R 1962 Applications of superconductivity to the generation and distribution of electric power Elect. Eng. 81 122–8 Meets R J 1974 Butt gap discharges in laminated polyethylene insulation impregnated with very cold helium gas Proc. 3rd Int. Conf. on Gas Discharges, IEE Conf. Publication 118 419–25 Patel S, Chen S, Haugan T, Wong F and Shaw S T 1995 Effect of bending on Jc −e characteristic of silver-sheathed oxide superconducting tape with sausaging Cryogenics 35 257–62 Schauer F 1991 Superconducting power transmission Superconducting Technology (Singapore: World Scientific) Swift D A 1975 Dielectric design for superconducting a.c. cable with solid insulation Rev. Gen. Electr. 84 741–7 Von Dollen D, Metra P and Mujibar Rahman 1993 Design concept of a room temperature dielectric HTS cable Proc. Am. Power Conf. (Chicago, 1993) vol 55–11, pp 1206-11 Weedy B M and Swingler S G 1987 Review of tape materials for cable insulation at liquid nitrogen temperatures Cryogenics 27 667–72 Wimmershoff R 1974 VDI + VDE Ges. für Meb- und Regeltechnik ( Karlsruhe, 1974 ) lecture Wimmershoff R 1979 Termination of a d.c. bulk power cable with superconductors Proc. 3rd Int. Symp. on High Voltage Engineering ( Milan, 1979 ) pp 23.21 1–4

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Further reading Baylis J A, Lewis K G, Male J C and Noe J A 1974 AC losses in a composite tubular superconductor for power transmission Cryogenics 14 553-6 Beales T P, Friend C M, Le Lay L, Mölgg M, Dineen C, Jacobson D M, Hall S R, Harrrison M R, Hermann P F, Petitbon A, Caracino P, Gherardi L, Metra P, Bogner G and Neumuller H W 1995 Conductor development suitable for HTSC cables Supercond. Sci. Technol. 8 909-13 Bochenek E and Voigt H 1974 Supraleitendes flexibles Hochleistungs-Gleichstromkabel Elektrotech. Z. B 26 215-9 Croitoru Z, Lacost A and Deschamps L 1974 Techniques nouvelles de transport d’energie par cables souterrains. Le moyen et le long terme Rev. Gen. Electr. 126-45 Forsyth E B 1990 The high voltage design of superconducting power transmission systems IEEE Elect. Insul. Mag. 6 7-16 Gauster W F, Freemann D C and Long H M 1964 Applications of superconductivity to the improvement of electrical energy economics Proc. World Power Conf. ( Lausanne, 1964 ) paper No 56, pp 1954-72 Gerhold J 1976 Optimization of refrigerator-cooled current leads for superconducting devices Cryogenics 16 401-8 Gerhold J 1981 On the layout of low temperature bushings for high ac voltage Cryogenics 21 426-30 Holler D W and Radke U 1995 Elektrische Netzanforderungen and supraleitende Komponenten der Energieversorgung VDI-Ber. 1187 205-220 Jefferies M J and Mathes K N 1970 Insulation systems for cryogenic cables IEEE Trans. Power Appar. Syst. PAS-89 2006-14 Keilin V E, Kovalev I A and Pozvonkov M M 1975 Temperature distribution along superconducting power transmission lines Cryogenics 15 323-6 Klaudy P 1972 Über tiefstgekiihlte, besonders supraleitende Kabel Electrotech. Maschin. 89 93-110 Knaak W and Reiss H 1993 Wechselstromtaugliche Hochtemperatur-Supraleiter für Kabel, Strombegrenzer, Magnete und Transformatoren ABB Technik 3-93 9-18 Kubo I, Yamada O and Shiseki N 1974 Development of 154 kV cryogenic resistive cable Fujikura Tech. Rev. 58-67 Kurti N 1967 Electric power at low temperatures New Sci. 7 604-6 Maddock B J, Cairns D N H, Sutton J, Swift D A, Cottrill J E J, Humphries M B and Williams D E 1976 Superconducting ac power cables and their applications CIGRE 1976 Session 21-05, pp 1-11 Mauser S F, Burghardt R R, Fenger M L, Dakin T W and Meyerhoff R W 1976 Development of a 138-kV superconducting cable termination IEEE Trans. Power Appar. Syst. PAS-95 909-14 Meets R J 1977 The impulse voltage flashover of dielectric spacers in a helium-insulated superconducting cable model Cryogenics 17 77-80 Meyerhoff R W 1974 Development of a rigid ac superconducting power transmission line Advances in Cryogenic Engineering vol 19 (New York: Plenum) pp 101-8 Mizuno Y, Ohe T, Nomizu K, Muneyasu H, Nagao M, Kosaki M, Shimizu N and Horii K 1991 Evaluation of ethylene propylene rubber as an electrical insulation material of superconducting cables Proc. 3rd ICPADM (Tokyo, 1991) pp 317-21 Muller A 1975 Mechanical properties of insulating tapes at cryogenic temperatures Rev. Gen. Electr. 84 568-72 Schauer F 1981 Transport of cryogenic liquids or gases between installations at different electrical potentials Cryogenics 21 735-9 Schauer F 1984 A capacitance-graded cryogenic high voltage bushing for vertical or horizontal mounting Cryogenics 24 90-6 Swift D A 1975 Dielectric design for a superconducting a.c. cable with solid insulation Rev. Gen. Electr. 84 741-7 Tang Y J, Kato T, Hayakawa N, Yokomizu Y, Matsumura T, Okubo H, Kito Y, Miyake K and Kumano T Development of the prospective power transmission model system integrated under superconducting environment—PROMISE IEEE Trans. Appl. Supercond. AS-5 945-8

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H6 Fault current limiters

T Verhaege and Y Laumond

H6.0.1 Fault current limiters: why? Electricity transmission and distribution networks have occasionally to withstand line, phase-to-phase or ground short-circuits. One of the major causes of short-circuit is lightning which, by air ionization, ignites an arc. As the current supplied by the network can maintain the arc indefinitely, it is absolutely necessary to break the circuit rapidly. As generators are comparable to voltage sources, the short-circuit current can reach extremely high values, with the following prejudicial consequences for the line, transformers or generators: ( i ) thermal damaging, caused by an intense Joule effect; ( ii ) mechanical damaging, caused by intense electromagnetic forces; ( iii )catastrophic sequence, if the capacities of the circuit-breaking equipment are exceeded. Thermal damage can usually be avoided if the circuit-breaking equipment actually operates after a few half-waves, but the circuit-breaking equipment does not prevent mechanical damage which may result from the first half-waves. A device limiting the short-circuit current might then be needed to eliminate any risk of mechanical damaging and/or to avoid a failure by overload of the circuit-breaking equipment. All the equipment present in existing networks (generators, transformers, circuit-breakers, lines…) is designed and dimensioned in order to withstand a given maximum short-circuit current. Although actual short-circuit currents nearly reach the maximum values there still exists a pressing demand for higher powers to be conveyed and an increased interconnection between networks (as explained below). This demand generally cannot be satisfied, as it would make the short-circuit currents unacceptable. The interconnection between networks is used to increase the stability of the production-consumption balance. However, this may result in many sources supplying the same short-circuit. The remote sources have little effect because of the line inductance, but the situation becomes particularly critical if several neighbouring power plants supply a nearby conurbation. The limitation of the fault current is therefore a means to allow increasing network interconnection and to guarantee voltage stability. It represents a considerable saving in the case where it allows the existing lines to be operated under conditions for which they were not designed. It is also worth pointing out that short-circuit currents can, in the most unfavourable circumstances, reach or exceed the breaking power of the best existing circuit breakers, and this proves the major interest of a system capable of reducing them. Copyright © 1998 IOP Publishing Ltd

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Primarily short-circuit current limitation is of interest to electricity producers and distributors as revealed by the efforts they devote to the subject, but it is also of interest to industry for the protection of production equipment and to manufacturers and users of electrical equipment, because the short-circuit conditions are decisive for the design, dimensioning and cost of the equipment and electrical machines are usually over-dimensioned with respect to normal operation because of the accidental short-circuit conditions. H6.0.2 Fault current limiters: why superconducting? There are different means of mastering fault currents, each having advantages and drawbacks. ( i ) Ultra-rapid circuit-breakers (10 to 20 ms) allow the thermal effects due to over-currents to be reduced but cannot prevent risks of mechanical damage; they are themselves subject to certain limitations concerning their breaking power. ( ii ) Fuses, which present a thermal inertia, have the same drawbacks in addition to their mono-blow feature requiring manual replacement. ( iii )Static (electronic) circuit-breakers are operational for medium voltage applications; their use for high voltages is theoretically possible, but expensive and not 100% secure, because of the large number of components that have to be associated in series and because of the dependence on a triggering system. ( iv )Reduction of the network interconnection degree actually allows fault currents to be decreased but to the detriment of the stability of the production-consumption balance. ( v ) Generator and transformer leakage inductance can be increased, or complemented by series reactances; the electricity distributors do not favour this solution, because of the active power consumed and the subsequent voltage drop which varies with the load: compensation using capacitances is possible, but particularly costly. ( vi )The use of an inductance with an iron core saturated in the on-state and unsaturated in the off-state is mentioned in section H6.0.5. This solution makes it possible to combine criteria of low impedance in the on-state and high impedance in the off-state. A superconductor can be used to saturate the iron core, in order to minimize losses. The major drawbacks of this solution are the excessive size and cost of the device. ( vii ) The last solution to be considered is the use of superconductor quenching: superconductors have a high commutation power, bound to the great contrast between their on-state (or superconducting) and their off-state (nonsuperconducting and therefore resistive). The two quantities which determine their commutation power P ( W m−1 ) are their current density Jon (A m−2 ) accepted in the on-state and their resistivity ρo f f (Ω m) in the off-state, according to: P = ρ o f f Jo n2 . Passage to the resistive state can be obtained by different means: ( i ) a temperature rise beyond the critical temperature Tc, obtained by heating devices, for heat-controlled switches or valves ( ii ) the magnetic field increased beyond the critical field Bc 1 (type I) or BC 2 (type II), for magnetically controlled valves ( iii )the current increased beyond the critical current IC in the case we are dealing with here: the current limiters. The expression ‘resistive limiter’ (see section H6.0.6) is used when the superconductor is directly installed in series on the line to be protected; ‘inductive limiter’ (see section H6.0.7) is traditionally used when the superconductor is associated with an inductance electrically or magnetic-to-magnetic coupling: quenching then imposes the change-over from a low-inductance scheme to a high-inductance scheme,

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where the developed resistance unbalances the current distribution, and thus generates important flux variations. The superconducting current limiters are currently the only ones capable of associating an on-state of very low impedance to a rigorous and almost instantaneous limitation. The technology of this type of device still requires further development and techno-economic studies, in order to estimate the savings which can be made in the future. H6.0.3 Low-Tc or high-Tc superconductors? The choice of superconducting material used in a current limiter is highly dependent on the fixed time limits and on the type of structure chosen. ( i ) Conductors using low-Tc superconductors are already operational for applications such as in current limiters, which require low losses at 50 or 60 Hz and a highly resistive matrix: strands with niobium-titanium submicrometre filaments in a cupro-nickel matrix are produced industrially: they can withstand the 50 Hz operation with losses of about 10−5 W A−1 m−1 and have a resistivity of 4 × 10−7 Ω m in the normal state. Their quenching is accompanied by a rapid temperature rise, which makes them irreversible, and by protection problems for the conductor, which are, however, overcome quite well. Their use does, however, necessitate a low-temperature cryogenic environment (liquid helium at 4.2 K). ( ii ) Conductors with high-Tc superconductors (YBCO, BSCCO, etc) have not reached the same degree of development and can therefore only be considered for projects of industrial current limiters in a more distant future. Their major advantages are simplified cryogenics (liquid nitrogen at 77 K) and better stability. ( iii )The performances (resistivity, critical current density) obtained using small high-Tc samples are sufficient to be applied to the current limiter, the main difficulty being to reproduce these performances on industrial conductors. In particular, the use of long conductors raises problems because resorting to a metal sheath of low resistivity (e.g. silver) would compromise the limiting effect. The use of massive conductors (small length and large cross-section) seems more likely. The required length is defined in the resistive concept, by the ratio Vn /ρo f f Jo n of the rated voltage to the resistivity in the on-state. In the state of the art, hecto- or kilometric lengths are usually necessary, and this is incompatible with the use of massive conductors. This difficulty can be bypassed in resistive-inductive concepts (see section H6.0.7), which use a transformer effect to make the superconductor operate under low voltage and high current. Nevertheless, we will see in section H6.0.7 that the size and the weight of such limiters are penalizing. H6.0.4 Typical specification The efficiency of the current limiter can be measured according to the following criteria: ( i ) it must have, in normal operation, sufficiently low active and reactive loss levels (a few per thousand maximum for active loss, a few per cent for reactive loss); ( ii ) it must accept without quenching various overloads that may occur in normal operation, e.g. 50% overload for 20 min, 3-4 In current peaks on transformer closing, etc: under these very rare conditions, a high loss level can be tolerated; ( iii )it must avoid, whatever the external conditions, any current exceeding a certain threshold (e.g. 5 In ), which is far below the short-circuit current obtained without limiter;

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( iv )it must recover the on-state as quickly as possible after the fault has disappeared; in some cases, this may require a few tenths of a second, which is a hindrance leading to the use of two independent superconducting windings; ( v ) it must have a small size, a small weight and a moderate cost in comparison with the savings brought by the limiter on the network; ( vi )it must have extremely low cold losses: cryogenic machines typically consume 500 electric watts per watt at 4.2 K (liquid helium), or 10 electric watts per watt at 77 K (liquid nitrogen). Therefore, cold losses can hardly exceed ≈2 ×10−6 p.u. for a low-Tc superconducting limiter and 10−4 p.u. for a high-Tc limiter; ( vii )it must be highly reliable for minimum maintenance, which necessitates, in particular, extensive automation of the detection functions, circuit-breaker control and cryogenic supply. H6.0.5 The saturated iron core concept A current limiter with a saturated iron core, unlike the other configurations, does not use the superconductor quenching to create the off-state. The superconductor is used here for its ability to supply permanent ampere turns, almost without losses. The principle is shown in figure H6.0.1.

Figure H6.0.1. The saturated iron core concept.

Figure H6.0.2. (B-H): the evolution in a saturated iron core fault current limiter.

An iron core is saturated-at least in the on-state-by a superconducting winding supplied with direct current. The a.c. line on which the limitation operates, comprises a conventional winding (copper) around the iron core. The operating mode is shown in figure H6.0.2: the curve ( B-H ) of the iron core is schematized by a nonsaturated zone, where B is proportional to H, and a saturated zone, where B is independent of H. In the on-state, the a.c. amplitude is insufficient to take the iron core out of its saturated state (AA’ )- As a first approximation, the assembly behaves as if there were no iron core: flux variations are low, so that the voltage drop in the a.c. winding is low, and the superconducting winding keeps its d.c. without a.c. loss. When the a.c. amplitude is sufficient to unsaturate the iron core in a part of the cycle (sector BS of the BB’ evolution), the inductive voltages appear (in BS); these voltages are able to slow the evolution of the current and then to reduce its peak value. The limitation can be compared to that of a reactance made up of the transformer primary and magnetic core, provided that the secondary current is kept constant. In fact, the magnetic coupling tends to increase the secondary current, which reduces the apparent inductance of the primary and thus restricts the limitation. Quenching can also be feared from this rapid increase in the secondary current, if the conductor works close to its critical current and/or does not withstand the high rate of change of the magnetic fields. It is therefore necessary to reduce the magnetic coupling, for example by connecting an annexe superconducting winding in series with the secondary or by exploiting the natural inductance of the d.c. supply.

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The system is made heavier because of the need to associate two similar components, saturated by opposite fields +Hd c and −Hrd c , in series, because each of these components has a limiting effect on one half-wave only. The advantages of the saturated iron core current limiter are relative simplicity and operating conditions which do not stress the superconductor (d.c. operation without quenching) too much. However, its major drawbacks are its large dimensions and weight which are penalizing for commercial applications. H6.0.6 The resistive concept H6.0.6.1 Definition Resistive current limitation uses the great difference, for certain superconducting conductors, between an on-state with high critical current density ( Jc (A m−2 )) and an off-state with high resistivity ρo f f (Ω m). Quenching from the on-state to the off-state is obtained when the conveyed current density J ≥ Jc . Quenching is strongly irreversible when the product ρoffJc2 is high, because it is then accompanied by a rapid temperature increase. In the contrary case, quenching can be partial and reversible, the energy dissipated being then discharged by the cryogenic fluid, without excessive temperature rise; in this case, the limitation is bound to the conductor current-voltage characteristic beyond the critical current, at the helium-bath temperature (figure H6.0.3).

Figure H6.0.3. Typical irreversible (1) and reversible (2) current limitations.

In its simplest concept, the resistive limitation is obtained by mounting the superconducting conductor in series on the line to be protected. When the necessary conductor length justifies it, the conductor can be wound, usually in the ‘noninductive’ mode, which minimizes the reactive loss in the on-state. In order to protect the limiter, a rapid circuit-breaker must usually be added. Thanks to the limiter, the breaking power of the circuit-breaker can be reduced to a moderate value, much lower than the theoretical short-circuit current. Finally, the electrical scheme can include, in parallel with the superconductor, an impedance Zs h u n t whose function and definition are explained in section H6.0.6.3. H6.0.6.2 Conductor dimensions Section S of the conductor is determined by the required quench current Iq and the critical current density Jc , according to S = Iq /Jc . In order to limit the current below a specified threshold I < Im a x , the conductor length I must exceed

Therefore, the necessary conductor volume is larger than or equal to

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A high device compactness is therefore synonymous with a high ρo f f Jc2 product. However, the very high values of the product ρo f f Jc2 raise a problem of conductor protection: the high power density dissipated by the Joule effect cannot be discharged by the cryogenic fluid, and causes rapid temperature rise in the conductor. A fast circuit-breaker (typical breaking time: 20 ms) is used in order to limit the temperature to an acceptable level; when this is not sufficient, the conductor length is extended above lm i n , in order to increase the off-state resistance and thus reduce the residual current and the dissipated power. When this criterion is applied, i.e. when the product ρo f f Jc2 approaches or exceeds 1010 W m−3, the necessary conductor length is defined by

where ∆t( s ) is the breaking time of the circuit-breaker and Qm a x (J m −3) is the maximum acceptable value of the dissipated power density ( Qm a x can vary from 108 to 109 J m−3, according to the conditions). These rules regarding dimensions usually lead to lengths of the order of hectometres or kilometres, thus imposing a conductor arrangement in a multilayer coil. The directions of rotation of the different layers can be alternated in order to reduce the magnetic field and the reactive losses in the on-state. H6.0.6.3 Shunt impedance The Zs h u n t impedance (see figure H6.0.4) can become necessary when the superconducting winding impedance is higher than the normal load of the network in the off-state. This arrangement makes it possible to reduce the current Is u p of this winding during the limitation to a value significantly lower than the quench current lq , in order to limit the power dissipated in the cryogenic environment and the conductor temperature rise. The function of the ZS h u n t impedance is then to avoid over-voltages at the limiter terminals during quenching and, if such are the specifications, to allow passage of the rated current during the limitation period (i.e. ZS h u n t + Is u p = In ) The result is obtained by choosing ZS h u n t such that

if Rm a x is the resistance of the winding after complete quenching. The Zs h u n t impedance can be produced by a resistor or an arrester. The use of an inductance raises problems, because the impedance obtained is then dependent on the rapidity of the transient phenomenon for passing to the off-state. The ZS h u n t impedance, which has only to operate during short periods of limitation before breaking, is pulsatile and of short dimensions. In the on-state, the terminal voltage of ZS h u n t is very low, and its consumption can be ignored.

Figure H6.0.4. The basic scheme of the resistive fault current limiter. Copyright © 1998 IOP Publishing Ltd

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H6.0.6.4 Losses and cryogenic requirements The reactive losses of the resistive limiter can be reduced to a very low level (0.1-1%) if the winding is designed so as to have a very low inductance. This result can be reached with a multilayer winding. This very-low-inductance arrangement is also useful to minimize the eddy current losses in the metal structure elements (current leads, etc) and in the wall (if also metallic) of the cryostat. The very-low-inductance arrangement also minimizes the magnetic field to which the superconductor is subjected, and therefore the a.c. losses. The ideal case, which is never exactly reached, is the restriction of these losses to the self-field losses of the superconducting strand. These losses can be decreased to approximately 10−7 the line power if low-Tc superconducting strands, optimized for the 50 Hz operation, are used; this corresponds to a cryogenic cost (electricity consumption of the liquefier) lower than 10∼4 the line power. This result is obtained because of the very high product ρo f f Jc2 (which reduces the necessary superconductor volume) and the use of transposed thin strands. The cryogenic balance would be less favourable for high-Tc superconductors at their present performance level, in spite of the higher operating temperature. The resistive limiter also shows active losses and cryogenic needs at the current leads, i.e. the conductors which ensure the changeover from ambient temperature to cryogenic temperature. Current leads for a.c. operation are not commonly used. With respect to the leads for d.c. or slightly variable operation, additional precautions must be taken to minimize the consequences of skin effect and eddy currents. It is theoretically possible to approach the performance of d.c. leads shown in table H6.0.1. For a low-Tc current limiter operating at liquid helium temperature (4.2 K), the use of high-Tc . leads would allow a substantial reduction of the cryogenic requirements: the thermal transition from 4.2 K to ≈77 K (liquid nitrogen) is ensured by a bulk high-Tc superconductor, and a metallic conductor ensures the complementary transition from 77 K to 300 K (see chapter D10).

The reference for a classical 1 kA metallic d.c. lead is a heat deposition of ≈1 W at 4.2 K, which evaporates 1.4 1 h−1 of helium, the vapour of which still extract ≈75 W from the lead between 4.2 K and 300 K; this heat extraction precisely compensates the Joule losses of the lead. In the case of a high-Tc lead, helium consumption is practically negligible, and the cooling of the metallic part of the lead is ensured by nitrogen which is considerably easier to liquefy than helium. The consumptions, which are proportional to the rated current and independent of the voltage, become prohibitive for low-voltage applications, for which the resistive-inductive limiter should be preferred to the purely resistive limiter. They are acceptable beyond a few kV, and secondary for high-voltage applications. To sum up, the electric and cryogenic consumption in the on-state can be, for some applications, reduced to a very moderate level. The cryogenic consumption during a limitation is always important, but its economic impact is low as it does not happen frequently. H6.0.6.5 Protection The superconductor quenching generates rapid temperature rises and intense electric fields by the Joule effect. Therefore, there are risks of damage by conductor overheating and/or dielectric breakdown. The

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mastery of these risks is all the more delicate as the rated voltage is higher. Active protection consists of ensuring a rapid detection of any quenching, whether spontaneous or caused by a short-circuit, so as to order the opening of a rapid circuit-breaker. If the total quenching is a detectable phenomenon, a local incipient quenching occurring in a small part of the winding with little measurable effect at the limiter terminals cannot, nevertheless, be excluded. A highly sensitive detection system is therefore necessary. However, the response time of the existing circuit-breakers (=20 ms) makes thermal damaging of the conductor possible, if the product ρo f f JC2 of the latter is particularly high. It is then necessary to complete the active protection with a type of passive protection, which ensures rapid current decrease after any incipient quenching. To do this the resistance of the winding after complete quenching, dimensioned so as to sufficiently reduce the residual current, is used. Any local incipient quenching must therefore be transformed rapidly into a total quenching. In particular, such a result can be reached by using assembled conductors including some nonsuperconducting strands with an appropriate definition (Verhaege et al 1991), ensuring a rapid propagation followed by a mass quenching of the winding. The current limiter in the on-state is exposed to the phase voltage, and must be correctly insulated from the ground. Dielectric constraints are more severe during the limitations, when important electric fields can appear locally, in the current leads and ends of the winding layers in particular. The transitory period during which the coil is partially quenched is particularly critical, because of the inhomogeneous distribution of the resistive voltage drops. H6.0.6.6 Recovery For a resistive limiter of high power, it is difficult to reduce the on-state recovery time to less than a few seconds, given the great amount of energy released. We will see in section H6.0.7 that the same holds true for resistive-inductive limiters. Two limiters must be used when the specifications require a rapid re-connection. H6.0.7 The resistive-inductive concept H6.0.7.1 Definition The term ‘inductive limiter’ is often used for designating various structures of limiter involving reactances for the limitation. Most of these structures are, however, based on the quenching of a superconductor, i.e. its passage to the resistive state. In spite of the diversity of the proposed structure, it can be considered that they have the following common features: ( i ) the on-state is obtained by an electrical scheme with low equivalent inductance, assuming current flows through a superconducting component; ( ii ) the off-state is obtained from the previous scheme which shows a high inductance as soon as a sufficient resistance appears in the superconducting component. It is possible, by the transformer effect, to modify the current-voltage couple which must be supported by the superconductor, without modifying the commutation power required from it. Maintaining the commutation power, which is always resistive, therefore justifies the ‘resistive-inductive structure’ label. We will briefly describe here a few types of resistive-inductive structure, knowing that they have many problems in common with the resistive structure, which has already been tackled in section 6.0.6 (figure H6.0.5).

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Figure H6.0.5. Association of a resistive limiter and a shunt inductance.

The scheme shown in figure H6.0.5 connects in parallel the low-inductance superconductor with a normal or superconducting inductance. The latter does not take part in the limitation directly, but allows, if such are the specifications, the limited current to be kept after breaking of the branch including the superconductor. It is not very efficient to limit over-voltages at the limiter terminals, given the high impedance of an inductance towards transient phenomena. The dimensions of the triggering superconducting winding of the resistive-inductive limiter are the same as those of the resistive limiter, but the dimensions and cost of the inductance are high, in comparison with the advantages in performance. H6.0.7.2 A mixed transformer The scheme in figure H6.0.6 shows the limitation principle using a superconducting secondary transformer in short-circuit: in the on-state, the secondary masks the magnetic flux variations, so much so that the apparent inductance at the primary is low. When the primary and secondary currents reach a certain threshold, the secondary quenches and no longer masks the flux variations, so much so that the apparent inductance at the primary becomes high and ensures limitation.

Figure H6.0.6. A mixed transformer limiter.

Versions with or without iron core can be designed. Using a transformer with a normal primary and a superconducting secondary presents certain advantages. ( i ) There are no more current leads, which results in a technological simplification and a cryogenic saving which is significant in the case of low-voltage applications. ( ii ) It is possible to choose the transformation ratio so as to meet the characteristics of the superconductor available. In particular, the number of secondary turns can be reduced to unity, in order to use a high-Tc ceramic in the form of a solid tube, which is easier to manufacture than flexible conductors. ( iii )It is theoretically possible to lighten considerably the dielectric strength problems in a cryogenic environment, including the case where there are more turns at the secondary than at the primary. In fact, as the secondary is closed up on itself it is not exposed to any voltage, even in the limitation phase: the resistive and inductive voltage sources perfectly make up for one another according to R 2 I 2 + L 2 dI 2 /dt + MdI l /dt = 0 where I1, I2 are the primary and secondary currents, R1 is the resistance of the quenched secondary, L2 is the inductance of the secondary and M is the primary-secondary mutual inductance. Copyright © 1998 IOP Publishing Ltd

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So that this overall balance is repeated locally, thus avoiding excessive electrical fields appearing inside the winding, the whole conductor length should be uniformly exposed to the same magnetic field conditions and quench. Such conditions are never reached exactly, but can be approached in the case of a toroidal winding. The major disadvantage of the transformer limiter, which is the same for all the inductive versions, concerns its size and its weight: the inductive limitation is obtained by an inductance capable of absorbing a comparable power to that of the network; it comprises a copper winding, which has to withstand the rated current permanently and the limited current in pulsatile operation. Using an iron core does not greatly induce the size of the windings, for two reasons: ( i ) the length of the secondary conductor must be sufficient to obtain a sufficient resistance in the quenched state; ( ii ) the voltage per turn is limited by the saturation field of iron, which imposes the turn number. The resulting primary winding, when operating in fault conditions (i.e. three to five times the rated current, without flux compensation by the secondary), completely saturates the iron core, so that the magnetic field produced is moderately higher than it would be in air. The version without an iron core can be considered, especially in the case of a low-Tc current limiter, where the core losses must be diverted from the cryogenic area with technological complications. Nevertheless, the necessary amount of copper is rather high (e.g. 3 t for 45 MV A). The second disadvantage is a permanent resistive load on the line and therefore heat dissipation which will have to be discharged. Finally, another disadvantage is the complexity of the cryostat: the cryostat separates the primary from the secondary and is subject to a variable magnetic field. As a metallic cryostat would generate unacceptable losses, a composite cryostat must be used, but its technology is much more delicate. In addition, the thermal insulation distances cause poor magnetic coupling between the primary and the secondary and this results in high reactive losses in the on-state (a few per cent). There are no significant differences to the resistive version with respect to the superconductor volume, the energy dissipated or the recovery time after a limitation. H6.0.7.3 A superconducting transformer As a variant of the preceding scheme (figure H6.0.6), it is possible to make the transformer primary superconducting, with the following advantages: ( i ) increased compactness because of high current densities; ( ii ) better coupling between the primary and the secondary; ( iii )no more copper primary losses. Nevertheless, current leads are necessary as in the resistive version. With respect to the resistive current limiter, the advantage of the superconducting transformer is to allow the limited current to flow continuously in the superconducting primary, thus avoiding the circuit-breaking in the case of a nonpermanent fault. The disadvantages of a superconducting transformer are that the dimensions are much more important, and it has increased active and reactive losses and increased cryogenic needs. H6.0.7.4 Other variants Other variants of resistive-inductive limiters have been examined or proposed by various authors (Tixador et al 1992). In particular, a slightly more complex scheme (figure H6.0.7) has been proposed, comprising a nonsuperconducting transformer, whose primary and secondary are connected in parallel. A superconductor is mounted in series on the secondary.

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Conclusions

Figure H6.0.7. A variant resistive-inductive limiter.

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version

of

a

Figure H6.0.8. Another variant version of a resistive-inductive limiter.

Given the transformation ratio for the example shown in figure H6.0.7, the superconductor operates with low current. On the other hand, it must produce a high resistive voltage in order to ensure efficient limitation. The use of a saturable transformer allows this voltage to be reduced to an acceptable level. The dimensions required for the system are penalizing. Another variant (figure H6.0.8) involves coupling the three phases of a three-phase network on the same iron core (Shimizu et al 1992). This device is not only a limiter. It tends to balance the currents of the three phases, at the cost of a certain imbalance of voltages. In particular, the fault currents between one phase and the ground are limited without superconductor quenching. H6.0.8 Conclusions The superconducting current limiter may constitute a key element in the protection of electrical equipment, in the increase of power conveyed on existing networks and in the extension of the degree of network interconnection and service quality. Apart from rare exceptions, the phenomenon exploited is the passage from the superconducting state to the resistive state, obtained when the current reaches a certain threshold. It is used in various ways, according to the structure proposed. The basic version remains the ‘resistive limiter’ which is by far the most compact. Although many laboratory demonstrations have already been carried out, significant progress must still be made to reach the objective of industrial products. Further reading Damstra G C, Greenwood A, Sabot A and Schramm H H Superconducting technology for current limiters and switchgear CIGRE Fleishman L S, Bashkirov Yu A, Munger H, Brissette Y and Cave J R 1992 Design considerations for an inductive high Tc superconducting fault current limiter IEEE Trans. Appl. Supercond. AS-3 570 Gray K E and Fowler D E 1978 A superconducting fault-current limiter J. Appl. Phys. 49 Ikegami T, Yamagata Y, Ebihara K and Nakajima H 1992b Application of high-Tc superconductors to current limiting devices IEEE Trans. Appl. Supercond. AS-3 566 Ito D, Tsurunaga K, Tada T, Hara T, Ohkuma T and Yamoto T 1992a Development of 6.6 kV/1.5 kA-class superconducting fault current limiter ICEC (Kiev, 1992) (London: Butterworths) Ito D, Yoneda E, Fujioka T and Tsurunaga K 1990 Development of superconducting a.c. fault current limiter Adv. Cryogen. Eng. 35 653 Lindmayer M and Schubert M 1992 Resistive fault current limiters with HTSC-measurements and simulation IEEE Trans. Appl. Supercond. AS-3 884 Meerovich V, Sokolovsky V, Jung G and Goren S 1992 Development of high-Tc superconducting inductive current limiters for power systems IEEE Trans. Magn. MAG-28

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Pham V D, Collet M, Laumond Y, Bekhaled M, Verhaege T and Fevrier A Limitation du courant par transition supraconductrice CIGRE 90 Raju B P, Parton K C and Bartram T C 1982 Fault current limiting reactor with superconducting d.c. bias winding CIGRE 82 Shimuzu S, Tsukamoto O, Ishigohka T, Uriu Y and Ninomiya A 1992 Equivalent circuit and leakage reactances of superconducting 3-phase fault current limiter IEEE Trans. Appl. Supercond. AS-3 578 Slade P G et al 1992 The utility requirements for a distribution fault current limiter IEEE Trans. Power Delivery PD-7 507 Thuries E, Pham V D, Laumond Y, Verhaege T, Fevrier A, Collet M and Bekhaled M 1990 Towards the superconducting fault current limited IEEE 1990 - 90SM 412–7 PWRD Tixador P, Brunei Y, Leveque J and Pham V D 1992 Hybrid superconducting a.c. fault current limiter principle and previous studies IEEE Trans. Magn. MAG-28 Verhaege T, Agnoux C, Tavergnier J P, Lacaze A and Collet M 1991 Protection of superconducting a.c. windings IEEE Trans. Magn. M-28 751 Verhaege T, Cottevieille C, Estop P, Quemener M, Tavergnier J P, Bekhaled M, Bencharab C, Bonnet P, Laumond Y, Pham V D, Poumarede C and Therond P G 1996 Experiments with a high voltage (40 kV) superconducting fault current limiter Cryogenics 36 Verhaege T, Tavergnier J P, Agnoux C, Cottevieille C, Laumond Y, Bekhaled M, Bonnet P, Collet M and Pham V D 1992 Experimental 7.2 kVr m s /1 kAr m s /3 kAp e a k current limiter system IEEE Trans. Appl. Supercond. AS-3 574 Verhaege T, Tavergnier J P, Février A, Laumond Y, Bekhaled M, Collet M and Pham V D 1989 25 kV superconducting fault current limiter MT-11 (Tsukuba, 1989) Yoneda E S, Tasaki K, Yazawa T, Maeda H, Matsuzaki J, Tsurunaga K, Tada T, Fujisawa A, Ito D, Hara T, Nakade M and Ookuma T 1994 The current status of superconducting fault current limiter development Cryogenics 34 (ICEC suppl.)

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Energy storage

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H7.1 Small and fast-acting SMES systems H W Lorenzen, U Brammer, M Harke and F Rosenbauer

Since the last century, which created revolutionary developments in the field of the generation and application of electrical energy, the use of electrical energy has rapidly grown, and this even though electrical energy has one disadvantage: methods of storing it are very limited. So in general there is the problem of guaranteeing the power supply even if the demand varies strongly. Load fluctuations in the range of minutes, hours or even longer may be balanced by means of conventional technology. However, fast alterations in the load or short circuits in the power network can cause voltage sags with a duration of some 100 ms to seconds. In former times this did not matter much, but nowadays many modern plants are based on voltage sensitive machinery (e.g. semiconductor, paper, rubber and plastics industries). Voltage drops may entail an interruption of production, resulting in an expensive loss. It is estimated that in the USA the annual costs of such voltage sags amount to some billions of dollars. In order to compensate for these temporary bottlenecks in power supply an energy storage system offering a high power and a very short access time is expedient. One alternative is superconducting magnetic energy storage (SMES) which provides a very high power despite its comparatively low storage capacity. In principle SMES can be regarded as a pulse magnet. The magnet consists of one or several coils wound from a superconductor wire. The energy is stored in the magnetic field produced by the superconducting current. Since superconductors show no d.c. resistance, for the SMES at rest no ohmic losses occur in the superconducting cable (even with maximum operation current). In the past the use of SMES has been considered in a poor light by power suppliers. The reasons were probably the high effort of cooling and a traditional opposition to new technologies. Thus, apart from SMES for research purposes, very few SMES systems have been installed so far. Since the beginning of the 1990s the situation has been changing gradually. The rising industrial demand for a high-quality power supply has resulted in a growing interest in SMES. Indeed, in the future an increasing number of SMES systems will be installed in utility grids or near customer sites. Further important developments will be initiated by superconducting cables made of high-temperature superconductors. So SMES may be viewed as a technology with a high future potential.

H7.1.1 SMES in comparison to other energy storage Table H7.1.1 compares characteristic qualities of different types of energy storage. The storage types mentioned are electrical storage (capacitors), hydraulic storage (pumped storage plants), magnetic storage (coils) and electrochemical storage (batteries). The comparison of the storage types is based on the energy density, the relative specific costs (reference values are the costs for a medium-sized SMES system), relative access time (with respect to electrochemical storage) and critical or limiting quantities. From table H7.1.1 the advantages of small SMES systems become evident. Within the storage systems that reveal a short relative access time they have the lowest price. The cycle efficiency ηc y c of magnetic Copyright © 1998 IOP Publishing Ltd

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energy storage

where ∆Wm is the energy exchange, Pv the losses, tc y c the cycle time or cycle period, shows that small, fast-acting SMES systems are particularly suitable for pulsed operation, e.g. to level the load noise in power supply systems. In addition, according to equation (H7.1.1) a small, fast-acting SMES system can represent the fast-acting component of a hybrid storage system that consists of two (or more) different types of energy storage. Yet the high costs of SMES systems are a considerable disadvantage. H7.1.2 Aspects for the design of small, fast-acting SMES systems In order to guarantee low losses and thus a high cycle efficiency, the coils of small, fast-acting magnetic energy storage systems have to be made of superconducting material. As a result of this a series of factors for the conception, design and construction of SMES can be deduced. By assuming that the coils are fabricated from low-temperature superconductors four points are important. ( i ) Cooling of the storage coils: •

since the superconductors applicable show very low critical temperatures, only liquid or supercritical helium is suitable as coolant;

• due to thermal losses the coils have to be placed in a vacuum vessel; • for economical reasons a thermal shield cooled by gaseous helium or liquid nitrogen is

favourable;

• in order to reduce the thermal losses the current leads may be made of a high-temperature

superconductor;

• the kind of cooling (direct or indirect) has to be chosen with regard to the thermal heat

transfer by conduction.

( ii ) Optimum coil design: • • • • • •

minimum amount of superconducting material; maximum storage capacity; minimum surface of the coils; minimum a.c. losses; high cycle efficiency; low magnetic stray field.

( iii )Choice of the appropriate superconducting cable: • multifilament wire to guarantee low a.c. losses; • sufficient (copper) matrix for reasons of stability; • maximum current with respect to the converter.

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( iv )Suitable quench protection: • appropriate operation of the SMES to avoid quenches as far as possible; • in case of a quench, fast discharge of the SMES by means of an external resistor; • if necessary, bypass of the quenching coil.

The above aspects, which may be individually stressed according to the specific application, govern the optimum and most economical design of SMES. The design of SMES is discussed in detail in the next subsection. The different ways of handling a quench are analysed in a subsequent subsection. H7.1.3 Design of small, fast-acting SMES plants This subsection deals with the design of small, fast-acting SMES units for which eddy currents and superconductor losses are a severe problem. Emphasis is laid on the electromagnetic compatibility of SMES because this subject will strongly influence the public acceptance of SMES plants in densely populated regions. Finally, two SMES designs are discussed to illustrate the main conclusions derived in this subsection. Components of an SMES plant Figure H7.1.1 shows the main components of an SMES plant: • • • •

SMES refrigeration system power conversion system (PCS) control system.

Figure H7.1.1. Components of an SMES plant.

The SMES plant consists of one or several coils that are contained in a cryostat (or dewar). In general the coils are electrically connected in series. The current leads are the interface between the superconducting coils and the PCS outside the cryostat. The PCS transfers the energy from the power system or grid to the coils and vice versa. In section n H7.1.4 the PCS for SMES plants is briefly described. For a detailed discussion please refer to the literature about power converter technology (e.g. Mohan et al 1990). The refrigeration system cools down the coils and keeps them at the operating temperature. Usually liquid helium serves as coolant. Fast load cycles generate high a.c. losses in the coils which require a high refrigeration capacity. Thus the cycle efficiency of a fast SMES system is mainly determined by the power consumption of the cryogenic plant. Refrigeration systems are thoroughly explained in chapter D4. Copyright © 1998 IOP Publishing Ltd

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The control system supervises the whole SMES plant. It has to operate the unit in a long-range ‘intelligent’ way so that quenches are avoided. Furthermore it controls the energy flow via the converter and thus the efficiency of the SMES unit with respect to the specific application. H7.1.3.1 SMES design The design of SMES may be divided into the steps listed below: ( i ) specification of the maximum stored energy Em a x and the maximum power Pm a x ; ( ii ) specification of the constraints (e.g. maximum current and voltage of the converter, maximum spatial dimensions of the coils); ( iii )selection of the appropriate superconducting cable and the cooling technology (e.g. a.c. or d.c. cable, direct or indirect cooling, kind of quench protection); ( iv )choice of the basic SMES type (solenoid or torus); ( v ) optimal coil design; ( vi )development of the PCS and the control system. The design of the single components of an SMES unit cannot be regarded as independent problems (as the above list might indicate). Each step of the design influences several components, so often the SMES design is a kind of iterative process. The starting point of SMES design is the specification of Em a x and Pm a x . The goal is to find a design which meets these conditions. This problem has no unique solution and, in general, there is no design which is ‘best’ for all purposes. Thus it is reasonable to optimize the SMES design with respect to the specific application. Some common optimization criteria are cost, reliability (even in the case of a quench) and environmental compatibility. The latter two items may be viewed from the point of view of economics as well. Costs can be divided into costs for the design, the construction and the installation of the SMES plant and costs for the operation. For small fast SMES units the cycle efficiency and thus the operational cost are mainly due to the a.c. losses. Therefore the minimization of the a.c. losses will be a major aspect in the following discussion. Note that SMES plants with a large storage capacity usually have far longer charge and discharge times, so the a.c. losses are of no special concern which may result in a completely different design from that for small and fast units. The a.c. losses are greatly reduced by using an a.c. superconducting cable instead of a d.c. one. The selection of the appropriate a.c. superconducting strand together with the cooling technology is thoroughly explained in chapter B8. The development of a suitable PCS is not discussed here, because it would exceed the scope of this book. For this reason the topic of SMES design is restricted to a comparison of the basic SMES types and the optimal coil design. H7.1.3.2 Basic SMES types SMES systems are designed as solenoids or tori. A brief comparison of both is given in table H7.1.2. For a solenoid the stored energy per unit length of superconductor is roughly twice as high as for a torus. Since the superconductor losses are proportional to the length of the superconducting cable, toroidal SMES systems suffer from considerably higher superconductor losses. However, if the solenoid is pool cooled, the benefit of fewer superconductor losses decreases: the solenoid and the stainless steel helium vessel behave like a transformer and the heat generated by eddy currents is absorbed by the liquid helium†.

† Without pool cooling (e.g. in the case of forced-flow internal superconductor cooling) no inner vessel is needed so eddy currents can only occur in the warm vacuum enclosure.

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A great problem concerning toroidal geometry is caused by the large Lorentz forces acting on the superconductor. Similar to a solenoid the radial forces on each coil of the torus result in a radial tensile stress. In addition there is a net attractive force on the coils towards the centre of the torus. These forces have to be compensated by a mechanical structure. As a result of eddy currents, materials with a high electrical conductivity should be excluded from regions of high magnetic fields. Hence the mechanical structure might be made of glass-fibre reinforced plastic (GRP), GRP, however, involves a considerable risk, because it is brittle at low temperatures. During charging and discharging the forces on the GRP vary, which may result in a rapid fatigue of the mechanical structure. These problems become even worse if a coil is bypassed in the case of a quench (see section H7.1.5). In this situation the symmetry of the magnetic field is destroyed, so that the delicate balance of the forces between neighbouring coils is disturbed. The two coils on both sides of the bypassed coil experience large torques which try to break the toroidal arrangement. This effect requires a further mechanical support. The main advantage of a torus is the very low magnetic stray field compared with a solenoidal SMES of the same storage capacity†. Since there is a growing public interest in the electromagnetic compatibility of technical systems, the toroidal geometry is becoming more and more attractive to SMES designers. There is also no doubt that a solenoid can be shielded effectively. However, as pointed out in another part of this section, shielding a solenoid usually cancels the advantages of this geometry. For large SMES units further advantages of a toroidal construction are the comparatively small coils and the modular design. H7.1.3.3 Optimum coil design The main part of the SMES design is the optimization of the coil geometry. This procedure is inevitably based on a numerical field computation. Some optimization criteria are: ( i ) a small amount of superconducting material to diminish the superconductor losses and the cost of the superconductor; ( ii ) a simple coil geometry to reduce the cost of the coil fabrication; ( iii )a small surface of the SMES which enables choice of a small vacuum vessel and guarantees a low radiation heat transfer to the coils; ( iv )low a.c. losses during rapid charge or discharge; ( v ) small tension stress at the superconducting cable, small forces between the coils of a multicoil arrangement; ( vi )reliable quench protection; ( vii ) low magnetic stray field (although this is primarily determined by the choice of the basic SMES type). It is impossible to optimize the coil geometry with respect to all of the aspects listed above. Furthermore, the design constraints have to be taken into account. The major constraints are Em a x , † Note that if a torus coil is bypassed in the case of a quench, temporarily the leakage of the magnetic field might be quite high.

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Pm a x and the maximum current and voltage of the converter. Besides, the current leads, the electrical insulation of the superconducting winding and the helium (in the case of pool cooling) must yield some voltage limits. As mentioned above, low a.c. losses are essential for small fast SMES systems. Unfortunately the calculation of the superconductor losses strongly depends on the load cycle. It is quite reasonable but nevertheless uncertain whether the optimum coil shape with respect to one specific cycle is the best solution for all cycles possible. Moreover, the equations for the numerical computation of the superconductor losses may be not available for the superconducting cable selected and may be restricted to special applications or may depend on material parameters that are roughly estimated from experimental measurements. As a result of these difficulties it is not recommended to focus too much on the exact calculation of the superconductor losses. It is more expedient to be aware of some general laws: ( i ) the superconductor losses are directly proportional to the length of the superconducting cable; ( ii ) among other things the superconductor losses depend on the time-varying component of the magnetic field. Therefore SMES with a comparatively low magnetic field at the superconducting winding favours low superconductor losses. On the other hand a low magnetic field implies a high current which is limited by the PCS. Another important optimization criterion is the amount of superconductor material necessary to store the energy specified. This aspect also favours low superconductor losses. The criterion can be easily evaluated so that it is frequently used for coil design. In a mathematical sense many coil-shape optimizations are exclusively based on this criterion and treat additional aspects as constraints. Most designers consider coils with a superconducting winding of rectangular cross-section only. Other cross-sections may yield similar or slightly better results (Hassenzahl 1989). Yet these small improvements usually do not justify the far more expensive fabrication of the coils. For this reason the following analysis is restricted to coils with rectangular cross-sections. (a) Solenoids Figure H7.1.2 shows the simplest kind of solenoid, a circular cylindrical winding of rectangular cross-section.

Figure H7.1.2. A simple solenoidal winding.

The geometry of a solenoid is completely described by its inner radius Ri and the ratios α = R0/Rj , β = L/R j . An optimization with respect to the stored energy per unit length of superconductor results in a thin-walled solenoid (α → 1) (Hassenzahl 1989, Wesche 1992). For a 5.5 GWh SMES system a minimum at β = 0.2 was found (Hassenzahl 1989). A more detailed parameter study for a 50 kWh SMES system came to the following conclusions (Wesche 1992). ( i ) For a thin-walled solenoid the β dependence of the winding volume is weak, but values of β < 1 are advantageous. Copyright © 1998 IOP Publishing Ltd

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( ii ) A small magnetic field at the conductor corresponds to a high current density, a large inner radius Ri and a thin winding. ( iii ) Decreasing the current density reduces the maximum hoop stress in the winding. Summing up an optimization with respect to the stored energy per unit length of superconductor results in a thin-walled solenoid that is characterized by a comparatively low energy density, a high current and a high hoop stress. The optimum solenoidal geometry differs markedly from the Brooks coil (α = 2, β = 0.5) which reveals the highest inductance per unit length of conductor (Murgatroyd 1986). This is due to the bad superconductor utilization of the Brooks coil: the magnetic field at the superconducting winding is highly nonuniform and there is a sharp peak that determines the maximum current. Most of the superconductor, however, is operated far below the maximum current allowed by the local field level. In contrast, the optimization with respect to the stored energy per superconductor volume leads to a very high superconductor utilization and a comparatively high field uniformity. Another way of improving the efficiency of the superconductor utilization is a cross-sectional adaptation of the superconductor winding to the local magnetic field (tapped winding). For that purpose the winding is subdivided into a number of concentric sections with each operating at the maximum possible current. This may be done by feeding each section with a different current. It is usually better, however, to wind the sections from wires of different diameters. So they all take the same current and may be connected in series but nevertheless run at different overall current densities (Wilson 1990). (b) Toroids The basic geometry of a torus composed of solenoidal coils is illustrated in figure H7.1.3.

Figure H7.1.3. A torus composed of solenoidal coils.

Toroids are characterized by the toroidicity ε or the aspect ratio A: ε = I/A = (R0 - Ri )/(R0 + Ri ) (Birkner 1993, Komarek 1995). Sometimes the aspect ratio is defined as A = Ri/R0 (Hassenzahl 1989). The two different definitions of A can be distinguished easily, because the first one leads to values A > 1 and the second one to A < 1. An idealized torus is obtained by rotating the coil shape around the z axis whereas any real toroidal SMES system is composed of several coils. For simplicity the optimal coil shape is usually deduced from an idealized torus. This is reasonable, if the spatial distance between the coils is small compared with the inner radius Ri Based on this approach the optimization with respect to the energy stored per superconductor volume results in a shape called the ‘Shafranov D-shape’ (Shafranov 1973). Searching

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for a shape that guarantees a constant tension in the superconducting cable leads to a shape called the ‘Princeton D-shape’ (File et al 1971). Both D-shapes are equivalent. This means that the mechanical optimum corresponds to the maximum magnetic energy of a torus coil. It is also possible to find a shape that minimizes the surface and thus the radiation heat input (Shafranov 1973). Based on a fixed coil shape the toroidicity may be optimized. In contrast to solenoidal SMES the results are far more dependent on the storage capacity. For a 5.5 GWh SMES system with a circular coil shape an optimization with respect to the stored energy per unit volume of superconductor yields a broad maximum at A = 0.6 and Ri = 0.6RO respectively (Hassenzahl 1989). An analysis that is restricted to small SMES is given by Birkner (1993). This time the optimization leads to broad maxima at ε = 0.68 (Shafranov D) and ε = 0.60 (circle), which is equivalent to Ri = 0.25RO in the latter case. By referring to the circular shape the maximum energy stored by the Shafranov D exceeds the value of the circular shape by 18%. Yet it has to be carefully considered whether the benefits of D-shaped coils justify the higher cost for the fabrication. Both optima show a low magnetic field and a high current which is typical for SMES with minimal material input and low a.c. losses. Similar to a solenoid, the disadvantages of the optima are low energy density, high current and high mechanical tension in the cable. This does not matter much as long as the stored energy is small.

H7.1.3.4 Electromagnetic compatibility (a) Threshold values for humans in electromagnetic fields In several countries the lowest threshold values for humans in electromagnetic fields apply to people with heart-pacemakers. A heart-pacemaker may be affected by the magnetic stray field of an SMES system in two ways: ( i ) the d.c. component may activate a reed switch of the device; ( ii ) the a.c. component induces a voltage in the pacemaker which might be erroneously interpreted as a heart signal. An overview of some European draft guidelines is given by Fleischer (1995). In the USA the federally accepted level for a static magnetic field is limited to 0.5 mT. According to German preliminary standards, people with heart-pacemakers must not be subjected to fields exceeding 0.5 mT root mean square (rms) for frequencies up to 1 Hz. For higher frequencies the threshold value is indirectly proportional to the frequency. By referring to the maximum dB/dt of a sinusoidal magnetic field, a limit of dB/dt = 11 mT s−1 may be derived for the a.c. component above 1 Hz. During operation an area around the SMES plant must be prohibited for public access. To estimate the extent of this region to comply with German standards, two extreme cases have to be compared: the SMES system at rest with maximum current (at the lowest temperature) and the fastest load cycle. The first one is straightforward. The lowest achieveable temperature of the superconductor is determined by the temperature of the coolant (e.g. the temperature of liquid helium in the case of pool cooling). So the maximum current is fixed and the lines of constant magnetic flux density can be evaluated by means of a magnetic field computation. The second case is somewhat critical, because German standards are applicable to SMES systems in periodic operation only. The fastest discharge, however, occurs in the case of a quench and is a highly nonperiodic event. To make things easy one might use the threshold value for dB/dt proposed above. So the maximum dI/dt is calculated with the help of the equation Um a x = LS M E S dI/dt or, if the SMES is discharged via a resistor R in the case of a quench, with the help of the equation dI/dt = R/LS M E S Im a x (the index max refers to the maximum values of the converter). Since the relation I ∼ B holds for every SMES without a ferromagnetic shield, the lines of constant dB/dt can be derived from the static field computation.

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(b) Magnetic shielding If the magnetic stray field of the SMES system is a problem, magnetic shielding of the SMES system might be necessary. For a torus magnetic shields are not advisable, because the fringe field is diminished effectively by reducing the spatial distance between neighbouring coils. The stray field of a solenoid, however, reveals only a weak dependence on the design. Significant reductions require magnetic shielding. The principle of active shielding (additional shielding coils with opposite magnetic moment) has been successfully applied to magnetic resonance imaging (MRI) magnets. It is proposed to shield solenoidal SMES systems in this way as well. The coils may be arranged concentrically as for MRI magnets (Schoenwetter and Gerhold 1995, Vollmar and Altpeter 1990) or grouped in a multipole assembly like a stack of wood (Komarek 1995). The analysis of Schoenwetter and Gerhold (1995) for a 50 kWh SMES shows that the active shielding roughly doubles the amount of superconductor necessary to store the specified energy. Besides, the forces are more critical (tensile stress and forces between the coils). Finally, the shielding efficiency is greatly reduced, if a coil is bypassed in the case of a quench. It is likely that the same problems occur for a multipole assembly. Hence an active shielding, in general, cancels the advantages of the solenoidal design over the toroidal design. This is not surprising, since from the viewpoint of a single torus coil the other coils might be regarded as shielding coils. Magnetic shields made of superconducting materials are used, for example, for SQUIDS where a small volume is to be shielded against a low field. Unfortunately both conditions (small region, low field) do not hold for SMES. A simple solution is a ferromagnetic shield. By means of the example discussed below for small and fast SMES systems the following general conclusions can be drawn. ( i ) A ferromagnetic shield neither increases nor decreases the superconductor losses and Lorentz forces (tensile stresses in the superconducting winding) appreciably. Thus the shield preserves the advantages of the solenoidal geometry. ( ii ) In the shield, eddy current losses are of no importance, because they are very low and are produced outside the cryostat. ( iii )The magnetic field of the eddy currents may be a serious problem for fast SMES systems. It reduces the flux-carrying capacity and thus the shielding efficiency of the ferromagnetic shield. In general the flux reduction is negligible, if the local skin depth does not exceed the lamination thickness. If the required lamination in the r and z directions cannot be guaranteed, the cross-section of the shield must be sufficiently increased. ( iv )For small solenoids a simple cylindrical shield is not effective with respect to the weight. A shape optimization may reduce the weight of the shield by one third compared to the cylindrical shield†. Nevertheless the weight of the shield usually is quite a deterrent, so that ferromagnetic shields may be restricted to very small SMES systems. Example: Electromagnetic compatibility of a 1.4 MJ/1 MW SMES of solenoidal and toroidal geometry This case study compares the electromagnetic compatibility of a toroidal SMES to a solenoidal one. An existing torus, the 1.4 MJ/1 MW Munich Pilot Plant (see section H7.1.9), serves as an example. The hypothetical solenoid is designed equivalently to the Munich torus. At 5.0 K the solenoid stores 394 J per unit length of superconductor which is 1.7 times the value of the Munich Pilot Plant. The magnetic stray field will be characterized by the German threshold values for humans with heart-pacemakers. Despite the Munich SMES system being specified as fast acting, the fastest load cycle is not the worst case with respect to heart-pacemakers but the SMES system at rest with maximum current is. Figure H7.1.4 shows the lines of constant magnetic flux density for the maximum current of 1467 A at the lowest temperature of 4.6 K. Because of symmetry the lines are restricted to one quadrant. The magnetic

† An optimization is almost useless for solenoids with a large aspect ratio b. In this case almost all magnetic flux is collected by the front plate and then carried by the cylindrical jacket which is the far larger part of the shield.

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Figure H7.1.4. The Munich SMES system and lines of constant magnetic flux density in the torus midplane for the maximum current at 4.6 K.

flux density falls below the critical value of 0.5 mT at a distance of 2.47 m, measured from the centroid of the torus. The magnetic stray field of the equivalent solenoid falls below the threshold value of 0.5 mT at a distance greater than 7.13 m (in the plane z = 0) and 8.95 m (on the z axis), measured from the centroid of the solenoid. At a distance of 2.47 m—in case of the torus the limit for heart-pacemakers—the magnetic flux density rises from 12.5 mT (in the plane z = 0) to 22.5 mT (on the z axis). This clearly demonstrates that the stray field of a torus is far lower than that of a solenoid. Thus, if a fringe field is a problem, the toroidal SMES must be preferred unless the solenoid can be shielded effectively. Usually an actively shielded solenoid is not superior to a torus. For this reason the equivalent solenoid with a ferromagnetic shield will be considered. A commonly used shield—a hollow cylinder with walls of constant thickness—is placed around the cylindrical cryostat (see ‘initial shield’ in figure H7.1.5). Ordinary iron is chosen as the ferromagnetic material (saturation magnetic flux density Bs a t = 2.17 T at H = 900 A/m) because of its low cost. The thickness of the shield is 27.5 cm so that the magnetic flux density remains below 0.5 mT at distances greater than 2.47 m, measured from the centroid of the solenoid. This results in a volume of 3.52 m3 and a mass of 27.4 t respectively. The question arises of whether the same shielding effect can be achieved with shields of other shapes and less mass. This problem is tackled by means of a shape optimization (Brammer and Lorenzen 1995). The optimum that was found by using a stochastic and a deterministic optimization method is shown in figure H7.1.5. The volume of the optimized shield is 2.29 m3 which is equivalent to a reduction of 34.8%. The mass of the shield is 17.9 t.

H7.1.4 Operation of SMES systems For the operation of an SMES system there are characteristic alternating modes of charging, stand-by and discharging. The SMES system is charged by applying the supply voltage Us u p to its terminals. If Us u p is constant, the current grows linearly with the rate of current rise di/dt = Us u p /LS M E S because of the nonexistent resistance of the coil LS M E S . Having reached the intended storage current, IS M E S , the coil is shorted. The current running in the superconductive coil is maintained, until a voltage with inverse sign is applied for discharging. If the discharging voltage is kept constant, the current decreases linearly as well. Copyright © 1998 IOP Publishing Ltd

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Figure H7.1.5. An equivalent solenoid and ferromagnetic shield optimized with respect to the mass, with magnetic flux density vectors.

H7.1.4.1 Cycle efficiency Although the storage current remains constant in the rest period, the storage device does not work without some losses. Actually the power necessary for cooling and operation of the plant has to be set up as a loss. That is why for the evaluation of SMES the so-called cycle efficiency η is defined (see equation (H7.1.1)). Obviously high cycle efficiencies require short cycle times. In applications such as load levelling, the SMES system is permanently charged and discharged alternating quickly between the two modes of operation. In this case good cycle efficiencies are expected. If the SMES is employed to smooth transient voltage drops, it runs in the stand-by mode most of the time. Considerations about the efficiency of the plant step into the background compared with the desire for a high voltage quality.

H7.1.4.2 Connection to the grid with a flux pump In order to connect the SMES system to the supplying source there are two physically completely different possibilities. The first one is the use of a flux pump. The principle of operation of a flux pump will be explained with reference to figure H7.1.6. The whole system and its function can be described by the following circuits: in the storage circuit with the inductance LS M E S and the ideal switch Ss there is the storage current iS M E S The pump circuit with the current ip consists of the ideal switch Sp and the inductance Lp which is coupled magnetically with the charging circuit. The charging circuit contains the inductance LL , which is magnetically coupled with the pump circuit, the ideal switch SL and the supplying source UQ and is passed through by the current iL. At the beginning the switch Ss is closed and in the SMES unit there is a current iS M E S 1. All the other switches are opened. If SL is closed, the charging current iL generates a magnetic field. As a result a magnetic flux Φ is interlinked with the pump circuit. The switch Sp is now closed and SL is opened. In the pump circuit the current ip - Φ/Lp develops. If the switch Ss is opened, the whole magnetic flux Φg e s = LS M E S iS M E S 1 + Lpip spreads through the inductances Lp and LS M E S : Φg e s = (LS M E S + Lp )iS M E S 2 . If Ss is closed and Sp is now opened, the current iS M E S 2 > iS M E S 1 runs in the SMES system, and the next charging cycle may start. The advantage of connecting the SMES system to the supplying source by means of a flux pump lies in the galvanic separation of the superconducting coil and the supplying Copyright © 1998 IOP Publishing Ltd

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Figure H7.1.6. The basic structure of a flux pump.

source, so there is no current in the charging circuit during the stand-by mode. There is no longer a necessity to provide reactive power. However, flux pumps work quite slowly. For the connection of SMES only flux pumps with rectifier circuits can be taken into account. Hereby the switching losses have to be considered. All the flux pumps used so far have only had a small rated power. For that reason the storage and the supplying source of real plants have always been connected galvanically. This is done by means of converter connections. H7.1.4.3 Converter connections The choice of the converter circuit is strongly influenced by the storage device employed. An essential application is feeding an inductive load with pulses of extremely high power, as they are used, for example, in fusion research or in particle physics. This requires a converter circuit that is able to join two inductances—the storage and the load coil—with each other. A whole series of converter circuits has been developed particularly for this purpose (Ehsani and Kustom 1988). Generally, they can be divided into two groups, the nonuniform switching circuits and the inductor converter bridges. The nonuniform switching circuits require continuous change of switching time intervals. Their topology is usually quite simple. The single shunt capacitor circuit with one to four switches belongs to this group of converters as well as the single or dual flying capacitor bridge. The circuits mentioned above often require a forced turn-off capability, so that the use of gate turn-offs (GTOs) or insulated gate bipolar transistors (IGBTs) is recommended. In fact, inductor converter bridges, which can be designed as one- or multiphase (see figure H7.1.7), offer better possibilities for the control of energy transfer. The principle of inductor converter bridges is to couple two bridge circuits on the a.c. side. This side is buffered by star-connected capacitors. Then one bridge works as an inverter, the other as a rectifier.

Figure H7.1.7. Circuit diagram for a three-phase inductor converter bridge. Copyright © 1998 IOP Publishing Ltd

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On the other hand all applications in the field of power economics demand a connection of the SMES to the three-phase system. The most simple way is the use of a conventional six-pulse Graetz bridge. However, there are considerable disadvantages like the lack of freewheel in the dead intervals and the reactive power being dependent on the firing angle. An alternative was the so-called B6N-bridge, whose additional thyristors connecting the anode and cathode terminal to the neutral point could take over the freewheel or the sector control with turn-off capable switches. B6-bridges with interphase commutation, as they are used to supply induction motors, consume capacitive reactive power dependent on the firing angle. If a B6-bridge with interphase commutation is connected in parallel with a conventional B6-bridge on the d.c. side (figure H7.1.8), the rates of reactive power neutralize each other if each bridge participates in the conduction with the same current. Depending on which bridge takes over the bigger amount of storage current, a capacitive or inductive reactive power results. So this circuit offers the possibility to vary the reactive power independently of the real power. However, the voltage-link a.c. converter shown in figure H7.1.9 represents the most versatile way of connection. A self-commutated pulse-width-modulation inverter allows one to adjust active and reactive power independently of each other. The actual supplying voltage follows from the link voltage by a d.c. chopper. In stand-by mode only the chopper works: the power system is completely relieved. Additionally the voltage link offers the possibility of combining several kinds of energy storage, e.g. SMES, batteries and motor-generated flywheels.

Figure H7.1.8. Parallel connection of a B6-bridge with interphase commutation and a conventional B6-bridge.

Figure H7.1.9. A voltage link a.c. converter.

When rating the converter, independent of the choice of the type, attention must be paid to the fact that its nominal power and thus its costs tend to the product of maximum voltage and maximum current. By assuming a constant charging and discharging power, the supplying voltage is lowest at the maximum current or the supplying voltage is highest at the minimum current to which the storage is discharged, respectively. Hence the necessary over-dimensioning of the converter depends on the energy exchange ∆WM of the storage, which therefore should not be too big. The ratio of installation and discharging Copyright © 1998 IOP Publishing Ltd

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power can be calculated with the help of the maximum stored energy WM. m a x

So at an energy exchange of ∆WM = 0.5WM the converter has to be over-dimensioned by about 40%. For applications in electrical power networks the SMES plant must be connected to the respective voltage level via a transformer. The coupling may be done in series or in parallel as shown in figure H7.1.10. For the parallel coupling another transformer is connected to the busbars of medium voltage whose secondary side is combined with the SMES. In this case the storage plant is to be considered as a current source which delivers an auxiliary compensating current. Regarding the series connection, a transformer with in-phase regulation is shifted into the busbars which are connected to the SMES. Here the storage plant represents a voltage source which delivers an auxiliary compensating voltage. The topology of the grid and the special purpose of the storage device determine whether serial or parallel coupling requires less compensating power. Control is certainly easier for series coupling. Nevertheless parallel coupling is the most common solution so far. The installation is simple and existing systems are easy to expand. At a failure of the SMES system the other system components remain unaffected, whereas the in-phase regulator has to be bridged in this case.

Figure H7.1.10. Parallel and series coupling of the SMES system.

This answers the question of how the power system behaves in the case of a disturbance of the SMES system. Far more difficult are the problems that occur inside the storage device during a failure. The following section is dedicated to this subject. H7.1.5 Protection systems H7.1.5.1 Concept The most dangerous fault in an SMES system is a quench. This quench causes a temperature rise in the superconductor that may melt the solder of a multifilamentary conductor or cause other irreversible damage to the coil (see also chapter C3). The best protection against such damage is to prevent a quench by means of an operating system. This operating system monitors the current electrical and thermal state of the SMES system. From these data the possible thermal reserves can be calculated and the future operating conditions, such as maximum allowable current or maximum allowable loading and unloading frequency, can be limited. Also the design of the SMES can prevent a quench or limit its effect. The use of a stabilized superconductor is an example of such a passive quench protection. Then, however, other problems like higher expenditures for copper or higher a.c. losses in the (now larger) copper core of the superconductor may arise. Nevertheless the risk of a quench must be considered. If the design of the SMES is not inherently safe, the SMES has to be equipped with an active quench protection system. This system must detect the quench and react with the appropriate countermeasures.

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H7.1.5.2 Quench detection For this purpose there are basically two physical effects that can be used. Both methods rely upon the fact that the quench generates normal-conducting zones that cause ohmic resistance and therefore ohmic losses. If possible both methods should be taken into consideration to have a back-up. The two methods are given below. (a) Thermal quench detection The ohmic losses caused by the quench lead to a warming up of the superconductor. If the SMES has a forced cooling system this heat can be measured indirectly via the temperature of the cooling medium at the coil’s outlet. In the case of indirectly cooled coils the temperature of the coils can be measured directly. Here the temperature sensor has to be placed near the point where the quench is most probable, i.e. where the magnetic field has its maximum. For a toroidal SMES system this is at the inner winding towards the centre of the torus. In both cases a quench is detected if the temperature exceeds a certain threshold. The ideal sensors for this purpose are GaAlAs-diode thermometers. They are very sensitive in the temperature range of interest, very small and immune to high magnetic fields. Though this quench detection method is a thermal one it is relatively fast due to the very low thermal capacities at low temperatures. (b) Electrical quench detection In this case the voltage caused by the ohmic resistance is observed. For this purpose each single coil must carry some additional windings that form an auxiliary coil. The voltage across the main coil, which consists of an ohmic and a reactive part, and the voltage across the auxiliary coil (which is only proportional to the reactive part) are measured. By taking the right transformation ratio a into account, the ohmic voltage can be calculated using

If this difference uo h m i c exceeds a certain threshold a quench is reported. To prevent the system from being too sensitive, the threshold has to be chosen carefully. This is especially the case if the SMES is fed by a converter. The advantages of this method are its very fast reaction time and compatibility with any cooling system. Another method is the use of a centre tap of the coil whereby the voltages across the first and the second half are compared. Again a quench is detected when the difference exceeds a threshold. A disadvantage is that a quench near the centre tap cannot be detected. This method is very dependent on the self- and mutual inductance that may change asymmetrically during normal operation and could imitate a quench. Additionally the signals must be tuned very accurately. H7.1.5.3 Countermeasures If a quench is detected a set of countermeasures has to be performed (figure H7.1.11). First of all the coil(s) must be protected from thermal destruction: the current in the coils must be reduced very fast. This means that the energy stored in the SMES has to be dumped into an external resistor or, if possible, even better into the grid. This resistance is limited by the maximum allowable discharging time and insulation voltage

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Figure H7.1.11. An SMES system divided into single coils with a protection system.

If an SMES system consists of several coils like a toroidal SMES, another important measure is the bridging of a quenched coil. As a result of the lower inductivity of a single coil, the current declines faster in the affected coil. If possible, the SMES system can even continue its operation. To avoid losses caused by the additional current leads, which have to be rated for the full current, this bypass has to be placed in the cryogenic area. For the protection of accelerator magnets this task is performed by diodes. If the voltage across the coil (ohmic and reactive) exceeds the forward voltage the current commutates automatically into the bypass. This is not possible for an SMES system because the reactive voltage during the fast loading and unloading during normal operation is far higher (some hundreds of volts) than the forward voltage of the diode (some volts). So this leads to the need for an easily switchable bypass, which is ideally a power semiconductor that works at low temperatures. Bipolar power semiconductor devices like thyristors are not suitable for this purpose because their operation is based on thermal charge carrier generation. These devices lose their switchability below a certain temperature level. For the desired power range only hybrid devices like IGBTs remain. Up to now only one IGBT with a special layer arrangement and doping seems to be usable for this purpose (Rosenbauer and Lorenzen 1996). It is switchable at a temperature of 5 K and has a low on-state resistance. This allows a fast commutation into the bypass and low losses during operation. When selecting the device not only does the current rating of the device have to be observed but also the blocking ability has to be considered. It is lower than the rating at room temperature. The measures already discussed affect only the reduction of the current in the coil to limit the maximum coil temperature. Another problem occurs if only a part of the coil quenches and the remainder stays superconducting. The quenched part warms up quickly but the superconducting part still is at low temperature. To homogenize the temperature distribution within the coil and protect it from thermal stress, quench heaters have to be provided. These are capacitors which feed a short but high current pulse into an auxiliary winding or (via a centre tap) directly into the coil. This initiates an intentional quench caused by high magnetic field strength, high current and especially high a.c. losses within the superconductor. Due to the very short pulse, current leads with a small diameter are sufficient. The control of the whole protection system can be performed by a microcontroller. This device Copyright © 1998 IOP Publishing Ltd

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includes the ability of analogue-to-digital conversion for the measurement of temperatures and voltages, numeric calculations (like the nonlinear voltage-temperature characteristics of the GaAlAs diodes) and output ports to trigger the countermeasures. It can act as a subsystem of the already mentioned operating system of the SMES unit.

H7.1.6 Characteristic data of SMES systems The characteristic data of SMES systems obtained by measurements or calculations are sparse. So the data of table H7.1.3 are to be interpreted as approximate values that refer to small optimized SMES systems.

In order to cool the coils at constant operating current, small fast SMES systems with a storage capacity up to several megajoules require a cooling power of the order of 10 W (at the liquid-helium level). During pulsed operation the heat input in the liquid helium increases considerably, since hysteresis and eddy current losses have to be covered. Usually small refrigeration plants consume 500–1000 W electrical power to remove a helium heat input of 1 W. The specific costs of SMES are subject to great uncertainties. Table H7.1.4 gives some estimated values for the investment costs per unit of stored energy. The cyle efficiency of an SMES system depends on the storage capacity and the load cycle. The data presented in table H7.1.5 are valid for an almost constant current only. Note that the SMES systems referred to in tables H7.1.4 and H7.1.5 are mostly design studies and that the cycle efficiencies are not necessarily calculated according to equation (H7.1.1).

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H7.1.7 Applications of SMES As already mentioned the ideal applications for SMES are those problems where high active power at very short response times is required. In most applications reactive power has to be provided as well. The converter, an essential component of an SMES plant, additionally can undertake this task and work as the reactive power source. In the following, typical applications of SMES are presented.

H7.1.7.1 Power levelling This subject covers the levelling of a variable energy supply and the levelling of a variable energy demand. The first one may be caused by the use of regenerative energy, e.g. wind power. When the demand is lower than the supply, the surplus energy is stored in the SMES system. With this energy the storage can cover the gap, if the supply is not sufficient. The latter is relevant at hammer mills, electric arc furnaces or rolling mills for example. Here the ratio of peak power to average power and the frequency of the changes between peak and low power are very high. The cost for the supply of electric energy is a function of the energy that has been consumed and the maximum power needed. The latter is due to the fact that the maximum power governs the design of the infrastructure, e.g. the power plants and transformer stations. The SMES system can cut the peaks off and fill up the valleys. Thus from an external point of view the high peak power demand is transformed to a cheaper continuous power demand. The energy consumed remains constant.

H7.1.7.2 Uninterruptable power supply (UPS) With the rising use of computers and computer-controlled devices the voltage quality becomes more and more important. Voltage quality demands a continuous supply of electricity with constant frequency. This may be affected when lightning strikes cause short circuits or earth faults in overhead lines. To neutralize such a fault the whole overhead line or only the affected phase is switched off for a very short time. This time is too short to be noted by normal consumers but long enough to disturb sensitive devices. The short time voltage drop can be compensated for by a fast-acting storage method like SMES. Depending on the layout of the network and the sensibility of the consumer the SMES system can be located in a substation or adjacent to the consumer’s installation.

H7.1.7.3 Operation of the grid A third major field for SMES is the optimization of the operation of the grid. This includes the damping of oscillations of synchronous generators and the improvement of the dynamic stability of power supply systems. Another point may be the use in the primary control (frequency-power control) or immediate reserve. At present this is done by means of gas turbines or partially throttled steam turbines. The idea is to operate the thermal power plants at constant level with their best efficiency and use a storage system for the remainder. This reduces both energy and maintenance costs, because the stress of the thermal infrastructure of the power plant will be reduced.

H7.1.7.4 Example: load noise levelling in a single-phase network The German Railways operate their own single-phase 162/3 Hz, 110 kV grid. This feeds the numerous transformer stations, which for their part feed the actual 15 kV overhead lines. The metropolitan area of Munich is a point of concentration of the consumption of electric energy. Nevertheless there is no

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equivalent source nearby. Several long distance and regional railway lines meet here. Additionally there is an extensive commuter network. This rapid transit system is operated by phase-controlled electrical multiple units, which are very powerful to perform a fast acceleration between the stations, which are close together. Unfortunately these multiple units have a very poor power factor of 74%. The largest part of this electrically operated network is fed by a single substation in the west of Munich with a total rated apparent power of 45 MVA. Additionally there is a 12.7 Mvar capacitor bank installed to compensate for the reactive power needed. The numerous trains require high power for the short time of starting up only. When the trains run at constant speed, the power demand is moderate. This results in a high load noise, as measurements taken in the substation illustrate (figure H7.1.12). When a high current is drawn, a corresponding voltage drop caused by the grid’s internal impedance can be noted. This voltage drop is sometimes so extreme that the undervoltage protection of the multiple units responds. As a consequence, the start-up current per multiple unit has to be limited during peak hours. This increases the running schedule and decreases the punctuality.

Figure H7.1.12. Recorded voltage (top) and current (bottom) at the busbar.

It was asked whether an SMES system could solve these problems. The first step included the development of a control system (Kaerner 1995). This system must make sure that on the one hand the storage capacity of the SMES system is not exceeded, but on the other hand that there is always sufficient energy stored. The whole control system was simulated using the recorded data as input. The simulations showed that a 300 kW h storage is sufficient for this purpose. The use of this storage would reduce in our example the maximum current at the secondary busbar from 3676 A to 2334 A and increase the voltage from 12.6 kV to 14.6 kV (figures H7.1.13, H7.1.14). In this way the load noise can be reduced significantly. Subtracting the energy needed to operate the storage (cooling, vacuum, etc), the SMES system would save about 800 kW h per day. The better compensation of reactive power would achieve additional savings. These specifications led to the basic design of an SMES plant, the key data for which are shown in table H7.1.6. Copyright © 1998 IOP Publishing Ltd

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Figure H7.1.13. Simulated voltage (top) and current (bottom) at the busbar. The SMES is active between 05.15 and 22.00 h.

Figure H7.1.14. Energy stored in the SMES system.

H7.1.7.5 Example: balancing asymmetric fast transient voltage drops LEMG (Kaerner and Lorenzen 1992) did another case study on behalf of an electricity distribution company. This company operates their 110 kV overhead lines with solid earthing. A major customer, which produces staple fibres, operates many extruders and spinning frames fed by converters. The customer (35 MVA, 28 MW) is connected via a single 110 kV line only. About five to eight times per 100 km and per year there is a serious earth fault in an overhead line network, which may be caused by lightning strikes or birds. To neutralize the fault, the affected phase is switched off for up to 200 ms. This results in an asymmetric voltage drop. During this time Copyright © 1998 IOP Publishing Ltd

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the undervoltage protection of the converters breaks. The spinning frames stop and the already produced footage of fibre is worthless. The problem can be solved by a serial- or parallel-connected compensator with or without an active power source. The calculation results for different control strategies and connection types can be seen from table H7.1.7. The serial-connected compensator with active power is the optimal solution with regard to the converter layout. In this case additionally the control strategy is very simple, because the compensator acts as a voltage source (figure H7.1.15). The other control strategies require very complicated algorithms.

For the supply of active power a fast-acting storage of 1.5 MJ is required. This is ideally an SMES system. As this is approximately the size of the Munich Pilot Plant, data for a suitable device may be taken from section H7.1.9.

H7.1.8 An overview of SMES projects This section starts with a brief review of the history of SMES. It continues with an overview of SMES projects that are already (or likely to be) realized. The projects mentioned are mainly restricted to SMES for electric utility use. This does not imply that SMES for research applications and SMES projects that are not described are of less technical importance. Copyright © 1998 IOP Publishing Ltd

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Figure H7.1.15. A serial-connected compensator to neutralize earth faults.

H7.1.8.1 History of SMES When the intermetallic superconductors such as Nb-Ti were discovered in the early 1960s, it became possible to construct large superconducting magnets. The design and the construction of such magnets demonstrated that they would be able to store large amounts of energy. In 1971 it was proposed to use a multiphase Graetz bridge in order to provide an interface between a superconducting magnet and the electric utility grid. This was the starting point for the development of several SMES concepts for utility applications. Cost analyses showed that only very large units, in excess of 1000 MW h, would be cost competitive with conventional alternatives. So some design studies on very large-scale SMES systems were initiated. This strategy contradicts common technical developments that start with small units and then scale them up step by step. The most detailed studies were performed in the USA and in Japan during 1985 and 1986. In Japan a 5000 MW h/1000 MW SMES system was designed for utility load levelling (Masuda and Shintomi 1987). In the USA a team headed by Bechtel developed a 5000 MW h/1000 MW SMES concept for the same application. The results of this baseline study led to two design-improvement and cost-reduction studies, also performed by Bechtel. Shortly after the birth of the Strategic Defense Initiative (SDI) the Engineering Test Model (ETM) project to build a 20 MW h/400 MW SMES system for military applications was launched. The SMES-ETM project was terminated in August 1993 when the SDI was cancelled. Since the beginning of the 1990s the rising industrial demand for a high-quality power supply has resulted in the development of small and fast SMES units which are to be installed near the customer sites. In contrast to the early activities, several test units are now being built or have been successfully tested. There is even a 1 MJ/1 MW SMES system which is commercially available. H7.1.8.2 Hardware projects (a) Overview Compared with the many design studies, until 1996 very few SMES had been constructed and operated. Several small units for research applications have since been built, e.g. SMES for fusion magnets or plasma heating. Some of the current hardware projects for utility applications are listed in table H7.1.8. Copyright © 1998 IOP Publishing Ltd

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(b) Bonneville Power Administration Project The first SMES unit which was installed in a utility grid was a 30 MJ/10 MW SMES system ordered by the Bonneville Power Administration (BPA). The aim was to damp oscillatory current components on a large transmission line operating between the Pacific Northwest of the USA and Southern California. The SMES system consisted of a solenoidal coil made of Nb—Ti superconductor. The dewar had an outer diameter of 3.6 m and a height of 2.7 m. The coil was first energized in February 1983. From November 1983 to March 1984, with the exception of brief staged tests, over 1200 hours of operation with modulation were accumulated. The efficiency of the SMES unit measured was about 86% (Rogers et al 1985). The unit never operated as successfully as expected because of problems with the refrigeration system and inadequate funding. (c) SSD of Superconductivity Inc. At the end of the 1980s there was growing interest in small SMES units to improve the power quality in electric grids. This resulted in the first commercially available SMES unit offered by Superconductivity Inc. (SI). The SI’s superconducting storage device (SSD) provides up to several seconds of back-up power to protect the load from voltage sags or momentary power losses. At present this SMES is the only real industrial application. The first field test of an SSD took place in the summer of 1990 in cooperation with Iowa Electric Company. By the end of 1994 another demonstration unit and five commercial installations were in operation. The storage capacity of the SSD systems ranges from 1 to 3 MJ; the maximum power is 1.4 MVA. The SSD is based on a solenoidal coil of approximately 89 cm height and 46 cm diameter. The coil which is made of a.c. Nb—Ti cable is placed in a liquid-helium cryostat. The cryostat is 203 cm long with an outside diameter of 76 cm. It contains a reservoir of liquid helium to continue operation for up to 60 h if the refrigerator fails. The principle of operation of an SSD system is explained by means of figure H7.1.16. Under normal conditions the load is connected directly to the utility grid (solid-state isolation switch SI is closed), and the SSD is isolated from the load. Switch S2 is in the right position, so that the magnet current circulates through the magnet and switch S2. In this way the load is protected if something happens to the SSD system. When a voltage sag or loss of power is detected, the solid-state isolation switch S1 disconnects the load from the utility grid and switches in the SSD. The magnet is discharged by connecting the magnet to the d.c.-d.c. converter (switch S2 in middle position). The capacitor across the converters is charged until the voltage reaches a preset level. Switch S2 then is opened and the load is fed via the d.c.-a.c. converter and the transformer, thereby causing the capacitor to discharge. When the voltage reaches a preset minimum value, switch S2 is closed (switch S2 is in the middle position), and the cycle begins

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Figure H7.1.16. A simplified block diagram of SI’s SSD.

again. It is essential that the magnet is discharged by means of current pulses. As the magnet discharges, the current decreases while the capacitor guarantees a constant voltage. Therefore the duration of the pulses increases to maintain a constant power level. The switchover to the SSD begins within 450 µs and the full power is reached in 2–4 ms. After the utility grid has resumed normal power conditions the solid-state isolation switch SI reconnects the load to the grid. In order to charge the magnet, the magnet is connected to the magnet power supply (switch S2 in the left postion). The charging procedure lasts several minutes to prevent overload of the grid. All SSD systems have been delivered in mobile trailers and installed on site. During operation SI provides all maintenance for the SSD system. The system is monitored remotely at SI’s Wisconsin headquarters, so that the unit can be repaired in the case of a fault without involving the customer (Dewinkel et al 1995). (d) Project of B&W and ML&P A 1800 MJ/30 MW solenoidal SMES system is currently being developed by Babcock and Wilcox (B&W) in cooperation with Anchorage Municipal Light and Power (ML&P). The motivation for this project was the unscheduled loss of generating capacity in the Anchorage utility. The SMES system is supposed to support the utility’s system frequency during the period required for a hydroelectric plant to increase its power output. This unit will be the first commercial mid-sized SMES system capable of delivering full four-quadrant real and reactive power. The first tests were planned for late 1997. The magnet consists of several sections which are assembled into a long solenoid of 17.7 m length and an aspect ratio of 6.4. A cryostable Nb-Ti Rutherford cable and liquid-helium pool cooling were chosen. As pointed out in section H7.1.3 the solenoid and the helium vessel act like a transformer. Thus the a.c. losses in the helium vessel come up to one third of the total a.c. losses (Huang et al 1995). Copyright © 1998 IOP Publishing Ltd

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(e) German projects As a consequence of the low magnetic stray field, European and Japanese SMES projects favour toroidal geometries. Two European hardware projects are located in Germany. The Munich 1.4 MJ/1 MW SMES system is described in another section. In addition to the Munich project a 100 kJ/100 kW SMES system for compensation of fast pulsing loads was developed by Forschungszentrum Karlsruhe in cooperation with the Universities in Karlsruhe and Aachen. Ten coils with an outer diameter of approximately 35 cm are arranged as a torus of about 36 cm radius. The low-loss superconducting cable is based on Nb-Ti. A pool of liquid helium serves for cooling. The whole unit is transportable so that field tests for the demonstration of applicability and reliable operation at a three-phase supply are possible at different sites (Spaeth et al 1995). It is expected the project will finish in mid-1998. First tests were performed on six coils in a solenoidal assembly (see footnote of table H7.1.8). In May 1996 this 250 kJ/80 kW solenoidal SMES system proved to compensate for a fast pulsing load varying by 300 kW s−1. By using another PCS the same solenoidal SMES achieved 1 MW in 2 ms pulses at a repetition rate of 10 Hz[!—fnrH7.104—> † (Juengst and Salbert 1996). At the end of the tests the repetition rate was increased from 10 to 30 Hz and the magnetic field change from 60 to 90 T s−1. (f) Japanese projects In Japan two long-term SMES projects to develop SMES systems for load levelling in electric utility networks exist. The Kyushu team is currently constructing a 1 kWh/1 MW module-type SMES system. The toroidal SMES consists of six coils made of Nb-Ti cable. Pool cooling is chosen. It is planned to start the tests in an actual power system in June 1997. This small experimental device is the first step in constructing a 1-10 MW h/100 MW SMES for load levelling (Tsutsumi et al 1995). Besides the Kyushu project there is a national Japanese programme to develop all components required for a 100 kW h/40 MW toroidal SMES. The programme was launched in 1991 by the Ministry of International Trade and Industry in Japan (MITI). It is managed by the International Superconductivity Technology Center (ISTEC). Originally scheduled for six years it was extended to eight years later. After the successful completion of this project the construction of a pilot demonstration plant is expected (Masada et al 1995). (g) HTS micro SMES of ASC In June 1996 the American Superconductor Corporation (ASC) delivered the first SMES system made of a high-temperature superconductor. The storage capacity is 5 kJ which suffices for research applications. The solenoidal coil is constructed from a Bi-2223 conductor. In contrast to SMES systems made of Nb-Ti no coolant is used but a single-stage Gifford-McMahon cryocooler is used instead. The operating current is 100 A at 25 K. The magnet can be ramped from zero to 100 A in 2 s and back to zero in 2 s. Although the storage capacity is very small, this might be the starting point for the development of larger units. H7.1.9 The Munich Pilot Plant The construction, operation and protection of SMES systems have been discussed in former paragraphs and a few fields of application have been presented in detail. In this final section an established plant will be studied in more detail. The Munich Pilot Plant, which is to be finished soon, serves as the example. The aim of this research project is to build up a small SMES system with high power, which allows fast alterations of storage current, possesses a high cycle efficiency and is suitable for indoor installation (Lorenzen et al 1995). Conceivable purposes and employment for the testing plant have already been specified in section H7.1.7.

† These additional tests are part of a second project: an SMES-based power modulator for the TESLA accelerator at DESY, Germany.

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Figure H7.1.17. Cross-section and horizontal projection of the Munich Pilot Plant. VT— vacuum tank; WS— thermal shield; SP— superconducting coil.

In all those applications the plant is situated either directly at the consumer’s site, in a transformer substation or in the power station. The available room is often limited, in particular no significant environmental influences must occur even in the immediate neighbourhood. Therefore only the torus is taken into account for this type of construction. However, ideal toroids are difficult to manufacture. Grouping six single flat coils to form a star of coils seems to be the best compromise between the approach to the torus shape and simplification of manufacture. For reasons of manufacturing, the field coils are constructed circularly and not in the so-called Shafranov shape optimized with regard to the stored magnetic energy and the mechanical stresses. In order to reduce the eddy currents as far as possible the whole support construction is made of GRP. The essential geometrical data can be taken from figure H7.1.17 and table H7.1.9.

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The choice of the superconductor is discussed in detail in volume 1. As the technical risk of the design should be limited— lately the reliability of such a device can be verified— Nb-Ti and the cable LCT10 + 1 from ABB, which has already been used successfully in one coil of the Large Coil Task project, were chosen. Certainly this cable is not optimal with respect to the a.c. losses, but it was put at our disposal by ABB. It consists of ten multifilamentary conductors of the type BBC-S300 which are twisted around a cable core made of copper. One conductor contains 318 filaments with a diameter of 17 µ m. The cable with a diameter of 2.21 mm was braided with glass filaments for insulation and rolled up onto the winding support in seven layers of 55 windings. Each layer is again insulated with a glass filament. To ensure mechanical resistance the winding was vacuum impregnated. Figure H7.1.18 shows the winding cross-section.

Figure H7.1.18. Cross-section of the superconducting coil.

The SMES system is indirectly forced-flow cooled by supercritical helium. This method of cooling was chosen for several reasons (Schoettler 1994): ( i ) design and manufacture are simpler for indirectly cooled coils than for pool-cooled ones; ( ii ) pool-cooling of the whole torus in one vessel would require an enormous supply of helium; if there were separate vessels for each coil the eddy current losses arising would not be acceptable (vessels in the high field area); ( iii )the coolant remains in the single-phase state, so calculations are more reliable and, in particular, the heat transfer is better known; ( iv )due to shorter and wider cooling pipes compared with internally cooled or cable-in-conduit superconductors, stationary pressure drop and pressure rise during coil quenching are lower. The supercritical helium is pumped through cooling pipes, which are wound up on the inner and outer side of the winding in one layer each (see figure H7.1.18). Figure H7.1.19 shows the complete cooling scheme. As the magnetic induction increases to the inner winding layers, their temperatures determine the critical current. That is why the innermost layer should be particularly well cooled. The connection of the six inner cooling layers to an inner circuit by heat exchangers serves this purpose. After having passed the inner cooling circuit, the coolant is piped through the six outer cooling layers. As the temperature is less critical here no heat exchangers are needed. Leaving the last coil at about 0.55 MPa and 5.0-5.9 K, the helium is expanded in a Joule-Thomson ( JT) valve, where it liquefies partially. This fills up the small pool in the dome. From this pool the current leads as well as three of the heat exchangers vaporize helium for cooling. Any excess of liquid and gaseous helium flows through an overflow tube to a big external dewar where the other three heat exchangers are located. The helium is supplied with a mass flow rate of 5.7 g s−1 by a refrigeration plant, Linde Standard I, which has a rated refrigeration capacity of 60 W. Even in ordinary operation the total heat input can be up to 20 W per coil. In this case the external dewar represents a buffer, which allows operation for up to one hour with charge cycles, the losses of which are nearly double the refrigerator capacity. The whole torus is hung up in an evacuated tank. The heat input as a result of radiation would be so high that the refrigeration plant could hardly cope. Therefore a thermal shield is provided which is cooled

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Figure H7.1.19. Cooling circuit for the SMES device.

by liquid N2 down to a temperature of about 80 K. This shield is also made of GRP to avoid eddy current losses. The worst failure in the operation of an SMES system is quench. Even in this case none of the components must be destroyed, and after the failure has been rectified, the plant should be activated immediately. For that purpose a protection system which recognizes the quench in time and takes appropriate measures for the protection of the equipment is necessary. In order to prevent a quench right from the beginning, a safeguard is provided in the control, which determines the maximum admissible current from measured values of temperature and magnetic induction by means of the characteristic IC = ∫ ( B, T ). Table H7.1.10 shows the relation between temperature, storage current, magnetic induction and stored energy for the Munich Pilot Plant.

Two different conceptions of measurement, which serve for the identification of quenches, have been realized. The first one is to record the temperature at the critical point of the winding. If certain temperature Copyright © 1998 IOP Publishing Ltd

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limits are exceeded it is interpreted as a quench. The second one is the comparison of the voltage across the main coil with the voltage across an auxiliary winding. The ‘sick’ coil must be bridged instantly, because the whole storage cannot be de-excited fast enough. This is done by semiconductor switches. The switches have to be installed in the cryogenic area, so only IGBTs of the non-punch-through type can be used. At the same time a current pulse is shot into the coil by the quench heater. This makes sure that the whole coil becomes normal-conducting. So unacceptable mechanical stresses owing to high temperature differences are avoided. Those would occur if the quench remained locally limited. Additionally only a sufficiently high coil resistance produces commutation of the storage current to the bridging switch. Finally the whole storage is discharged to a protection resistance. For safe operation of the plant it is necessary to measure a series of completely different parameters, e.g. temperatures, pressures, magnetic fields, voltages, He mass flow and level in the dome tank. All these measurements require sensors inside the vessel whose connections must be led through to the storage cryostat. Further bushings are necessary for the control wires of the semiconductor switches for coil bridging as well as for the quench heaters. Most of the electrical terminal leads are led to the storage cryostat through the dome. All the necessary helium and nitrogen lines must be connected by the dome, which also contains the small liquid-He reservoir with three heat exchangers. So the dome represents an element particularly sophisticated in construction. The essential data for the design and construction of the SMES system are summarized in table H7.1.11.

The tests and investigations planned on the SMES system are divided into three phases which differ by the way the SMES system is supplied. The first tests will verify the fundamental function of the SMES system. In this phase motor generators feed the SMES. In contrast to a supply with converters there are practically no harmonics in current and voltage, which is desirable for the first operation because of the EMC sensitivity of the measuring equipment. The second step is a connection to the grid as visible in figure H7.1.20. The SMES is fed by an uncontrolled B6-bridge and a modified d.c. chopper. To apply the rectified voltage Ud to the SMES, T1 and T2 are fired. In the stand-by mode T2 is fired; the storage circuit is closed over D1. If neither of the thyristors is fired; D2 becomes conducting and the SMES is discharged by the dump resistor Rd c h . By operating T1 in the switching mode (T2 fired) the supplying voltage can be varied steplessly between 0 and Ud . In the same manner the apparent dump resistor can be varied between 0 and R d c h by operating T2 in the switching mode. In this testing phase several predefined charging cycles are rehearsed and the former

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Figure H7.1.20. Experimental set-up.

calculations on the thermal behaviour and on the determination of superconductor losses are verified on the basis of the test results. Tests with thermal overload by fast cycles, the investigation of the thermal limits of the plant, functional tests of the current safeguard, the protection system and the central control as well as experiments on the quench behaviour are all relevant to this subject. Finally, the assembly in the last testing phase corresponds to the equipment for future applications. The SMES system is connected to the grid by a pulse-width-modulation inverter, a voltage link and a d.c. chopper. In this testing phase experience in the operation of SMES in the following different applications should be obtained: ( i ) smoothing symmetric and asymmetric voltage drops; ( ii ) load levelling; ( iii )generating high-power pulses; ( iv )damping of generator oscillations; ( v ) working with generators in separate networks. Additionally another concept for connecting the SMES system to the power system should be examined as it provides a resonant current converter and water-cooled pulse-width-modulated GTOs serve as switches. By using a regenerative snubber and an optimized trigger logic, switching frequencies of 1 kHz are obtained. To enable the flow of current in the storage coil in cases of failure, an additional independent safety circuit is provided. This converter has the advantage of requiring minimum effort in semiconductor devices and therefore lower losses. However, in the power range of interest there is no experience in operating with this technique. Besides, this method makes it impossible to upgrade an existing plant with other storage components. H7.1.10 Critical evaluation and prospects In general energy storage systems are economically and technically favourable if they: ( i ) overcome economically serious situations in the short term; ( ii ) delay investments in the medium term or make them completely unnecessary; ( iii )are imperative due to the time-varying supply of energy.

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In table H7.1.12 several applications of energy storage systems are shown which are suited to SMES systems as well. Hence small, fast-acting SMES systems are most likely to be chosen for applications characterized by a limited storage capacity and a short reaction time: ( i ) levelling the load noise of consumers that have a rapidly changing power demand within a period of 0.5-5 s; ( ii ) balancing of short time, symmetrical and asymmetrical voltage drops resulting in reaction times of 0.2-1 s; ( iii )perhaps improving the stability in power supply systems.

References Birkner P 1993 On the design of superconducting magnetic energy storage systems IEEE Trans. Appl. Supercond. AS-3 246–9 Brammer U and Lorenzen H W 1995 Magnetic shielding of small high power SMES IEEE Trans. Appl. Supercond. AS-5 329–32 Dewinkel C C, Billmann J and McCann K K 1995 Micro-SMES and power quality: technology, applications, markets Use of Superconductivity in Energy Storage ed K P Juengst, P Komarek and W Maurer (Singapore: World Scientific) pp 328–38 Ehsani M and Kustom R L 1988 Converter Circuits for Superconductive Magnetic Energy Storage 1st edn (College Station, TX: A&M University Press) File J, Mills R G and Sheffield G V 1971 Large superconducting magnet design for fusion reactors IEEE Trans. Nucl. Sci. NS-18 277–82 Fleischer T 1995 Technology assessment of superconducting magnetic energy storage Use of Superconductivity in Energy Storage ed K P Juengst, P Komarek and W Maurer (Singapore: World Scientific) pp 351–66 Hassenzahl W 1989 A comparison of the conductor requirements for energy storage devices made with ideal coil geometries IEEE Trans. Magn. MAG-25 1799–802 Huang X, Kral S F, Lehmann G A, Lvovsky Y M and Xu M 1995 30 MW Babcock and Wilcox SMES program for utility applications IEEE Trans. Appl. Supercond. AS-5 428–32 Juengst K P and Salbert H 1996 Fast SMES for generation of high power pulses IEEE Trans. Magn. MAG-32 2272–5 Kaerner J F 1995 Load levelling in railway substations using high power energy storage Elektrische Bahnen 93 49–57 (in German) Copyright © 1998 IOP Publishing Ltd

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Kaerner J F and Lorenzen H W 1992 Converter for connecting SMES to electrical power supply systems Proc. 1992 Int. Symp. on Power Electronics (Seoul, 1992) pp 189–97 Komarek P 1995 High Current Application of Superconductivity (Stuttgart: Teubner) (in German) Lorenzen H W, Brammer U, Rosenbauer F and Schoettler R M 1995 Small, high-power 1 MJ SMES Use of Superconductivity in Energy Storage ed K P Juengst, P Komarek and W Maurer (Singapore: World Scientific) pp 120–30 Masada E, Myohi H, Ogiso K and Nonaka F 1995 Development of elementary technologies for 100 kWh/40 MW toroidal SMES Use of Superconductivity in Energy Storage ed K P Juengst, P Komarek and W Maurer (Singapore: World Scientific) pp 97–109 Masuda M and Shintomi T 1987 The conceptual design of utility-scale SMES IEEE Trans. Magn. MAG-23 549–52 Murgatroyd P N 1986 The Brooks inductor: a study of optimal solenoid cross-sections IEE Proc. 133 309–14 Rogers J D, Boening H J and Schermer R J 1985 Operation of the 30 MJ superconducting magnetic energy storage system in the Bonneville Power Administration electrical grid IEEE Trans. Magn. MAG-21 752–5 Rosenbauer F and Lorenzen H W 1996 Behaviour of IGBT modules in the temperature range from 300 to 5 K Advances in Cryogenic Engineering vol 41 (New York: Plenum) pp 1865–72 Schoenwetter G and Gerhold J 1995 Design of SMES with reduced stray field IEEE Trans. Appl. Supercond. AS-5 336–9 Schoettler R M 1994 Cooling of small SMES system with forced flow supercritical helium Cryogenics 34 825–31 Shafranov V D 1973 Optimum shape of a toroidal solenoid Sov. Phys.-Tech. Phys. 17 1433–7 Spaeth H, Steinhart H, Komarek P, Juengst K P, Maurer W, Haubrich H-J and Tischbein T 1995 100 kJ/100 kW SMES for compensation of fast pulsing loads Use of Superconductivity in Energy Storage ed K P Juengst, P Komarek and W Maurer (Singapore: World Scientific) pp 131–40 Tsutsumi K, Tokunaga T, Irie F, Okada H, Ezaki T and Takeo M 1995 Development of a 1 kWh/1 MW module type SMES Use of Superconductivity in Energy Storage ed K P Juengst, P Komarek and W Maurer (Singapore: World Scientific) pp 328–38 Vollmar H E and Altpeter R 1990 Current SMES-designs and stray field considerations Superconductivity in Energy Technologies (Duesseldorf: VDI) pp 140–51 Wesche R 1992 Optimization studies of solenoidal windings for superconducting magnetic energy storage Cryogenics 32 578–83 Wilson M N 1990 Superconducting Magnets (New York: Oxford University Press) p 24

Further reading Brammer U and Rasch P 1996 Optimization of ferromagnetic shields for solenoidal SMES IEEE Trans. Magn. MAG-32 1274–7 Juengst K P, Komarek P and Maurer W (eds) 1995 Use of Superconductivity in Energy Storage (Singapore: World Scientific). (These conference proceedings of an IEA symposium present a broad survey of present SMES projects.) IEEE Trans. Appl. Supercond. (IEEE Trans. Appl. Supercond. publish the latest papers related to the field of applied superconductivity. The June volumes of uneven years may be of special interest, since they contain the proceedings of the biennial Applied Superconductivity Conference (ASC).) Komarek P 1995 High Current Application of Superconductivity (Stuttgart:Teubner) (in German). (This book offers delightful reading on high current applications of superconductivity. The main topics are theory of superconductivity, superconducting cables, magnet design, research magnets, superconductivity in power engineering and nuclear fusion and economical aspects. However, there is no English translation.) Mohan N, Undeland T M and Robbins W P 1989 Power Electronics: Converters, Applications and Design (New York: Wiley). (This book gives a broad insight into power electronics. Besides the detailed description of fundamental circuits it explains a multitude of applications in various fields of electrical engineering. Additionally the physics of all common power semiconductors is discussed and some practical hints for the design are given.) Wilson M N 1990 Superconducting Magnets (New York: Oxford University Press). (This standard work covers all aspects relevant for the design and the fabrication of superconducting magnets. The topics are thoroughly discussed in a precise but nevertheless easily understandable manner.)

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H7.2 The impact of SC magnet energy storage on power system operation E Handschin and Th Stephanblome

H7.2.1 Introduction In order to maintain reliable, economical and high quality electric energy systems in the future, it will be increasingly necessary to take the following aspects into account: ( i ) increased requirements on the part of the consumers with respect to voltage quality and frequency stability in large interconnected power systems is now an important consideration in many parts of the world ( ii ) limits on the reinforcement of existing networks ( iii )environmentally sustainable, efficient use of the available primary energy sources. Energy storage devices can make an important contribution to meeting these requirements. Significant developments in the field of materials science means that there will be considerable future scope for the use of superconducting magnetic energy storage (SMES). The direct storage of electric energy in the field of a superconducting coil (SC coil) allows access times in the range of milliseconds, while the duration and the number of charge/discharge cycles (in contrast with other storage devices) have no influence on the lifetime of an SMES. Furthermore, a charge/discharge efficiency of up to 95% can be obtained. The energy stored in an SMES system is given by the integral over the volume of the field distribution according to

(All SMES variables are characterized by the subscript SM.) The principle of the SMES construction can be represented by the structure shown in figure H7.2.1. The transformer and the converter represent the interface between the grid and the SC coil. According to the voltage US M adjusted by means of the firing angles of the thyristors at the converter, the SC coil will be charged or discharged, while the firing angles are set by an SCR (silicon-controlled rectifier) control circuit, which gets its set value from a controller

The voltage USM is limited by the insulation resistance of the used thyristors as well as by the admissible rate of change of current |dISM /dt| with respect to the losses within the coil. Apart from this

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Figure H7.2.1. Block diagram of an SMES system.

the stability of the superconductor can be characterized by means of the maximal rate of change of active power |dISM /dt|. From equation (H7.2.2) the actual achievable power of an SMES system can be found

Equation (H7.2.3) shows that the achievable power is always proportional to the square root of the stored energy WS M . The cooling system guarantees the required operating temperature of the SC coil, which is about 4 K for metallic superconductors and 20 K or higher for ceramic superconductive material. In the case of a quench—the uncontrolled change of the SC coil from a superconducting to a normal-conducting state—a coil protection ensures that the SC coil discharges with maximal speed to prevent the destruction of the system. In contrast to the active power allocation, which is limited by the actual energy content WSM of the SMES, the allocation of reactive power is not limited in time or in quantity. A complete four-quadrant operation of the SMES becomes possible by using GTO (gate turn off) thyristors. In this case there is no need for a reactive power compensation. The limitations of the operating area shown in figure H7.2.2 are determined by the maximal active power PS M , m a x of the SC coil and the maximal apparent power SS M , m a x of the converter unit. By means of these limitations the SMES active power control loop and the SMES reactive power control loop are coupled. In the case of an apparent power demand outside the operating area shown in figure H7.2.2 the provision of reactive power should be subordinated with respect to the provision of active power. Nevertheless reactive power control by means of SMES should be possible even if PS M , m a x ≥ PS M , m a x For this reason PS M , m a x < PS M , m a x is assigned as shown in figure H7.2.2. Depending on the different power system operational requirements the use of SMES with a capacity in the range of a few MW h as well as the use of SMES with a capacity in the range of about 1000-5000 MW h has been discussed (Bayer et al 1994b, Komarek 1991). With respect to the possibilities of SMES the following requirements may be met by SMES with a low or medium storage capacity: ( i ) allocation of seconds reserve power ( ii ) improving the dynamic and transient stability because of the immediate allocation of synchronizing damping power by the SMES ( iii )an improved voltage quality by means of reactive power compensation of the converter

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Figure H7.2.2. The operating area of an SMES system.

( iv )lifetime prolongation of the components of primary—controlled power plants by smoothing the operation of the primary control ( v ) damping of low—frequency power oscillations ( f < 1 Hz) on the transmission lines ( vi )increasing the maximal stable transmission of load with long transmission lines. An additional feature of a large SMES is: ( vii )smoothing of the daily load level by consuming and providing electrical energy with very high efficiency. The following only deals with SMES with a capacity of a few megawatt hours according to the requirements mentioned before, because the operation of such units seems to be technically and economically reasonable in the near future. To achieve the maximum possible degree of economic attractiveness, the use of the SMES should be planned on the basis of the existing power system control concept with the aim of replacing various control components in as efficient a way as possible. A feasibility study of SMES shows that in comparison to conventional storage devices an SMES system seems to be economical when either the energy content of the storage device is very high (and thus the specific storage costs drop due to the related cost reduction) or when a large power increase is required just for a short time (Bayer et al 1994a). The latter applies in particular when providing seconds reserve power. A further increase in economical attractiveness can be achieved by using the SMES simultaneously for the provision of seconds reserve power and for the other possibilities mentioned above. Therefore a system engineering approach (figure H7.2.3) taking the technical and economical properties of SMES into account is presented. The system engineering approach for integration of SMES into the power system control is applied to the model power system shown in figure H7.2.4. Due to the interconnected operation a certain amount of seconds reserve power has to be allocated within the model power system. For economical reasons (Bayer et al 1994a) seconds reserve power in the model power system shown in figure H7.2.4 is provided exclusively by SMES. On the other hand dynamic and transient stability problems can arise according to the entire system load P1, and the contribution of the interconnected system in the short—circuit power SS C . I S at the bus where the model power system is connected to the interconnected system. Copyright © 1998 IOP Publishing Ltd

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Figure H7.2.3. The system engineering approach.

Figure H7.2.4. A model power system with a load of P1.0 = 1.2 GW in the border system. Pr 1, Pr 2, Pr 3: rated power of the plants.

on:

The use of SMES within the model power system shown in figure H7.2.4 will be illustrated based

( i ) the economical provision and allocation of seconds reserve power ( ii ) the damping of electromechanical oscillations ( iii )the improvement of voltage quality. Starting from these requirements the dimensions and location of the SMES will be determined and the control scheme will be designed. H7.2.2 Modelling In order to study the short— and mid—term impact of SMES on power system dynamics the power system and the SMES are modelled with respect to this time range. Concerning the power system well known models are used for the generators (Anderson et al 1990), the voltage control circuits including the power system stabilizer (PSS) (Handschin and Wohlfahrt 1988, IEEE Committee Report 1981) and the turbine control circuits valid for this time range (Bongers et al 1981). In the time range considered constant firing is assumed and hence the secondary control is not modelled. Furthermore the steady—state network equations are used; load modelling is based on the use of constant load admittances.

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The SMES model applied to the system engineering approach under investigation results from the power engineering requirements of the SMES. Thus a detailed modelling of all the SMES components shown in figure H7.2.1 does not take place. The power flow between the superconducting coil and the grid is determined by the firing angles of the thyristors within the converter unit. Because the time constants of the thyristors as well as those used in the calculation of the firing angles are in the range of a few 10–3 s a consideration of the calculation of the firing angles, the thyristors and the converter circuit is not necessary with respect to investigations of the impact of SMES on short— and mid—term dynamics. As a result active and passive filters for reducing the harmonic content of the voltage and current caused by the converter unit are not taken into consideration. The SMES model applied to the system engineering approach under investigation is shown in figure H7.2.5.

Figure H7.2.5. The SMES model.

Two first—order time delay elements represent the dynamic performance of the provision of active power PS M and reactive power QS M By means of the time constant for the active power, TP S M , the time delay of the converter unit as well as the maximal rate of change of active power, |dPSM /dt|m a x , limited by the SC coil itself are taken into account. The time constant for the reactive power, TQ S M , represents the time delay occurring in connection with the reactive power compensation of the converter. H7.2.3 System engineering approach H7.2.3.1 Dimensioning of SMES In electrical energy systems, spontaneously obtainable electric reserve power must be available for the case of an unforeseeable power deficit (e.g. as a result of a loss of a power plant unit). In this case each partner within the European interconnected network (UCPTE) has to provide 2.5% of the current network load as seconds reserve power. To illustrate the activation of this power we will focus on the methods used in Germany. The situation in other countries varies in detail but not in quality. According to the recommendations of the Deutsche Verbundgesellschaft (DVG: association of German interconnected power companies), one half of this seconds reserve power must be activated within 5 s and the remainder in a further 25 s. The seconds reserve power must be maintainable until it is replaced by the minutes reserve power activated by the secondary control (DVG 1992) (figure H7.2.6).

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Figure H7.2.6. DVG recommendation (DVG 1992).

Figure H7.2.7. Possible methods of activating seconds reserve power (—) and minutes reserve power (—. —) in a steam power plant and in a decentralized SMES: T—increase in steam generator thermal output; H—dethrottling of high-pressure control valves; V—opening of overload valve; I—dethrottling of low—pressure control valves; HS—steam—side trip of high—pressure feedwater heaters; HW—water—side bypassing of high—pressure feedwater heaters; C—condensate flow shut—off (closure of condensate isolating valve); LS—steam—side trip of low—pressure feedwater heaters.

Within a power system the dispatcher has at its disposal the possibilities for providing seconds reserve power as shown in figure H7.2.7. Its use is planned according to power plant availability, dynamic capabilities of the individual unit types and costs incurred. Large thermal power plants are designed for sliding pressure operation. The currently most frequently applied method of provision of seconds reserve power is throttling of the turbine control valves, i.e. ‘modified sliding pressure operation’ (figure H7.2.7). Throttling causes a constant heat consumption increase of about 0.3% in a power plant unit, if this unit provides 2.5% seconds reserve power related to its rated power output (Kürten 1986). A further procedure for the provision of seconds reserve power is the condensate stop, hereafter more accurately referred to as preheated interrupting. The low—pressure feedwater preheaters are interrupted by fast—acting control valves in the extraction lines (figure H7.2.7, label LS) and at the same time the condensate flow is stopped (figure H7.2.7, label C). This quickly achieves a power increase in the turbine stages downstream of the steam extraction. Because of this interruption by the preheater it is possible to reduce throttling. This discontinuous measure for power increase is Copyright © 1998 IOP Publishing Ltd

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activated in combination with the dethrottling in the event of major faults only. SMES could provide the entire required seconds reserve power or could be used in combination with preheater interruption or other measures. The economic attractiveness of SMES has been investigated for different operating constraints in comparison with the conventional provision of seconds reserve power (Bayer et al 1994a). Figure H7.2.8 summarizes the most important results of this preliminary study.

Figure H7.2.8. Present value of the specific costs for the provision of seconds reserve power.

The present value method was used to calculate the specific reserve costs of providing seconds reserve power; current figures for the interest rate, bank discount rate and price increase rate were taken into account (Martin and Naser 1986). For purposes of comparison with other measures, not only the operating costs but also the so—called relocation costs have been taken into account; they are incurred when the optimal power plant schedule calls for the unit’s full output, which cannot, however, be provided owing to the seconds reserve power which must be kept available. Thus, the investment costs involved in the installation of substitute power had to be applied in addition to the supplementary costs of the increased heat consumption. For the throttling, this resulted in a present value of the specific reserve costs of more than 5 million DM MW-1 (DM: Deutsche Mark) of reserve power, assuming mean specific power plant costs of 2.4 million DM MW-1 (figure H7.2.8). On the other hand, the specific costs for the combination of throttling with preheater interruption can be reduced compared with pure throttling (figure H7.2.8). As the response time is in the milliseconds range, SMES is technologically extremely suitable for the provision of seconds reserve power. As the specific costs of SMES decrease with increasing storage capacities, there was also a substantial drop in specific costs for the seconds reserve power with increasing power. Figure H7.2.8, curve 3 shows the present value of the specific reserve costs as a function of the SMES power, assuming a discharge time of 5 min for operation of the SMES at full power. For these efficiency calculations, the energy—related SMES costs (approximately 50% of total costs), such as those for the coil, cryostat and cooling system, were interpolated between magnetic resonance tomograph list prices in the kilowatt hour range and estimated costs (from the USA) in the gigawatt hour range. For the case of seconds reserve power discussed here in the megawatt hour range, this cost estimate has to be carefully checked by additional system designs and cost calculations. The calculation of the SMES costs has taken into account both the energy—related low—temperature losses (through the cryostat wall) and the power—related losses (current leads and converters). The electrical energy for cooling the superconducting coil and also the converter and transformer losses have been priced at 0.1 DM (kW h)-1. The number of charge/discharge cycles has only minimal influence and does not displace the cost relations between the various measures. The storage costs increase slightly as the number of cycles becomes higher, as here the losses increase. The losses in throttling decrease slightly, as the power plant is operating closer to its optimum.

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In comparison with throttling as well as in comparison with a combination of throttling with preheater interruption, SMES produces cost advantages as shown in figure H7.2.8. The outcome of this is that the entire seconds reserve power, which has to be allocated within the model power system, should be provided by one SMES exclusively. Thus the allocation of seconds reserve power by SMES is the criterion for the dimensioning of the SMES. In order to meet the DVG recommendation within the model power system shown in figure H7.2.3, the rated power PS M. m a x of the SMES has to be

The SMES has to provide this power until it is replaced by the minutes reserve. The energy, WS W , can be calculated according to the characteristics shown in figure H7.2.6:

Assuming that the SMES is on average only about one half charged and that the SMES is not discharged by more than one tenth of its maximum storage capacity for technical reasons, the SMES capacity WS M. m a x can be calculated

The converter is to be dimensioned to allow for |QS M| = |PS M| even if PS M = P S M. m a x . This yields SS M. m a x = 141 MV A. It must be emphasized that an SMES system with such an oversized converter is still highly economically attractive with regard to providing seconds reserve power. Concerning the dynamic performance of SMES the time constants TP S M and TQ S M (figure H7.2.5) are assigned with respect to Mitani etal (1990) and Handschin et al (1994): TP S M = 0.03 s; TQ S M = 0.015 s. H7.2.3.2 Location of SMES In respect of economical installation, the SMES location must be determined with regard to providing seconds reserve power and achieving maximal improvement of short term dynamics. Because of the existing transmission capacities within electrical supply systems, the provision of seconds reserve power causes no restrictions with respect to the SMES location. Thus the SMES location can be exclusively determined with respect to an improvement of short-term dynamics. To obtain an optimal coordination of power plant and SMES control, it is suggested in this study that the SMES should be located in the vicinity of an equivalent power plant (figure H7.2.4). In order to determine the optimal location of SMES with respect to a maximal improvement of stability, a sensitivity analysis of the eigenvalues of the system is carried out. By means of a sensitivity analysis, the optimal location of SMES with respect to the three equivalent power plants (figure H7.2.4) can be determined such that the subsequent controller optimization entails maximum improvement of short-term dynamics (Feliachi 1990). Referring to the damping of power and hence rotor oscillations, the SMES active power control as well as the PSS have an influence on the system eigenvalues correlated with these oscillations. A PSS represents a supplementary excitation control for damping oscillations caused by negative damping of high-speed excitation systems. In the case of simultaneous use of a PSS and SMES at the same power plant it is possible that the fast voltage control of the SMES reactive power control compensates for the voltage modulation of the generator terminal voltage by the PSS. With respect to the advantages of the four-quadrant operation of SMES the simultaneous use of SMES and a PSS at one power plant will not be examined. Thus the following concept for locating SMES and a PSS within the model power system

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shown in figure H7.2.4 is taken into consideration: one power plant equipped with SMES / two power plants equipped with PSS. The eigenanalysis used for determining the power plants which have to be equipped with SMES and PSS starts by linearizing the nonlinear model of the system shown in figure H7.2.4 around a given operating point not taking SMES and PSS into account. The linearized, closed-loop system can be described by a set of linear, time invariant (LTI) differential equations in the state space form

where A is the [n, n] system matrix B is the [n, m] input matrix and C is the [m, n] output matrix. There are differences concerning the effect of each eigenvalue of the system matrix A on the dynamic performance of the system. An eigenvalue analysis according to equation (H7.2.8) represents a reliable method for calculating those eigenvalues which are dominant with regard to power and rotor oscillations (Feliachi 1990). In the following the abbreviation λi , is used for those dominant eigenvalues. The eigenvalues λi , are those eigenvalues which participate most in the so-called criticality index

The criticality index is a value that describes the dynamic performance of a system after a certain excitation of the system. Thereby the dynamic performance of the system after a certain excitation (usually a step or pulse function) is called the response of the system. In equation (H7.2.8) ym(t) represents the response of the systems, if there is a pulse shape change of load in the vicinity of power plant m. The system is excited by this impulse-shaped change of load because of the frequency spectrum of this excitation. The Q matrix has a diagonal structure

Taking into account the structure of the output vector y the variables qk k are determined such that the criticality indices with respect to the rotor oscillations of the jth generator

as well as the criticality indices with respect to the power oscillations of the jth generator

can be examined. Starting from the n = 38 eigenvalues of the model power system shown in figure H7.2.4 the subsequent eigenvalues can be characterized as being dominant with respect to the rotor and power oscillations of the three generators within the model power system: λ1 = −0.2146 ± j5.1441; λ2 = −0.6761 ± j9.6098; λ 3 = −1.081 ± j10.4377. In order to achieve a concept of SMES and PSS location which is as independent as possible of the operating point we have to examine whether the dominant eigenvalues characterized above are still Copyright © 1998 IOP Publishing Ltd

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dominant even if the operating point of the system is changed. This examination is performed for a variation of the entire system load P1 (P1.0 ≤ P1 ≤ 1.5P1.0 ) and a variation of the contribution of the interconnected system in the short-circuit power SS C. I S at the bus where the model power system is connected to the interconnected system (4 GV A ≥ SS C. I S ≥ 2 GV A) It is shown by Stephanblome (1994) that the three eigenvalues characterized above are still dominant with respect to the rotor and power oscillations even if P1 and SS C. I S are varied within the defined limits. With regard to the controller synthesis, the configuration of the dominant eigenvalues as a function of the different operating points characterized by, (P1.S S C. I S ) has to be taken into consideration (table H7.2.1).

Figure H7.2.9. Linearized system with static feedback.

In respect of the evaluation of the optimal location of SMES and a PSS a sensitivity analysis of the dominant eigenvalues with respect to static feedback (figure H7.2.9) is applied. With regard to the static feedback Kp j (figure H7.2.9) the variables yp j and up j , respectively, are elements of the output vector y and the input vector u while the output matrix C and the input matrix B according to equation (H7.2.7) are transformed into the vectors Cp j and Bp j . For calculating the sensitivity dλi /dKp j of the eigenvalue λi with respect to the static feedback Kp j the system shown in figure H7.2.7 is represented using the left-hand-side eigenvector ui (equation (H7.2.12)) and the right-hand-side eigenvector wi (equation (H7.2.13)):

Considering equation (H7.2.12) and (H7.2.13) as well as a normalization of the eigenvectors

the sensitivity δ λi /δ Kp j of the eigenvalue λi with respect to the static feedback Kp j can be calculated

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The sensitivity δ λi /δ Kp j is a complex variable. To obtain an evaluation and a comparison of the damping effects of different feedback schemes, the real part Re[δ λi /δ Kp j ] < 0 of the sensitivity according to equation (H7.2.15) is used. The sensitivity approach should be applied to the location of the SMES and PSS. The output of the PSS serves as an input uj = uP S S. j for the generator voltage control loop. Concerning the SMES there is an impact of the SMES active power control on the torque balance of the jth generator. The linearized system equations (H7.2.7) are extended by a first-order time delay element (TP S M. j = 0.03 s, see section H7.2.3.1) acting on the torque balance of every generator within the system while the input variable of this first-order time delay element will be u j = PS M. s e t. j . This modification of the linearized system represented by equations (H7.2.7) does not affect the dominant eigenvalues of the system under consideration (Feliachi 1990). With respect to the rotor and power oscillations which have to be damped, a feedback of the accelerating power yj = Pb j , the generator power yj = PG j and the generator speed y = wG j have to be taken into account. The feedback of the accelerating power Pb j yields a maximization of the absolute value of the real part Re[δ λi /δ Kp j ] which does not depend on the operating point, the dominant eigenvalue under consideration or the use of SMES or PSS. Thus the following factors concerning for the location of the SMES and PSS are taken into account: ( i ) SMES at power plant 1 / PSS at power plants 2 and 3 (input variable of SMES and PSS is Pb j ) ( ii ) SMES at power plant 2 / PSS at power plants 1 and 3 (input variable of SMES and PSS is Pb j ) ( iii )SMES at power plant 3 / PSS at power plants 1 and 2 (input variable of SMES and PSS is Pb j ). The sensitivity according to equation (H7.2.15) characterizes the impact of the static feedback existing in the system on the shifting of the eigenvalueλ According to the simultaneous use of SMES and PSS the term

is calculated for each possible location factor and for the two operating points B1 (Pl = P1.0 ; Ss c. i s = 4 GV A) and B2 (P1 = 1.5P1.0 ; Ss c. i s = 2 GV A) representing the limits of the operating area under consideration. From figure H7.2.10 it can be seen that the use of SMES at power plant 2 and the use of a PSS at power plants 1 and 3 yields the optimal damping of the rotor and power oscillations while the result does not depend on the actual operating point. H7.2.3.3 SMES control scheme The control of the SMES located at power plant 2 (section H7.2.3.2) has to provide the seconds reserve power as well as improve the short-term dynamics. Whereas for allocation of the seconds reserve power the SMES has to be adapted to the other primary controllers within the power system, for the improvement of the short-term dynamics a suitable allocation of active and reactive power is necessary. Therefore an event driven control scheme (figure H7.2.11) yields optimal operation conditions (Stephanblome 1994). Load changes are combined with low-frequency or aperiodic changes of system frequency. In this case the system frequency is a useful input signal for the SMES controller II shown in figure H7.2.11, whereas improving the short-term dynamics requires immediate and sufficient allocation of active power and reactive power. Due to this and with respect to the results of section H7.2.3, Pb 2 and the generator terminal voltage UG 2 (see (a) below) are used as input signals of controller I.

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Figure H7.2.10. (a) SMES at power plant 1; (b) SMES at power plant 2; (c) SMES at power plant 3.

Figure H7.2.11. Event-driven SMES control scheme.

(a) Synthesis of the SMES controller II So that SMES can provide a predefined part of the seconds reserve power according to the agreements of the interconnection partners it has to be adapted with a suitable control to the behaviour of the other primary controllers used in the interconnected system. With respect to the coordinated provision of seconds reserve power by the interconnected network and the SMES according to the DVG recommendations the SMES controller II is realized by means of a proportional controller. The gain factor arises from the provision of the maximal power PSM. max of the SMES at the moment of the loss of the biggest plant within the system on the one hand and by the drop of the other primary controllers within the system on the other hand. According to the existing DVG recommendations, the maximal allocation of seconds reserve power, PS M. m a x = 100 MW, by means of the SMES has to take place 30 s after an active power change ∆P1.I S = 0.05(Pr 1 + Pr 2 + Pr 3 + Pr. I S) = 2600 MW. One half of this power should be allocated within 5 s. There will be a frequency deviation ∆fD V G. s t a t depending on the amplitude ∆P1.I S of the load change. Neglecting secondary control effects and assuming a unique drop s = 5% of all power plants in primary Copyright © 1998 IOP Publishing Ltd

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control of the interconnected system, the frequency deviation ∆fD V G. s t a t can be calculated

Thus the gain factor KS R P of the SMES controller I can be determined

In figure H7.2.12 the allocation of seconds reserve power by means of the power plants in primary control of the interconnected system is illustrated for a load change ∆P1.I S = 2600 MW within the interconnected system.

Figure H7.2.12. Allocation of seconds reserve power ∆Pt, I S by the power plants in primary control of the interconnected system (∆P1. I S = 2600 MW).

If the SMES is not taken into account, the real power ∆Pt, I S is provided solely by the power plants in primary control of the interconnected system. If the SMES is taken into account the power plants in primary control of the interconnected system are relieved from the allocation of seconds reserve power according to the ratio between the rated power capacities of the border system on the one hand and the interconnected system on the other hand. The SMES provides seconds reserve power due to its parametrization while meeting the DVG recommendation (t = 5 s; PS M = 50 MW). (b) Synthesis of the SMES controller I by means of pole placement Within this approach the requirements on SMES with respect to improving short-term dynamics are now concerned with the damping of the dominant eigenvalues of the system. Because of this the static feedbacks Kp j of the factors affecting SMES and PSS location, as determined in section H7.2.3.2, must be assigned such that all dominant eigenvalues are shifted to the left as far as possible. Because of the use of such locally available input variables it becomes possible to parametrize the SMES active control loop and the SMES reactive power control loop independently of each other (Stephanblome 1994). The SMES reactive power control loop should be subordinated to the SMES active power control loop. Consequently the parametrization of the static feedbacks Kp j of the PSS and the SMES active power control loop is done first and, taking into account the results of this parametrization, the parameters of the SMES reactive power control loop are assigned next. Copyright © 1998 IOP Publishing Ltd

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For determining the optimal SMES and PSS location as described in section H7.2.3.2 the real part Re[δ λ i /δ Kp j ] of the sensitivity is used. In order to achieve an optimal damping of the dominant eigenvalues a phase-shifting transfer function is also taken into consideration (figure H7.2.13).

Figure H7.2.13. Transfer function Hp j (s) of the feedback loop of the linearized system.

If we consider the optimal damping of the dominant eigenvalue λi the phase shifting by means of the transfer function Fp j (s ) must be

Hence the transfer function Hp j (s ) is realized by means of the lead-lag structure shown shown in figure H7.2.14.

Figure H7.2.14. The lead-lag structure.

Thus the PSS as well as the SMES controller II are realized by means of lead-lag structures according to figure H7.2.14. Taking the phase-shifting transfer function Fp j (s ) into account the sensitivity of the dominant eigenvalue λi with respect to the feedback transfer function Hp j (s ) can be calculated

If the approach for determinating the optimal location of SMES and PSS takes the sensitivity of the static feedback Kp j into consideration, the time constants T1 p j and T2 p j of the phase-shifting transfer function Fp j (s ) have to be calculated first. Because of this the frequency ωm a x has to be assigned with respect to the phase-frequency characteristic of Fp j (s ) (shown in figure H7.2.14) so that an optimal damping of all dominant eigenvalues λi under consideration can be achieved. Depending on the time constants T1 p j and T2 p j , the required phase shifting arg(Fp j (ωm a x )) becomes

Moreover the following expressions are valid

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Figure H7.2.15. (a) PSS at power plant 1, (b) PSS at power plant 2, (c) PSS at power plant 3.

The time constants T1 p j and T2 p j can be assigned according to equation (H7.2.21) and (H7.2.22). With respect to the model power system shown in figure H7.2.4 this approach is applied to the parametrization of the SMES located at power plant 2 and the PSS located at power plants 1 and 3 (section H7.2.3.2). The arguments of the sensitivities δ λi /δ Kp j of the SMES/PSS concept under consideration are shown in figure H7.2.15. The dominant eigenvalue λi has the most effect on the electromechanical oscillations within the systems, especially if the static stability of the system decreases (Stephanblome 1994). Thus the damping of this eigenvalue is most important. In order to achieve optimal damping by

the phase shifting of the PSS at power plant 1 should be assigned to: arg(FP S S. 1 (λ1. B 1 )) = −30°, arg(FP S S, 1(λ1, B 2 )) = −36°, with respect to the arguments of the sensitivities δ λi /δ Kp S S. 1 shown in figure H7.2.15(a). The parameter ωm a x is assigned by calculating the arithmetic mean value of Im[λ1, B 1 ] and Im[λ1, B 1 ] (see table H7.2.1). Taking into account the phase-frequency characteristic of Fp s s. 1(s) shown in figure H7.2.13 a phase shifting of arg(FP S S, 1(ωm a x )) = −33° produces a sufficient damping of the dominant eigenvalue λ1 within the operating area under consideration. When thinking about the parametrization of FP S S. 1(s ), the dominant eigenvalues λ2 and λ3 should be taken into consideration too although they are of secondary importance. In particular in the case of (FP S S. 1(s )) < 0° it can be obtained from figure H7.2.12(a) that the damping of the dominant eigenvalue λ2 deteriorates. Consequently the phase shifting is established to be arg(FP S S, 1 (ωm a x )) = −20° because such a parametrization yields an accuracy of 10% concerning the achievement of equation (H7.2.19) with respect to the dominant eigenvalue λ1 . Copyright © 1998 IOP Publishing Ltd

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According to equations (H7.2.21) and (H7.2.22) the requirements on the phase shifting by FP S S. 1 (s) lead to: T1.P S S. 1 = 0.1837 s, T2. P S S. 1 = 0.2572 s. The illustrated approach concerning the parametrization of the phase-shifting transfer function FP S S. 1(s ) is also applied to the parametrization of the phase shifting transfer functions FS M E S, 1 (s ) and FP S S. 3 (s ). From this it is found that the time constants of FS M E S. 1(s ) and FP S S. 3(s ) are T1, S M E S, 2 = 0.0681 s, T2. S M E S. 2 = 0.2997 s, T1. P S S. 3 = 0.1537 s, T2. P S S. 3 = 0.3074 s. The corresponding phase-frequency characteristics of FP S S. 1(s ), FS M E S, 2 (s ) and FP S S, 3(s ) are shown in figure H7.2.16.

Figure H7.2.16. The phase-frequency characteristic of FP S S, 1(s ), FS M E S, 2(s ) and FP S S, 3(s ).

Since the time constants T1 p j and T2 p j of the phase-shifting transfer functions Fp j (s ) are determined, the sensitivities δ λi /δ Hp j (s ) can be calculated according to equation (H7.2.20). By means of the static feedbacks Kp j the dominant eigenvalues should be shifted to the left as far as possible. Assuming a certain shifting |∆λi| of a dominant eigenvalue λi the gain factor of the static feedback Kp j can be calculated

Concerning the shifting |∆λi| of a dominant eigenvalue λi it must be taken into consideration that the transfer function Hp j (s ) also causes a shifting ∆λr of the nondominant eigenvalues λr of the system. This may yield a significant deterioration of the damping of certain eigenvalues and even instabilities because of a shifting of these eigenvalues into the right half-plane. Thus the shifting of those eigenvalues which are not dominant has to be limited and an adaptive stochastic algorithm (Krug and Schonfeld 1981) is applied in order to optimize the Kp j with respect to a maximal shifting |∆λi| of the dominant eigenvalues λi while limitations concerning the shifting ∆λr of the nondominant eigenvalues λr are taken into account. With regard to the model power system shown in figure H7.2.4 the target functions:

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have to be maximized with respect to the restrictions

During this optimization process the entire configuration of eigenvalues of the system is changed according to the parametrization of the static feedbacks Kp j . Assuming equations (H7.2.27) and (H7.2.28) it is shown by Feliachi (1990) that even if the entire configuration of the eigenvalues is varied the impact of the eigenvalues λr + ∆λr on the oscillations under consideration can be neglected with respect to the impact of the eigenvalues λi + ∆λi on those oscillations

With regard to the configuration of the nondominant eigenvalues the following assumptions yield a sufficient oscillatory characteristic of the closed-loop system (Feliachi 1990)

Due to this the parameters Γ and Y (equation (H7.2.26)) are assigned to

Applying the adaptive-stochastic algorithm yields KP S S. 1 = 0.98

KS M E S. 2 = 10.12

KP S S. 3 = 1.02.

The parametrization of the SMES reactive power control loop is done in the same way. First the dominant eigenvalue λU G 2 = 2.1 +j3.04 with respect to the generator terminal voltage UG 2 is identified. It can be shown by means of the mentioned sensitivity analysis that the generator terminal voltage UG 2 itself is the most suitable input signal with respect to the SMES reactive power control loop. Within the operating area under consideration the argument of the sensitivity δ λU G 2 /δ KQ− S M E S. 2 is about 180° ± 10° thus there is no need for a phase-shifting transfer function Fp j (s ) within the feedback loop and the SMES reactive power controller can be realized by means of a proportional acting controller. The adaptive stochastic algorithm is applied again for assigning the gain factor Kp 4 = KQ−S M E S. 2 of this proportional acting controller. The following target function is maximized taking into account the gain factors KP 1 = KP S S. 1 , KP 2 = KS M E S. 2 and KP 3 = KP S S. 3 calculated above

Concerning the shifting of the eigenvalues λr the restrictions formulated in equation (H7.2.26) are still valid. Furthermore the configuration of the dominant eigenvalues λi + ∆λi (i = 1, 2, 3) achieved by the optimization according to equations (H7.2.25) and (H7.2.26) should not be changed any more. The subsequent value of the gain factor KQ − S M E S. 2 is calculated KQ − S M E S. 2 = 5.21. Copyright © 1998 IOP Publishing Ltd

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Thus the parametrization of the SMES active control loop, the SMES reactive control loop and the PSS is completed. In order to study the impact of the SMES on transient stability the dynamic performance of the model power system shown in figure H7.2.4 after a three-phase short circuit on a double line (TSC = 0.18 s) with the following fault clearance by switching off the fault line will be analysed. Because of the choice of the fault position there are a considerable number of transient phenomena which have to be controlled by the power system control. The steady-state operating point B2 which is of relatively low static stability is used representing a peak load situation of the model power system P1 = 1.5P1.0 and a relative weak link to the interconnected power system (SS C. I S = 2 GV A). The short-term dynamics after the fault ares described by the frequency fG 2 and the terminal voltage UG 2 of generator 2 as shown in figure H7.2.17. It can be seen from figure H7.2.17 that without the use of SMES transient instability occurs after the three-phase short circuits with respect to the investigated operation point. During the short-circuit time TSC a large decrease of the terminal voltage occurs because of a relatively weak coupling to the interconnected system. Thus the load demand P1(U/U0 )2 of the voltage-dependent loads is reduced. With respect to the investigated peak-load situation and the constant driving torques of the generators an acceleration of the turbogenerator set occurs such that after switching off the fault line the synchronism cannot be restored and the system becomes unstable. At this operation point the additional destabilizing effect of the PSS in the transient time range is not essential for the system to show unstable behaviour. Therefore switching off the PSS does not guarantee the stabilization of the system in this special situation. Because of the fast transition between consuming and providing active and reactive power the use of SMES yields a significant improvement of the short-term dynamics. Within the case study under consideration when SMES is used the system becomes stable while without the use of SMES transient instability occurs. Figure H7.2.18 shows the active power PS M as well as the reactive power QS M provided by the SMES for the fault situation under consideration. During the fault time TS C on the one hand SMES reduces the imbalance between consumption and generation by means of consuming active power PS M (figure H7.2.18(a)). On the other hand the longitudinal voltage drop caused by the large reactive part of the short-circuit current can be decreased because the SMES allocates inductive reactive power QS M (figure H7.2.18(b)) and with this supports the voltage (figure H7.2.17(b)) during the fault time. When this voltage support occurs the SMES immediately further reduces the imbalance between power generation and consumption by the voltage dependent loads. Following the switching off of the fault line the SMES damps the electromechanical oscillations and the voltage oscillations by the allocation and charging of active power PS M and reactive power QS M respectively. In order to use SMES within the model power system the realization of the SMES voltage control loop will be investigated referred to the robustness of control quality depending on system load Pl and the contribution of the interconnected system in the short-circuit power SS C, I S at the bus where the model power system is connected to the interconnected system. Consideration with be given to the permanent variation of the parameters and changes of network topology caused by switchings within an electrical supply system, the maximal use of the capacity of existing generators and transmission lines and the constraints of planning of interconnections. Figure H7.2.19 shows the assessment of the transient stability and the voltage quality for the example of generator 2 and presents the time-dependent, squared and integrated deviation of the terminal voltage ∆UG 2 from the steady operation point (ITAE criterion) for the described three-phase fault dependence on Pl /Pl 0 and SS C. I S . Figure H7.2.19 illustrates that by means of SMES the use of generator and transmission capacities can be increased without instabilities even in the case of the largest assumed system fault. Moreover the use of SMES allows more degrees of freedom for planning the interconnection coupling of the border system. In this case the requirements resulting from stability aspects for the short-circuit power SS C. I S at

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Figure H7.2.17. (a) Frequency fG 2 of generator 2 as a result of a three-phase short circuit. (b) Terminal voltage UG 2 of generator 2 as a result of a three-phase short circuit.

the connecting nodes in the interconnected system can be reduced. Apart from this in the case of stable system operation without the use of SMES the application of SMES may lead to an increase of frequency and voltage quality. H7.2.4 Conclusions Through the use of SMES it becomes possible to meet requirements and quality improvements which cannot be realized by means of conventional techniques. This section has discussed the use of SMES for providing seconds reserve power and improving the short-term dynamics. The investigations have been Copyright © 1998 IOP Publishing Ltd

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Figure H7.2.18. (a) SMES active power PS M as a result of a three-phase short circuit. (b) SMES reactive power QS M as a result of a three-phase short circuit.

illustrated by means of an example based on a model power system. A system engineering approach has been used, which considers the requirements on SMES from the beginning of the design. Starting from this, the dimensioning, the location of the SMES and the synthesis of the controller have been described. The provision of seconds reserve power by SMES could be of economic interest. Hence, the use of SMES can contribute to an enhanced utilization of existing power plants and a more efficient use of the primary fuels with a resulting reduced emission. Therefore this provision of seconds reserve power becomes a criterion for the dimensioning of the SMES within the model power system. The factors affecting the location of SMES because of the existing transmission capacities cause no particular restrictions on the provision of seconds reserve power. Thus the SMES location can be exclusively determined with respect to an improvement of the short-term dynamics. A SMES/PSS concept Copyright © 1998 IOP Publishing Ltd

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Figure H7.2.19. ITAE-criterion of the deviation ∆UG 2 of the terminal voltage of generator 2.

has been developed because both the SMES and the PSS have a significant impact on those eigenvalues which are dominant with respect to the short-term dynamics under consideration. It has been shown that sensitivity analysis of the dominant eigenvalues is an appropriate method to determine the optimal SMES location as well as the optimal PSS locations. An event-driven control scheme has been represented which enables the coordinated allocation of the seconds reserve power by SMES and conventional primary controlled power plants as well as the improvement of the short-term dynamics by the SMES. The provision of seconds reserve power according to the DVG recommendations is achieved by a proportional controller which is parametrized with respect to the drops in primary control of the power plants of the interconnected system. Starting from the sensitivity analysis used for determining the optimal SMES location, the improvement of the short-term dynamics is realized by maximum shifting of the dominant eigenvalues to the left by applying an adaptive stochastic algorithm. It has been shown that within the use of SMES the dynamic system performance can be improved and transient instabilities may not occur. Furthermore the proposed SMES/PSS concept shows a high insensitivity against topological changes and load changes. It may therefore be possible to postpone the expansion of transmission line capacities, even if the system load is increasing.

References Anderson P M, Agrawal B L and Van Ness J E 1990 Subsynchronous Resonance in Power Systems (New York: IEEE) Bayer W, Bithin R, Kurten H, Radtke U, Taube W, Vollmar H E and Willmes H 1994a Supraleitende Energiespeichern zur Bereitstellung schneller Reserveleistung in der elektrischen Energieversorgung Elektrizitatswirtschaft 8 446-52 Bayer W, Kurten H, Vollmer H E, Radtke U, Taube W, Kleimaier M, Handschin E and Stephanblome Th 1994b Economical and technical aspects of SMES applications to power systems operation and control 1994 Session Papers of the GIGRE Conf. ( Paris, 1994) Bongers C, Grebe E, Handschin E and Haubrich H J 1981 Identification and control of the mid-term dynamic behaviour of electric power systems Contribution to the PSCC’81 Copyright © 1998 IOP Publishing Ltd

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Deutsche Verbundgesellschaft e.V. 1992 Operational Behaviour of Thermal Power Plants Necessary for a Reliable Power Supply (Heidelberg: Deutsche Verbundgesellschaft) Feliachi A 1990 Optimal siting of power system stabilisers IEE Proc. C 137 No 2 Hanschin E, Schroeder M and Stephanblome Th 1994 The use of SMES to improve the power system dynamics Proc. IEA Symp. on Use of Superconductivity in Energy Storage ( Karlsruhe, 1994) Handschin E and Wohlfahrt H 1988 Different power system stabilizer properties under unified short- and mid-term aspects Proc. IFAC Symp. on Power System Modelling and Control Applications (Brussells: Federation IBRA-BIRA) IEEE Committee Report 1981 Excitation system models for power system stability studies IEEE-PAS, vol 100, No 2, S. 994–509 Komarek P (ed) 1991 First Int. SMES Workshop Conf. Proc. ( Karlsruhe, 1991) Krug W and Schonfeld S 1981 Rechnergestutzte Optimierung fur Ingenieure (Berlin: VEB Technik) Kiirten H 1986 Provision and activation of active power seconds range reserve in thermal power plants, effectiveness and economic aspects 6th CEPSI Conf. on Electric Power Supply Industry ( Jakarta, 1986) Martin P and Naser W 1986 Wirtschaftlichkeit der verschiedenen Wirkeistungs Sekundenreserve MaβSnahmen ( VDI Report 582) (Dusseldorf: VDI) pp 241–75 Mitani Y, Tsuji K, Murakami Y, Hayashi S and Tanaka Y 1990 Stabilization of hydroelectric power systems with long distance power transmission by using superconducting magnetic energy storage (SMES) IEEJ 110 53-62 Stephanblome Th 1994 Integration von SMES in die Netzregelung Doctoral Thesis University of Dortmund

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PART I SUPERCONDUCTING ELECTRONICS

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I1 Josephson junctions

Horst Rogalla

I1.0.1 The Josephson effect Although the concept of tunnelling was long known theoretically and was already being used by 1928 for the explanation of experimental data for, e.g., the natural decay of certain heavy nuclei by α -particle emission (Gamov 1928), electron tunnelling in solid-state structures was observed only quite late. In 1958 Esaki observed electron tunnelling in narrow semiconductor p-n junctions (Esaki 1958) and was awarded the Nobel prize in 1973. Also in 1973, Giaever was awarded the Nobel prize for his demonstration of single-electron tunnelling through a thin oxide barrier (<5 nm) between metal electrodes (Giaever 1960). In particular, electron tunnelling between superconducting electrodes turned out to be a very interesting tool for investigating the electronic structure of superconductors. Also the tunnelling of Cooper pairs through barriers seemed to be possible, but the probability of it occurring seemed to be so small compared with the probability for single-electron tunnelling that its contribution to the total tunnelling current could be neglected (see e.g. Tinkham 1985). Unimpressed by this pessimistic prediction, Josephson published a paper (Josephson 1962) in 1962 in which he predicted the possibility and consequences of Cooper pair tunnelling for which he was awarded the Nobel prize, also in 1973. Anderson (1970) gave a very interesting description of how Josephson discovered this effect. In this chapter a compact overview of the basics of the Josephson effect and its realization in Josephson junctions will be given. Due to the restriction in length it is far from being a complete description and the reader interested in more information is referred to the standard textbooks. In his original paper Josephson used quantum-mechanical methods to calculate the effects of Cooper-pair tunnelling. A very elegant, quite simple and more illustrative method was used by Feynman (1965). His wave-mechanics method will be used here to calculate the two Josephson equations. Imagine two superconductors separated from each other by a small gap (e.g. vacuum) of distance d (see figure I1.0.1). The superconductors SI and S2 can be described by a macroscopic wavefunction Ψ1 . 2(r ). Let us vary the distance between the two superconductors. If the two superconductors were in contact (d = 0) the two wavefunctions would be the continuation of each other in the opposite superconductor. At the interface there would be no phase difference (ϕ2 − ϕ1 = 0). If the gap between the two superconductors were large, then the two wavefunctions would be independent and the phase difference at the interface would be ambiguous. Copyright © 1998 IOP Publishing Ltd

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Figure I1.0.1. Gedanken experiment for the Josephson effect.

In between these two extremes there exists an interval for d in which the two wavefunctions are more or less weakly coupled. The wavefunctions have the form used in the Ginzburg-Landau theory. They are normalized to the Cooper-pair density and their phase is position and time dependent. The energy of the two electrons constituting a Cooper pair is close to 2EF , EF being the Fermi energy.

The time evolution of the two wavefunctions is described by the Schrödinger equation:

Here E1 and E1 are the energy eigenvalues of the wavefunctions Ψ1 and Ψ2 respectively. We assume a symmetrical coupling of the two wavefunctions, represented by the scalar k. For simplicity we take the energy zero midway between E1 and E2 If we apply a voltage V between the two superconductors, the energy of superconductor S1 is, for example, decreased by eV and that of superconductor S2 increased by eV

Substituting the explicit form of the wavefunctions and separating for the real and imaginary part yields

The change in Cooper-pair density ∂ni /∂t is counterbalanced by an external current source connected to the superconductors. We deal here only with the supercurrents and assume that the current has the same Copyright © 1998 IOP Publishing Ltd

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value I in S1 and S2

Assuming identical Cooper-pair densities nS = nS 1 = nS 2 in both superconductors, we can write down the first Josephson equation making use of equations (I1.0.4) and (I1.0.5):

where ϕ = ϕ2 − ϕ1 is the phase difference across the junction. The pre-factor before the sine expression determines the maximum supercurrent I0 through the junction.

By subtracting equation (I1.0.7) from equation (I1.0.6) we obtain an equation describing the development of the phase difference across the junction in time:

The phase change is proportional to the applied voltage, with the proportionality factor being a quotient of the fundamental constants e and h. In addition to the supercurrent I0 sinϕ there also flows a normal current I N which can originate from tunnelling, but also from normal-conducting links between the superconducting electrodes. We add the normal current to equation (I1.0.11) and introduce a voltage-dependent resistance R( V )

I1.0.2 The Josephson tunnel junction Let us assume the ideal case of a tunnel junction with identical electrodes at T = 0. We connect it to a current source, slowly increase the current and measure the resulting voltage across the junction (see figure I1.0.2). For I ≤ I 0 the voltage across the junction is zero and a supercurrent flows across the junction. The phase changes from ϕ = 0 at I = 0 to ϕ = π/2 at I =I 0 . Above the critical current a voltage arises which resembles the quasiparticle current-voltage ( I−V ) characteristics. The voltage switches from V = 0 to V = Vg . The current can be varied between nearly I = 0 and I ≤ = I g with almost no voltage change (V ≈Vg ). For current values above I g the relation between current and voltage is ohmic with a resistance Rn = ∆V/∆I. For currents I ≈ 0 the junction switches back to the zero-voltage state. For real junctions at non-zero temperature the I-V characteristic is somewhat rounded and a subgap-structure is present. Qualitatively, the discussed dependence remains the same: the gap current Ig is always larger than the critical current and the differential resistance near the gap voltage is small compared with the practically temperature-independent normal resistance Rn . The temperature dependence of the gap voltage near Tc can be approximated by:

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Figure I1.0.2. Schematic current-voltage characteristic of a Josephson junction.

with Vg 0 = Vg (T = 0). This expression is useful over quite a wide temperature range. The exact temperature dependence for all temperatures can be calculated from the self-consistent gap equation in the Bardeen-Cooper-Scrieffer (BCS) theory (Bardeen et al 1957). The temperature dependence of the critical current I0 in tunnel junctions was calculated by Ambegoakar and Baratoff (1963)

For T = 0 we find a relation between I0 and Ig

I0 is always smaller than Ig , roughly by a factor of 3/4 with decreasing tendency for higher temperatures. For a more detailed discussion on single-electron tunnelling and Cooper-pair tunnelling see e.g. the textbooks of Wolf (1989) and Tinkham (1996). I1.0.3 The RSJ model Apart from coupling between the superconducting electrodes via tunnelling, there can also be coupling via a normal-conducting barrier. Cooper pairs can diffuse through a normal-conducting barrier and thus more or less weakly couple the phases in the superconducting electrodes (superconductingnormal-superconducting (SNS) junction). An electrically similar situation can be created for a superconducting tunnel junction if the junction is shunted by a sufficiently small resistor. A simple electronic circuit can model this situation, putting a shunt-capacitor and shunt-resistor in parallel to a Josephson circuit element (representing the first and second Josephson equations). This model is called the resistively shunted junction model (RSJ model). For a practical shunted tunnel junction the capacitance is mainly the capacitance formed by the electrodes separated by an insulating barrier. Capacitance can easily be included by expanding the RSJ model to the RCSJ model (see figure I1.0.3) by introducing a shunting capacitance C. Also, inductance can be included in the branches of the equivalent circuit. The model can be further extended to unshunted tunnel junctions if one allows for a voltage-dependent shunt resistor R( V ). It is common practice to refer to all the different versions of the model as the RSJ model. Using Kirchhoff ’s law, all the characteristic data of this circuit⎯such as I−V characteristics and time dependences of the currents and voltage⎯can

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Figure I1.0.3. Electronic scheme of the resistively shunted junction (RCSJ) model with a shunt capacitance.

be calculated

For zero capacitance a closed expression for the quasi-static I−V characteristic of the resistively shunted Josephson junction can be derived:

The resulting I−V characteristic is the standard nonhysteretic I−V characteristic for a junction in the zero-capacitance limit. It is depicted in figure I1.0.4. Using the second Josephson equation (I1.0.11) we can replace the voltage by the time derivative of the phase in (I1.0.16).

The equation can be written in a dimensionless form with the following parameters: characteristic frequency ωc = (2e/h )(Ic /G) McCumber parameter βc = ωc C/G normalized time τ = ωc t normalized current i = I /Ic

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Figure I1.0.4. Parabolic I−V characteristics for a Josephson junction calculated from the RSJ model for C = 0.

or

This set of parameters originates from McCumber (1986). Other choices are also possible, e.g. the Johnson and the Stewart parameter set. In the McCumber parameter set, the characteristic frequency ωc is the Josephson frequency at the characteristic voltage V0 = Ic R and βc is defined as the ratio between the capacitive susceptance of a junction at ωc and its normal conductance. This parameter determines the amount of hysteresis in a junction: a value of βc = 0 corresponds to a junction without hysteresis, βc = ∞ to a junction dominated by hysteresis. The second-order nonlinear differential equation (I1.0.19) can in general only be solved numerically. In conjunction with a drive term I = I(t ), the dynamic behaviour of the junction can be chaotic: Josephson junctions are an accepted model system for the study of chaotic behaviour, from both the physical and mathematical points of view. The junction dynamics can be modelled by the viscously damped motion of a particle of mass m in a tilted washboard potential. It follows from equation (I1.0.19) that the potential has the form:

then this motion can be described by

The model of the washboard potential (see figure I1.0.5) is especially useful in visualizing thermal activation, noise and the effect of non-sinusoidal current phase relationships as, for example, in µ bridges.

I1.0.4

High-frequency response

Josephson junctions respond in a characteristic way to high-frequency radiation. Therefore we will study Josephson junctions which can be described by the RSJ model and have negligible capacitance. First the Copyright © 1998 IOP Publishing Ltd

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Figure I1.0.5. The washboard potential.

voltage-driven case will be studied, meaning that source impedance is small compared with the junction impedance. We can make use of equation (I1.0.16) and assume a driving voltage of

Also neglecting the normal-conducting contribution, we find the following current through the junction

This is a standard problem in the analysis of the spectrum of a frequency-modulated oscillator. After expansion in Bessel functions of the first kind, the current through the junction can be written as:

In the time-average (quasi-static) characteristic there is a current contribution In for ωj = nωs , thus at a voltage Vn

At these voltages, current steps (called Shapiro steps (Shapiro 1963)) occur in the I-V characteristics (see figure I1.0.6). These steps can easily be observed in low-Tc and high-Tc Josephson junctions with small or no hysteresis. In only weakly shunted tunnel junctions the current steps appear as current spikes in the subgap region of the junctions. The maximum zero-voltage current decreases like J0 with increasing microwave amplitude in the junction while the first-order step slowly appears and finds its maximum shortly before the zero-voltage current vanishes. Both zero-voltage current and the steps show reentrant behaviour as function of the microwave amplitude as a predicted by the voltage model. Often the impedance of the Josephson junction is significantly smaller than that of the voltage source. In that case a current-driven model has to be applied. Starting from equation (I1.0.18) a time-dependent Copyright © 1998 IOP Publishing Ltd

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Figure I1.0.6. Constant-voltage current steps in a high-TC Josephson junction under the influence of microwave radiation. The junction is of the ramp type with a Ga-doped PrBaCuO barrier.

drive current replaces the current I

This equation can be solved in general only numerically. As in the voltage-driven case, current steps occur for the current-driven case at voltages Vn = n(hωs /2e). The dependence of the maximum zero-voltage current I0 on the microwave amplitude in the junction is more linear than in the voltage source model and also the microwave amplitude dependence of the heights of step n deviates from the dependence on the Bessel function Jn . I1.0.5 Magnetic field response of Josephson junctions If one applies a magnetic field to a Josephson junction so that it penetrates its barrier, the current distribution and the critical current of the junction are changed. In addition, currents through the junction generate magnetic fields on their own. The combination of both in ‘long Josephson junctions’ can lead to quite complicated current distributions and is outside the scope of this chapter. Here we will deal with junctions in which self-field effects can be neglected: ‘short Josephson junctions’. Let us for simplicity assume the junction geometry as depicted in figure I1.0.7. The magnetic field is applied in the z direction, the barrier lies in the x−z plane with a thickness d in the y direction. Following the Ginzburg-Landau theory (see e.g. Abrikosov 1988), the supercurrent density J is related to the wavefunction Ψ by

With Ψ = ns1/2 e i ϕ we can solve for the gradient of the phase:

Without transport current the phase between two adjacent points is

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Figure I1.0.7. Geometry for the discussion of the magnetic field response of a superconductor. Bs is the field inside the barrier which is equal to the external field.

With ϕ ( x ) = ϕ1 (x ) − ϕ2(x ) we denote the phase difference across the junction. The change of ϕ (x ) along the junction is The change along one side of the barrier can be found by line integration along the paths A1-B1-C1D1 and A2-B2-C2-D2 shown in figure I1.0.8

The integration along B1-C1-D1-E1 and B2-C2-D2-E2 does not contribute if the integration path is deep inside the superconductor. The integration around the full loop A1-B1-C1-D1-D2-C2-B2-A2A1 yields the flux through this area

Thus:

and

The current through a rectangular junction is given by

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Figure I1.0.8. Part of the barrier section of a Josephson junction for the calculation of the magnetic field response of the junction.

where CR denotes the cross-section of the junction, t the thickness of the electrodes, w the width of the junction and Jc the local critical current density of the junction, which we assume here for simplicity to be position independent. With Φe f f = Bz (2λL + d )w being the effective magnetic-field sensitive area of the junction and IC = twJC the critical current of the junction we find:

The maximum supercurrent through the junction is reached for ϕ0 = π/2

This equation is identical to the one describing the Fraunhofer diffraction pattern in optics. The dependence of the maximum Josephson current through a rectangular junction on the magnetic field (flux) is depicted in figure I1.0.9. With a more detailed analysis, the magnetic field dependence of more complicated junction geometries and for wide Josephson junctions can also be calculated.

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Figure I1.0.9. Magnetic field dependence of the maximum Josephson current IC (Φ) through a rectangular Josephson junction.

I1.0.6 Practical Josephson junctions During the years since the first experimental verification (Anderson and Rowell 1963) of the Josephson effect, many different forms of Josephson junctions have been developed, different in electrode and barrier materials, different in the coupling mechanism between the electrodes and different in junction geometry. There is no room here to review all the low-Tc and highTc realizations; just four examples will be discussed to show some of the most relevant junction types: Nb/Al/Al2O3 /Al/Nb tunnel junction Nb/Cu/Nb proximity-effect junction YBa2Cu3O7-δ grain-boundary junction YBa2Cu3O7-δ /PrBa2Cu3O7-δ /YBa2Cu3O7-δ ramp-edge junction.

In principle Josephson junctions consist of two electrodes and a region in which a weak coupling between the electrodes is realized (see e.g. the schematic representation in figure I1.0.1). The challenge lies in the design of the weak-coupling area. The realizations are quite different for low-Tc and high-Tcjunctions. In the following both groups of junctions will be discussed. I1.0.6.1 Low-Tc Josephson junctions The first low-Tc Josephson junctions were fabricated mainly from aluminium (-Tc = 1.18 K). Al forms a very dense oxide with very good dielectric properties just by oxidation in air or even better in pure oxygen. Sandwiched tunnel junctions, like the one sketched in figure I1.0.10 with S as a superconducting Al electrode and I as an Al2O3 insulating barrier, show very good I-V characteristics with nearly ideal dependence of the subgap current on the temperature. The disadvantage of Al junctions is their low operating temperature: they are good for operation at a temperature below 1 K, as are junctions making use of a Coulomb blockade (e.g. single-electron devices). For applications in liquid helium at 4.2 K, lead junctions or junctions made from lead compounds such as Pb-Bi were used for quite a long time. Their weakness is obvious: lead is a soft material and not stable over long periods in a normal environment containing water vapour. The junctions degrade quite quickly. This was one of the major reasons for the failure of the IBM supercomputer project. On the other hand the preparation of these junctions is quite easy and the electrical characteristics are good. So in a number of laboratories this technique is still used for superconducting electronic circuits of low complexity. For a number of years the workhorse of low-Tc electronics has been the Nb/Al/Al2C3 /Al/Nb junction (Gurvitch et al 1983). It combines the advantages of a naturally grown Al2C3 barrier with the high Tc of niobium: 9.3 K. The preparation makes use of the fact that Al deposited onto clean Nb surfaces wets the surface

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Figure I1.0.10., Sandwiched Josephson junctions, (a) Josephson tunnel junction and (b) Josephson proximity-effect junction.

and thus forms a very low-resistance contact to the Nb electrode. At 4.2 K Nb is superconducting and also a thin layer of Al (typically 5 nm) covering the niobium becomes superconducting via the proximity effect. Thermal oxidation is then used to form a naturally grown Al2O3 barrier of high quality. Then another layer of Al and the Nb counter-electrode is deposited. The Al layers thus form a tunnel junction whose critical temperature is enhanced by the proximity effect due to the Nb electrodes. The density of Cooper pairs through the cross-section of such a junction is depicted in figure I1.0.11. The reproducibility

Figure I1.0.11. A schematic representation of the Cooper-pair density through the cross-section of an Nb/Al/Al2O3 /Al/Nb Josephson junction.

of these junctions is very good; complexities of more than 105 junctions on one chip have been reached with reasonable yield (Müller et al 1997). Critical current densities of up to 104 A cm−2 are standard. Higher current densities are possible but at the cost of the junction quality. Characteristic of this type of junction is the ‘proximity-effect knee’ in the I−V characteristic at the top of the gap (see figure I1.0.12) which occurs as a small hysteresis in the ‘knee-area’ in current-driven junctions. The superconducting banks of electrodes in Josephson junctions can also be coupled by a normal conductor (N in figure I1.0.10(b)). In this case one obtains a pure proximity-effect junction which can be fully described by the RSJ model without the addition of a voltage-dependent normal current describing the subgap characteristics of a superconducting tunnel junction. In proximity-effect Josephson junctions copper is often used as the normal metal since pure copper has quite a large decay length ξ of about 50 nm. The decay length is the characteristic length for the exponential decay of the Cooper-pair density into the normal conductor. These junctions have often been used in the study of arrays of Josephson junctions because of the ease of preparation, the perfect RSJ-like electrical characteristic of the individual junction (see figure I1.0.13) and, as a result, the good accordance between model calculations and experiments.

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Figure I1.0.12. Typical I-V characteristic of an Nb/Al /Al2O3 /Al /Nb tunnel junction.

Figure I1.0.13. Typical I-V characteristic of an Nb-Cu-Nb proximity-effect Josephson junction (from Benz 1990).

Recently the classical SNS junction found a renaissance in the work on programmable Josephson voltage standards (Benz and Hamilton 1996). In these junctions PdAu is used as the proximity barrier material.

I1.0.6.2 High-Tc junctions High-Tc Josephson junctions can be divided into three major classes: intrinsic Josephson junctions, grain-boundary junctions (GBJs) and junctions with artificial barriers. Whereas the intrinsic junctions show tunnel-junction-like I-V characteristics, the latter two classes exhibit more or less pronounced RSJ-like I-V characteristics and an Ic Rn product at 4.2 K of the order of 1-10 mV. As an example, one major representative of the GBJs and of the junctions with artificial barriers will be discussed in the following-the bicrystal junction and the ramp-edge junction. That grain boundaries in high-Tc superconductors act as Josephson junctions was already known Copyright © 1998 IOP Publishing Ltd

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shortly after the first bulk materials were synthesized. This property was used for the fabrication of the first Josephson junctions and SQUIDs (Gough et al 1987). Grain boundaries also occurred in early high-Tc thin films prepared by a two-step method. Josephson junctions could be separated from the bulk of the film by etching and in this way the first thin-film Josephson junctions and d.c. SQUIDs were prepared (Koch et al 1987). They showed very irreproducible characteristics and strong noise. Controlled grain boundaries can be fabricated in epitaxial films of YBa2Cu3O7-δ on bicrystals of SrTiO3. By way of structuring a bridge across the grain boundary, individual Josephson junctions can be prepared (see figure I1.0.14). The junctions show an RSJ-like I-V characteristic and the critical current density of these junctions is adjustable over a wide range by changing the misfit angle between the crystals (see e.g. Dimos et al 1990, Gross 1994) (see figure I1.0.15). This type of junction is widely used today, mostly fabricated on a bicrystal with a misfit angle of 24°. They show a critical current density of about 105A cm−2 and an IcRn product of about 100 µV, both at 77 K. Bicrystal GBJs have been described widely in the literature (Gross et al 1977) and are very important in simple applications like SQUIDs (see e.g. Clarke 1996).

Figure I1.0.14. Schematic drawing of a bicrystal junction. SrTiO3-I and SrTiO3-II are single crystals with the orientation of one of the main axes rotated by a certain angle, e.g. 24°.

The grain boundaries are restricted to a simple geometry and have only a very few straight lines. Somewhat more complex logic circuits are difficult or impossible to design on such simple topologies. In addition, the spread in the parameters of junctions, even prepared on the same grain boundary, is quite large and circuits with about ten Josephson junctions are difficult to realize. Apart from the low Ic Rn product at 77 K, these junctions are very well suited for application in d.c. SQUIDs. Bicrystal d.c. SQUIDs with an integrated flux transformer and pickup loop or in flip-chip designs have been fabricated by a number of groups and a field sensitivity of the order of 10 fT Hz−1/2 has been demonstrated (Drung et al 1996). A great many articles about GBJs have been published and further details on the properties of GBJs can be found in the literature (see e.g. Gross 1994). One major disadvantage of the GBJs-the uncontrolled properties of the grain boundary and thus of the barrier of the junction-can be overcome by junctions with artificial barriers, the most important representative of which is at present the ramp-type junction. Ramp-type junctions are based on the idea that a flat ramp etched into a c-axis-oriented high-Tc film gives access to the copper oxide planes and nevertheless allows for epitaxial growth of a barrier and counter-electrode (see figure I1.0.16). Many different barrier materials can be employed as long as they grow epitaxially on the base electrode and support epitaxy for the counter-electrode. This allows barriers with a wide range of conductance to be used, from high-resistance ones (e.g. Ga-doped PrBaCuO) to low-resistance ones (e.g. Co-doped YBaCuO). The latter are especially useful for operation at 77 K. Both types are already quite far developed, especially

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Figure I1.0.15. (a) The I−V characteristic of a 24° GBJ (from Dimos et al 1990). (b) The dependence of the critical current of GBJs as a function of the misfit angle ϕ (from Gross 1994).

Figure I1.0.16. A schematic drawing of a ramp-type junction.

the low-resistance barriers at Conductus and Northrop Grumman (Antognazza et al 1995) and the high-resistance barriers at Twente University (Verhoeven et al 1996). A disadvantage of the Co-doped barriers is their limited temperature range: the barrier becomes superconducting below a certain Co-concentrationdependent temperature. On the other hand ramp-type junctions with a Ga-doped PrBaCuO barrier have Copyright © 1998 IOP Publishing Ltd

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a relatively high normal resistance over the whole temperature range (Verhoeven 1996). Their drawback is the lower maximum operating temperature which excludes operation at 77 K. Weighing up these two types of ramp-type junctions, the Co-YBaCuO barriers are especially good for not too complex circuits at 77 K because of the higher bit-error rate at this temperature in logic circuits. At lower temperature, e.g. 40 K, the PrBaCuO barriers are of advantage due to their quite high IcRn product and because critical current and resistance of the junctions are adjustable independently. Interestingly the supercurrent transport through the barrier occurs via direct tunnelling whereas the quasiparticle transport occurs via resonant tunnelling in an independent channel. When applying these junctions they have the advantage of an independent choice of Jc via barrier thickness and Rn via Ga doping. Naturally this also has an influence on the IcRn product (see figure I1.0.17). For fixed barrier thickness Ga doping increases the IcRn product. Values of nearly 10 mV have been achieved in junctions with RSJ-like characteristics.

Figure I1.0.17. Dependence of Jc on Arn for ramp-type junctions of different Pr barrier thicknesses and Ga-doping levels. Lines of constant IcRn product have been included.

References Abrikosov A A 1988 Fundamentals of the Theory of Metals (Amsterdam: North-Holland) ch 17 Ambegoakar V and Baratoff A 1963 Tunneling between superconductors Phys. Rev. Lett. 10 468; erratum 11 104 Anderson P W 1970 How Josephson discovered his effect Phys. Today 23 23 Anderson W and Rowell J M 1963 Probable observation of the Josephson superconducting tunneling effect Phys. Rev. Lett. 10 230 Antognazza L, Berkowitz S J, Geballe T H and Char K 1995 Proximity effect in YBa2Cu3O7−δ /YBa2 (Cu1-xCox )3O7-δ/YBa2Cu3O7−δ junctions: from the clean limit to the dirty limit with pair breaking Phys. Rev. B 51 8560 Bardeen J, Cooper L N and Schrieffer J R 1957 Theory of superconductivity Phys. Rev. 108 1175 Benz S P 1990 Dynamical properties of two-dimensional Josephson junction arrays PhD Thesis Harvard University Benz S P and Hamilton C A 1996 A pulse-driven programmable Josephson voltage standard Appl. Phys. Lett. 68 3171 Clarke J 1996 SQUID Sensors: Fundamentals, Fabrication and Applications ( NATO ASI Series E: Appl. Sciences 329) ed H Weinstock (Dordrecht: Kluwer) Dimos D, Chaudhari P and Mannhart J 1990 Superconducting transport properties of grain boundaries in YBa2Cu3O7 bicrystals Phys. Rev. B 41 4038 Copyright © 1998 IOP Publishing Ltd

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Drung D, Ludwig F, Müller W, Steinhoff U, Trahms L, Koch H, Shen Y Q, Jensen M B, Vase P, Hoist T, Freltoft T and Curio G 1996 Integrated YBa2Cu3O7-x magnetometer for biomagnetic measurements Appl. Phys. Lett. 68 1421 Esaki L 1958 New phenomenon in narrow germanium p-n junctions Phys. Rev. 109 603 Feynman R P 1965 Lectures on Physics vol 3 (New York: Addison-Wesley) ch 21 Gamov G 1928 Z. Phys. 51 204 Giaever I 1960 Energy gap in superconductors measured by electron tunneling Phys. Rev. Lett. 5 147; 1960 Phys. Rev. Lett. 5 464 Gough E, Colclough M S, Forgan E M, Jordan R G, Keene M, Muirhead C M, Rae A I M, Thomas N, Abell J S and Sutton S 1987 Flux quantization in a high-Tc superconductor Nature 326 855 Gross R 1994 Interfaces in High-Tc Superconducting Systems ed U S L Shind and D A Rudman (New York: Springer) Gross R, Alff L, Beck A, Froelich O M, Koellen D and Marx A 1977 Physics and technology of high Tc superconducting Josephson junctions IEEE Trans. Appl. Supercond. 7 2929 Gurvitch M, Washington M A and Muggins H A 1983 High quality refractory Josephson tunnel junctions utilizing thin aluminum layers Appl. Phys. Lett. 42 472 Josephson B D 1962 Possible new effects in superconductive tunneling Phys. Lett. 1 251 Koch R H, Umbach C P, Clark G J, Chaudhari P and Laibowitz R B 1987 Quantum interference devices made from superconducting oxide thin films Appl. Phys. Lett. 51 200 McCumber D E 1986 J. Appl. Phys. 39 3113 Müller F, Pöpel R, Kohlmann J, Niemeyer J, Weimann T, Grimm L, Dunschede F-W and Gutmann P 1997 Optimized 1 V and 10 V Josephson series arrays IEEE Trans. Instrutn. Meas. 46 229 Shapiro S 1963 Josephson currents in superconducting tunnelling: the effect of microwaves and other observations Phys. Rev. Lett. 11 80 Tinkham M 1985 Introduction to Superconductivity (Malabar: Krieger) footnote p 192 Tinkham M 1996 Introduction to Superconductivity 2nd edn (New York: McGraw-Hill) Verhoeven M A J 1996 High-Tc superconducting ramp-type junctions PhD Thesis University of Twente Verhoeven M A J, Gerritsma G J, Rogalla H and Golubov A A 1996 Ramp-type junction parameter control by Ga doping of PrBa2Cu3O7-barriers Appl. Phys. Lett. 69 848 Wolf E L 1989 Principles of Electron Tunneling Spectroscopy (New York: Oxford University Press)

Further reading In the following textbooks more information can be found on superconductivity in general and on Josephson junctions and their applications Barone A and Paterno G 1982 Physics and Applications of the Josephson Effect (New York: Wiley) Likharev K K 1986 Dynamics of Josephson Junctions and Circuits (New York: Gordon and Breach) Orlando T P and Delin K A 1991 Foundations of Applied Superconductivity (Reading, MA: Addison-Wesley) Poole C P Jr, Farach H A and Creswick R J 1995 Superconductivity (San Diego, CA: Academic) Tilley D R and Tilley J 1990 Superfluidity and Superconductivity 3rd edn (Bristol: Institute of Physics) Tinkham M 1996 Introduction to Superconductivity 2nd edn (New York: McGraw-Hill) Van Duzer T and Turner C W 1981 Principles of Superconductive Devices and Circuits (New York: Elsevier)

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I2 SQUID sensors

J Flokstra

I2.0.1 Introduction This chapter can be considered as an introduction to the principal concepts of SQUIDs and the application of these devices in a variety of measuring systems (there are many excellent review papers available; see, for instance, Clarke (1993). Many further valuable papers can be found in the book edited by Weinstock (1995)). SQUID is an acronym for superconducting quantum interference device. These devices are the most sensitive detectors for magnetic flux currently known. All physical quantities that can be converted to a magnetic flux can be measured with extreme sensitivity. SQUIDs can be used, for instance, for measuring magnetization, magnetic susceptibility, magnetic fields, current, voltage and small displacements. To mention a few applications, SQUID systems are used for detecting the very small magnetic fields due to spontaneous or evoked brain activity (Hämäläinen 1993). These fields are of the order of 10−14 T. SQUIDs are also applied for observing the very small displacements of a resonant mass antenna at millikelvin temperature after the passage of a gravitational wave (for a review see Blair (1993)). Displacements of 10−19 m can be detected and even 10−21 m will be reached in the near future. The SQUID is based on two physical phenomena, namely Josephson tunnelling and flux quantization. The first key element is the Josephson junction which is characterized by a limited critical current at zero voltage and by switching to the voltage state above a current threshold. The second basic element of the SQUID is a closed superconducting loop for which the flux is quantized in units of the flux quantum Φ0 = h/2e = 2.07 × 10−15 Wb. There are two main versions of the SQUID. The d.c. SQUID consists of a superconducting loop interrupted by two junctions which are each damped by a resistor and it is biased by a current source. The output is the voltage across the parallel junctions. The radiofrequency (RF) SQUID has only one junction in the loop and the read-out uses a resonant LC circuit inductively coupled to the superconducting loop. The average voltage across the resonant circuit is the output of the RF SQUID. In both cases the output is a periodic function of the magnetic flux applied to the loop with period Φ0. Apart from the two basic configurations for the SQUID, there are also some modified versions. The relaxation oscillation SQUID (Adelerhof 1994), for instance, has the d.c. SQUID configuration but the junctions are not damped. Instead the loop is shunted by an inductor and resistor in series. The output of a suitably biased device is a pulse train with a frequency that depends on the applied flux. The device is in principle a flux-to-frequency converter. Copyright © 1998 IOP Publishing Ltd

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SQUIDs have been realized in low-Tc and high-Tc materials. In general, low-Tc . SQUIDs are based on niobium (Tc = 9.2 K) and the junction area consists of oxidixed aluminium. In the case of high-Tc YBaCuO is the most frequently used material (Tc = 92 K) for the SQUID. There are several methods to prepare the junction as is described in chapter I1. An important sensor aspect of the SQUID is the method of applying the magnetic flux to the loop. Often use is made of a superconducting flux transformer loop transferring flux picked up with a coil to the SQUID loop. Also the linearization of the device is of importance, which is realized by applying feedback in order to arrive at a flux-locked loop configuration. The organization of this chapter is as follows. In section I2.0.2 attention is paid to basic theory describing the SQUIDs. Section I2.0.3 deals with the hardware realization of low- and high-Tc SQUIDs and with SQUID instrumentation. Various SQUID applications are described in section I2.0.4. I2.0.2 General theory of SQUIDs The basic theory of Josephson junctions, d.c. SQUIDs and RF SQUIDs has been extensively reviewed elsewhere, e.g. Clarke (1993) and Weinstock (1995). The theories were established for low-Tc devices but the approach is mostly also applicable for high-Tc materials. I2.0.2.1 Josephson junctions The Josephson junction, being the key element of the SQUID, consists of two weakly coupled superconductors, often in the form of thin films. The weak coupling is realized by a very thin insulating barrier sandwiched between the two superconductors permitting the tunnelling of Cooper pairs and quasiparticles. The pair tunnel current I is given by I = I0 sin δ with I0 the maximum supercurrent and δ the difference between the superconducting phases on both sides of the barrier. When I exceeds the value of the maximum critical current I0 , a voltage V across the junction appears according to V = ( h/2e ) dδ / dt The current-voltage (I−V) curve is in general hysteretic and has two branches, one for the Cooper pair tunnelling (V = 0) and one branch for the quasiparticles (V≠ 0). In order to observe the hysteresis the barrier resistance Rj of the junction has to be so large that the Stewart-McCumber parameter βc exceeds 1

βC = 2πI0Rj2C / Φ0 with C the capacitance of the junction. The hysteresis can be removed by shunting the junction with an additional resistor Rs such that βc < 1, replacing Rj by the parallel resistance of Rj and Rs . The I-V curve for this resistively shunted junction model (RCSJ) has a shape similar to that of the curve for n Φ0 presented in figure I2.0.1. The curve is rounded near I0 due to thermal fluctuations. The noise-rounding parameter Γ is the ratio between the thermal energy kBT and the junction coupling energy I0Φ0/2π. In practice, Γ should be smaller than about 0.1. I2.0.2.2 D.c. SQUIDs A schematic representation of the d.c. SQUID is given in figure I2.0.2. Two resistively shunted junctions are connected in parallel in a superconducting loop with total inductance L, which is split into equal parts

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Figure I2.0.1. I-V and V-Φ curves of a d.c. SQUID. Ib is the bias current. The curves in the right-hand figure are for increasing Ib . The curves have been calculated for β = 1.

Figure I2.0.2. A schematic representation of a d.c. SQUID. The bias current is given by Ib and is chosen somewhat above 2I0. The noise of the junctions is represented by the current sources In 1 and In 2.

for the two branches of the loop. The two junctions are generally described by the RCSJ model. The equations for the two junctions are

where Vi , δi and Ii are, respectively, the voltage and phase difference across and the current through junction i. The bias current is given by Ib and the application of an external flux results in a circulating current J = (I2 −I1 )/2 in the loop. The Nyquist noise current associated with the shunt resistor R5 is represented by In . The spectral density of the noise current is equal to 4kT/R5 . The flux quantization condition of the loop can be expressed as

where the total flux Φt is the sum of the external flux Φe and the flux due to the screening current J. It is seen that the voltage-phase relations of the junctions are coupled by the quantization condition. The coupling of the phases describes the interference of the superconducting wavefunctions of the junctions and it is to this effect that the ‘I’ in the word SQUID refers. The voltage across the device can be obtained from the above equations. In general the solution can only be numerically obtained. It is a fast oscillating voltage with many harmonic frequency components. Copyright © 1998 IOP Publishing Ltd

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The d.c. voltage is the time-averaged value and figure I2.0.2 shows the characteristic behaviour of the d.c. SQUID in the I-V plot and the V versus Φ curve for a range of d.c. bias currents. An important parameter is the screening factor β which is defined as

The two outer I-V curves are obtained for applied fluxes of n Φ0 and (n + 1/2)Φ0. where n is an integer. The critical current modulation depth at V = 0 can be well approximated by

The V-Φ curve changes periodically in flux with period Φ0. The curve becomes sinusoidal at higher voltages. The maximum voltage modulation depth can be approximated by

An important parameter is the flux to voltage transfer δV/δ Φ. This coefficient is maximum for Φ = (n + 1/4)Φ0 and equals

The d.c. SQUID is operated at the points where the transfer is maximal. A modulation signal with a frequency of typically 100 kHz and an amplitude Φ0/2 is applied to use the SQUID at the points where the slopes are alternately maximal positive and negative. With β = 1, the ideal transfer is equal to πR5 /4L. The voltage noise spectral density of the SQUID is given by

where the first term represents the thermal voltage across the SQUID and the second term is due to the fluctuations of the circulating current generated by the thermal current of the resistors. For β = 1, Sv ≈ 1.6(2kBTR5 ). It is common to express the noise in the flux noise spectral density SΦ , which in the case of β = 1 is

Finally, the energy sensitivity is defined as

and equals in the presented simplified theory

A more sophisticated approach leads to the numerical factors of 9 for 2.64 and 16 for 4.68 in the above equation. In order to obtain a low energy sensitivity one has to design SQUIDs with a low inductance and a low junction capacitance and operate it at low temperatures. It is common to express the energy sensitivity in units of h (≈10−34 J s). The limit of the SQUID sensitivity is equal to h.

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Figure I2.0.3. A schematic representation of an RF SQUID. The SQUID loop is coupled to a tank circuit consisting of inductor Lt and capacitor Ct and to an input coil Li. The mutual inductances are given here by Mt and Mi , respectively. The tank circuit is fed by the RF current source IR F. The averaged voltage across the tank circuit is given by Vt .

I2.0.2.3 RF SQUIDs The RF SQUID consists of a superconducting loop with inductance L interrupted by a single Josephson junction with critical current I0 and characterized by a nonhysteretic I-V curve. A schematic picture of the RF SQUID is given in figure I2.0.3. The flux quantization condition for the loop can be expressed as

δ + 2πΦt / Φ0 = 2πn where Φt is the total flux entering the loop. The supercurrent in the loop is again given by I = I0 sin δ The total flux consists of two contributions, namely an external applied flux Φe and the flux due to the circulating current in the loop. It holds that Φt = Φe − L I0 sin(2πΦt / Φ0 ). The relation between the total flux and the applied flux is given in figure I2.0.4 for the case LI0 = 1.25Φ0 Only the regions with positive slope are stable. The consequence is that an increase or decrease of the applied flux leads to jumps in the total flux at different Φe values so that hysteresis occurs when the flux is swept up and down. This leads to an energy dissipation of the order of I0Φ0 .

Figure I2.0.4. The relation between the total flux Φt , and the externally applied flux Φe for an RF SQUID. The curve is calculated for LI0 = 1.254Φ0 .

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Figure I2.0.5. The characteristic figures of an RF SQUID. Left: Ir f−Vt curve; there are nearly constant levels in the voltage at distinct current intervals. Right: Vt − Φ curve; the curve is triangularly shaped with period Φ0.

The RF SQUID is inductively coupled (mutual inductance M) to the coil of a resonant LC circuit as indicated in figure I2.0.3. An RF current source (frequency ωR F ) supplies a current to the resonant circuit (typically 20 MHz) inducing a flux in the SQUID loop. The voltage across the LC tank circuit depends on the magnitude of the RF current and on the value of the external flux that is applied to the SQUID loop. The characteristic I−V curves for the RF SQUID are given in figure I2.0.5 for flux values equal to nΦ0 and (n + 1/2)Φ0. The curve exhibits nearly constant voltage levels and when the RF current is biased at a fixed value, the voltage turns out to be a triangular-shaped curve versus flux with amplitude ωR F Lt Φ0 /2M and period Φ0. The resulting flux-to-voltage transfer is then ∂V / ∂ Φ = ωR F Lt / M Lt being the self-inductance of the tank circuit. In general the RF SQUID is operated at the first level of the step-like curve and use is made of a modulation flux of typically 100 kHz with amplitude just as in the case of the d.c. SQUID. The noise of RF SQUIDs is not only determined by the SQUID loop itself but also by the contributions of the tank circuit and the preamplifier. Detailed descriptions can be found in the literature (see e.g. Clarke (1993) and Weinstock (1995) and references therein). In general RF SQUIDs are much less sensitive than d.c. SQUIDs although comparable sensitivity can be obtained if the operating frequency is increased to the gigahertz range. I2.0.2.4 Other SQUID configurations Apart from the two basic SQUID configurations, some other designs have also been developed. One oi these is based on relaxation oscillations. Schematic diagrams of the relaxation oscillation SQUID (ROS) and the double ROS (DROS) are given in figure I2.0.6. The junctions are not shunted and are therefore hysteretic. Shunting the (D)ROS with an inductor and resistor in series and applying a suitable bias current causes the voltage across the device to oscillate in time. In both cases the device follows the I-V curve, alternately switching to the voltage state at the critical current and then returning to the zero-voltage state at low current through the junctions. Typical outputs of the ROS and DROS are given in figures I2.0.7(a) and I2.0.7(b), respectively. The ROS output is a series of pulses with a maximum amplitude equal to the gap voltage and a frequency dependent on the flux applied to the SQUID loop. The ROS is in fact a flux-tofrequency converter. The DROS output is taken in between the two SQUID loops and the average

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Figure I2.0.6. Schematic diagrams of a ROS (left) and a DROS (right). The SQUIDs are shunted by an inductance Ls h and a resistance Rs h. An additional resistor Rd is added for adequate damping of the oscillations.

Figure I2.0.7. (a) Typical output signal of a ROS. The current through the SQUID increases exponentially until the critical current is reached. Then a voltage peak occurs across the SQUID. The ROS switches back to the V = 0 state at a minimum return current. The result is a series of pulses in time, (b) Typical output signal of a DROS. The V−Φ curve has steep slopes between the states V = 0 and V ≠ 0.

voltage at that point is zero or equal to a finite voltage, depending on the critical current of the two SQUIDs. The resulting V-Φ curve has almost a block shape. The maximum flux-to-voltage transfer ∂V/∂ Φ is typically 1 to 10 mV/Φ0 (Adelerhof 1994), which is a factor of 10 to 100 larger than that of the standard d.c. SQUID. Although the noise level is comparable to that of the d.c. SQUID, the read-out of the device is simpler because less sophisticated electronics is needed. The improvement of the transfer function of a SQUID has also been obtained by using a series array of SQUIDs or by using a SQUID in a special positive feedback mode (see also section I2.0.3). I2.0.3 Realization of SQUIDs In this chapter the hardware realization of standard d.c. and RF SQUIDs as well as some system aspects will be described. Both low- and high-Tc SQUIDs will be treated. Some attention will be paid to the Copyright © 1998 IOP Publishing Ltd

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realization of a few alternative SQUID configurations. I2.0.3.1 Low-Tc d.c. SQUIDs D.c. SQUIDs are designed according to the theory presented in the previous section. In order to obtain a low value for the energy sensitivity, the SQUID inductance L should be low and the junction size (determining C) small. The values for L , I0 , C and R5 are chosen such that the conditions for β and βc are fulfilled. The junctions used for low-Tc d.c. SQUIDs are made in multilayers of thin films, see for instance Takada and Koyanagi (1994). Although various materials have been used, the most successful combination is based on niobium and aluminium. Sputter deposition has been used to prepare in situ a trilayer consisting of a base electrode of Nb (typically 300 nm thick), a barrier layer of Al of 5-10 nm which is thermally oxidized to form an insulating Al2O3 top layer (∼1 nm), and a counter-electrode of Nb (300 nm). This material combination is stable in time and is not affected by repeated thermal cycles. Lift-off techniques or reactive ion etching are used to prepare the small lateral dimensions of the junction. The junctions have typically dimensions of 5µm × 5 µm and the thin insulating barrier leads to critical current densities of typically 100 A cm−2. The critical current of the junction is then 25 µA. The junctions are shunted by a thin-film resistor to remove the hysteresis. The resistor is a small metal strip of, for example, palladium with a resistance of about 1 Ω. The presented SQUID parameter values may vary greatly for different SQUIDs. The superconducting loop of the SQUID is commonly shaped as a square washer structure as depicted in figure I2.0.8. The loop is interrupted by a slit bridged by the two junctions. The self-inductance is given by L ≈ 1.25µ0d where d is the inner dimension of the SQUID hole. The total width of the washer has to be larger than about 3d. Typical dimensions of the hole are 25 × 25 µm2, whereas the outer dimensions are 1 × 1 mm2. The self-inductance is in this case 40 pH and, with I0 equal to 25 µA, the screening parameter β has the optimal value of 1. The washer design allows a spiral input coil to be put on top (see figure I2.0.8). The signal to be measured creates a current in the input coil. The resulting flux is coupled to the SQUID hole. Assuming perfect coupling, the mutual inductance equals nL, where n is the number of turns of the input coil. The self-inductance of the input coil is equal to n 2L. Using 50 turns an inductance of 0.1µH is obtained. Finally, the SQUID has a modulation coil on top of the washer structure (not shown in figure I2.0.8).

Figure I2.0.8. A schematic representation of a washer-type d.c. SQUID. The two junctions bridge the slit in the washer. The SQUID hole has a side length of d, whereas the washer dimension is w. The input coil is on top of the washer. Also a modulation coil is placed on top (not shown).

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This coil is used to apply an additional sinusoidal or square-wave flux to the SQUID hole at a frequency of typically 100 kHz (see below). The working principle of the d.c. SQUID is illustrated in figure I2.0.9. Using an offset current and a modulation current the SQUID is biased alternately to the points of the V-Φ curve with maximal slope. The amplitude of the modulation is thus equal to 1/2Φ0. The SQUID output is amplified by a step-up transformer circuit and connected to a low-noise preamplifier (figure I2.0.10). Phase detection takes place with the lock-in amplifier at the modulation frequency. The integrated signal is fed back as a flux to the SQUID so that the device is used as a null detector. In this way linearization of the system is obtained. The transfer from flux in the SQUID to output voltage is determined by the feedback circuit and has typically a value of 1 V/Φ0

Figure I2.0.9. Operational principle of a d.c. SQUID. The SQUID signal is modulated with a square-wave signal of 100 kHz. The amplitude equals 1/2Φ0. The output signal is zero in the symmetric case (dotted lines). As soon as a flux is applied the modulation curve shifts (full curve) and an output signal of 100 kHz occurs.

Figure I2.0.10. The electronic configuration of a d.c. SQUID. An output transformer is applied to amplify the weak SQUID signal. The preamplifier has a typical noise level of 1 nV Hz−1/2. The output is fed back to the SQUID by a resistor. Rf b and Mf b determine the overall transfer from input flux to output voltage VF L L.

The SQUID chip is adequately shielded from the disturbing environment by placing the chip in a superconducting enclosure. The system is used in this case as a current meter. In many cases, the input coil is connected to a pickup coil. This loop is superconducting and is called a flux transformer. The transfer of the flux transformer is independent of the signal frequency and with the frequency-independent

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behaviour of the SQUID itself (apart from 1/f noise at low frequencies), a unique device is available for the d.c. and low-frequency range. This frequency property is of extreme importance for many applications. The pickup coil can be a simple coil so that the magnetic field component is directly sensed. In this case also noise fields are measured. Often the pickup coil consists of several coils in a gradiometer configuration. A first-order gradiometer consists of two coils with opposite turn directions such that the homogeneous field contribution is cancelled. It thus measures just the field gradient. A second-order gradiometer consists of three coils in a 1-2-1 configuration and now the homogeneous field and first-order gradient field contributions are cancelled. To obtain an optimal transfer the inductance of the pickup coil has to be in general equal to the input inductance. Field sensitivities down to a few fT Hz−1/2 have been obtained. Below a certain frequency, typically 1 Hz, 1/f noise limits the sensitivity of the device. Bias current reversal can be used to reduce the 1/f noise contribution from critical current fluctuations (Koch et al 1983). The energy sensitivity of standard d.c. SQUIDs is typically of the order of 10−30 J s. SQUIDs with a much better resolution have been realized even down to the quantum limit of 10−34 J s. In the latter case the SQUID is not suitable for signal coupling and thus not practical. The best practical d.c. SQUIDs have reached an energy sensitivity of 35h ( Jin et al 1997). I2.0.3.2 High-Tc d.c. SQUIDs Various types of high-Tc junction have been developed as mentioned elsewhere in this book. The majority of the junctions are prepared from YBaCuO. The reproducibility of the high-Tc junctions is not as good as for the low-Tc junctions. In many SQUID configurations junctions are used on the basis of bicrystal grain boundaries or ramped edges. The weak-coupling layer is then formed by the grain boundary induced in the superconducting layer by the bicrystal or by an intermediate layer of nonsuperconducting material such as PrBaCuO. The junctions can be structured with argon ion etching. No resistive shunt is usually necessary to remove hysteresis, as high-Tc junctions are intrinsically resistively shunted with βc < 1. The washer configuration for the SQUID loop is also frequently used for high-Tc SQUIDs. It is a challenge to integrate the SQUID chip with an input coil on top, but several groups have succeeded in preparing these fully integrated devices (David et al 1995, Hilgenkamp 1994). However, often an input coil on a separate substrate and a SQUID are placed face-to-face in a flip chip combination (Nilsson et al 1995). It is also common to use directly coupled SQUIDs in high Tc. A schematic picture of this configuration is given in figure I2.0.11 The SQUID loop is directly connected with a parallel loop that acts as a signal pickup loop. The external flux entering this loop induces a current that is led almost completely around the SQUID hole consisting of a slit in a superconductor area. The width of the pickup coil turn is optimized for obtaining maximal field sensitivity. Typically, white noise levels down to 20 fT Hz−1/2 are obtained. The 1/f noise can be largely eliminated (Koelle et al 1993) by applying the earlier mentioned bias current reversal technique. I2.0.3.3 Low-Tc RF SQUID The low-Tc RF SQUID is equipped with an input coil and a coil that is part of the resonant circuit. The latter coil is also used as the feedback coil and for applying a flux modulation. A drawing of a typical RF SQUID is presented in figure I2.0.12 (please refer to BTi and Quantum Design). The SQUID is made from a solid niobium body with the Josephson junction in the centre. The two coils are toroids in separate toroidal spaces prepared in the niobium body. A current flowing through a coil induces a current in the SQUID body that also runs through the Josephson junction. Cross-talk between the two coils is eliminated

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Figure I2.0.11. A schematic diagram of a directly coupled high-Tc d.c. SQUID. The flux coupled to the large pickup loop (dimensions l and w) generates a current that runs along the slit-shaped SQUID hole. This current induces a flux in the SQUID.

Figure I2.0.12. A cutaway drawing of a BTi RF SQUID. Reproduced by permission of Biomagnetic Technologies Inc (BTi), 9727 Pacific Heights Boulevard, San Diego, CA 92121-3719, USA.

in this way. An external pickup coil can be connected by screwing it to niobium blocks that are attached to the ends of the input coil. The energy sensitivity of an RF SQUID is typically 5 × 10−29 J Hz−1. The 1/f point is typically at 1 Hz. The sensitivity of RF SQUIDs can be improved by using higher resonance frequencies and applying preamplifiers based on cold electronics (Muck et al 1995). Systems have been developed at operation Copyright © 1998 IOP Publishing Ltd

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frequencies up to 3 GHz and using high-electron-mobility transistors (HEMTs) energy sensitivities of 3 × 10−32 J Hz−1 have been obtained. I2.0.3.4 High-Tc RF SQUlDs High-Tc RF SQUIDs have been successfully developed operating at 77 K. In one example a conventional washer-type RF SQUID is integrated with a superconducting half-wavelength microstrip resonator which serves as a tank circuit. The YBaCuO washer contains a step-edge junction forming the SQUID loop (Braginski 1995, Zhang et al 1994). A flux noise level of 10 µ Φ0 Hz−1/2 is obtained for a 50 pH SQUID operating at 150 MHz. A characteristic feature of the device is the frequency-independent behaviour of the noise down to 1 Hz. The field sensitivity is, however, rather poor. Several approaches have been followed to improve on this. One way is the application of superconducting side wings to enlarge the flux focusing of the resonator strip (see figure I2.0.13). Also directly coupled RF SQUID configurations have been developed.

Figure I2.0.13. Drawing of a high-Tc RF SQUID integrated into a l/2 microstrip resonator. The right-hand side is a magnified detail of the encircled part of the left-hand side. Reproduced from Zhang et al (1994) by permission.

I2.0.3.5 A few other configurations Many other SQUID configurations have been developed. As an example of such a system a DROS is presented in figure I2.0.14 (van Duuren et al 1996). The sensing SQUID is realized in a gradiometric configuration which means that the SQUID consists of two washers connected in opposite directions. The SQUID is equipped with an input coil that also couples to both SQUID holes. The second, so-called reference SQUID is a single washer and the critical current is set by a small coil on top of the washer. The inductances and the resistors have been indicated in the figure. The large flux-to-voltage transfer permits the use of simplified read-out electronics. A so-called cartwheel SQUID is shown in figure I2.0.15. The SQUID has been shunted by several loops forming a cartwheel. The SQUID itself is placed at the centre. Cartwheel SQUIDs have been fabricated for low- and high-Tc use. With a high-Tc version, field sensitivities of 18 fT Hz−1/2 have been reached in the white noise region (Ludwig et al 1995). I2.0.4 Application of SQUIDs Low-Tc RF and d.c. SQUIDs have been commercially available since the beginning of the 1970s and 1980s, respectively (e.g. from Biomagnetic Technologies Inc, San Diego, CA or Quantum Design, San Diego, CA). The sensors are equipped in such a way that standard measurements (e.g. magnetization and Copyright © 1998 IOP Publishing Ltd

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Figure I2.0.14. The hardware configuration of a DROS.

Figure I2.0.15. The configuration of a cartwheel SQUID. There are eight loops parallel to the DROS. The DROS is situated at the centre of the picture and cannot be seen.

susceptibility) can be performed in an easy manner. Also more complicated sensors consisting of a series of SQUIDs to improve the output signal are available from industry (e.g. from Hypres, Elmsford, NY). Nowadays high-Tc d.c. SQUIDs can also be obtained commercially (e.g. from Conductus Inc, Sunnyvale, CA). Apart from separate sensors whole SQUID-based instruments are offered by the industry. Among these systems are rock magnetometers, susceptometers and multichannel biomagnetometers. In many laboratories research is performed on improving the SQUID characteristics with respect to sensitivity and 1/f noise properties and special applications of SQUIDs are mainly developed in the research laboratories. The aim of this section is to present some typical applications of SQUIDs. The list is far from Copyright © 1998 IOP Publishing Ltd

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complete and for obtaining an extended overview of the application areas of SQUIDs reference is made to the literature (Clarke 1993, Gallop 1991, Weinstock 1995). I2.0.4.1 Biomagnetism Biomagnetism is one of the main areas for the application of SQUIDs. In particular, magnetocardiography and magnetoencephalography have been studied extensively using multichannel low-Tc SQUID systems in specially shaped cryostats. High-Tc multichannel systems have also been developed. The reader is referred to chapter G2.3 in this book for more information. I2.0.4.2 Nondestructive evaluation SQUID systems can be used for nondestructive evaluation (NDE) of a variety of materials and structures in order to detect defects (Donaldson et al 1993). The studied object is usually at room temperature and an important parameter is then the lateral spatial resolution that can be obtained. The maximal resolution is of the same order of magnitude as the distance between the cold inner cryostat wall and the warm outer cryostat wall. A typical set-up of a SQUID system for nondestructive evaluation is given in figure I2.0.16 (Donaldson et al 1993). A magnet is used to magnetize a test subject (e.g. a plate) which is displaced just below the pickup coils of a flux transformer coupled to a SQUID. Cracks in the plate or variations in the magnetic concentration will disturb the magnetic field pattern. The flux change due to this is detected by the SQUID. The advantage of the SQUID is that these flux variations can be measured with unchanged sensitivity in a relatively large background field. This method of detection is called remote magnetometry. It has been used, for instance, for detecting surface cracks in ferromagnetic steel plates. The sensitivity of the SQUID is such that even diamagnetic materials can be studied. Drilled holes in a Plexiglass (Perspex) plate could be detected due to the variation in diamagnetic susceptibility. A similar technique is used for remote galvanometry in which currents flow through the object: and the generated field is mapped. These currents may be directly injected, due to induction or caused by galvanic processes in the material. In the case of induction, the field coil in figure I2.0.16 is used for

Figure I2.0.16. A typical measurement set up for NDE. Reproduced from Donaldson et al (1993) by permission of Kluwer Academic Publishers. Systems have been developed for low Tc (liquid-helium cryostat) and high Tc (liquid-nitrogen cryostat).

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Figure I2.0.17. A schematic representation of a rock magnetometer with horizontal access (4) for sample supply. A pickup coil (9) is connected to the SQUID (6), which is cooled via a bottom plate (3) attached to the helium reservoir (2). For details see ter Brake et al (1984).

applying an a.c. field which induces eddy currents in the sample. Remote galvanometry has been used to detect defects in pipelines and metal plates as well as to sense electrolytic corrosion. Just as in biomagnetic measurements, the inverse problem is also a reality in remote sensing. It is in general not possible to arrive at a unique solution for the source distribution on the basis of the field map. A rather new remote magnetometry area is the scanning high-Tc SQUID microscope (Black et al 1995). Magnetic images at frequencies ranging from zero to 200 GHz have been obtained for thin films and bulk metallic samples with a spatial resolution of about 30µm. At present NDE with high-Tc SQUIDs attracts a lot of attention (Proc. Applied Superconductivity Conf. 1997) for detection of cracks in, for instance, aircraft wings. Even hand-held systems are under development. I2.0.4.3 Geophysics SQUID systems play an important part in determining the magnetic properties of the earth (Clarke 1983). This concerns on the one hand the characterization of specific earth samples, and on the other the mapping of the earth’s magnetic field and electromagnetic impedance. The ability to measure in background fields without sensitivity decrease, and with a large dynamic range and bandwidth, make the SQUID a valuable detecting instrument. Instead of using liquid helium in a low-Tc system, the apparatus can be equipped with a cryocooler. In this case special attention has to be paid to the suppression of mechanical vibrations that would cause a deterioration in the performance of the SQUID. One of the oldest applications is in the determination of the magnetic properties of rock. Figure I2.0.17 shows a schematic picture of a rock magnetometer with a horizontal access tube for sample supply at room temperature (ter Brake 1984). The magnetic moment can be traced in three perpendicular directions as Copyright © 1998 IOP Publishing Ltd

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Figure I2.0.18. A typical system for exploring gravitational waves. An aluminium bar of 2270 kg (Explorer) has a resonant disc transducer connected to the end of the bar. The small displacements of the disc can be measured by the SQUID via a capacitive coupling of the disc-to-flux transformer circuit. For details see Astone et al (1993).

Figure I2.0.19. A schematic diagram of a superconducting gravity gradiometer. The system consists of two test masses suspended with a cantilever spring. A gradient in the gravity results in a small relative displacement of the two masses, leading to current changes in the inductive superconducting network that is coupled to the SQUID. The figure on the left presents the concepts, whereas the right-hand figure gives the configuration. Reproduced from Paik (1994) by permission.

a function of the position in the cylindrical sample rod. In this way static and dynamic properties of earth samples are studied. In paleomagnetism (Collinson 1983) the history of the earth’s field is studied. Subjects are, for instance, the position and polarity of the earth’s magnetic poles in time. Also continental drift has been studied. Copyright © 1998 IOP Publishing Ltd

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A second group of applications in geophysics is the direct inspection of the magnetic properties of the earth. In magneto-telluric research the surface impedance of the earth is studied. The horizontal components of the electromagnetic waves coming from the ionosphere (frequency range from 0.1 mHz to 100 Hz) are measured in order to obtain information about the earth’s resistivity. These data result in an indication of the location of oil fields, mineral resources and geothermal sources. For the elimination of disturbances, use is often made of a reference SQUID system placed some kilometres away from the detecting SQUID system. Many other topics could be dealt with in this section about geophysics such as the study of internal ocean waves, hydrocracks and tecto-, piezo- and seismo-magnetism; however, instead the reader is referred to the literature (Clarke 1993, Gallop 1991, Weinstock 1995). I2.0.4.4 Gravity SQUID systems are used for measuring gravitational forces. Typical topics in this area are gravity systems for mineral surveying, inertial navigation, general relativity verification, deviations from 1/r 2 and gravitational waves. In this section attention will be paid to gravitational wave detectors and gravity gradiometers Gravitational waves are emitted by bodies when the mass distribution varies nonspherically (rotating double stars, collapse of stars). The radiation causes a longitudinal strain which can be detected by a massive antenna in the shape of a bar or a sphere. The resulting strain ∆l/l, however, is extremely small and typically much smaller than 10−19. Therefore systems are necessary for detecting strains of the order of 10−21. A typical configuration is given in figure I2.0.18 (Astone et al 1993). The vibrations of the antenna are amplified by a resonant mass transducer and the displacements induce a current in a superconducting transformer that is coupled to a very sensitive SQUID system. The antenna has to be cooled down to the millikelvin range to reduce the mechanical noise of the system and it also needs a very high quality factor. The SQUID has to approach the quantum limit with respect to the sensitivity. At present a number of bar antennas with strain sensitivities of about 10−18 have been fabricated and are in use for detecting gravitational waves but have been unsuccessful so far. There are a few initiatives for making 3 m diameter spherical detectors with a sensitivity of about 10−21†. Gravity gradiometers are in principle also displacement sensors. They are a combination of two accelerometers where the displacement of the masses is sensed by a SQUID system in a manner very similar to that of the gravitational wave antenna. A typical configuration is presented in figure I2.0.19 (Paik 1994). The gravity gradient is a tensor and expressed in Eötvös (1 Eötvös is 10−9 s2. For mineral surveying and navigation one needs sensitivities of 1-10 Eötvös in the frequency range from d.c. to a few hertz. Several space-borne instruments have been proposed with sensitivities of 0.001 Eötvös down to the millihertz range. References Adelerhof D J, Nijstad H, Flokstra J and Rogalla H 1994 (Double) relaxation oscillation SQUIDs with high flux to voltage transfer, simulations and experiments J. Appl. Phys. 76 3875-86 Astone P, Bassan M, Bonifazi P et al 1993 Long-term operation of the Rome ‘Explorer’ cryogenic gravitational wave detector Phys. Rev. D 47 362-75 Black R C, Wellstood F C, Dantsker E, Micklich A H, Koelle D, Ludwig F and Clarke J 1995 High- frequency magnetic microscopy using high Tc SQUID IE EE Trans. Appl. Supercond. AS-5 2137-41 Blair D G (ed) 1993 The Detection of Gravitational Waves (Cambridge: Cambridge University Press)

† There are two initiatives to develop spherical gravitational wave detectors with a diameter of about 3 m: TIGA (W O Hamilton and W W Johnson-a US collaboration) and Grail (a Dutch collaboration).

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Braginski A I 1995 Fabrication of high temperature SQUID magnetometers SQUID Sensors: Fundamentals, Fabrication and Applications ed H Weinstock (Deventer: Kluwer) pp 235–88 Clarke J 1983 Geophysical applications of SQUIDs IEEE Trans. Magn. MAG-19 288–94 -1993 SQUIDs: theory and practice The New Superconducting Electronics ed H Weinstock and R W Ralston (Deventer: Kluwer) p 123–80 Collinson D W 1983 Methods in Rock Magnetism and Paleomagnetism. Techniques and Instrumentation (London: Chapman and Hall) David B, Grundler D, Krumme J P and Doessel O 1995 Integrated high Tc SQUID magnetometer IEEE Trans. Appl. Supercond. AS-5 2935–8 Donaldson G B, Cochran A and Bowman R M 1993 More SQUID applications The New Superconducting Electronics ed H Weinstock and R W Ralston (Deventer: Kluwer) 181–220 Gallop J C 1991 SQUIDs, the Josephson Effects and Superconducting Electronics (Bristol: Hilger) Hämäläinen M S, Hari R, Ilmoniemi R J, Knuutila J and Lounasmaa O V 1993 Magnetoencephalography- theory, instrumentation and applications to noninvasive studies of the working brain Rev. Mod. Phys. 65 413–95 Hilgenkamp J W M, Brons G C S, Soldevilla J G, Ijsselsteijn R P, Flokstra J and Rogalla H 1994 Four layer monolithic integrated high Tc dc SQUID magnetometer Appl. Phys. Lett. 64 3497–9 Jin, Amar A, Stevenson T R, Wellstood F C, Morse A and Johnson W W 1997 35h two-stage SQUID system for gravity wave detection IEEE Trans. Appl. Supercond. AS-7 2742–6 Koch R H, Clarke J, Goubau W M, Martinis J M, Pegrum C M and van Harlingen D J 1983 Flicker ( 1/f ) noise in tunnel junction dc SQUIDs J. Low Temp. Phys. 51 207–24 Koelle D, Micklich A H, Ludwig F, Dantsker E, Nemeth D T and Clarke J 1993 dc SQUID magnetometers from single layers of YBa2Cu3O7-x Appl. Phys. Lett. 63 2271–3 Ludwig F, Dantsker E, Kleiner R, Koelle D, Clarke J, Knappe S, Drung D, Koch H, Alford N McN and Button T W 1995 Integrated high Tc multiloop magnetometer Appl. Phys. Lett. 66 1418–20 Mück M, Becker Th and Heiden C 1995 Use of super Schottky diodes in a cryogenic radio frequency superconducting quantum interference device read out Appl. Phys. Lett. 66 376–8 Nilsson P A, Ivanov Z G, Stepantsov E A, Hemmes H K, Hilgenkamp J W M and Flokstra J 1995 Noise Characterization of YBa2Cu3Oj Flip Chip SQUID Planar Gradiometers ( Inst. Phys. Conf. Ser. 148) vol 2, ed D Dew-Hughes (Bristol: Institute of Physics) pp 1537–40 Paik H J 1994 Superconducting accelerometry: its principles and applications Class. Quantum Grav. 11 A133–44 1997 Proc. Applied Superconductivity Conf. 1996 IEEE Trans. Takada S and Koyanagi M 1992 Fabrication of integrated dc SQUID with refractory tunnel junctions Principles and Applications of Superconducting Quantum Interference Devices ed A Barone (Singapore: World Scientific) pp 151–88 ter Brake H J M, Ulfman J A and Flokstra J 1984 SQUID-magnetometer with open-ended horizontal room-temperature access J. Phys. E: Sci. Instrum. 17 1024–9 van Duuren M J, Lee Y H, Adelerhof D J, Kawai J, Kado H, Flokstra J and Rogalla H 1996 Multichannel SQUID magnetometry using double relaxation oscillation SQUIDs IEEE Trans. Appl. Supercond. AS- 6 38–44ärf> Weinstock H (ed) 1995 SQUID Sensors: Fundamentals, Fabrication and Applications (Deventer: Kluwer) Zhang Y, Mück M, Braginski A I and Topfer H 1994 High-sensitivity microwave rf SQUID operating at 77 K, Supercond. Sci. Technol. 77 269–73

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I3 Single-flux quantum electronics

K Nakajima

I3.0.1 Introduction Josephson integrated circuits have attracted considerable interest because of their high-speed operation with very low power dissipation as compared with conventional semiconductor circuits (McDonald et al 1980). Recently, many Josephson large-scale integrated (LSI) circuits have been fabricated by using integration technology based on the superconducting materials of Nb and NbN (Nakagawa et al 1991). The operation modes of these LSI circuits fall into two broad categories: voltage and phase. The first category has a close relation with the hysteretic current-voltage (I-V) characteristics of a Josephson junction. The second category is related to quantum mechanical quantization of magnetic flux. The flux quantization(h/2e) is a basic characteristic of superconductors and is represented in terms of the phase of a macroscopic quantum mechanical wavefunction. Let us discuss the fundamentals of single-flux quantum electronics starting from Josephson junction behaviours caused by the nonlinear relation between the current and phase difference across the junction. I3.0.2 Equivalent circuits The low-frequency form of the Josephson equations relating the phase difference φ to current I and voltage V is ( Josephson 1965)

where Ic is the critical current in the junction and e and h are the usual fundamental constants. Since the availability of accurate models for the junctions facilitates the analyses of superconducting circuits, the phenomenological RCSJ (resistively and capacitively shunted junction) model is commonly used (McCumber 1968, Stewart 1968). It is a simplification of the microscopic model and adequate for electrical time responses down to a few picoseconds (Werthamer 1966). A Josephson junction is driven by a current source, because its impedance is usually much smaller than the source impedance. The circuit equation for the conventional case with a current source is

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where the first term describes supercurrent flow in a junction and the second and the third terms are resistive and capacitive shunts to account for normal and displacement current flow at finite voltages. Much of the analysis to be discussed here will involve junction behaviour at voltages below the energy gap, in which case Ic , G and C are all considered to be constant. Now we can devise an equivalent circuit for a junction as shown in figure I3.0.1.

Figure I3.0.1. Equivalent circuit for a Josephson junction.

If we compare equation (I3.0.3) with the fundamental equation for a rigid pendulum, it is easy to see that the analogue of the phase difference is the angle, that of voltage is the angular velocity, that of capacitance is the moment of inertia, that of conductance is the damping constant, that of the critical current is the maximum gravitational torque and that of the source current is the applied torque. Therefore we can use a rigid pendulum to get an insight into the behaviour of a Josephson junction (Sullivan and Zimmerman 1971). I3.0.3 Parallel arrays of junctions Let us consider the parallel array of Josephson junctions shown in figure I3.0.2. The Josephson equation relating the phase difference φi to the magnetic flux L Ii is (Josephson 1965)

Figure I3.0.2. A parallel array of Josephson junctions (discrete Josephson transmission line).

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where L and Ii are the loop inductance and the circulating current of the loop i respectively. If every junction has the same electric parameters, then taking account of Kirchhoff ’s current law for the i th junction we can write the nonlinear differential difference equation

The mechanical analogue of a parallel array of junctions consists of a series of pendula connected by springs (Scott 1969). If we can assume that the phase difference changes only slightly over a distance ∆x, equation (I3.0.5) can be written as a partial-differential equation. The partial-differential equation has the same form as the sine-Gordon equation with damping and acceleration terms. The sine-Gordon equation (Rubinstein 1970)

has soliton solutions which correspond to flux quanta (fluxons) as discussed in the following. I3.0.4 Fluxon oscillation and spatial distribution Analytical solutions to the resulting circuit equations (I3.0.3), (I3.0.5), (I3.0.6), etc are, however, only rarely obtainable and numerical simulations must be employed. Unfortunately, numerical simulations often yield little physical insight into dynamic behaviour, and it is therefore desirable to find approximate analytical treatments which can give such insight i.e. use exact analytical solutions rather than numerical simulations depending on the circumstances. Let us now assume a solution of equation (I3.0.3) for a d.c. voltage with a small sinusoidal variation with time

where ω = dφ/dt = (2e/h )V and ω0 » ω1. Substituting (I3.0.7) into equation (I3.0.3),

neglecting ω1/ω0 compared with unity, we obtain

where R = 1/G. It should be noted that the amplitude of a.c. voltage decreases with increasing product C R, and that the integration of voltage V over the time period T = 2π/ω0 leads to a flux quantum h/2e. If the shunt conductance is so large as to short completely the capacitance, we can neglect the second-derivative term in equation (I3.0.3). It corresponds to a situation where damping effects dominate inertial effects in a mechanical model. Under this situation equation (I3.0.3) is analytically solvable. It can be seen from the analytical solution that the a.c. voltage having a high number of harmonics is like an array of solitary waves (solitons) for I close to Ic and with increasing current I voltage becomes closer to a simple sinusoidal oscillation with a d.c. component nearly equal to I R (e.g. Barone and Paterno 1982, Van Duzer and Turner 1981). Copyright © 1998 IOP Publishing Ltd

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Next let us consider a spatial distribution of magnetic field in a Josephson junction. In the continuous limit ∆x → 0 in figure I3.0.2, one obtains a differential equation which determines φ in the absence of a current bias

where Lu and Jc are the inductance and the critical current density per unit length respectively. Equation (I3.0.11) describes the spatial change of φ in a Josephson junction itself, if Lu = µ 0(2λ + d)/a where µ0, λ, d and a are the permeability, the penetration depth of the superconductor, the insulator thickness and the junction width respectively. For a uniform junction of small dimension or low critical current, the magnetic field due to the tunnelling current can be neglected. In this case, the maximum current variation with applied magnetic field Ha follows a Fraunhofer diffraction pattern (Rowell 1963). This result is derived from the uniform distribution of magnetic field in the junction

Let us now assume a solution of equation (I3.0.11) for uniform magnetic field with a small spatially sinusoidal variation

where k = dφ/dx, k0 = 2e LuHa /h and k0 » k1. Substituting equation (I3.0.13) into equation (I3.0.11) and neglecting k1/k0, we get

Therefore, equation (I3.0.13) can be written as

It should be noted that the magnetic field Hy smooths out as Ha increases, and that the integration of Lu Hy over the wavelength λ0= 2π/K0 leads to a flux quantum h/2e. Namely, the spatial variation decreases as the number of fluxons in the junction increases. Equation (I3.0.11) can be solved in terms of elliptic functions, including boundary conditions (Owen and Scalapino 1967). It can be seen from the analytical solution that the waveform has a high number of harmonics for a small number of fluxons, and as more fluxons fit into the junction they not only have a smaller dimension, but become more sinusoidal in shape as well. The spatial variation of φ causes the variation of the maximum Josephson current. The experimental observation of the spatial distribution of the maximum Josephson current in the tunnel junctions is performed by using a laser-scanning technique (Scheuermann et al 1983) or low-temperature scanning electron microscopy (Bosch et al 1985). Multiplying equation (I3.0.11) by dφ/dx one can integrate equation (I3.0.11) and obtain the single-fluxon solutions (e.g. Scott 1970), which are shown in figure I3.0.3

where λJ = (h/2eLu Jc )1/2 and the signs + or - indicate the sense of magnetic field and correspond to a fluxon or antifluxon. Substituting equation (I3.0.16) into equation (I3.0.1), we obtain the supercurrent Copyright © 1998 IOP Publishing Ltd

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Figure I3.0.3. Soliton solutions of the sine-Gordon equation.

distribution (vortex) around a single fluxon. The vortex structure is in this case one dimensional, which is different from Abrikosov vortices in type 2 superconductors. The nature of the vortex is also different because it has no normal core. If we ask for solutions of the form φ = φ (ξ ) where ξ = x − ut, the sine-Gordon equation (I3.0.6) becomes

Hence, one can also obtain the soliton solutions propagating with a constant velocity u (e.g. Scott 1970)

where u < 1. If we measure distance in units of λJ and time in units of τJ = (h Cu /2e Jc )1/2, where Cu is the capacitance per unit length, in the parallel array in figure I3.0.2, the soliton equation (I3.0.19) corresponds to a single fluxon propagating on the Josephson transmission line (JTL) without damping and acceleration. In this case the propagating velocity υ of a fluxon cannot be higher than the limiting velocity c = 1/(Lu Cu )1/2 = v/u. I3.0.5 Vortex transitions in threshold curves A superconducting quantum interference device (SQUID) is considered as a basic device for superconducting electronics including single-flux quantum circuits. A SQUID consists of one or more inductive loops each of which contains more than one Josephson junction. In order to deal with the range in operating currents for which Josephson junctions in the SQUID may remain in the zero-voltage state, threshold curves in the current plane are commonly investigated. There are existence regions in which a Copyright © 1998 IOP Publishing Ltd

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given quantum state corresponding to the storage of a number of fluxons remains unchanged. By crossing the boundary (threshold curve) of these regions, the device switches into a different quantum state. To obtain threshold curves, Gibbs free energy, Lagrange multiplier formalism, etc, are used, since the equations are nonlinear and therefore admit solutions that do not correspond to physically stable states (Schulz-Du Bois and Wolf 1978). On the other hand, one may utilize linearized equations as an approximation. It works well for SQUIDs in which the product LIcof loop inductance L times Josephson critical current Ic per junction is of the order of a flux quantum h/2e and the approximation becomes increasingly better for LIc » h/2e. Even where it is not so good, at LIc » h/2e, linearization may be used for quickly getting threshold curves of limited accuracy on SQUID states. Besides providing quick information, it may be used as a starting estimate which can subsequently be improved upon by other numerical techniques. Let us consider only an isolated single loop (two-junction SQUID or d.c. SQUID) in the parallel array of junctions in figure I3.0.2. If the feed point for the bias current I is symmetrically placed at the centre of L and the feed of the control current Ic, is through L as in the conventional manner for a d.c. SQUID, the flux-quantization condition for the loop is

where n is an integer. According to Lagrange’s method, the threshold curves are obtained by looking for the maximum (or minimum) of the bias current

which is compatible with equation (I3.0.20). The broken line in figure I3.0.4 shows the threshold curves for the d.c. SQUID of LIc = 0.5h/2e.

Figure I3.0.4. Threshold characteristics of a two-junction SQUID. The broken and solid lines denote the results obtained by using Lagrange multiplier formalism and the linearization method respectively.

Linearization is introduced by replacing (Schulz-Du Bois and Wolf 1978)

where Ii is the circulating current in the loop. Substituting equation (I3.0.22) into equation (I3.0.20) Ii is obtained. The linearization in equation (I3.0.20) causes inaccuracy, but for φ = ±π/2, 0 there is no error Copyright © 1998 IOP Publishing Ltd

Josephson sampling system

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at all. Each quantum state distinguished by the index n in equation (I3.0.20) satisfies both of the following conditions

The solid line in figure I3.0.4 shows the resulting contour for LIc = 0.5h/2e. In the overlap regions two states corresponding to the values n and n + 1 for the number of fluxons are possible. Therefore, d.c. SQUIDs act as memory cells. The device operation in the first overlapping region is described as the 0 and 1 fluxon states which correspond to the binary representation of the two memory states 0 and 1. A voltage spike across the device associated with the transition between the two different states in the cell occurs as soon as the threshold curve is crossed. This voltage decays within the order of tens of picoseconds depending on the specific device parameter if the bias current is below a certain critical current. The operation across the threshold curve above the critical current brings on a voltage transition of the junction in the loop. The voltage transition provides a possible method of reading in the memory system (Zappe 1974). I3.0.6 Josephson sampling system In order to observe single-fluxon propagations, voltage spikes associated with the operation across the threshold curve, etc, the measurement system requires special tools compatible with the cryogenic environment and capable of measuring a few microamp and tens of microvolt signals having rise and fall times of the order of 10 ps. A direct measurement of travelling fluxons has been performed by using a wide-bandwidth amplifier, an ordinary sampling oscilloscope and a minicomputer system (Matsuda and Uehara 1982). However, such measurements utilizing conventional room-temperature techniques are difficult to perform in general because of the loss of bandwidth for signal transmission to room temperature. It is therefore desirable to provide for high-speed measurements directly on the superconducting chip. A biased Josephson junction can be utilized as an amplitude comparator for the signal to be measured. The case for a cryogenic sampling scheme based on Josephson junctions is argued in this section. Figure I3.0.5 shows the basic principle of the sampling technique utilizing a Josephson junction. Let us first consider the feed of an unknown repetitive signal waveform Iu(t ) and a known d.c. bias current Ib into the Josephson junction as a current comparator (Hamilton et al 1979). At some time T, the sum of these currents exceeds the critical current Ic of the junction and it switches to the voltage state. The time of this switching occurrence can be accurately measured by observing the voltage across the junction with a conventional sampling oscilloscope. The amplitude of the unknown signal at time T is determined as

By changing Ib and repeating the process, one can determine the value of the unknown signal at other times. This process is readily automated so that the unknown signal waveform is observed directly on a sampling oscilloscope. This technique enables observations of the rising edge of signal waveforms even though it is ultimately limited by the time jitter of the room-temperature sampling equipment. An improved technique is accomplished by supplying a very short sampling pulse Ip in addition to the bias current for the sampling gate. Hence, the current flowing through the junction comprises three components: ( i ) the unknown signal Iu ( t ) which is to be sampled, ( ii ) a sampling pulse Ip , occurring at time T, which can be delayed with respect to the unknown signal; ( iii ) a bias current Ib . The Josephson

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comparator is part of a feedback loop which adjusts the bias current so that the maximum of the sum of the three currents is exactly at the critical current of the junction

as shown in figure I3.0.5.

Figure I3.0.5. A sampling technique utilizing a Josephson junction.

Figure I3.0.6. A current pulse generator.

Since the threshold level and the amplitude of the sampling pulse are constant, the bias current will follow the unknown signal waveform at the time of the sampling pulse. The sampling pulse is arranged to have a variable delay T which spans the region of interest of the repetitive unknown waveform. The entire waveform can be mapped out as a function of the delay T. The maximum level of the unknown signal is restricted so that the signal alone never overcomes the critical current of the sampling gate. The current pulse generator shown in figure I3.0.6 consists of a two- or three-junction interferometer (SQUID) Q, a single Josephson junction J and a resistor R (Faris 1980). The threshold current of the

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Josephson sampling system

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interferometer Q is controlled by the control current Ic t which is provided by an external pulse generator. The bias Ib p is a d.c. current. The zero-field threshold current of the interferometer is much higher than the critical current of the single junction J. Initially, Q and J are in the zero-voltage state and the bias current Ib p is flowing through Q. As the control current Ic t reduces the threshold current of Q, Q switches to the resistive state. Since R is small, the current Ip through J rises and quickly reaches the critical current. Here J also switches to the resistive state and the current Ip drops very sharply since the resistance of J is much higher than that of Q. Thus a current pulse Ip having a short duration is produced. A feedback unit is used to find the threshold value of Ib in figure I3.0.5 (Tuckerman 1980). It is difficult to find a precise threshold value, because of noise. Instead, for each value of Ib there is a nonzero probability that the Josephson comparator will switch. The probability is near zero if Ib is below the theoretical threshold, and near one at or above that threshold. The value of Ib required to maintain a given switching probability (usually chosen to be 50%) differs from the theoretical threshold value by a constant. This constant depends on the amount of noise, which presumably does not change during a waveform measurement. The output of the comparator is amplified, then a fixed reference voltage Vr is subtracted from this signal and the difference is integrated over time. The integrator output will stay approximately constant for one specific switching probability (determined by Vr ). This integrator output is used to control the bias current Ib , completing the feedback loop. The comparator is repeatedly enabled with a period Ts , and the integration rate is slow enough to make Ib , fluctuate very little when the system is in a steady state. In other words, t1 Ts , where t1 is the time constant of the closed-loop system. This feedback scheme requires no lock-in amplifier.

Figure I3.0.7. A block diagram of the complete sampling system.

The block diagram of the complete system is shown in figure I3.0.7. Apart from the sampler chip, there are an external trigger and delay generator and a feedback circuit. The output of a pulse generator with a subnanosecond rise time is split equally by using a matched power splitter. One of the resulting pulses is applied to one of the experiments and the other one is applied through an adjustable delay line to the on-chip pulse generator. Now by using the sampling system, we can observe the voltage spike and the voltage transitions of the d.c. SQUID associated with the operation across the threshold curve which was discussed in the preceding section. Figure I3.0.8 shows the experimental result of the voltage transitions for three bias currents (Fujimaki et al 1987a). The observed waveform consists of roughly two parts. One is the

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Figure I3.0.8. The voltage transition waveform of a two-junction SQUID in which the critical current of each junction is 0.32 mA.

pulselike waveform which has 8 ps half-width and 85 µA pulse height and corresponds to a single-fluxon transition. The other is the transition part to the voltage state. The leading pulselike waveform goes into the transition region when the bias current is increased. I3.0.7 Single-fluxon propagation As we have already seen in the previous section, the sine-Gordon equation (I3.0.6) has a travelling soliton solution with a constant velocity. The form of equation (I3.0.6) is invariant to the Lorentz transformation of the independent variables from (x, t ) to (x ’, t’ )

and hence

Substitution of equation (I3.0.28) into equation (I3.0.6) leaves the form unchanged. This invariance to the Lorentz transformation leads to a contraction of both space and time by the factor (1 – u 2 )1/2 just as in the special theory of relativity. The effect is clearly evident in the soliton solution of equation (I3.0.19). We can observe the contraction effect in a JTL, because the voltage pulse associated with a traversing fluxon is obtained through the relation of equation (I3.0.2). Let us discuss the steady-state propagation of a single fluxon when bias currents and losses are included in the model which is represented by the partial-differential form of equation (I3.0.5). So far analytic solutions of this equation have not been found. An approximate result can be obtained by a first-order perturbation calculation. Kirchhoffs laws imply that the power dissipated by the shunt conductance must be balanced by the power supplied from the bias source as the pulse propagates along the line. This can be expressed in a power balance equation as (Scott 1970)

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Single-fluxon propagation

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where φ is the soliton solution for the lossless case given in equation (I3.0.19), distance and time are measured in units of λJ and τJ respectively, Γ = g(h/2eJc C )1/2, γ = jb /Jc , and jb and g are the bias current density and the shunt conductance per unit length of the JTL, respectively. By integrating equation (I3.0.29), the fluxon velocity υ becomes

For junctions of finite length l » λJ and small damping (4Γ/πγ )2 « 1, the fluxon is reflected from the end with a change in sign of its magnetic flux. The fluxon reflection is equivalent to the reflection of an electromagnetic pulse from an open-ended strip line. Hence, the fluxon travels repeatedly up and down the length of the junction by this reflection, changing sense at each end. One may calculate approximately the shape of the I−V characteristics associated with this process (Fulton and Dynes 1973). The time-average voltage is V = ch/2el, because at any given point a voltage pulse of a fluxon is experienced on the passage both of the fluxon and of the antifluxon. One may expect that the time-average voltage is V = nch/2el for n fluxons which are simultaneously performing this back-and-forth motion. These processes involve fluxon-antifluxon collisions occurring within the junction. The collision will be discussed in the next section. The zero-field step, which is observed in the I−V characteristics without external applied magnetic field, can be explained in terms of the time-average voltage associated with the fluxon motion (Chen et al 1971). Sustituting equations (I3.0.19) and (I3.0.30) into equation (I3.0.2), we can obtain an approximate solution υp to describe the voltage pulse waveform of the travelling fluxon

We can observe the voltage pulse waveform by using the sampling system discussed in the preceding section. Figure I3.0.9(a) shows the fluxon waveform propagating on the discrete JTL which is equivalent to the model shown in figure I3.0.2 (Fujimaki et al 1987b). The JTL comprises 31 lead-alloy junctions each of which is 4 µ m × 4 µ m and shunted with an additional resistor. The interval ∆x between two adjacent junctions is 60 µm, and hence the total length of the JTL is 1800 µ m. The critical current density

Figure I3.0.9. (a) The travelling waveform of a fluxon. (b) The pulse height and width of the fluxon as a function of the normalized bias current.

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of the junction and the L IC product of the single loop are 2.625 kA cm−2 and 0.2h/2e, respectively. The sampling gate (comparator junction) is connected at the centre of the JTL through a resistor of 3.4 Ω. The observed pulselike waveform as well as agreeing with the result of numerical simulations has a second small peak and the trailed ripple structure because of the discreteness of the JTL. The amplitude and width of the observed waveform are shown in figure I3.0.9(b) as a function of the total bias current applied to the JTL. The result seems to show the Lorentz contraction. A logic element operating on the principle of a single-fluxon storage was originally proposed by Anderson. The so-called flux shuttle has been widely discussed in connection with its application as a shift register. In this system each fluxon represents a bit of information. A few tens of picoseconds per single shift should be expected with an energy dissipation of the order of 10−19 J/shift (Fulton and Dunkleberger 1973). Let us discuss under what conditions an array of fluxons is statically stable against fluxon-fluxon repulsion in this discrete limit. Increasing L to h/2elc obviously increases the stability. Numerical estimates put the stability limit at LIc /(h/2e) = 0.8-0.9. The upper limit on LIC is less critical but it is useful to require that LIC ≤ 1.25h/2e, so that two fluxons cannot co-occupy a single loop (Fulton et al 1973). I3.0.8 Fluxon-antifluxon collisions An important aspect of fluxon dynamics is the study of the interactions between fluxons and antifluxons. For the lossless sine-Gordon system, one can derive both analytic expressions for soliton-soliton and for soliton-antisoliton collisions. We are interested in the soliton-antisoliton collision given by (Scott 1969)

because fluxon and antifluxon have opposite propagation directions to each other on the JTL with loss and driving (bias) current. The soliton in equation (I3.0.32) separates from the antisoliton in both the distant past and the distant future. It is interesting and important to notice that the soliton and the antisoliton can pass through each other without mutual destruction. However, an investigation of equation (I3.0.32) shows that even in the lossless case, the soliton colliding with the antisoliton will experience a phase shift δ (spatial advance) given by (Pedersen et al 1984)

This phase shift is due to the soliton-antisoliton attraction. The other two-soliton solution consists of a soliton-antisoliton pair bound together into an oscillatory state called a ‘breather’ which is also a result of the soliton-antisoliton attraction (McLaughlin and Scott 1978). It cannot decompose into a soliton and an antisoliton as t → ±∞ because its energy is less than the rest energy of the soliton-antisoliton components. These properties can be investigated through the inverse scattering transform method (Scott et al 1973). This method can be viewed as a nonlinear generalization of the Fourier-transform method wherein the Fourier components are replaced by solitons, antisolitons, breathers and radiation. For the JTL with loss and bias current, the collisions between fluxons and antifluxons can occur either as nondestructive collisions or as fluxon-antifluxon annihilation. The threshold driving term γt h corresponding to annihilation may be calculated by expressing the incident fluxon energy as being equal to the energy loss plus the soliton rest energy. The approximate result γt h is (2Γ)3/2 (Pedersen et al 1984). In a similar way, the line terminations are divided into ‘reflective’ and ‘reflectionless’ varieties according

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Fluxon-antifluxon collisions

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to whether an incident fluxon is reflected as an antifluxon or disappears at the termination. Since the boundary condition at a termination (dφ/dx = 0) can be satisfied by assuming collision with a virtual antifluxon, the critical values of line parameters that distinguish between passing through and annihilation are identical to those for reflective and reflectionless terminations (Akoh et al 1987). A nondestructive collision between a fluxon and antifluxon introduces some delay time instead of a phase shift δ. Indeed, the velocity during the interaction decreases to a minimum value after which it increases again so that the fluxon and the antifluxon emerge with the initial velocity of equation (I3.0.30). In a destructive collision the velocity falls to zero and the fluxon-antifluxon become a breather which decays subsequently and finally disappears. In order to observe the fluxon-antifluxon collision waveforms over the whole JTL, which is the same JTL as used to observe the travelling fluxon waveform in the preceding section, electrical delays are introduced to the fluxon and the antifluxon generators which are connected at each end of the JTL. The delay ∆T for fluxon generators shifts their collision point over the whole JTL. Hence, the delay ∆T indicates the space coordinate. On the other hand, another delay ∆S which controls the sampling gate set at the centre of the JTL indicates the time coordinate. The delay ∆S is fixed during the observation of waveforms in the space coordinate by the sweep of ∆T. We can obtain the time evolution of the waveform by changing ∆S. The experimental result of a fluxon-antifluxon annihilation is shown in figure I3.0.10 (Fujimaki et al 1987b). The horizontal axis represents the space coordinate along the JTL. In the top waveform in figure I3.0.10 a fluxon and an antifluxon are propagating independently toward the centre of the JTL. After 8 ps these fluxons start to interact, trailing ripple waves behind them. After 16 ps the amplitude of the colliding fluxons is negative. The negative amplitude indicates that the fluxon and the antifluxon are in a breather decay mode in which the fluxon-antifluxon pair is bound together in a damped oscillatory state. The oscillatory wave in the breather mode decays, dissipating the energy at the shunt resistors. After 24 ps

Figure I3.0.10. The spatio-temporal waveform of a fluxon-antifluxon annihilation. Copyright © 1998 IOP Publishing Ltd

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the collision is almost complete and the two fluxons annihilate. For a higher bias current, we can observe nondestructive collisions and phase shifts (time delay) which agree with the numerical simulations but are different from the spatial advance δ in a lossless sine-Gordon system. For the highest bias current, many fluxon-antifluxon pairs are generated at the collision point (Nakajima et al 1990). The pair creation may be explained as follows. The potential energy (magnetic energy) of fluxons is converted into kinetic energy (electric energy) at their collision. The higher-voltage part receives more input power from the bias current, because the input power is proportional to the product of the voltage and the bias current. The increase of the input power causes an instability (local voltage transition) in the discrete JTL so that an additional fluxon-antifluxon pair is created, satisfying a quantum condition inside the JTL. Once a new fluxon-antifluxon pair is created, the creation of infinite pairs is induced because of inertia (capacitance), and then they spread from the centre of the line to both ends. Figure I3.0.11 shows the numerical results for the fluxon-antifluxon interaction. The annihilation, the passing through of a fluxon pair and the pair creation of fluxons are shown in figure I3.0.11 (a), (b) and (c) respectively.

Figure I3.0.11. Simulation results of fluxon-antifluxon collisions. Annihilation (a), passing through (b) of a fluxon pair and pair creation of fluxons (c).

I3.0.9 Logic circuits Several families of fluxon logic circuits have been demonstrated. These logic families utilize nonhysteretic Josephson junctions, which suggests the possibility of using nonhysteretic high-Tc junctions, and are characterized by very low power dissipation because of the small amplitude and short duration of signal voltages. In spite of small amplitude and short duration, the signal does not decay because of flux quantization. Copyright © 1998 IOP Publishing Ltd

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Here we discuss representative fluxon logic families that use pulses associated with a travelling fluxon or latched current associated with a rest fluxon to designate logic values. To represent the pulse logic families, we describe phase-mode (PM) circuits (Nakajima et al 1991) and the rapid-singleflux-quantum (RSFQ) circuits (Likharev and Semenov 1991). The representative for current-latching logic is the quantum flux parametron (QFP) (Hosoya et al 1991). Figure I3.0.12 shows the basic circuits of these logic families. The first circuit, shown in figure I3.0.12(a), which belongs to both PM and RSFQ families, is called a T-branch (PM) or pulse splitter (RSFQ) and can either perform the logic ‘AND’ operation for low-current biasing, or provide fan-out into both connecting branches for high-current biasing.

Figure I3.0.12. The basic circuits of PM (a, b), RSFQ (a, c) and QFP (d) logic families.

The second circuit, in figure I3.0.12(b), which belongs to the PM family, is called an S-branch and can either perform the switching of fluxon outputs, or provide fan-in by inverting the travelling direction of fluxons, depending on current biasing to connecting branches. The third circuit (figure I3.0.12(c )) belongs to the RSFQ family and is called the buffer stage. It permits fluxon propagation only in the arrowed direction. The fourth circuit (figure I3.0.12(d)) belongs to the QFP family and is called simple QFP. It makes use of a two-junction SQUID (two coupled single-junction SQUIDs) with inductive coupling of an excitation current. If there is a current in the input logic signal (vertical centre line) when the excitation current is supplied, the element will divert a large current to the load (vertical inductor), resulting in a current gain. The logic levels are represented by the sign of the currents. An ‘INHIBIT’ gate which is a combination of the T- and S-branches is proposed as the basic element for the PM logic family (Onomi et al 1995). Since INHIBIT is a universal operation in terms of Boolean algebra, it is clear that all logical functions can be performed by combinations of INHIBITS. The INHIBIT gate can be easily configured to perform as a full adder. A complete computer system has been also proposed. For the RSFQ logic family, the ‘confluence buffer’ which is a generalization of the buffer stage, RS (reset set) and T (trigger) flip-flops are proposed as the basic elements. The RS and T flip-flops are combinations of the buffer stages and two-junction SQUIDs. Many kinds of logic circuit comprising a few

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thousand Josephson junctions each have been fabricated and measured experimentally at clock frequencies up to 20 GHz (Mukhanov and Kirichenko 1995, Polonsky et al 1994), for example, a shift register which consists of a buffer stage and a storage stage (a two-junction SQUID) (Mukhanov 1993), a counting-type analogue-to-digital converter (Rylov and Robertazzi 1995) and a digital signal processor (Mukhanov et al 1995). Simple circuits of this family, which is one of the most aggressive research subjects at present, have been estimated from the measurement of the d.c. voltage across the junction to operate at frequencies up to 370 GHz (Bunyk et al 1995). The fundamental principles of the pulse logic families (PM and RSFQ) are as follows: first, the transmission of information using fluxons and, second, the storage and recall of information in the interferometer loops via fluxons entering and leaving these loops. Signal propagation can be accomplished through amplifying JTLs or superconducting microstrip transmission lines to propagate the picosecond and microvolt pulses (Polonsky et al 1993). The PM or RSFQ logic schemes can drive their own clock or reset inputs, and therefore can be used in an asynchronous or synchronous system. A simple QFP combination called a D-gate is proposed as the basic element for the QFP logic family to improve margins for the circuitry. Useful functional gates can be made easily from the D-gate, for example majority gates, parity gates, etc. A full adder can be designed by putting a majority gate and parity gate together, sharing the same inputs. The majority gate’s output is the carry signal and the parity gate’s output is the sum. The QFP circuits cannot drive their own clock inputs and require an externally applied clock phase for each logic level. However, the input must be valid before the clock input, requiring each logic level to be pipelined. To guarantee forward propagation of logic in QFP circuits, a multiphase clocking scheme is used, which lends itself to an extremely highly pipelined system (Fleischman et al 1991). For example, the operation of a one-bit QFP shift register by a four-phase clock up to 36 GHz has been demonstrated (Hosoya et al 1995).

I3.0.10 Superconducting neural networks Recently, there has been increasing interest in the application of artificial neural networks for parallel and intelligent information processing. Superconducting circuits with a fluxon stream technique have much potential for neural very-large-scale-integrated (VLSI) analogue-type chips. Let us consider two discrete JTLs connected in series with an inductor L and a resistor R. We may consider fluxon pulses as neural impulses with soliton characteristics. When a fluxon propagating on the first JTL reaches the LR loop, it is trapped there because of the loss of the resistor and the threshold characteristics of the neuron junction which is the first junction of the second JTL. The circulating current to hold the fluxon, which corresponds to the neural potential level, decreases with the time constant L/R. The spatial summation of neural impulses is accomplished to connect plural JTLs to the neuron junction. When the temporal and spatial summation of the current flowing into the neuron junction exceeds the critical current, a fluxon stream begins to flow into the second JTL. However, this neuron circuit cannot avoid having a self-connection because of the incomplete input-output isolation. To reduce the incompleteness of the input-output separation, a coupled SQUID circuit is proposed. The quantum state (n = 0 or 1) of the single-junction SQUID is used as a neuron state. The quantum state can be read out by using a biased two-junction SQUID coupled to it. The output current of the two-junction SQUID is the output of the neuron. A synapse circuit having a variable connecting weight can be accomplished by using multiple two-junction SQUIDs as variable current sources. The fundamental operations of these elements and a neural-based analogue-to-digital converter have been demonstrated (Mizugaki et al 1994).

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References

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References Akoh H, Sakai S and Takada S 1987 Direct observation of fluxon reflection in a Josephson transmission line Phys. Rev. B 35 5357–60 Barone A and Paterno G 1982 Physics and Applications of the Josephson Effect (New York: Wiley) Bosch J, Gross R, Koyanagi M and Huebener R P 1985 Direct probing of the spatial distribution of the maximum Josephson current in a superconducting tunnel junction Phys. Rev. Lett. 54 1448–51 Bunyk P I, Oliva A, Semenov V K, Bhushan M, Likharev K K, Lukens J E, Ketchen M B and Mallison W H 1995 High-speed single-flux-quantum circuit using planarized niobium-trilayer Josephson junction technology Appl. Phys. Lett. 66 646–8 Chen J T, Finnegan T F and Langenberg D N 1971 Anomalous dc current singularities in Josephson tunnel junctions Physica 55 413–20 Faris S M 1980 Generation and measurement of ultrashort current pulses with Josephson devices Appl. Phys. Lett. 36 1005–7 Fleischman J, Feld D, Xiao P and Van Duzer T 1991 Evaluation of flux-based logic schemes for high-Tc applications IEEE Trans. Magn. MAG-27 2769–72 Fujimaki A, Nakajima K and Sawada Y 1987a Direct measurement of the switching waveform in a DC-SQUID Japan. J. Appl. Phys. 26 74–80 Fujimaki A, Nakajima K and Sawada Y 1987b Spatiotemporal observation of the soliton-antisoliton collision in a Josephson transmission line Phys. Rev. Lett. 59 2895–8 Fulton T A and Dunkleberger L N 1973 Experimental flux shuttle Appl. Phys. Lett. 22 232–3 Fulton T A and Dynes R C 1973 Single vortex propagation in Josephson tunnel junctions Solid State Commun. 12 57–61 Fulton T A, Dynes R C and Anderson P W 1973 The flux shuttle—a Josephson junction shift register employing single flux quanta Proc. IEEE 61 28–35 Hamilton C A, Lloyd F L, Peterson R L and Andrews J R 1979 A superconducting sampler for Josephson logic circuits Appl. Phys. Lett. 35 718–9 Hosoya M, Hioe W, Casas J, Kamikawai R, Harada Y, Wada Y, Nakane H, Suda R and Goto E 1991 Quantum flux parametron: a single quantum flux device for Josephson supercomputer IEEE Trans. Appl. Supercond. AS-1 77–89 Hosoya M, Hioe W, Takagi K and Goto E 1995 Operation of a 1-bit quantum flux Parametron shift register (latch) by 4-phase 36-GHz clock IEEE Trans. Appl. Supercond. AS-5 2831–4 Josephson B D 1965 Supercurrents through barriers Adv. Phys. 14 419–51 Likharev K K and Semenov V K 1991 RSFQ logic/memory family: a new Josephson-junction technology for sub-terahertz-clock-frequency digital systems IEEE Trans. Appl. Supercond. AS-1 3–28 Matsuda A and Uehara S 1982 Observation of fluxon propagation on Josephson transmission line Appl. Phys. Lett. 41 770–2 McCumber D E 1968 Effect of ac impedance on dc voltage-current characteristics of superconductor weak-link junctions J. Appl. Phys. 39 3113–8 Mcdonald D G, Peterson R L, Hamilton C A, Harris R E and Kautz R L 1980 Picosecond applications of Josephson junctions IEEE Trans. Electron Devices ED-27 1945–65 McLaughlin D W and Scott A C 1978 Perturbation analysis of fluxon dynamics Phys. Rev. A 18 1652–80 Mizugaki Y, Nakajima K, Sawada Y and Yamashita T 1994 Implementation of superconducting synapses into a neuron-based analog-to-digital converter Appl. Phys. Lett. 65 1712–3 Mukhanov O A 1993 RSFQ 1024-bit shift register for acquisition memory IEEE Trans. Appl. Supercond. AS-3 3102–13 Mukhanov O A, Bradley P D, Kaplan S B, Rylov S V and Kirichenko A F 1995 Design and operation of RSFQ circuits for digital signal processing Extended abstracts of ISEC’95 (Nagoya: Nagoya University) pp 27–30 Mukhanov O A and Kirichenko A F 1995 Implementation of a FFT radix 2 butterfly using serial RSFQ multiplier-adders IEEE Trans. Appl. Supercond. AS-5 2461–4 Nakagawa H, Kurosawa I, Aoyagi M, Kosaka S, Hamazaki Y, Okada Y and Takada S 1991 A 4-bit Josephson computer ETL-JC1 IEEE Trans. Appl. Supercond. AS-1 37–47

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Nakajima K, Mizusawa H, Sawada Y, Akoh H and Takada S 1990 Experimental observation of spatiotemporal wave forms of all possible types of soliton-antisoliton interactions in Josephson transmission lines Phys. Rev. Lett. 65 1667–70 Nakajima K, Mizusawa H, Sugahara H and Sawada Y 1991 Phase mode Josephson computer system IEEE Trans. Appl. Supercond. AS-1 29–36 Onomi T, Mizugaki Y, Nakajima K and Yamashita T 1995 Extended phase-mode logic-circuits with resistive ground contact IEEE Trans. Appl. Supercond. 5 3464–71 Owen C S and Scalapino D J 1967 Vortex structure and critical currents in Josephson junctions Phys. Rev. 164 538–44 Pedersen N F, Samuelsen M R and Welner D 1984 Soliton annihilation in the perturbed sine-Gordon system Phys. Rev. B 30 4057–9 Polonsky S V, Semenov V K and Kirichenko A F 1994 Single flux, quantum B flip-flop and its possible applications IEEE Trans. Appl. Supercond. AS- 4 9–18 Polonsky S V, Semenov V K and Schneider D F 1993 Transmission of single-flux-quantum pulses along superconducting microstrip lines IEEE Trans. Appl. Supercond. AS-3 2598–600 Rylov S V and Robertazzi R P 1995 Superconducting high-resolution A/D converter based on phase modulation and multichannel timing arbitration IEEE Trans. Appl. Supercond. AS-5 2260–3 Rowell J M 1963 Magnetic field dependence of the Josephson tunnel current Phys. Rev. Lett. 11 200–2 Rubinstein J 1970 Sine-Gordon equation J. Math. Phys. 11 258–66 Scheuermann M, Lhota J R, Kuo P K and Chen J T 1983 Direct probing by laser-scanning of the current distribution and inhomogeneity of Josephson junctions Phys. Rev. Lett. 50 74–7 Schulz-Du Bois E O and Wolf P 1978 Static characteristics of Josephson interferometers Appl. Phys. 16 317–38 Scott A C 1969 A nonlinear Klein-Gordon equation Am. J. Phys. 37 52–61 Scott A 1970 Active and Nonlinear Wave Propagation in Electronics (New York: Wiley) pp 247–52 Scott A C, Chu F Y F and McLaughlin D W 1973 The soliton: a new concept in applied science Proc. IEEE 61 1443–83 Stewart W C 1968 Current-voltage characteristics of Josephson junctions Appl. Phys. Lett. 12 277–80 Sullivan D B and Zimmerman J E 1971 Mechanical analogs of time dependent Josephson phenomena Am. J. Phys. 39 1504–17 Tuckerman D B 1980 A Josephson ultrahigh-resolution sampling system Appl. Phys. Lett. 36 1008–10 Van Duzer T and Turner C W 1981 Principles of Superconductive Devices and Circuits (New York: Elsevier-North-Holland) Werthamer N R 1966 Nonlinear self-coupling of Josephson radiation in superconducting tunnel junctions Phys. Rev. 147 255–63 Zappe H H 1974 A single flux quantum Josephson junction memory cell Appl. Phys. Lett. 25 424–6

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I4 Josephson voltage standards

J Niemeyer

14.0.1 Introduction This chapter describes the historical development of the voltage standard and the properties of the version of the Josephson voltage standard at present in use. The 200 year history of standard cells abruptly ended with the introduction of quantum standards based on the Josephson effect which reduces the calibration of a d.c. voltage to the determination of a frequency and the knowledge of the ratio of twice the elementary charge and Planck’s constant 2e/h = (Φ0 )−1 where Φ0 is the flux quantum ( Josephson 1962). As the frequency is finally controlled with extreme accuracy by the standard atomic caesium clock, only a convention on the value of Φ0 was needed to obtain a much better reproducibility for the voltage calibration than was obtained before with electrochemical cells. Moreover, complicated international comparisons for the adaptation of voltage standards were no longer required. As a result of the technical difficulties in the production of reliable Josephson junctions, in the 1970s single junction standards with a d.c. Josephson voltage of only a few millivolts were developed. The 1 V level was reached with the help of voltage dividers which limited the accuracy to a few parts in 10−8 (for a review see Petley 1983). With the progress achieved in junction technology it became possible to realize large series arrays of tunnel junctions which were used to directly generate Josephson voltages of up to more than 10 V. Calibrations with a reproducibility better by two orders of magnitude than in the case of the single-junction standards are performed now (for reviews see Kautz 1992, Niemeyer 1989, Pöpel 1992). Today, computer-controlled series array standards are used worldwide as primary standards in national calibration laboratories. Even industrial companies use these standards without the accuracy being reduced. As Josephson standards have no competitors in semiconductor electronics, they represent an important success in the application of superconducting electronics. 14.0.1.1 Development of standard cells and electronic standards In the beginning, the main purpose of electrochemical cells was to provide reliable sources of current. Following Galvani’s ideas (Galvani 1791), Volta built the first practical electrochemical cells in the year 1794 (Ostwald 1896), and these were the only available sources of electrical current for more than 50 years. Only much later, after the introduction of the Daniell cell (Zn-Cu) in 1836, were electrochemical cells used as stable voltage sources for maintaining and disseminating the unit volt. The first cell sufficiently Copyright © 1998 IOP Publishing Ltd

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stable to be used as a standard cell was the Clark cell (Zn-Hg) (Clark 1872). It was officially introduced at the International Congress of Electrical Engineers in Chicago in 1883 as a standard for maintaining the unit volt. The output voltage of the Clark cell was determined to be 1.434 international volts at 15 °C. In 1908, at the Internationa] Conference on Electric Units and Standards in London, the Clark cell was replaced by the international Weston standard cell (Cd-Hg) which had superior long-term stability, a smaller temperature coefficient and less hysteresis (Weston 1892). Moreover its output voltage of 1.0186 V was a better approximation to 1 V. The improved realization of the voltage induced the Rayleigh Committee to recommend in Washington, in 1910, the adoption of a more precise value for the maintenance of the volt. Similar improvements for the ampere and the ohm had been adopted at the London Conference two years earlier. The newly recommended units were called ‘International Units’. In January 1948 International Units were replaced by Absolute Units—metre, kilogram, second, ampere (MKSA). In this system, voltage is a derived quantity and its unit must be determined by an experiment, for example, by a voltage balance, which links it with the MKSA system. Accordingly, a standard cell became only a laboratory realization of the unit of voltage. The system of absolute units was adopted unchanged by the Systeme International (SI) which was instituted in 1960 by the Conference Generale des Poids et Mesures. The development of reliable standard cells formed the basis of international comparisons of voltage which began at the Rayleigh Conference in Washington in 1910. Figure I4.0.1 shows the results of the international voltage comparisons carried out between 1910 and 1980 and coordinated by the Bureau International des Poids et Mesures (BIPM) in Paris. In the late 1960s, the relative deviation of the output voltages of the single national standards could be determined with an uncertainty of less than 1µ V, but the values differed from each other by as much as 10 µ V (Melchert 1979). Between 1972 and 1980, a new quantum effect—the Josephson effect—led to much better calibration of the standard cells and, as a result, to strikingly smaller drifts and deviations of the various international standards. This clearly shows how a quantum standard can be superior to a standard based on an artefact such as an electrochemical cell. The discovery of the Josephson effect ended the role of the Weston cell as a primary voltage standard. At the beginning of the 1980s, electronic Zener voltage references replaced standard cells as transfer standards (Bachmair 1988), primarily because they are less sensitive to mechanical shock and electric charging. The output voltages of Zener references are typically 1 V, 10 V, 7 V and 1.0186 V, and their long-term drift is generally less than 1 × 10−6 per year at 10 V. For a more detailed description of the historical developments see Bachmair (1993). 14.0.1.2 Introduction of quantum standards As pointed out above, the realization of a physical unit by a material artefact implies a number of serious drawbacks: ( i ) they may be damaged by improper handling; ( ii ) the realized unit depends on external parameters; ( iii )intercomparisons can be difficult due to transport problems; ( iv )only central laboratories are able to maintain such systems. In contrast, quantum devices reduce the realization of the physical units to the determination of fundamental constants which are—as far as we know—independent of time and space and directly linked with basic physical theories whose consistency can be checked by a precise determination of the fundamental constants. Moreover, a quantum ‘recipe’ for the realization of a physical unit allows the calibration to be decentralized because the unit can be reproduced with fundamental accuracy in every laboratory. Regular intercomparisons are no longer necessary.

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Figure I4.0.1. Results of international voltage comparisons between 1910 and 1980, performed by six national calibration laboratories. The deviations of the output voltages of the six national standards from the average of the National Physical Laboratory (NPL) and the National Bureau of Standards (NBS) values are recorded.

In the past few decades, many efforts were therefore made to realize the physical units by quantum devices instead of material artefacts. Success has often resulted using physical systems where a large number of microscopic particles behave coherently. In particular, progress has been achieved by the laser-supported realization of the units of length and time and in the realization of electrical units on the basis of several macroscopic quantum effects: ( i ) the Josephson effect (Josephson 1962); ( ii ) the quantum Hall effect (von Klitzing 1986, von Klitzing et al 1980); ( iii )single-charge tunnelling (Geerligs et al 1990, Likharev 1988). The Josephson effect reduces the determination of a d.c. voltage to the precise counting of single flux quanta, Φ0 = h/2e. This laboratory realization is by orders of magnitude more precise than the SI realization of the unit volt. To ensure that all standard laboratories can take full advantage of the small uncertainty in the Josephson volt in the same way, the Consultative Committee of Electricity (CCE) has recommended the following exact value for the Josephson constant, (h/2e)−1 = (Φ0 )−1 ≡ KJ (Quinn 1989),

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where e is the elementary charge and h is Planck’s constant. This value has been in use since 1 January 1990, but only for maintaining the laboratory unit of the volt. It is equal to the SI value of (Φ0 )−1 within the accuracy with which it can be determined independentl within the SI laboratory, for example with the help of a voltage balance (Funck and Sienknecht 1991 Kose 1989, Taylor 1987, Taylor and Witt 1989). For the sake of completeness, it should be mentioned that a similar development took place in the realization of electrical resistance on the basis of the quantum Hall effect. The von Klitzing constant

was defined by the CCE in 1990 to be R K − 90 = 25812.802 Ω only for maintaining the practical laboratory unit. The determination of a current by precise counting of single charge quanta e has been under development since 1990. The method still suffers from the enormous difficulties in fabricating and measuring the nanocircuits needed for this experiment. The relations between electrical units according to their modern quantum definition are demonstrate in the so-called metrological triangle (figure I4.0.2).

Figure I4.0.2. The metrological triangle for electrical units. Josephson effect and single electron tunnelling relate the voltage and the current via the fundamental constants Φ0 and e to the frequency f. Voltage and current are related by the von Klitzing constant RK = h/e 2.

The principal analogy between the determination of voltage and current—i.e. the counting of quanta—has its roots in the fact that charge and flux are dual quantities from the viewpoint of quantum mechanics (Legett 1980, Prance et al 1991, Widom 1979). The figure suggests how important it would be for the consistency of the definitions to close the triangle experimentally by a precise determination of the current with the help of a single electron tunnelling experiment in analogy to the Josephson voltage standard on the basis of single-flux-quantum transfer. This could be of importance for the redefinition of the international system of physical units and might clarify discrepancies in the determination of the fine structure constant (Taylor 1991, Taylor and Cohen 1991). 14.0.2 Physics and development of the Josephson voltage standard 14.0.2.1 Josephson effects A Josephson junction consists of two superconducting electrodes weakly coupled by a barrier which, in the case of the voltage standard currently used, is an insulating barrier thin enough to let Cooper pairs tunnel through. The same effects are found, however, if the electrodes are separated by a normal-metal layer or

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a small constriction. When the junction is connected to an external source, the current flow between the two electrodes is in the ideal case determined by two equations ( Josephson 1962)

where j denotes the phase difference between the two macroscopic wavefunctions of the superconducting electrodes. The first equation implies that a current can flow without d.c. voltage across the junction if it is less than the critical current Ic (d.c. Josephson effect). When Ic is exceeded, an alternating current of frequency fJ flows due to the regular transfer of flux quanta through the Josephson element (a.c. Josephson effect). The fact that the frequency fJ is proportional to the d.c. voltage U or that the rate of transferred flux quanta equals the d.c. voltage forms the basis of modern voltage references. A real Josephson junction can be considered as a parallel connection of the ideal junction, a junction capacitance C and a shunt resistance R (McCumber 1969, Stewart 1968). In a voltage standard, the junction is biased by a current source with a d.c. component I0 and an a.c. component I1 sin ωt (figure I4.0.3).

Figure I4.0.3. The equivalent circuit of a real Josephson junction. Iq is the quasi-particle current, Id the displacement current, I the Josephson current and I0 + I1 sin wt the bias current.

The total bias current must equal the sum of the Josephson current I = Ic sin ϕ , the dissipative quasi-particle current Iq = U/R and the displacement current Id = CdU/dt. If U = (h/4πe)dϕ/dt is inserted, a complete description of a current-driven Josephson junction is obtained:

This model is correct when the current is uniformly distributed over the junction area: Ic = Jc wl, where w is the junction width, l the junction length in the direction of current flow and Jc the critical current density. For small angles, sin ϕ can be replaced by ϕ and the ideal junction equation can be written as U = (h/2eIc )dI/dt = LdI/dt. Within this limit, the Stewart-McCumber model is a resonator circuit with a resonance frequency 2πfp = ωp = (LC )−1/2 = (2eIc /hC )1/2, the plasma frequency. The quality factor of the resonator is defined by Q = R(C/L)1/2. For Q values much larger than ½ the junction is underdamped, and for values smaller than ½ it is overdamped. When the Josephson oscillator is phase locked to an external oscillator, the supercurrent is forced to oscillate at the frequency f or at higher harmonics nf of the external oscillator over a range of d.c. bias currents. This effect generates a number of constant-voltage steps in the d.c. characteristic of the junction (Shapiro 1963) at the voltages

The current width ∆In of the nth step is described by the nth-order Bessel function Jn :

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where VR F denotes the amplitude of the radiofrequency (RF) voltage across the junction. This equation is correct if the RF voltage across the junction is sinusoidal, as it is for the tunnel junctions used in today’s voltage standards. The maximum current width of an arbitrary step can be calculated by choosing the argument that maximizes the Bessel function. The equation for Un forms the basis of modern Josephson voltage standards. It has been proven experimentally with extreme precision (Kautz and Lloyd 1987, Niemeyer et al 1986, Tsai et al 1983) so that only quantum principles should limit the exactness (Kose and Niemeyer 1988). In practical experiments, fundamental limits are set by the reproducibility of the frequency of the external microwave source, which is only 1 part in 1012 when modern caesium clocks are used. Figure I4.0.4 shows the constant-voltage steps for a highly damped superconductor-normal-metalsuperconductor (SNS) Josephson junction and figure I4.0.5 for a strongly underdamped superconductorinsulator-superconductor (SIS) tunnel junction. The tunnel junction shows a strong hysteresis in the d.c. characteristic, which is the result of low damping and high capacitance. The strong increase in current at the gap voltage of about 2.5 mV and the extremely small current at lower voltages are due to the

Figure I4.0.4. Current-voltage characteristic of a superconductor-normal-metal-superconductor (Nb-PdAu-Nb) junction: (a) without microwave radiation; (b) with microwave radiation of 10 GHz. Horizontal axis: 0.4 mA div−1, vertical axis: 20 µV div−1.

Figure I4.0.5. Current-voltage characteristic of an Nb-Al2O3-Nb tunnel junction: (a) without microwave radiation; (b) with mirowave radiation of 70 GHz. Horizontal axis: 125 µA div−1; vertical axis: 1 mV div−1.

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superconducting energy gap, which must be exceeded before quasi-particles cross the barrier. This type of characteristic leads to so-called zero-current constant-voltage steps which cross the voltage axis (see figure I4.0.5). 14.0.2.2 Single junction standards Soon after the discovery of the Josephson effect, the voltage-frequency relation was experimentally verified with high precision. By comparing the output voltages of different junctions driven by the same microwave source it was shown that the value of 2e/h is independent to better than one part in 108 of the type and geometry of the junction material (Clarke 1968), as well as of environmental parameters like temperature, magnetic field and microwave power (Parker et al 1969). These results were confirmed with improved accuracy by Bracken and Hamilton (1972), Finnegan et al (1971) and Harvey et al (1972). To perform precise measurements of 2e/h, the small d.c. output voltage of the Josephson junction must be compared with the relatively large output voltage of the primary Weston cell standards. As figure I4.0.5 suggests, the maximum output voltage of a single tunnel junction at zero current is only about 1 mV. Even at larger microwave power this value does not exceed a few millivolts independent of whether the junction is highly damped as it would be in point contacts, microbridges and SNS junctions or strongly underdamped as in SIS junctions. Therefore, precise voltage dividers were developed and connected with a single Josephson junction to enhance the Josephson output voltage to the level of 1 V (Gallop and Petley 1974, Harvey and Collins 1973, Kose 1976, Kose et al 1974, Sullivan 1972, Sullivan and Dzuiba 1974). While performing these experiments it was found in the national calibration laboratories that, instead of measuring 2e/h, these instruments could be much better used to control the as-maintained laboratory voltages with a very high reproducibility if a fixed value of 2e/h were adopted. Consequently, in 1972 the CCE suggested using a value of 2e/h = 483 594.0 GHz V−1 only for voltage comparisons (Terrien 1973). In 1990 this value was corrected to 2e/h = 483 597.9 GHz V−1 and officially recommended as KJ − 90 (Quinn 1989). Since that time, the national voltage standards have been controlled by single-junction Josephson voltage standards and the role of the Weston cell as a primary voltage standard has ended. Besides their relative insensitivity against environmental influences, the new Josephson voltage standards had a much better reproducibility of a few parts in 108, which finally led to the enormous reduction in the spread of the standard voltages of the different national laboratories (figure I4.0.1; for a review see Petley 1983). On the other hand it was mainly the imperfect and difficult calibration procedure which limited the precision of the single junction standard. Moreover, because of the fixed ratio of the voltage divider and the relatively small tuning range of the microwave frequency, this standard could not be used as a potentiometer over a wide range of reference voltages. To overcome these difficulties, attempts were made in the past to use series arrays of Josephson junctions to obtain larger reference voltages (Endo et al 1983, Finnegan et al 1971). At that time junction fabrication processes had not yet been developed to guarantee a junction parameter spread small enough to bias the junction array by a single current source. The most developed multibiased standard with 20 junctions integrated in a microwave stripline resonator generated reference voltages up to 100 mV with an uncertainty of a few parts in 109 (Endo et al 1983), but the complexity of the bias system made the standard difficult to handle. Also an extension of this operation principle to higher output voltages would be impractible. It should be mentioned that there are at present extreme technological difficulties in fabricating reliable Josephson junctions from high-Tc superconductors with a sufficiently small parameter spread. In this way the situation is comparable to that in the 1960s where similar problems existed for traditional superconductors. To take advantage of the higher operation temperatures of ceramic superconductors not only for SQUIDs but also for the voltage standard, a transportable 10 GHz single-junction standard with an uncertainty of one part in 108 has been developed in consequence of the problematic junction fabrication (Tarbeyev et al 1996). However, the high-Tc version of a single-junction standard as well as the multibiased small series array standards do

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not overcome the difficulties as regards handling, precision and versatility of the voltage divider, which will only be possible with direct generation of large Josephson reference voltages by series arrays with a number of junctions. 14.0.2.3 Series array standards Faced with this situation, in 1977 Levinsen et al suggested using junctions with zero-current steps in series arrays to increase the output voltage. Such an array can tolerate a much wider spread of junction parameters than an array of highly damped junctions because it can be operated at zero current where all the junctions have steps. This simple mode of operation and the fact that semiconductor fabrication methods were modified to manufacture large arrays have made Levinsen’s idea a success. (a) Single junction parameters As a conditio sine qua non for the generation of stable zero-current steps, the phase lock between the strongly nonlinear junction oscillator and the external microwave oscillator must be maintained during the calibration procedure, not only for a single junction but also for thousands of junctions in the series arrays currently used. This means that no spontaneous switching should occur for many hours per single junction. This is of special importance in the case of overlapping steps because losing phase lock can result in switching to another voltage step, not an automatic return to the original reference voltage, a in overdamped junctions. For this reason, very good filtering of external noise is necessary, and chaotic behaviour of the coupled oscillators must be avoided to a very high degree. On the basis of the Stewart-McCumber model, Kautz analysed how the single-junction parameter must be defined to maximize locking strength in the face of chaos and phase variations across the junctio area (for an overview with a complete list of references see Kautz 1996). ( i ) The frequency of the external oscillator must be larger than the plasma frequency

where Jc is the critical current density and Cs the specific capacitance of the junction. ( ii ) To ensure homogeneous RF and d.c. current distribution across the junction area, the junction length and width must meet the following conditions

µ0 being the permeability of vacuum, s the thickness of the barrier; λL1,λL2 are the London penetration depths of the superconducting junction electrodes and n is the number of the constant-voltage step (Kautz and Costabile 1981, Vollmer 1984). To avoid chaotic behaviour, the microwave frequency f should be at least three times larger than the plasma frequency (Kautz 1980) which leads to a restriction of the critical current density Jc m a x = ( f/3)2 (πhCs /e). Together with the limitations of junction length and width, this results in a maximum value of the critical current Ic m a x = lm a x wm a x Jc m a x and thus in a maximum step width of ∆In m a x = 2Ic m a x| Jn(2eVR F /hf )|m a x , over which stable operating conditions for a voltage standard can be achieved. The calculated optimum junction parameters depend on the properties of the dielectric barrier and superconducting materials owing to the role played by the specific capacitance and the London penetration depths. The smaller the penetration depths the larger the junction area can be for a fixed value of the current density, and the larger the achievable current width of the constant voltage steps. In spite of their

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large gap voltages of 5 mV, pure Nb-N junctions with very large penetration depths are therefore no optimally suited to generate broad steps. If the Nb-N electrodes are backed by Nb, the step size can be the same as for Nb-Al2O3-Nb junctions. Nb-Nb2O5-PbBi junctions reach a step size two thirds that of Nb-Al2O3-Nb junctions but with a smaller junction size which may be advantageous for circuits with a very large number of junctions operated at lower microwave frequencies. However, the PbBi top electrode is not as durable as that made of Nb. The lack of durability is an even more severe problem for circuits made entirely of lead alloys. In table I4.0.1, the optimum junction parameters for different materials art listed for an external microwave frequency of 70 GHz. It is assumed that Un = 1 mV, which is about the seventh step or half the gap voltage. The material parameters have been taken from Kautz (1989) and Niemeyer et al (1984a, 1985a, 1989). The equation for Jc m a x shows that a large drive frequency plays an important role in the stable operation of a voltage standard. Figure I4.0.6 shows this dependence for two junction types, Nb-Al2O3Nb and Nb-Nb2O5-PbBi, used successfully in 10 V standards. The curves were determined under the same conditions as for table I4.0.1. It is also assumed that the dependence of the specific capacitance on the barrier thickness for differen current densities can be neglected. The curves clearly show that high frequencies are very advantageous because they provide large-amplitude steps at a relatively small junction size.

Figure I4.0.6. Dependence of the optimum junction parameters on the frequency of the external microwave. The full lines represent the values for Nb-Al2O3-Nb circuits, the broken lines are for Nb-Nb2O5-PbBi circuits.

From the curves for Nb-Nb2O5-PbBi circuits it can be seen that drive frequencies down to 25 GHz would still lead to a tolerable junction size even for 10 V arrays. This has been shown experimentally at least for 1 V circuits (Müller et al 1990). A more rigid calculation of the frequency condition (Frank and Meyer 1992) on the basis of the microscopic Werthamer model (Werthamer 1966) furnishes the result that the frequency of the external oscillator is allowed to approach the plasma frequency more closely provided the drive amplitude is sufficiently strong. Experimental results with Nb-Nb2O5-Pb alloy junctions support this result (Meyer et al 1996), but it was found for Nb-Al2O3-Nb junctions that an increase of the microwave frequency to a value of even 6fp still improves the stability of the standard (Müller et al 1997). This and the fact that at

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low frequencies the optimum junction size gets rather large makes it impractical to operate a zero-current voltage standard at 10 GHz, which is desirable from the viewpoint of using convenient and relatively inexpensive microwave equipment but compromises stability and easy handling of the standard. For Nb-Al2O3-Nb circuits in particular, it seems impossible to construct a 10 V array for 10 GHz operation. Even 1 V arrays at 10 GHz are extremely difficult to handle and fabricate (Hebrank et al 1995). The apparent sensitivity of Nb-Al2O3-Nb circuits to chaotic behaviour can be reduced by using junctions with moderate damping (Kim and Niemeyer 1995), but intrinsically shunted junctions with a highly resistive shunt are difficult to fabricate with a sufficiently low spread of parameters. (b) Microwave circuit First attempts at increasing the zero-current Josephson output voltage by means of a series array of tunnel junctions were not satisfactory because the RF drive frequency and the junction parameters were not chosen correctly (Kautz 1980, Kautz and Costabile 1981). Moreover, the microwave design for a homogeneous distribution of the microwave was not optimum. The correct junction parameters for Pb-alloy technology were found experimentally for f = 70 GHz (Niemeyer et al 1984b) and introduced into a new circuit design developed at Physikalisch-Technische Bundesanstalt (PTB) (Niemeyer et al 1984a, 1985a, b) and first realized in a cooperative effort between PTB and the National Institute of Standards and Technology (NIST) (formerly NBS) (Hamilton et al 1985, Niemeyer et al 1984a). The basic design of arranging the junction array in a superconducting microstrip line over a superconducting ground plane was very successful not only for effective and uniform microwave coupling to single junctions and voltage-standard circuits with more than 15000 junctions for reference voltages of more than 10 V (Lloyd et al 1987, Pöpel et al 1990), but also for Josephson oscillators which use series arrays to enhance the output power. The most recent version of the microstrip line is shown in figure I4.0.7 (Müller et al 1997). The series array on top of the silicon wafer forms a periodic line which is covered by a superconducting ground plane. Following Hinken (1988), Niemeyer et al (1984a) and Vollmer (1984), the main properties of the periodic stripline structure can be described by the simplified equivalent circuit model in figure I4.0.7. The admittance Y of the Josephson tunnel junction is determined by the junction capacitance C, connected in parallel to the subgap resistance Rs g and the admittance YJ of the sinϕ term. At zero d.c. current the RF power in the sinϕ term PR F = Re(YJ )VR2 F /2 generates a d.c. power of Pd . c . = Un2/Rs g . On the assumption VR F ≈ Un which describes the experimental situation for the generation of zero-current steps, one obtains Re(YJ ) ≈ 2/Rs g . Together with the shunt resistance, this results in the real part of the total junction admittance of ℜ(Y) = 3/Rs g . The imaginary part of Y is mainly determined by the junction Copyright © 1998 IOP Publishing Ltd

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Figure I4.0.7. A cross-section of a junction stripline together with a simplified equivalent circuit of the periodic stripline structure.

capacitance because the imaginary part of YJ is much smaller than ω C. By this one obtains for the total junction admittance Y = 3 / R + jω C. It shows that the RF current through the junction is mainly capacitive because ω C » 3/R. The subgap resistance is about a factor of ten larger than the normal-state resistance of the junctions. The lower cut-off frequency of the stripline is fl = (1/2π) [ Z0lk( µ0∈0∈r s ) 1/2 (C/2 )]−1/2 with Z0 = ( µ0/∈0∈r s )1/2ds /ws , lk the length of the stripline period, ds the thickness of the stripline insulator, ws the stripline width and ∈r s the dielectric constant of the stripline insulator. The wave impedance of the junction array is then Zk = Z0 (1 − [ fl /f ]2 )1/2. For the Nb-Al oxide-Nb circuits with an SiO2 dielectric developed by PTB (Müller et al 1997), the following numerical data may be used: f = 70 GHz, ∈r s = 3.7, lk = 24 µ m, ds = 1.7 nm, ws = 54 µ m, w = 50 µ m, l = 20 µ m, Cs = 6 µ F cm−2, R = 150 Ω and Z0 ≈ Zk = 6 Ω. The smallest attenuation measured for a stripline containing 3000 junctions was about 6 dB or 10−3 dB per stripline period containing one junction. Experiments show that this small value allows the maximum d.c. voltage output of 1 mV per junction to be reached in a single stripline including about 5000 junctions. This design therefore made it possible to arrange 14000 junctions for a 10 V standard chip in only four folded striplines connected in parallel to the microwave supply (figure I4.0.8). The d.c. characteristic of such a circuit is shown in figure I4.0.9. Stable output voltages of I4 V were reached at frequencies between 70 GHz and 73 GHz and microwave powers between 15 mW and 10 mW at the cold antenna input (figure I4.0.10). The antenna, Copyright © 1998 IOP Publishing Ltd

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Figure I4.0.8. Design of a 10 V standard chip.

Figure I4.0.9. The d.c. characteristic of a standard chip with 14000 junctions.

a fin-line taper (Hinken 1983, Hinken et al 1986) inserted into a slit in a waveguide, is connected with the four stripline microwave paths by three microwave dividers and two blocking capacitors designed as quarter-wavelength transformers. Each microwave path is terminated by a lossy line to prevent microwave reflections from the end, which would lead to standing-wave patterns along the lines and thus to an inhomogeneous microwave distribution. With respect to the d.c. connections, the four arrays are connected in series. With some variations, this basic design is used by all manufacturers worldwide for the standard circuits used in practice (Hamilton and Burroughs 1995, Kaplan 1995, Murayama et al 1995, Park et al 1995).

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Figure I4.0.10. A magnified part of the same characteristic under microwave radiation ( 73 GHz). The step region between -13 V and +13 V can be clearly seen. Vertical axis: 5 V div−1; horizontal axis: 5 mA div−1.

I4.0.3 Fabrication of Josephson series arrays The technology currently used for the preparation of large series array standard chips is based on junctions with Nb electrodes separated from each other by a thin (1.5 nm) Al2O3 barrier (Gurvitch et al 1983). After the development of a fabrication technology similar to that of semiconductors (Nakagawa et al 1986), it has become possible to apply the Nb-Al2O3-Nb technology to the preparation of voltage standard chips (Niemeyer et al 1986). Because of its long-term stability against material deterioration, it is, to our knowledge, now used by all laboratories for the production of complex Josephson circuits, including large series arrays. The most recent simplified production method developed at PTB (Müller et al 1997) avoids problems such as flux trapping and occasional shorts across the quarter wavelength transformers and is briefly described in the following. The first step is the deposition of a thin Al film onto the polished surface of a 3 in (∼7.6 cm) Si wafer, followed by the preparation of the Nb-Al2O3-Nb sandwich for the tunnel junctions. The smooth Si substrate covered by an Al film only 20 nm thick guarantees reproducibility and high junction quality. The d.c. magnetron-sputtered Nb base film of the sandwich is 150 nm thick and the top Nb film is 75 nm thick. Between the two Nb electrode layers, a thin Al film is deposited and thermally oxidized for 16 h at a temperature of 40 °C under an O2 pressure of 250 mbar. These oxidation parameters lead to a junction current density of 10–20 A cm−2 and, as a result, to a rather low plasma frequency of 11–16 GHz. Then the junction base electrode and half the fin-line structure are cut out of the sandwich by reactive ion etching in a CF4 plasma. This process stops at the bottom Al layer which serves as a conducting layer for the junction front-edge anodization (see figure I4.0.7). This additional edge insulation prevents flux trapping even in very large arrays. After removal of the bottom Al layer, the dielectric for the quarter-wavelength transformer is patterned by lifting off an SiO2 layer 200 nm thick, followed by the deposition of the Nb wiring layer of 450 nm thickness. The next step is the formation of the patterns for the junction wiring and the junction top electrodes. The complete array is covered by the stripline dielectric SiO2 layer 1.7 µ m to 2 µ m in thickness. Finally, a superconducting Nb layer (450 nm) which forms the groundplane and the second half of the fin-line and a resistive Pd layer (150 nm) which forms the groundplane for the matched loads are deposited and patterned by reactive ion etching and lift-off respectively. A complete 10 V chip fixed to a chip carrier is shown in figure I4.0.11.

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Figure I4.0.11. 10 V chip bonded to a chip carrier.

Figure I4.0.12. The calibration curve of a standard Weston cell with an output voltage of 1.018115 240 V. The calibration peak of 60 nV results from tuning the drive frequency by an equivalent amount.

I4.0.4 Precision measurements and standard calibration It is impossible to directly prove the accuracy of a Josephson voltage standard because there is no other voltage source of comparable precision that can be used for a comparison. Even at the level of one part in 1012, the accuracy to which a frequency can be determined, such a comparison cannot be made. In practice it turns out that the instrument to be tested limits the accuracy of the calibration. For this reason, the first calibrations of a standard Weston cell made against a Josephson array standard are still the most precise measurements. Figure I4.0.12 shows a 1 h calibration curve of a Weston cell with an output voltage of 1.018 115 240 V with a noise figure of less than 10 nV peak to peak. When such measurements were performed at different currents on the same voltage step, it was shown with a relative root mean square (rms) uncertainty to better than 5 × 10−10 that there is no current dependence of the step voltage (Niemeyer et al 1985b). A comparable calibration curve of a modern electronic Zener reference standard shows a noise figure of about 250 nV peak to peak at its 10 V output (and 25 nV at 1 V) (figure I4.0.13). To obtain these results a laboratory version of the series array standard with a klystron as a microwave source was used. The klystron was locked by two standard phase-lock loops to a 10 MHz quartz time base Copyright © 1998 IOP Publishing Ltd

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Figure I4.0.13. The calibration curve of a Zener reference standard with a 10 V Josephson standard. The voltage level (∆V = 0) is 9.999968 716 V. The calibration peak of 1 µV results from tuning the drive frequency by an equivalent amount.

which guaranteed a short-term frequency stability of better than one part in 1010. The long-term stability was controlled by the time signal of the PTB’s longwave radio transmitter DCF 77 (Grimm 1988, Pöpel et al 1991a). The noise figures of both measurements described in figures I4.0.12 and 14.0.13 can be explained by the noise of the output resistances of the secondary standards under test. To determine the noise contribution of the d.c. equipment of the Josephson standard, two of the klystron-driven standards were directly compared. In this case the frequency of one klystron-driven standard was controlled by an EIP counter to a few hertz. At the zero-voltage level the noise figure measured by means of an electronic nanovoltmeter (EM, Nla) as null detector was only 2 nV. The bandwidth of the nanovoltmeter was 0.1 Hz. At the level of 1 V, this figure increased to about 3 nV. The noise is equivalent to the voltage noise of a 100 Ω resistor and is mainly caused by the nanovoltmeter. These values correspond to a relative rms uncertainty of about 5 × 10−10 and may be an upper practical limit for Josephson series array standards with the d.c. instrumentation currently used (Pöpel et al 1991b). In commercially available instruments the klystron is usually replaced by a Gunn oscillator controlled by a frequency counter (EIP). In this case the wider frequency spectrum of the Gunn oscillator cannot be synchronized as well as the comparatively small one of the klystron. A direct comparison therefore results in a larger peak-to-peak voltage noise of about 9 nV measured with the nanovoltmeter at a bandwidth of 0.1 Hz. An account of the relative uncertainties for typical calibrations of a single Weston cell, a series connection of nine Weston cells and the 1 V or 10 V output of a Zener reference with such a voltage standard is given in table I4.0.2. Figure I4.0.14 shows the block diagram of a voltage standard used in practice. The frequency of a Gunn oscillator with a maximum output power of about 100 mW is controlled by a stable quartz oscillator with a phase-locked loop or by a frequency counter. The microwave power is guided through a waveguide system to the series array which is cooled to 4.2 K in an ordinary He transport Dewar. Standard metallic waveguides have an attenuation of about 5 dB in this convenient Dewar arrangement which requires relatively long waveguides. Flexible dielectric waveguides allow an attenuation of only 2 dB to be reached. This can be further reduced by using small Dewars or cryocoolers with a short distance between the array and the external oscillator. The sample holder at the end of the waveguide must be covered by a cryoperm shield to prevent a reduction of the junction’s critical currents by the earth’s magnetic field. The d.c. supply of the array consists of a current source to record the d.c. characteristic and the constant voltage steps of the array. An additional voltage source is used to lock the bias on a desired step during calibration of the device under test. All leads to the array have to be carefully filtered. The thermal electromotive forces on the voltage leads of 200 to 300 nV can be compensated by an auxiliary voltage source, or they can simply be measured by reversing the polarity of the array voltage. Copyright © 1998 IOP Publishing Ltd

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Figure I4.0.14. Block diagram of a Josephson voltage standard. The oscilloscope photograph shows a multiple exposure of the constant-voltage steps close to the voltage to be calibrated when the array is biased by the voltage source.

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I4.0.5 Future developments I4.0.5.1 Programmable Josephson voltmeters When strongly overlapping zero-current steps are used, as in large series array standards, the rapid selection of a particular voltage step is problematic. The use of highly damped junctions, such as externally shunted tunnel junctions or SNS junctions (see figure I4.0.4), in a series array avoids this problem because the steps do not overlap. To rapidly select an arbitrary voltage, nonhysteretic junctions can be arranged in a binary sequence of array sections with 1, 2, 4, 16, 32, etc, junctions, each of which is supplied by a current source which can be switched to zero or to the bias current for the first Shapiro step. In this case an arbitrary voltage is selected by controlling the current sources with a programmable switch and it is even possible to synthesize a.c. voltages with the accuracy of the d.c. standard (Benz 1995, Hamilton et al 1995). The disadvantage of such a device is the large number of junctions needed, because only the first voltage step contributes to the sum voltage. For arrays with externally shunted tunnel junctions, where 70 GHz can still be used, the junction number must be increased by a factor of about seven compared with a zero-current array. For SNS junctions, where a lower drive frequency is advantageous (Kautz 1994), the number of junctions needed for a 10 V circuit could be greater than 500000. To fabricate such a series array and distribute the microwave power homogeneously is a real challenge. I4.0.5.2 Integrated versions Another problem of the voltage standard is the rather complicated and expensive microwave equipment. To overcome this disadvantage, experiments have been made to integrate a cold semiconductor oscillator with the Josephson array. A frequency-controlled 10 GHz metal-semiconductor field effect transistor (MESFET) oscillator was successfully operated at 4.2 K and connected by hybrid integration with Nb-Al2O3-Nb series arrays (Hebrank et al 1995). Constant-voltage steps could be generated, but the stability at 1 V was not sufficient for a calibration due to chaos problems connected with the low frequency of operation. A 70 GHz version is under development. In this case, the short stripline connections between the oscillator and the series array are very advantageous as the losses and the impedance mismatch can be kept relatively small. Yet another improvement would be to integrate a single-flux-quantum logic circuit for dividing the frequency down to the level of a quartz oscillator to simplify frequency control. I4.0.5.3 High-Tc superconductors At present, underdamped tunnel junctions with highly hysteretic d.c. characteristics cannot be fabricated from high-Tc superconductors. Even if such junctions existed, it would be difficult to operate them at 77 K in a voltage standard. It has been shown (Kautz 1981) that for this junction type the average time for which a stable phase-lock is maintained between the external oscillator and the junction is determined by

τ = τ 0 exp(E/kT )

with E = h∆In /4πe and τ0 = (2IC /∆In )1/2/2fp .

A precise calculation of τ is difficult because no data on high-Tc tunnel junctions are available. An estimation made on the basis of the assumption that the barrier is similar to an Al2O3 barrier produced the result that the average lifetime of a phase-lock is easily reduced to values smaller than one second at 77 K (Niemeyer 1989). Furthermore, due to the large parameter spread in high-Tc arrays, the resulting ∆In is significantly smaller than that calculated for a single junction taking the Bessel function dependence into account. For these reasons, it seems doubtful whether high-Tc superconductors can be used to build a voltage standard on the basis of zero-current steps for operating temperatures near 77 K. In contrast, the construction of a programmable standard with highly damped junctions may be possible. Highly damped junctions can be fabricated with high-Tc superconductors, and large-amplitude Copyright © 1998 IOP Publishing Ltd

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constant-voltage steps can be generated even at elevated temperatures. However, the problem of fabricating an extremely large number of junctions with a small parameter spread has not yet been solved, but, in contrast to the difficulties faced with the underdamped junctions, this is not a fundamental problem. I4.0.6 Conclusions As the Josephson voltage standard is based on a macroscopic quantum effect, its accuracy is far better than that of standards represented by material artefacts. In practice the noise figure of the device to be calibrated therefore limits the precision of the calibration. With the CCE definition of KJ − 90, the Josephson series array voltage standard is accepted as the primary voltage standard for legal metrology worldwide. It is in practical use by the BIPM, most of the national calibration laboratories and a number of companies. Today’s voltage standards are made of series arrays with up to 20000 junctions in durable all-niobium technology. They generate Josephson voltages of up to more than 15 V. A compact computer-controlled version has been developed (Kupferman et al 1996). Secondary standards like Weston cells and Zener references are routinely calibrated with a relative uncertainty of up to one part in 109. Moreover, linearity measurements of digital voltmeters (Giem 1991, Goeke et al 1989, Popel et al 1991b) and resistance ratio measurements (Kohlmann et al 1993) can be made with high precision. A number of international comparisons between Josephson series array standards have been made (Eklund and Pajander 1995, Henderson et al 1992, Jensen 1995, Kim et al 1996, Lo-Hive et al 1995, Niemeyer et al 1990, Reyman et al 1992, 1996, 1997, Reyman and Witt 1993, Rodriguez and Huntley 1995). It has not been their aim to prove the Josephson relation for the d.c. voltage or the precision of the array output voltage but to test the proper operation of the equipment required for the comparison of the Josephson voltage with the output voltage of the room-temperature devices to be calibrated. All these experiments have confirmed the superior reproducibility and precision of the series array voltage standard which has no serious competitor in standard semiconductor electronics. The main problems still to be solved are achieving a more rapid selection of a certain voltage step without losing the precision and increasing the step stability, of the 10 V arrays in particular. Although great progress has been achieved in junction fabrication, a further improvement of the circuit technology would be helpful to increase the step stability and selectivity and to open up possibilities of manufacturing very large arrays of highly damped Josephson junctions for extremely stable standard voltages. References Bachmair H 1988 Elektronische Spannungsquellen ab Spannungs und Tranfernormale PTB-Bericht E-24 1–206 Bachmair H 1993 100 Jahre Normalelemente in der PTB/PTR—Ihre Bedeutung für die Darstellung und Bewahrung der Einheit der elektrischen Spannung PTB-Mitteilungen 103 395–404 Benz S P 1995 Superconductor-normal-superconductor junctions for programmable voltage standards Appl. Phys. Lett. 67 2714–6 Bracken T D and Hamilton W O 1972 Comparison of microwave-induced constant voltage steps in Pb and Sn Josephson junctions Phys. Rev. B 6 2603–9 Clark L 1872 Proc. R. Soc. 20 144 Clarke J 1968 Experimental comparison of the Josephson voltage-frequency relation in different superconductors Phys. Rev. Lett. 21 1566–89 Eklund G and Pajander H 1995 The SP Josephson array voltage standard. Swedish National Testing and Research Institute, Physics and Electrotechnics SP Report 1995 47 Endo T, Koyanagi M and Nakamura A 1983 High-accuracy Josephson potentiometer IEEE Trans. Instrum. Meas. IM-32 267–71 Finnegan T F, Denenstein A and Langenberg D N 1971 ac-Josephson-effect determination of e/h : a standard of electrochemical potential based on macroscopic quantum coherence in superconductors Phys. Rev. B 4 1487–522

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References

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Frank B and Meyer H-G 1992 Suitability of gap frequencies for the Josephson voltage standard J. Appl. Phys. 72 2973–7 Funck T and Sienknecht V 1991 Determination of the volt with the improved PTB voltage balance IEEE Trans. Instrum. Meas. IM-40 158–62 Gallop J C and Petley B W 1974 Operational experience with a cryogenic volt monitoring system IEEE Trans. Instrum. Meas. IM-23 267–71 Galvani L 1791 De Virebus Electricitatis in Motu Musculari Commentarius Geerligs L J, Andereeg V F, Holweg P A M, Mooji J E, Pothier H, Esteve D, Urbina D and Devoret M H 1990 Frequency-locked turnstile device for single electrons Phys. Rev. Lett. 64 2691–4 Giem J I 1991 Sub-ppm linearity testing of a DMM using a Josephson junction array IEEE Trans. Instrum. Meas. IM-40 329–32 Goeke W C, Swerlein R L, Venzke S B and Stever S D 1989 Calibration of an 8 1/2- digit multimeter from only two external standards Hewlett-Packard J. 40 22–30 Grimm L 1988 Reproduzierung der Spannungseinheit mit dem atomuhrgesteuerten Spannungsnormal der PTB Elektronische Spannungsquellen als Spannungs- und Transfemormale ( PTB-Bericht E-34) ed H Bachmair pp 15–34 Gurvitch M, Washington M A, Muggins H A and Rowell T M 1983 High quality refractory Josephson tunnel junctions utilizing thin aluminum layers IEEE Trans. Magn. MAG-19 791–4 Hamilton C A and Burroughs C J 1995 The performance and reliability of NIST 10-V Josephson arrays IEEE Trans. Instrum. Meas. IM-44 238–41 Hamilton C A, Burroughs C J and Kautz R L 1995 Josephson D/A converter with fundamental accuracy IEEE Trans. Instrum. Meas. IM-44 233–4 Hamilton C A, Kautz R L, Steiner R L and Lloyd F 1985 A practical Josephson voltage standard at 1 V IEEE Electron. Dev. Lett. EDL-6 623–5 Harvey I K and Collins H C 1973 Precise ratio measurements using a superconducting ratio transformer Rev. Sci. Instrum. 44 1700–2 Harvey I K, MacFarlane J C and Frenkel R B 1972 Monitoring the NSL standard of EMF using the AC Josephson effect Metrologia 8 114–24 Hebrank F X, Vollmer E, Funck T, Gutmann P and Niemeyer J 1995 Development of a hybrid integrated Josephson voltage standard operated at 10 GHz Proc. Eur. Conf. on Appl. Superconductivity (Inst. Phys. Conf. Ser. No. 48) (Bristol: Institute of Physics Publishing) vol 2 pp 1653–6 Henderson L C A, Reyman D and Witt T J 1992 NPL/BIPM comparison of Josephson voltage standards Meas. Sci. Technol. 3 1011–3 Hinken J H 1983 Simplified analysis and synthesis of fin-line tapers Archiv für Electronik und Übertragungstechnik Electronics and Communication AEÜ 37 375–80 Hinken J H 1988 Supraleiter Elektronik (Berlin: Springer) Hinken J H, Niemeyer J and Popel R 1986 E-band transformer from waveguide to superconducting, low impedance, antipodal fin-line NTZ Archiv 8 215–22 Jensen H D 1995 Josephson array voltage standard intercomparisons: Euromet Project 199 IEEE Trans. Instrum. Meas. IM-44 211–21 Josephson B D 1962 Possible new effects in superconductive tunneling Phys. Lett. 1 251–3 Kaplan S B 1995 Technology transfer and the Josephson voltage standard Supercond. Industry 8 25–30 Kautz R L 1980 On a proposed Josephson-effect voltage standard at zero current bias Appl. Phys. Lett. 36 386–8 Kautz R L 1981 The ac Josephson effect in hysteretic junctions: range and stability of phase lock J. Appl. Phys. 52 3528–41 Kautz R L 1989 Design and operation of series-array Josephson voltage standard Proc. Int. School of Physics ‘Enrico Fermi’ Course CX Kautz R L 1992 Design and operation of series-array Josephson voltage standards Metrology at the Frontiers of Physics and Technology ed L Grovini and T J Quinn (Amsterdam: North-Holland) pp 259–96 Kautz R L 1994 Quasipotential and the stability of phase lock in nonhysteretic Josephson junctions J. Appl. Phys. 76 5538–44 Kautz R L 1996 Noise, chaos, and the Josephson voltage standard Rep. Prog. Phys. 59 935–92

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Kautz R L and Costabile G 1981 A Josephson voltage standard using a series array of 100 junctions IEEE Trans. Magn. MAG-17 780–3 Kautz R L and Lloyd L 1987 Precision of series-array Josephson voltage standards Appl. Phys. Lett. 51 2043–5 Kim K-T and Niemeyer J 1995 Damping effect on the radio frequency induced voltage steps in a Josephson tunnel junction Appl. Phys. Lett. 66 2567–9 Kim K-T, Sakamoto Y and Sakuraba T 1996 Comparison of 10 V calibrations between KRISS and ETL Conf. on Precision Electromagnetic Measurements Conf. Dig. (Piscataway, NJ: IEEE) pp 236–7 Kohlmann J, Gutmann P and Niemeyer J 1993 Ratio standard for DC resistance using a Josephson potentiometer IEEE Trans Instrum. Meas. IM-42 255–7 Kose V 1976 Recent advances in the Josephson voltage standards IEEE Trans. Instrum. Meas. IM-25 483–9 Kose V 1989 Beschlusse der Meter-Konvention über die Weitergabe elektrischer Einheiten PTB-Bericht E-35 3–18 Kose V, Melchert F, Engiland W, Pack H, Fuhrmann B, Gutmann P and Warnecke P 1974 Maintaining the unit of voltage at PTB via the Josephson effect IEEE Trans. Instrum. Meas. IM-23 271–5 Kose V and Niemeyer J 1988 Superconducting quantum measures—possibilities and limits The Art of Measurement ed B Kramer (Weinheim: VCH) pp 249–61 Kupferman S L, Hamilton C A, Naujoks G and Vickery A 1996 A compact transportable Josephson voltage standard Conf. on Precision Electromagnetic Measurements Conf. Dig. pp 146–7 Legett A J 1980 Macroscopic quantum systems and the quantum theory of measurement Suppl. Prog. Theor. Phys. 69 80–100 Levinsen M T, Chiao R Y, Feldman M J and Tucker B A 1977 An inverse ac Josephson effect voltage standard Appl. Phys. Lett. 31 776–8 Likharev K K 1988 Correlated discrete transfer of single electrons in ultrasmall tunnel junctions IBM J. Res. Dev. 32 144–58 Lloyd F L, Hamilton C A, Beall J A, Go D, Ono R H and Harris R E 1987 A Josephson array voltage standard at 10 V IEEE Electron Dev. Lett. EDL-8 449–50 Lo-Hive J-P, Reyman D and Geneves G 1995 Comparison of 10 V Josephson array voltage standards between the BNM/LCIE and the BIPM IEEE Trans. Instrum. Meas. IM-44 230–3 McCumber D E 1969 Effect of AC impedance on DC voltage current characteristics of superconductor weak link junctions J. Appl. Phys. 39 3113–8 Melchert F 1979 Darstellung der Spannungseinheit mit Hilfe des Josephson-Effektes Tech. Messen 2 59–64 Meyer H-G, Wende G, Fritzsch L, Thrum F and Kohler H-J 1996 Microwave circuits for Josephson voltage standards at low operation frequencies fabricated in a modified all-niobium technology IX Trilateral German-Russian-Ukrainian Seminar on High Temperature Superconductivity (Gabelbach, 1996) (Düsseldorf: VDI Technologiezentrum) Müller F, Kohler H-J, Weber P, Blüthner K and Meyer H-G 1990 A 1-V series-array Josephson voltage standard operated at 35 GHz J. Appl. Phys. 68 4700–2 Müller F, Pöpel R, Kohlmann J, Niemeyer J, Meier W, Weimann T, Grimm L, Dünschede F and Gutmann P 1997 Optimized 1 V and 10 V Josephson series arrays Conf. on Precision Electromagnetic Measurements 1996; 1997 IEEE Trans. Instrum. Meas. to be published Murayama Y, Sakamoto Y, Iwasa A, Nakanishi M, Yoshida H, Klein U and Endo T 1995 Ten-volt Josephson junction array IEEE Trans. Instrum. Meas. IM-44 219–22 Nakagawa H, Nakaya K, Kurosawa I, Takada S and Hayakawa H 1986 Nb/Al-oxide/Nb tunnel junctions for Josephson integrated circuits Japan. J. Appl. Phys. 25 L70–2 Niemeyer J 1989 Josephson series array potentiometer Superconducting Quantum Electronics ed V Kose (Berlin: Springer) pp 228–54 Niemeyer J, Grimm L, Hamilton C A and Steiner R L 1986 High-precision measurement of a possible resistive slope of Josephson array voltage steps IEEE Electron. Dev. Lett. EDL-7 44–6 Niemeyer J, Grimm L, Meier W, Funck T, Dünschede F W, Ausbüttel E, Holtoug J and Mygind J 1990 Comparison of Josephson series-array voltage standards Metrologia 27 41–5 Niemeyer J, Grimm L, Meier W, Hinken J H and Vollmer E 1985a Stable Josephson reference voltages between 0.1 and 1.3 V for high-precision voltage standards Appl. Phys. Lett. 47 1222–3

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Niemeyer J, Hinken J H and Kautz R L 1984a Microwave-induced constant-voltage steps at one volt from a series array of Josephson junctions Appl. Phys. Lett. 45 478–80 Niemeyer J, Hinken J H and Meier W 1984b Microwave-induced constant voltage steps at series arrays of Josephson tunnel junctions with near-zero current bias IEEE Trans. Instrum. Meas. IM-33 311–5 Niemeyer J, Sakamoto Y, Vollmer E, Hinken J H, Shoji A, Nakagawa H, Takada S and Kosaka S 1989 Nb/Aloxide/Nb and NbN/MgO/NbN-tunnel junctions in large series arrays for voltage standards Japan. J. Appl. Phys. 25 L343–5 Niemeyer J, Vollmer E, Hinken J H and Meier W 1985b Stable constant voltage steps between 0.1 and 1.5 V from new series arrays of Josephson tunnel junctions SQUID ‘85—Superconducting Quantum Interference Devices and Their Applications ed H D Hahlbohm and H Lübbig (Berlin: de Gruyter) pp 1163–6 Ostwald W 1896 Elektrochemie, ihre Geschichte und Lehre Park S I, Kim K-T and Lee R D 1995 All-niobium process for Josephson series array circuits IEEE Trans. Instrum. Meas. IM-44 241–4 Parker W H, Langenberg D N, Denenstein A and Taylor B N 1969 Determination of e/h, using macroscopic quantum phase coherence in superconductors Phys. Rev. 177 639–64 Petley B W 1983 Quantum metrology and electrical standards: the measurements of 2e/h and γ ’p Quantum Metrology and Fundamental Physical Constants ( NATO ASI Series B: Physics) vol 98, ed P H Cutler and A A Lucas (New York: Plenum) pp 293–311 Pöpel R 1992 The Josephson effect and voltage standards Metrologia 29 153–74 Pöpel R, Niemeyer J, Fromknecht R, Meier W, Grimm L and Dünschede F W 1991a Nb/Al2O3/NbJosephson voltage standards at 1 V and 10 V IEEE Trans. Instrum. Meas. IM-40 298–300 Pöpel R, Niemeyer J, Fromknecht R, Meier W and Grimm L 1990 1- and 10-V series array Josephson voltage standard in Nb/Al2O3/Nb technology J. Appl. Phys. 68 4294–303 Pöpel R, Niemeyer J, Grimm L, Dunschede F W and Meier W 1991b Direct comparison of two independent Josephson voltage standards at 1 V and precision measurements up to 10 V IEEE Trans. Instrum. Meas. IM-40 805–10 Prance R J, Prance H, Spiller T P, Clark T D, Ralph J and Clippingdale A 1991 Quantum dualities in superconducting weak link circuits Macroscopic Quantum Phenomena ed T D Clark et al (London: World Scientific) pp 43–55 Quinn T J 1989 News from the BIPM Metrologia 26 69–74 Reyman D, Iwasa A, Yoshida H, Endo T and Witt T J 1992 Comparison of Josephson voltage standards of the Electrotechnical Laboratory and the Bureau International des Poids et Mesures Metrologia 29 389–95 Reyman D, Kim K-T, Christian L A, Frenkel R B and Witt T J 1996 Comparisons of the Josephson voltage standards of the BIPM with those of the KRISS, the NSL and the NML Metrologia 33 75–9 Reyman D and Witt T J 1993 International comparisons of Josephson array voltage standards IEEE Trans. Instrum. Meas. IM-42 596–9 Reyman D, Witt T J, Eklund G, Pajander H and Nilsson H 1996 Conf. on Precision Electromagnetic Measurements Conf. Dig. Suppl. 3–4; 1997 IEEE Trans. Instrum. Meas. to be published Rodriguez K M and Huntley L 1995 US intercomparison of Josephson array voltage standards IEEE Trans. Instrum. Meas. IM-44 215–8 Shapiro S 1963 Josephson currents in superconducting tunneling: the effect of microwaves and other observations Phys. Rev. Lett. 11 80–2 Stewart W C 1968 Current-voltage characteristics of Josephson junctions Appl. Phys. Lett. 12 277–80 Sullivan D B 1972 A cryogenic voltage divider and null detector Rev. Sci. Instrum. 43 499 Sullivan D B and Dzuiba R F 1974 A low-temperature d.c. comparator bridge IEEE Trans. Instrum. Meas. IM-23 256–60 Tarbeyev Yu V, Krzhimovsky V I, Katkov A S and Koltik EDA 1996 Transportable Josephson voltage standard based on high-temperature superconductors Conf. on Precision Electromagnetic Measurements, Conf. Dig. pp 241–2 Taylor B N 1987 History of the present value of 2e/h commonly used for defining national units of voltage and possible changes in national units of voltage and resistance IEEE Trans. Instrum. Meas. IM-36 659–64 Taylor B N 1991 The possible role of the fundamental constants in replacing the kilogram IEEE Trans. Instrum. Meas. IM-40 86–91 Taylor B N and Cohen E R 1991 How accurate are the Josephson and quantum Hall effects and QED Phys. Lett. 153A 308–12

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Taylor B N and Witt T J 1989 New international electrical reference standards based on the Josephson and quantum Hall effects Metrologia 26 47–62 Terrien J 1973 News from the Bureau International des Poids et Mesures Metrologia 9 40–3 Tsai J-S, Jain A K and Lukens J E 1983 High-precision test of the universality of the Josephson voltage-frequency relation Phys. Rev. Lett. 51 316–8 Vollmer E 1984 Berechnung planarer Mikrowellenschaltungen für Spannungsnormale mit in Reihe geschalteten Josephson-Elementen Thesis for Diploma Institute for HF Technology, Braunschweig Technical University von Klitzing K 1986 The quantized Hall effect Rev. Mod. Phys. 519–31 von Klitzing K, Dorda G and Pepper M 1980 New method for high accuracy determination of the fine-structure constant based on quantized Hall resistance Phys. Rev. Lett. 45 494–7 Werthamer N R 1966 Nonlinear self-coupling of Josephson radiation in superconducting tunnel junctions Phys. Rev. 147 255–63 Weston E 1892 Elektrotech. Z. 13 235 Widom A 1979 Quantum electrodynamic circuits at ultralow temperature J. Low Temp. Phys. 37 449–97

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I5.1 Analogue processing by passive devices P Hartemann

I5.1.1 Introduction The applications of superconductors to analogue processing by passive devices are dependent on the very weak microwave power dissipation at the surfaces of these materials. Therefore, the quality factor of microwave resonators in bulk or surface geometry is much larger than that of equal-sized resonators made of normal conductors. Thus by using superconductors, cavities for accelerating particles exhibit a higher conversion efficiency and the transfer functions of planar filters are more attractive. However, the substitution of normal conductors in signal-processing components by superconductors does not allow full advantage to be made of the superconductivity. Components which do not operate suitably with normal conductors, can show excellent characteristics with superconductors. For instance, superconducting thin-film technology makes it feasible to produce planar filters with a narrow bandwidth (less than 5% in relative value). In this case the transmission loss is very low and the frequency response is quasi-rectangular shaped with a high rejection level. With normal conductors, only three-dimensional (3D) geometry is practical to get such a bandwidth without excessive loss and poor selectivity. Moreover, it is possible to fabricate low-loss planar waveguides with narrow superconducting strips. Therefore the superconductors advance the miniaturization of signal-processing devices and this advantage is more significant when the processing circuits are integrated to form subsystems. The interest in the superconducting devices has been greatly enhanced by the discovery of superconductors operating at the liquid-nitrogen temperature (77 K) and the outstanding microwave features of these new materials in epitaxial films. At this temperature, the cooling systems are greatly lightened and about 100 times more efficient than small coolers providing the liquid-helium temperature (4.2 K). The superconductor technique takes advantage of the improvements achieved in cryocoolers for infrared detector arrays. This simplification of refrigerators leads to a widespread acceptance of a cryogenic environment and allows us to predict actual uses of superconductors for commercial applications. In the first section, the main electrical characteristics of superconductors and normal conductors are compared. The surface impedance, the kinetic inductance and the general properties of planar waveguides are considered. The applications to delay lines, resonators and filters are obvious from these characteristics and are described in the other sections. I5.1.2 Microwave characteristics of superconductors The superconductive state is characterized by the pairing of charge carriers which can be electrons for low-critical-temperature (Tc) superconductors or holes for high-Tc oxides. In a superconducting material, the charge carriers are paired (Cooper pairs) or single (normal carriers or quasiparticles). At any nonzero temperature, the number of normal carriers remains finite. A direct current is formed by Cooper pairs Copyright © 1998 IOP Publishing Ltd

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which travel in a superconducting connection without dissipation and the resistivity is zero up to the critical current density. For an alternating current, an electric field must be induced to reverse the paired carrier path as a consequence of the pair inertia. This field also drives the normal carriers which transfer some current with Joule loss. Therefore at nonzero temperatures, a superconducting strip carrying an alternating current always exhibits a resistance. However, at temperatures well below the superconducting transition, the majority of carriers is made up of Cooper pairs and the resistivity is much less than that of normal metals in the same cryogenic conditions. I5.1.2.1 Surface impedance The distinction between paired and unpaired carriers leads to the two-fluid model of superconductors. The conductivity σ is complex (Van Duzer and Turner 1981)

where ρl is the resistivity corresponding to the normal carriers and λL , is the frequency-independent London penetration depth which is equal to the magnetic field penetration depth for London superconductors such as high-Tc oxides, the pair coherence length being much smaller than λL ;

where m* is the effective mass of paired carriers, ns the paired carrier density, e the charge of paired carriers, µ 0 the vacuum permeability and w the angular frequency. σ2 represents the behaviour of paired carriers. σ1 and σ2 are proportional to the density of normal carriers and Cooper pairs respectively. These two parts of the conductivity are temperature dependent

with t = T/Tc (reduced temperature); σn is the normal conductivity just above the transition temperature and λL(0) is the magnetic field penetration depth at zero Kelvin. For the usual operating conditions, σ2 is much larger than σ1. As an example, for c-axis oriented epitaxial high-Tc superconductor films of YBa2Cu3O7 (YBCO) at 77 K and 10 GHz, σ1 is close to 106 S m−1 whereas σ2 is equal to 2.62 × 108 S m−1. The surface impedance is defined as the ratio of the electromagnetic electric field over the magnetic field at the surface. For thick conducting materials, this surface impedance is equal to the bulk wave impedance Z given by

from Maxwell’s equations. The effective surface impedance corresponds to the wave impedance at the surface of any film and substrate stack. For a film, the intrinsic surface impedance Zs is equal to the effective surface impedance when, obviously, the film is much thicker than the penetration depth (δn or λL ) or when the effective wave impedance at the film-substrate interface is quasi-infinite. The last condition is rarely fulfilled and the effective or intrinsic surface impedances can be very different according to the structure; this is usually the case when a resonance phenomenon occurs (Hartemann 1992, Klein et al 1990).

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The main electrical characteristics of normal-conducting or superconducting films are expressed by the relationships summarized in table I5.1.1. The intrinsic surface resistance Rs of a normal conductor is proportional to the square root of the frequency (neglecting the anomalous skin effect) whereas, for a superconductor, Rs is proportional to the squared frequency.

As deduced from table I5.1.1, the intrinsic surface impedance Zs of the superconducting film is ZS = RS + jXS + Z coth ( jkd ). Then

Figure I5.1.1 shows the calculated intrinsic film surface resistance and reactance versus YBCO thickness at 10 GHz and 77 K according to the two-fluid model. For film thicknesses larger than about 500 nm, i.e. 2.27 λL , the film behaves as a bulk material. The Mattis-Bardeen method constitutes another theoretical approach to determine the microwave behaviour of superconductors. It is deduced from the BCS theory and involves the energy gap 2∆ (Van Duzer and Turner 1981). While, at present, it has not been demonstrated that the BCS theory is valid for the high-Tc superconductors, the surface resistances determined by the Mattis-Bardeen method are close to the experimental values measured at 77 K. According to this theory at low reduced temperatures (t < 0.5), the conductivity σ1 varies approximately as ( 2∆/k B T ) exp(−∆/(kBT )), kB being the Boltzmann

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Figure I5.1.1. Calculated intrinsic film surface resistance and reactance versus thickness of superconducting YBCO films at 77 K (two-fluid model).

constant. Moreover, for these temperatures the energy gap 2∆ and the penetration depth ∆L are quasi-independent of temperature. Then the surface resistance Rs changes as (ω 2/T ) exp(−∆/(kBT )). However, this exponential behaviour is not observed at very low temperatures and the two-fluid or Mattis-Bardeen model approaches are not valid. A temperature-independent residual resistance due to the superconducting material defects must be added to the values calculated according to the theoretical models. This residual surface resistance is the lowest attainable value of the surface resistance. For niobium, the best value of the residual surface resistance is close to 10−9 Ω −1 at 10 GHz whereas it reaches about 10−5 Ω −1 with YBCO epitaxial films. As shown in figure I5.1.2, the Rs values measured for epitaxial YBCO films at 77 K (t ≈ 0.83) are very close to the curve drawn for niobium at the same reduced temperature (Piel and Müller 1991). Therefore at the usual temperatures the qualities of niobium and YBCO are equivalent. For YBCO at 10 GHz the measured intrinsic surface resistance is 2 × 10−4Ω −1, that is 50 times smaller than that of pure copper cooled at 77 K. At 94 GHz and 77 K the surface resistance of YBCO is close to 2 × 10−2Ω −1 and about three times smaller than that of copper. The curvatures of graphs are due to the anomalous skin effect for copper and to the energy gap for niobium. I5.1.2.2 Kinetic inductance For a superconducting strip made up of a quasiparticle density small with respect to the Cooper pair density (σ1 « σ2 ), the internal inductance Li per unit length is equal to the sum of a magnetic inductance LM and a kinetic inductance LK which accounts for the kinetic energy of Cooper pairs. By calculating the conventional magnetic energy and the superconducting charge carrier kinetic energy per unit length, expressions of LM and LK may be found

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Figure I5.1.2. Intrinsic surface resistance versus frequency for copper, niobium and YBCO. These curves are deduced from experimental results, t is the reduced temperature T/Tc .

where W is the width of the strip and d its thickness. The internal inductance Li is given by

For d » µ L For d « µ L

In the last case the kinetic inductance varies as λ2L and is inversely proportional to the pair density. Then thermal, magnetic, optical or electronic means can be used to induce a kinetic inductance change by variation of the pair number. The electronic process would consist of injecting a biasing current in the strip to adjust the pair density by the depairing effect. I5.1.2.3 Characteristics of superconducting planar waveguides Planar electromagnetic waveguides have been fabricated using the configurations sketched in figure I5.1.3 with superconducting electrodes. This section is more particularly devoted to the transverse electromagnetic mode propagation conditions in the microstrip geometry. According to the standard theory, a transmission line exhibits a series impedance Z and a parallel admittance Y per unit length. The complex wavevector k is equal to -j(ZY)1/2 and the characteristic impedance Zc is given by (Z/Y )1/2

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Figure I5.1.3. The main geometries of superconducting planar waveguides.

where Re(Z/Zc ) denotes the real part of the ratio Z/Zc and Im(Z/Zc ) the imaginary part. Generally the series resistance, which is proportional to Rs, and the parallel conductance are small with respect to the series impedance Lω and the parallel admittance Cω respectively. L is the series inductance and C the parallel capacitance per unit length. In these conditions, the phase delay tl per unit length is equal to (LC )1/2 and the characteristic impedance is close to (L/C )1/2. L is equal to the sum of the geometric inductance and the electrode internal inductances (see figure I5.1.3 for definitions of H and W )

In this case the fringing effect is neglected (W » H)

The term vd is the phase free propagation velocity in the dielectric material with a relative permittivity ∈r (vd = (∈r∈0 µ 0 )−1/2 ). R denotes the slowing factor

If the thicknesses d and H are much larger than λL , the factor R is almost equal to 1. For thin films (d « λL). R is close to λL(2/Hd )1/2. It is enhanced when the dielectric thickness is also small. This wave-slowing is attributed to the influence of the kinetic inductance which becomes dominant for thin superconducting films. The characteristic impedance Zc of a transmission delay line depends on the ratio W/H and the relative permittivity ∈r of the dielectric material. For ∈r equal to 9.7 (MgO) the characteristic impedance is 50 Ω for W/H close to 1, the slowing effect being negligible. If the dielectric is lanthanum aluminate (∈r ≈ 24), the 50 Ω impedance is obtained for a ratio W/H of about 0.33. These two substrates are used to deposit epitaxial YBCO films. With superconducting electrodes, the characteristic impedance without taking into account the fringing effect (W » H) is

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where Zd is the wave impedance of the dielectric. As an example, for a microstrip line with a characteristic impedance of 44 Ω, a dielectric thickness of 40 nm (∈r ≈ 9.7) and a width W of 1 µ m the slowing factor R would be 9 using 30 nm thick YBCO electrodes at 77 K (λL = 220 nm). The attenuation per unit length due to the ohmic loss is kc″:

with Rl the guide resistance per unit length. The corresponding attenuation per unit delay time is kc″υg , υg being the propagation velocity in the guide (υg = υd /R)

The dielectric material introduces an additional propagation loss by the factor tan δ. The dielectric loss per unit length is kd″:

where Gl is the parallel conductance per unit length taking into account the dielectric loss. The corresponding dielectric loss per delay unit is kd″υg

For the usual dielectric materials cooled at 77 K, the loss factor tan δ is very small. For MgO, tan δ is close to 10−6 at 10 GHz and for LaAlO3 tan δ is about 3 × 10−6. Then the dielectric loss is generally negligible. The expressions for ohmic attenuations are summarized in table I5.1.2 for limit cases. The attenuation per unit delay time for thin electrodes does not depend on thickness. If the ratio W/H is smaller than about ten, the effective permittivity (∈r e f f ) taking into account the dielectric filling

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factor of the microstrip guide must be used instead of the material permittivity ∈r

Moreover the dielectric factor loss is also weighted by another filling factor to obtain the effective factor loss (Hoffmann 1987). These filling factors are not specific to superconducting microstrips. The loss reduction induced by the use of superconducting electrodes is particularly emphasized for narrow widths W. For a microstrip line with a width of 2 µ m and ratio W/H of 1, the ohmic loss would be close to 0.1 dB cm−1 for YBCO electrodes at 77 K and 10 GHz whereas this loss would be equal to 3 dB cm−1 with copper electrodes in the same conditions. Current density peaks occur along the strip edges as a consequence of the fringe field due to the finite dimension of the strip. The fringe field effect is an increasing function of the ratio H/ W. To a rough approximation the effective strip width is widened by a W/H-dependent fringe factor. The current is mainly carried by the dielectric-strip interface for substrates such as LaAlO3 with a high permittivity (∈r ≈ 24) used for depositing YBCO films. However, for any electrode thickness some of the current travels on the external surface of the strip. With an electrode thickness of 1 µ m the calculated ratio of the current densities carried by the two strip surfaces is greater than ten for W/H close to unity (El-Ghazaly et al 1992). To avoid losing the superconductive characteristics of materials, the current density peak level must not exceed the critical value, i.e. a few times 106 A cm−2 at 77 K. The same kind of consideration is valid for the microwave magnetic field. In consequence the microwave power-handling capability of superconducting planar waveguides is limited. For instance, the surface resistance measured on a stripline resonator with YBCO electrodes at 77 K and 1.5 GHz is almost constant up to a microwave peak magnetic field Hr f of 80 Oe at the edges of the centre strip and increases with respect to Hr f according to a quadratic law for stronger fields (Oates et al 1991). Then the main advantages of superconducting planar waveguides are as follows. (a) Low conduction loss The surface resistance of superconductors is smaller than that of cooled copper up to about 100 GHz for YBCO at 77 K. This crossover frequency would be raised by operating at lower temperature. (b) Frequency-independent propagation velocity The penetration depth λL, does not vary versus frequency. Then the guides are dispersionless. Very short pulses can propagate without shape degradation along relatively great lengths. (c) Slow-wave propagation for very thin electrodes This slowing effect is attributed to the increase of the kinetic inductance in very thin films. It can be adjusted electronically.

I5.1.3 Planar waveguide transmission line applications Planar waveguides are employed to delay a signal for processing or to transmit it between semiconductor chips with minimum degradation and crosstalk and in the shortest possible time. In the transmission line domain, superconductivity is of interest because it might be possible to fabricate micrometric linewidth waveguides without excessive insertion losses. For signal processing, methods to obtain the longest delay per unit area are being investigated. For interconnections in multichip modules, low-loss narrow linewidth connections involve a reduction of interconnect layer number and an increase of the module dimensions.

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Figure I5.1.4. Simple delay lines, (a) Mask of a YBCO 44 ns delay line on a 127 mm thick LaAlO3 substrate (5 cm in diameter). Reproduced from Liang et al (1993b) by permission of IEEE. (b) 25 ns coplanar waveguide delay line made of a thallium compound film (Tc » 100 K) on a 4.1 ´ 4.1 cm2 LaAlO3 substrate. Reproduced from Fenzi et al (1994) by permission of IEEE.

I5.1.3.1 Signal processing The elementary function necessary to process signals is the delay. Simple or tapped delay lines have been realized with low-Tc or high-Tc superconducting electrodes. (a) Simple delay lines Compact simple delay lines for temporary storage of microwave signals are designed with a spiral- or meander-shaped path (figure I5.1.4). A delay of 44 ns has been obtained by cascading two YBCO 22 ns delay line modules (Talisa et al 1996). The substrates are 250 µ m thick LaAlO3 wafers (5 cm in diameter). Two mirror-image spiral lines are stacked to make up a stripline structure. To obtain a delay of 22 ns, the 150 µ m wide line is 1.5m long. It exhibits an impedance of 27 Ω. A short coplanar line is inserted between the stripline and the coaxial connectors for impedance matching. At 77 K, the insertion loss of the 44 ns delayline is close to 1.5 dB between 2 and 5.2 GHz with an amplitude ripple of about 1 dB. By comparison, the same delay is obtained with a 10 m long coaxial cable with a diameter of 8.25 mm, the total insertion loss being 5 dB at 5 GHz and 300 K. A delay of 100 ns has been obtained by assembling four monolithic meander-shaped coplanar waveguide delay lines like that shown in figure I5.1.4(b). In this case the loss increases from 2.2 to 8.3 dB between 2 and 6 GHz (Fenzi et al 1994). To compact devices the linewidth must be narrowed. Then the substrates must be thinned to keep a practical characteristic impedance. A sapphire wafer is stronger than an LaAlO3 one and more easily thinned. Buffer layers (in CeO2 or MgO) have been deposited on sapphire to obtain high-quality YBCO films. However, for layers with micrometric thicknesses, the most promising technique would consist of depositing epitaxial superconducting and dielectric films on a thick substrate by sputtering, coevaporation or laser ablation, which is very difficult to achieve. (b) Tapped delay lines A transversal filter is made of a tapped delay line, tap-weighting elements and an adder. This architecture has been exploited to implement dispersive delay lines in stripline geometry with niobium (at 4.2 K) and YBCO (at 77 K) electrodes. The dispersion is induced by frequency-selective backward-wave couplers which are distributed along two spiral-shaped delay lines. Figure I5.1.5 shows the typical configuration of such a dispersive delay line. The coupler length is close to a quarter wavelength. The delay is a linear

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Figure I5.1.5. Typical YBCO dispersive delay line in stripline geometry. Reproduced from Ralston et al (1992) by permission of IEEE.

function of the frequency and a downchirp filter is performed. A weighting of taps is achieved by variation of the coupler strength through the spacing between the two lines. A delay change (∆τ ) of 11 ns over a bandwidth (∆f) of 2.7 GHz centred at 4.2 GHz was measured on a YBCO delay line at 77 K, the number of couplers being 48. With niobium electrodes at 4.2 K a delay change of 37.5 ns for a bandwidth of 2.6 GHz has been obtained, the product ∆f ∆τ being equal to 97.5. In this case the experimental frequency response in amplitude or phase follows the predicted curve with a better accuracy than for the YBCO lines. A linearly frequency-modulated pulse is expanded or compressed in time with such a dispersive delay line. A short pulse is supplied to the input of a first dispersive line with a flat weighting of taps. The impulse response of this line has a rectangular envelope and a downchirp frequency modulation with a bandwidth of 2.7 GHz (figure I5.1.6). Then this expanded signal is time-reversed by mixing it with the oscillations coming from a local oscillator, i.e. the part of the signal spectrum with the lowest frequencies, which are the most delayed at the mixer input, are the least delayed after the mixing. At the output of the time-reverser, the signal is upchirped. Then the signal is compressed in time by a second downchirp delay line. If this last line is flat weighted, it works as a matched filter. The compression ratio is equal to the product of the bandwidth ∆f and the delay change ∆τ. The compressed pulse shows a relative sidelobe level of —13 dB. This is too high for accurate signal processing. Therefore a bell-shaped weighting is introduced in the second tapped delay line (Ralston et al 1992). The sidelobe level is notably reduced and the compressed pulse width is enlarged (figure I5.1.6). The compressor is no longer a matched filter. The pulse compression technique is currently employed to improve the performance of radar. By this technique the range resolution is increased with the same maximum range proportional to the transmitted energy per pulse, or the maximum range is stretched with the same range resolution given by the time width of the processed received pulse. However, the ultra-wide bandwidth of superconducting dispersive delay lines is too great for radars. These lines have been designed for instantaneous spectrum analysis in military applications (countermeasurements). A typical block diagram of a real-time spectrum analyser based on a dispersive delay line is shown in figure I5.1.7. This analyser architecture is called MCM (M for multiplication, C for convolution). At the output, the spectrum of the input signal is spread in time which is proportional to the frequency. When only the amplitude of the spectrum is required, the last multiplication in an MCM configuration is suppressed. The dual architecture (CMC) is also used.

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Figure I5.1.6. Frequency responses of YBCO dispersive delay lines (∆ f = 2.7 GHz, ∆ t = 11 ns) and the corresponding compressed pulse. Reproduced from Ralston et al (1992) by permission of IEEE.

Another potential application of the superconducting tapped delay line is the time-integrating correlator (figure I5.1.8). The signal at the output of a tap is equal to the sum S1(t — T1 ) + S2(t — T2 ), T1 and T2 being respectively the propagation times between the considered tap and both ends of the delay line and t the time. After mixing by Schottky diodes or Josephson junctions and time integrating by conventional circuits, the signal (S3) forms a sample of the cross-correlation of the signal S1 with the reference S2 . The number of correlation function samples is equal to the tap number. With this architecture, it is possible to obtain the cross-correlation of two very long wideband signals without the necessity for high-speed peripheral electronics. With superconducting electrodes the bandwidth would be widened. Moreover, by lengthening the line, the allowed synchronization time shift between both input signals would be more relaxed and the number of samples would increase. This kind of device may be used for wideband spread-spectrum communication systems, spectrum analysis and goniometry. I5.1.3.2 Interconnections Another potential application of superconducting transmission lines consists of interconnecting semiconductor chips to implement modules for processor arrays in computers. In this case, the propagationinduced signal delay and degradation must be minimum. Generally the semiconductor components (CMOS, GaAs) are compatible with an operating temperature of 77 K. The delay of a CMOS logic gate is halved, Copyright © 1998 IOP Publishing Ltd

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Figure I5.1.7. Block diagram of an MCM configuration spectrum analyser with a wide instantaneous bandwith. VCO—voltage-controlled oscillator.

Figure I5.1.8. General configuration of a time-integrating correlator.

the operating temperature being lowered from 300 K to 77 K. Conventional silicon-based bipolar circuits do not operate suitably at low temperatures because of the carrier freeze-out phenomenon. However, for specially designed heterojunction bipolar transistors (HBTs) the gain is maximum at 8 GHz at around 50–85 K. At 4.2 K it drops slightly from the value measured at room temperature. Moreover, cryogenic high-electron-mobility transistors (HEMTs) show a remarkably low noise level up to about 20 GHz when they are cooled at 13 K. Copyright © 1998 IOP Publishing Ltd

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Therefore the use of high-Tc superconductors to interconnect semiconductor chips has been investigated, the maximum current imposed by the critical current density (≈106 A cm−2) being suitable. Superconducting connections with a width of a few micrometres are theoretically practical, whereas the minimum linewidth of connections in normal metal is close to 20 µm. The interconnection areal density per layer would be enlarged by taking advantage of superconductors and the number of wiring layers would be drastically reduced. For instance to interconnect about a hundred chips with normal-metal lines, a substrate made of 63 layers was fabricated. With superconducting interconnects, only a few layers would be necessary and the number of elementary circuits per module could be raised. However, high-quality YBCO films are deposited on expensive crystalline substrates with a high dielectric constant which introduces an excessive propagation delay. Nevertheless, YBCO two-layer interconnects and paths between levels by vias have been realized (Burns et al 1993). Moreover CMOS and YBCO devices operating at 77 K have been fabricated on the same sapphire substrate. High-Tc superconductors would be practical in interconnections, if it were possible to deposit films on large areas of cheap low-permittivity dielectric material. At present these conditions cannot be achieved. I5.1.4 Resonators Superconducting resonators are used in particle accelerators or in signal processing. They have a geometry with three dimensions (cavities) or two dimensions (planar waveguides). For accelerators, the cavities are formed by (super)conducting walls enclosing vacuum. For signal processing, the resonating volume in cavities or planar waveguides may be filled by gas or solid dielectric material. Any kind of resonator is specified by a fundamental characteristic—the quality factor Q

where We is the energy stored in the resonator and Pl the dissipated power. Another useful expression for Q is deduced from the previous one

where τ is the decay time required for dissipating or evacuating the stored energy when the resonator is no longer sustained. The reciprocal of the loaded quality factor (1/QL ) is equal to the sum of the reciprocals of the Q factors of different loss mechanisms

where Qc is the Q factor attributed to dissipation in conducting materials, Qd the Q factor of the dielectric material enclosed in the resonator ( Q d = 1/tan δ ), Qr the Q factor due to radiation, Qe the Q factor related to the transmission to the load through the external coupling network, f0 the resonance frequency and 2∆f is the 3 dB frequency bandwidth of the resonator response. I5.1.4.1 Superconducting cavities for signal processing The investigations are concentrated on the introduction of high-Tc superconductors in gas-filled cavities and dielectric resonators. Gas-filled cavities The best reported result concerns cavities made of a melt-processed YBCO thick film (about 40 µ m in thickness) deposited on polycrystalline yttria-stabilized zirconia substrates (Lancaster et al 1992). The

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technological process consists of coating the substrate by either screen printing, doctor blade or dipping techniques with an ink containing a YBCO powder. After drying and sintering at 1030-1050°C, the YBCO film exhibits a surface resistance of 1.09 mΩ −1 at 77 K and 5.66 GHz, i.e. 6.8 times lower than Rs of cooled copper. The quality of these YBCO films is inferior to that of epitaxial thin films (Rs ≈ 0.09 mΩ −1 ). However, large curved areas can be coated with superconducting thick films. Cylindrical or coaxial or helical cavities were realized according to this technique. An unloaded quality factor of 715 000 was measured at 77 K for a cylindrical cavity operating in the transverse electric mode TE011 at 5.66 GHz. A pure copper cavity under the same conditions would exhibit a Q factor close to 104000. (b) Dielectric cavities Ceramic dielectric resonators are currently utilized to implement filters or oscillators. At room temperature and 10 GHz, the typical dielectric quality factor is about 20 000 for ceramics and 2.6 × 105 for sapphire. At liquid-nitrogen temperature (77 K), the Q value of sapphire at 10 GHz is close to 60 × 106 and at 4.2 K it is 8 × 109. With normal-metal electrodes, it is impossible to benefit from this extremely high dielectric quality factor using the conventional resonating mode. For the usual TE011 mode, the Q value of the sapphire resonator is about 5 × 104 at 300 K and 1.8 × 105 at 77 K at a frequency of 10 GHz with copper electrodes. To limit the loss due to the conducting surroundings, two means can be used. ( i ) The whispering-gallery (WG) modes allow us to approximate the intrinsic Q values of sapphire with very weak degradation by losses in the surrounding metallic elements, the modal fields being confined within a small region near the resonator boundary. ( ii ) With low-loss high-Tc superconducting electrodes cooled at 77 K, the quality factor of the sapphire resonators reaches a few times 106 for the TE011 mode at 10 GHz (Shen et al 1992). It increases when the superconducting electrodes are enlarged. The configuration of such a resonator is sketched in figure I5.1.9. Tl2Ba2CaCu2O8 (Tc = 106 K) and YBa2Cu3O7 (Tc = 93 K) films have been employed as electrodes. A sapphire-loaded cavity (SLC) in superconducting niobium shows the same general configuration and exhibits a Q value greater than 109 at 4.2 K and 10 GHz. In this case a resonating mode with a large Q value (a WG mode) is used to obtain the highest confinement of fields in the sapphire resonator (Tobar and Blair 1991).

Figure I5.1.9. A cross-section of a dielectric resonator containing high-Tc superconducting electrodes ( Qu = 106 at 10 GHz and 77 K). The sapphire cylinder diameter is close to 9 mm. Copyright © 1998 IOP Publishing Ltd

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I5.1.4.2 Planar waveguide resonators The planar waveguide geometries shown in figure I5.1.3 have been utilized to realize microwave resonators. Several configurations has been used; they are: ( i ) ring resonators; ( ii ) half-wavelength resonators: the two ends of the resonating element are open; the resonator may be straight, bent or folded like a hairpin; ( iii )quarter-wavelength resonators: one end of the resonator is grounded, the other is open. From relation (I5.1.4) the conduction quality factor of a waveguide resonator is

kc″ and λg are respectively the attenuation coefficient due to ohmic loss and the wavelength in the guide. From equation (I5.1.2), for the microstrip geometry

wher λ0 is the wavelength in vacuum. Some expressions for Qc are summarized in table I5.1.3.

Table I5.1.4. emphasizes a great advantage of superconductors: the Q factor obtained with superconducting planar waveguide resonators is quite similar to that of 3D copper cavities cooled at the same temperature (77 K). At present it is not possible to realize gas-filled YBCO cavities with such a surface resistance. Only bulk YBCO or thick films allow us to fabricate cavities and in this case the surface resistance is much higher. However, using a sapphire resonator with flat epitaxial YBCO electrodes, as described in the previous paragraph, the Q value is comparable to that of hypothetic epitaxial YBCO gas-filled cavities. Resonators are the basic components of oscillators and filters. These applications are considered in the two following sections. I5.1.5 Resonator-based oscillators The simplest oscillating device is ‘active’. It consists of a loop formed by a resonator and a sustaining amplifier. ‘Passive’ frequency sources are often preferred. In this case an oscillator, not necessarily very stable, is stabilized by phase-locking on an external resonator working in reflection or in transmission.

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A typical schematic diagram of an oscillator stabilized by a resonator operating in transmission mode is shown in figure I5.1.10. Moreover an efficient technique to increase the short-term frequency stability can be implemented according to a more complicated system. In this case the resonator is used in transmission mode as a band-pass filter in the loop oscillator whereas in the reflection mode it serves as the dispersive element of a frequency discriminator, the simultaneous utilization of both modes being achieved by means of a circulator. A voltage-controlled phase shifter is introduced in the loop oscillator to lock the oscillating frequency to the resonant frequency of the resonator (Ivanov et al 1995).

Figure I5.1.10. Block diagram of a VCO stabilized by a resonator operating in transmission. The discriminator operates as a sensor for the automatic frequency control to reduce the VCO signal phase noise.

However, generally dual-oscillator phase-locked systems are used (figure I5.1.11(a)). A VCO is under the control of an oscillator containing a high-Q-value resonator. Currently the frequency of a reference oscillator based on a quartz resonator is much lower than the VCO frequency and a divider of the VCO frequency or a multiplier of the reference frequency is necessary to equalize both frequencies for mixing (Anastassiades and Aubry 1993). This frequency change introduces a noise level increase as emphasized in figure I5.1.1 1(b). With a reference oscillator frequency close to that of the VCO, the divider or multiplier is not necessary and the corresponding extra noise level does not exist. Reference oscillators operating at microwave frequencies are built with cavities or planar waveguide resonators. High-Q-factor resonators are employed in frequency sources to obtain continuous wave signals with a very narrow spectrum around the carrier frequency (short-term stability), and with a small frequency

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Figure I5.1.11. (a) A block diagram of phase-locked dual oscillators, the reference oscillator frequency being lower than that of the VCO; (b) typical corresponding noise spectra (N = 100, reference frequency = 10 MHz). Reproduced from Anastassiades and Aubry (1993) by permission of Chapman and Hall. Such a configuration allows us to take the advantages of each separate oscillator; a low noise floor, low noise close to the carrier and minimal aging.

shift with changing temperature and minimal aging (long-term stability). The oscillating circuit includes an amplitude limiter and the main origin of noise is the random phase fluctuations which exhibit a root mean square ∆Φr m s for a 1 Hz bandwidth. The phase fluctuations are specified by the single-sideband noise L( f ) or the spectral density SΦ( f ) versus the frequency offset ( f ) from the carrier. L( f ) is denned as the ratio of the single-sideband noise power measured on a spectrum analyser in a 1 Hz bandwidth ( f Hz away from the carrier frequency) to the total signal power

The lowest level of the phase-noise spectral density is given by the thermal white noise; it is equal to -174 dBc Hz−1 at 300 K and -180 dBc Hz−1 at 77 K below a signal power of 1 mW. For low-Q oscillators the spectrum base width is provided by the resonator phase transfer function and it is equal to Copyright © 1998 IOP Publishing Ltd

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Signal processing applications ∆ f = f0 / 2QL.

For high-Q oscillators the spectrum base width is given by the flicker noise introduced by the loop amplifier. If the signal frequency is multiplied by the factor N (N > 1) an extra noise of at least 20 log N in dB is added to the noise measured around the fundamental frequency. Then a high-quality continuous wave source requires: ( i ) a high loaded Q factor of the resonator to narrow the spectrum. ( ii ) a small frequency multiplication factor of the fundamental resonator frequency to avoid a large degradation of the noise level. These requirements are performed by sapphire resonators operating in TE or WG modes at microwave frequencies. Some characteristics of oscillators implemented with various resonators are summarized in table I5.1.5 for comparison.

To benefit from the ultra-high Q factor of sapphire resonators, the noise introduced by the associated semiconductor components must be minimized. Moreover, as pointed out previously, an outstanding phase-noise level is obtained by frequency locking for a frequency offset from the carrier smaller than about 10 kHz (-163 dBc Hz−1 at 1 kHz offset at 9 GHz and 77 K with a WG mode resonator). X-band feedback loop oscillators based on a sapphire-loaded niobium cavity cooled at 4.2 K were fabricated. In this case, the oscillator noise level reaches -120 dBc Hz−1 at a 1 kHz offset (Tobar and Blair 1992). Improvements seem possible. However, sapphire resonators excited in a WG mode inside normal-conducting cavities also allow us to produce X-band oscillators with a remarkable spectral purity as reported in table I5.1.5 (Taber and Flory 1995). Without selection, the mode density in this kind of resonator is very high. The wanted

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WG mode resonance generally exhibits a small amplitude with respect to that of the dominant spurious mode. Moreover, the frequency interval between the chosen WG resonance and the adjacent undesirable mode can be very narrow (a few megahertz). Then the resonator must be appropriately designed to obtain a clean frequency gap between adjacent modes as large as 250 MHz in the X-band (Taber and Flory 1995). With a sapphire puck operating in the TE011 mode, there is a single resonating frequency over a bandwidth wider than 1 GHz and the use of superconducting electrodes to excite this last mode allows us to limit the degradation of the resonator Q value which is caused by losses in the electrodes.

I5.1.6 Filters Bandpass or bandstop filters are implemented with lumped elements or distributed components. The low surface resistance of superconducting thin films is exploited to reach high Q values for spiral inductors (lumped elements) or planar resonators (distributed components). Superconductivity allows us to realize narrow-bandwidth and low-loss planar filters which cannot operate using normal conductors in the same configuration. Then the filters are miniaturized with respect to conventional filters based on 3D gas-filled or dielectric cavities and it may be possible to print several filters on the same substrate for channellized receiver applications. In spite of the cryogenic constraints, a significant saving in weight and volume can be achieved by high-Tc superconductor technology. This feature is very important for space-based communications equipment. In table I5.1.6, the predicted weight of a superconducting C-band (4-8 GHz) input multiplexer is compared with that of a conventional multiplexer made of dual-mode dielectric-resonator-loaded cavities.

A fourfold mass gain could be obtained using superconducting thin films. Moreover, thermal switches based on the superconducting-normal transition can be associated with a filter bank for tuning or switching, normal-metal lines being used to drive the switches by providing heat and magnetic field.

I5.1.6.1 Planar lumped-element filters Figure I5.1.12 shows a part of lumped-element filter including spiral inductors and capacitors printed on a high-Tc superconducting film (Swanson et al 1992). I5.1.6.2 Planar resonator filters Different configurations of microstrip bandpass filters are shown in figure I5.1.13. Generally the side-coupled resonator configuration is utilized. A filter is mainly characterized by its frequency response at low power and its behaviour with applied microwave power.

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Figure I5.1.12. Part of a typical planar lumped-element filter: (a) pattern printed on a superconducting film (microstrip geometry); (b) equivalent circuit.

Figure I5.1.13. Main configurations of strip resonator filters: (a) side-coupled resonator bandpass; (b) end-coupled resonator bandpass; (c) interdigital bandpass; (d) comb-line bandpass.

(a) Frequency response at low power For this kind of bandpass filter made of n identical resonators introducing n poles in the transfer function, the insertion loss at the band centre is A

where CK is a coefficient which depends on resonators and the frequency response shape, Bf is the relative bandwidth of the filter (Bf = ∆ff /f0 ) and Qu = f0 /∆fr is the unloaded quality factor of the resonators. For a given bandwidth and in-band ripple level, the flanks of the filter frequency response are more abrupt when the resonator number and Q value are enlarged. Moreover, to achieve a filter with a narrow bandwidth (B ≤ 1%), the resonator Q factor must be high and only superconducting planar resonators meet Copyright © 1998 IOP Publishing Ltd

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this requirement. Therefore by using superconducting resonators, it is possible to fabricate very selective and low-loss filters in planar geometry. This would be completely impossible with normal conductors. For example, a four-pole bandpass filter was produced, the relative bandwidth being 0.5% at 9 GHz (∆ff ≈ 50 MHz). This pseudo-interdigital filter was fabricated in microstrip geometry. With YBCO electrodes and a copper ground plane the insertion loss is close to 1.3 dB at 77 K, but the loss is 11.2 dB with copper electrodes at room temperature, the frequency response being very different from the required quasi-rectangular shape (Fathy et al 1993). With a double-sided YBCO-coated substrate, the loss is close to 0.6 dB. For larger relative bandwidths the difference between the losses of superconducting and normal-conducting filters is smaller. Figure I5.1.14 shows the calculated frequency response of a three parallel-coupled microstrip resonator filter with a relative bandwidth of 1%. With YBCO electrodes at 77 K the loss is close to 0.5, dB. The loss is 4.5 dB for copper electrodes at 77 K.

Figure I5.1.14. The calculated frequency responses of three microstrip resonator filters: full curve—YBCO at 77 K (Rs = 2.5 × 10−4 W −1 ); broken curve—copper at 77 K (Rs = 1.1 × 10-2 W −1 ).

Cellular communication base-stations may benefit from high-temperature superconducting filters. They require receiving front-end filters with a relative bandwidth less than 5% and a centre frequency spread from 0.8 to 2 GHz according to the communication systems. The low loss and high selectivity of planar multipole filters based on YBCO can induce an enhancement of the station sensitivity and capacity. The frequency response of such a filter operating at 0.9 GHz is shown in figure I5.1.15. The bandwidth is equal to 25 MHz with a minimum loss of 0.5 dB and an out-of-band signal rejection close to 60 dB, the average unloaded Q value of the 19 resonators being about 10 000 (Zhang et al 1995). For a relative bandwidth larger than 10% the loss of microstrip filters made of normal-conducting electrodes at 300 K is practicable. As an example, the loss of a four-pole filter is close to 1.8 dB at 14 GHz for a relative bandwidth of 14%. If this filter were cooled to 77 K, the loss would be smaller than 1 dB. In this case the advantage of superconducting electrodes is not obvious. (b) Power response The power response of superconductors at high frequencies is a limiting factor in many applications mainly when superconductors are employed in transmitter circuits. For high microwave power, the surface resistance (Rs ) of superconductors increases significantly. For epitaxial YBCO films, Rs behaves as the sum of the surface resistance measured at a very weak level and a second term approximately proportional

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Figure I5.1.15. The measured frequency response of a 19-pole microstrip resonator filter with YBCO electrodes on a 75 mm diameter LaAlO3 substrate. The bandwidth is equal to 25 MHz (2.79% in relative value) and the minimum insertion loss is close to 0.5 dB at 77 K. The pattern is shown in the inset. This filter was designed for cellular communication base-stations. Reproduced from Zhang et al (1995) by permission of IEEE.

to the squared magnetic field related to the microwave power. This second term may be attributed to the effect of grain boundaries constituting Josephson junctions and film defects which make microwave magnetic flux penetration easier (Gates et al 1995). Then the nonlinearity depends on the quality of films. In the absence of d.c. bias, odd harmonics (mainly the third harmonic) are generated and disturb the signal at the output. This nonlinearity also introduces an intermodulation between the signals applied to the device input. For instance, the nonlinear behaviour of a 28 µm wide coplanar waveguide with a length of 5 mm was characterized by a single-tone harmonic generation measurement (Wilker et al 1995). This waveguide operating at 1.3 GHz exhibits a third-order harmonic level of -45 dBm (3 × 10−8 W) for a fundamental frequency input power of 36 dBm (4 W). In this case the guiding strips are fabricated with Tl2Ba2CaCu2O8 films cooled at 80 K. With YBCO electrodes at 77 K, the same third-harmonic level is reached when the input power is close to 25 dBm (0.3 W). The harmonic generation capability of devices is often specified by the signal input level corresponding to the crossing of the extrapolated straight lines (plotted on a log-log scale) representing the output powers at the fundamental frequency (slope = 1) and at the third-order harmonic frequency (slope = 3) versus input power at the fundamental frequency. This third-order intercept (TOI) point for the previous waveguide corresponds to 62 dBm for YBCO electrodes and 75 dBm for the thallium compound. Multipole filters require superconducting films with a high current-handling capability. Because of the stored energy, large currents circulate along electrodes of high-Q resonators while the filter output power remains moderate. The nonlinear behaviour of a five-pole planar filter was characterized by a two-tone intermodulation measurement (Liang et al 1995). This filter designed for cellular communication systems exhibits a relative bandwidth of 1.2% at 2 GHz with five 3 mm wide parallel-coupled YBCO resonators. At a temperature of 45 K the output level (per tone) of the third-order mixing product is equal to -30 dBm (1 µ W) for an input power (per tone) of 32 dBm (1.6 W). At 77 K this input power is reduced to 28 dBm (0.63 W) for the same third-order level. Then the TOI point is equal to 62 dBm and 57 dBm at 45 K and

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77 K respectively. The same nonlinearity may be expressed differently: at 45 K the microwave power applied to the input of this filter is close to 36 W (45 dBm) for an excess attenuation of 0.15 dB at the upper band edge, the low-level transmission loss being less than 0.2 dB. According to these results the microwave power-handling capability of high-Tc superconductors is suitable for analogue signal processing in receivers and, at present, in transmitters radiating a relatively moderate microwave power. I5.1.7 Conclusion Superconductors offer the possibility of allowing a microwave power dissipation much lower than that of cooled normal conductors. This feature enables passive devices for analogue processing to achieve performances which could not be approached by other means. Thus low-Tc superconducting cavities to accelerate particles are used currently because of their capability to transfer energy with a very high conversion efficiency. Moreover, a very significant impact of superconductors on analogue signal processing is expected since the feasibility of epitaxial high-Tc superconducting films on large wafers has been demonstrated. These films allow the fabrication of devices exhibiting excellent microwave characteristics and, more particularly, they make practicable narrow-band multipole filters in surface geometry. High-Tc superconductors provide the opportunity to introduce superconducting signal processing elements in ground, airborne or space systems, the operating temperature being around the boiling point of liquid nitrogen at atmospheric pressure. Therefore the cryogenic refrigeration burden is drastically reduced with respect to devices cooled at 4.2 K which require a very large and heavy cryogenic environment. Moreover, the overall efficiency of closed-cycle refrigerators capable of cooling electronic circuits to 4.2 K for a long-life use is very poor (≈0.02%). Nevertheless several scientific satellites have been launched with devices cooled at 4.2 K or lower temperatures. In this case a tank containing liquid helium is integrated in the satellites. For instance, the satellite ISO, put into orbit in late 1995, includes a 2250 1 tank full of liquid helium which will give an in-orbit operational lifetime longer than the targeted 18 months. In space, passive radiative cooling techniques cannot provide temperatures lower than 90-100 K. Also, continuously operating components based on high-Tc superconductors such as YBCO require closed-cycle cryocoolers. However, at about 77 K the mass and volume of the cryogenic environment is greatly reduced and the cooler efficiency reaches about 3% for a surrounding temperature of 300 K. In consequence a programme known as the High Temperature Superconductivity Space Experiment (HTSSE) started in the USA to investigate the possibilities of incorporating high-Tc superconducting devices into space systems (Nisenoff et al 1993). Fifteen elementary passive components were mounted in the original small satellite. The list of these space-qualified devices mainly includes several planar resonators (ring or half wavelength) and various multiresonator filters. Unfortunately this satellite was destroyed during launching. A second satellite must demonstrate the functionality of advanced high-temperature superconducting devices and subsystems in the space environment. For instance, several multiplexers consisting of four channels, a complete microwave receiver front-end, an analogue signal spectrum analyser using a dispersive delay line, a 45 ns delay line and a digital instanteous frequency-measurement subsystem (Liang et al 1993a) have been built for integration in this satellite which was launched in 1997. Superconducting elements have been successfully introduced in receiver front-ends for wireless communication ground base-stations. These cold heads consist of a high-Tc superconducting narrow-band planar filters and low noise semiconductor amplifier. The cryogenic front end then provides a 2–4 dB noise figure reduction with respect to uncooled receivers in the same mounting (Simmons and Madden 1996). The noise figure is close to 0.7 dB when a cool head operating at 1.9 GHz is tower or mast mounted to greatly reduce the degradation due to the cable linking the antenna to the station. The decrease in the

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noise figure introduced by the superconducting front-ends involves an extension of the handset range and a reduction of the number of costly base-stations in a given area. However, this commercial application will be effective only if the reliability and the cost of small cryocoolers are acceptable. A breakthrough in high spectral purity microwave oscillators may be achieved by taking advantage of the intrinsic extremely high Q value of sapphire resonators. Sapphire WG mode resonators loading normal-conducting cavities compete with TE mode resonators with high-Tc superconducting electrodes. Oscillators exhibiting outstanding short-term frequency stability will be utilized in radars to detect slow-moving or low observable targets. While the refrigeration load is very much reduced at 77 K, it is not totally removed. Then superconducting devices will be effectively used in operational systems only if they exhibit unique characteristics and allow a substantial improvement of system performance. High-Tc . superconducting microwave passive devices will fulfil these conditions more especially as the processing circuits will be integrated and cooled in the same cryostat. Superconductivity pushes technology towards the miniaturization and integration of analogue signal-processing subsystems. References Anastassiades J and Aubry J P 1993 Frequency synthesizers The Microwave Engineering Handbook vol 2, ed B L Smith and M H Carpentier (London: Chapman and Hall) pp 118–38 Burns M J, Char K, Cole B F, Ruby W S and Sachtjen S A 1993 Multichip module using multilayer YBa2Cu3O7−δ interconnects Appl. Phys. Lett. 62 1435–7 El-Ghazaly S M, Hammond R B and Itoh T 1992 Analysis of superconducting microwave structures: application to microstrip lines IEEE Trans. Microwave Theory Tech. MTT-40 499–508 Fathy A, Kalokitis D, Pendrick V, Belohoubek E, Piqué A and Mathur M 1993 Superconducting narrow band pass filters for advanced multiplexers 1993 IEEE MTT-S Digest (New York: IEEE) pp 1277–80 Fenzi N, Aidnik D, Skoglund D and Rohlfing S 1994 Development of high temperature superconducting 100 nanosecond delay line SPIE Proc. 2156 143–51 Hartemann P 1992 Effective and intrinsic surface impedances of high-Tc superconducting thin films IEEE Trans. Appl. Supercond. AS-2 228–35 Hoffmann R 1987 Handbook of Microwave Integrated Circuits (Norwood, MA: Artech House) Ivanov E N, Tobar M E and Woode R A 1995 Advanced phase noise suppression technique for next generation of ultra low-noise microwave oscillators Proc. 1995 IEEE International Frequency Control Symp. (New York: IEEE) pp 314–20 Klein N, Chaloupka H, Müller G, Orbach S, Piel H, Roas B, Shultz L, Klein U and Peiniger M 1990 The effective microwave surface impedance of high-Tc thin films J. Appl. Phys. 67 6940–5 Lancaster M J, Maclean T S M, Wu Z, Porch A, Woodall L and Alford N NcN 1992 Superconducting microwave resonators IEE Proc.-H 139 149–56 Liang G C, Shin C F, Withers R S, Cole B F, Johansson M E and Suppan L P 1993a Superconductive digital instantaneous frequency-measurement subsystem 1993 IEEE MTT-S Digest (New York: IEEE) pp 1413–6 Liang G C, Withers R S, Cole B F, Garrison S M, Johansson M E, Ruby W S and Lyons W G 1993b High-temperature superconducting delay lines and filters on sapphire and thinned LaAlO3 substrates IEEE Trans. Appl. Supercond. AS-3 3037–42 Liang G C, Zhang D, Shih C F, Johansson M E, Withers R, Anderson A C, Gates D E, Polakos P, Mankiewich P, de Obaldia E and Miller R E 1995 High-temperature superconducting microstrip filters with high power-handling capability 1995 IEEE MTT-S Digest (New York: IEEE) pp 191–4 Nisenoff M, Wolf S A, Ritter J C and Price G 1993 Space applications of high temperature superconductivity: the High Temperature Superconductivity Space Experiment (HTSSE) Physica C 209 263–7 Oates D E, Anderson A C, Sheen D M and AH S M 1991 Stripline resonator measurements of Zs versus Hr f in YBa2Cu3O7−x thin films IEEE Trans. Microwave Theory Tech. MTT-39 1522–9 Oates D E, Nguyen P P, Dresselhaus G, Dresselhaus M S, Koren G and Polturak E 1995 Nonlinear surface impedance of YBCO thin films: measurements, modeling and effects in devices J. Supercond. 8 725–33

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Piel H and Mülier G 1991 The microwave surface impedance of high-Tc superconductors IEEE Trans. Magn. MAG-27 854–62 Ralston R, Kastner M A, Gallagher W J and Batlogg B 1992 Cooperating on superconductivity IEEE Spectrum 29 50–5 Safa H 1993 ”Cavités supraconductrices Compte Rendu des 4è Joumées d’Aussois ( France ) Shen Z Y, Wilker C, Pang P, Holstein W L, Face D and Kountz D J 1992 High Tc superconductor-sapphire microwave resonator with extremely high-Q values up to 90 K IEEE Trans. Microwave Theory Tech. MTT-40 2424–32 Simmons J P Jr and Madden J M 1996 Practical HTS/Croygen. Syst. Wireless Appl. Microwave J. 39 124–36 Swanson D G Jr, Forse R and Nilsson B J L 1992 A 10 GHz thin film lumped element high temperature superconductor filter IEEE MTT-S Digest (New York: IEEE) 1191–3 Taber R G and Flory C A 1995 Microwave oscillators incorporating cryogenic sapphire dielectric resonators IEEE Trans. Ultrason. Ferroelectr: Frequency Control UFFC-42 111–9 Talisa S H, Janocko M A, Meier D L, Talvacchio J, Moskowitz C, Buck D C, Nye R S, Pieseski S J and Wagner G R 1996 High temperature superconducting space-qualified multiplexers and delay lines IEEE Trans. Microwave Theory Tech. MTT-44 1229–39 Tobar M E and Blair D G 1991 A generalized equivalent circuit applied to a tunable sapphire-loaded superconducting cavity IEEE Trans. Microwave Theory Tech. MTT-39 1582–94 Tobar M E and Blair D G 1992 Phase noise of a tunable and fixed frequency sapphire loaded superconducting cavity oscillator IEEE 1992 MTT-S Digest (New York: IEEE) pp 477–80 Van Duzer T and Turner C W 1981 Principles of Superconductive Devices and Circuits (New York: Elsevier-North-Holland) Wilker C, Shen Z Y, Pang P, Holstein W L and Face D W 1995 Nonlinear effects in high temperature superconductors: 3rd order intercept from harmonic generation IEEE Trans. Appl. Supercond. AS-5 1665–70 Zhang D, Liang G C, Shih C F, Lu Z H and Johansson M E 1995 A19-pole cellular bandpass filter using 75-mm-diameter high-temperature superconducting thin films IEEE Microwave Guided Wave Lett. MGWL-5 405–7

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I5.2 Analogue-to-digital converters G J Gerritsma

I5.2.1 Introduction Analogue-to-digital converters (ADCs) find widespread use in consumer as well as professional electronics. Examples are digital tape recorders, digitizing oscilloscopes, transient recorders etc. To date only semiconductor ADCs are being used, but it is expected that due to the ever-growing demand for increase in speed and accuracy opportunities for their superconductor counterparts will present themselves. An example is the need in some radar systems to view both large and small signals simultaneously (Przybysz 1993). This requires the use of 16-bit converters having an input signal bandwidth of 10 MHz. Achieving this in semiconductor technology has so far remained elusive although steady progress is being made. Superconductor ADCs are based on the principle of flux quantization in a closed superconductor loop. What the circuit has to do is to find a solution to the equation ( m − l ) Φ 0 ≤ Φ < m Φ0 where Φ is the unknown magnetic flux, Φ0 = h/2e is the flux quantum and m an integer number being the result of the quantization process, i.e. the output of the ADC. In this way the input signal Φ can be determined to within an accuracy of Φ0/2. Given the fact that the quantization process relies on a fundamental physical constant it has an inherent accuracy that cannot be found in its semiconductor counterpart. In practice accuracy is determined by many factors so exploiting this advantage is a real design challenge. As the input signal of the ADC is usually not a magnetic flux, conversion to other analogue domains is required. Now any superconductor loop is also an inductor and by injecting current in it, the required conversion to flux is obtained through the relation Φ = LI. When the input signal is a voltage Ohm’s law V = RI is used. If we wish to maintain the inherent precision of the conversion of flux to the digital domain, the inductors as well as the resistors have to be highly linear and stable in time. Various ways exist to determine the number m, i.e. to perform the conversion from the analogue to the digital domain. Trade-offs have to be made between accuracy, speed and complexity in relation to the specific application for which the ADC performance has to be optimized. I5.2.2 Some ADC basics As its name implies, the task of an ADC is to convert a signal from the analogue to the digital domain. To that end the analogue signal x ( t ), which is both continuous in time as well as in amplitude, is sampled in time and quantized in amplitude. The first operation transforms the time-continuous signal into a time-discrete signal xi = x (ti ), where ti are instances of time at which the signal is sampled. Usually the time

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intervals between successive samples are equidistant and we may write ti = iT, where i is an integer number and T the sampling period. This time-discrete signal is still continuous in amplitude and may be represented by a real number x. Here the index i as well as the physical unit of the signal, for instance mV, have been dropped. The second operation involves the transformation of this real number x into a set of discrete numbers ym , most conveniently represented by the set of integer numbers m, i.e. ym = m, or to put it in mathematical language, to map the set of real numbers onto the set of integer numbers. This may, for instance, be conceived by mapping x ∈ [0, 0.5) onto y0 = 0, x ∈ [0.5, 1.5) onto y1 = 1, x ∈ [1.5, 2.5) onto y2 = 2, etc. For this particular mapping the spacing between the discrete output levels is 1. Its quantization error e, being the difference between input signal and output signal, is given by e ∈ [-0.5, 0.5). The physical units of input and output signal are considered to be the same. This results in a circuit with, on average, unity gain. For a sensitive ADC the actual level spacing may be as low as 1 µ V. The smallest step that can be made in the quantizing process, with or without the physical unit attached, is in the sequel denoted by LSB, which is an acronym for least significant bit. The error may equivalently be written as e = ±0.5 LSB. The points x = 0.5, x = 1.5, x = 2.5, etc, are transition points that are set by reference signals. A graphical representation of this mapping process is depicted in figure I5.2.1.

Figure I5.2.1. The basic analogue-to-digital conversion process assuming eight available output levels. The full lines represent the output signal, the dotted lines the quantization error.

In practice the range of the input signal x is of course limited to a finite domain and also only a finite number of output values ym are available. The number of output values being generated is usually a power of 2. If the exponent is n we have a so-called n-bit converter and the numbers 0 to 2n − 1 are being used. In a binary number system an n-bit word is required to represent the numbers 0 to 2n − 1. An arbitrary natural number x may be deconstructed as x020 + x121 + x222 +…, where x0 , x1 , x2 , etc, are either 0 or 1. Furthermore the input signal range may be varied by scaling in the analogue domain, i.e. by putting x’ = ax + b. If the input signal is in the range xm i n ≤ x < xm a x this scaling is used to transform it to the range 0 ≤ x’ < 2n − 1 and subsequently to map this onto the integer numbers 0 to 2n − 1. This automatically takes care of input signals having a negative value. The resulting binary output code has an added offset, which is the customary way of dealing with bipolar signals. If the sampling clock that determines the time instances ti is not correlated with the input signal, the quantization error may be treated as white noise. Its root-mean-square (rms) value can easily be calculated leading to er m s = 1/ . For an n-bit converter the peak-to-peak amplitude Ap p = 2n − 1. If we assume that the input signal is

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sinusoidal the rms value of its amplitude is given by Ar m s = App/ . The signal-to-noise ratio (SNR) is by definition the so-called dynamic range of the ADC and is equal to Arms/erms. In conformity with analogue signal conventions this may also be expressed in decibels (dB). For n ≥ 5 it may be approximated by SNR = 6.02n + 1.76 dB. For a 16-bit converter this leads to a dynamic range of 98.1 dB as is found in digital consumer electronic systems like compact-disc players and digital tape recorders. The actual dynamic range of these systems is usually somewhat lower. Besides the quantization error, errors may also arise in the analogue part of the converter circuit resulting in an inaccuracy in the scaling constants a and b. Furthermore, noise, interference and hysteresis effects play a role as well. Yet another source of error results from parallel processing of the input signal, i.e. the conversion process is performed by several comparators that act simultaneously on the input signal, each having a different reference signal level. As a result of various delays in the circuit, the parallel digital outputs are not available at the same time. If one waits long enough this poses no problem, but if one is in haste the resulting error may be large. However, this type of error may be minimized by using the so-called Gray code. This code together with straight binary code and decimal code is depicted in table I5.2.1. From this table it can be seen that if a transition is made from quantizer level 3 to 4 in binary code the two least significant bits have to be turned off whereas the most significant bit has to be turned on. As in practice this does not occur simultaneously, intermediate, or metastable, code words appear. The circuit designer must make sure that reading the output data is delayed until the final code word has stabilized, otherwise serious quantization errors may result. This does not happen if the Gray code is being used. Two subsequent words differ by one bit only so no intermediate words exist. As a result the error that is made if the output data are read too fast is never more than one bit. However, what can never be corrected is if the comparators sample at different instances of time.

As we mentioned in the introduction, in a superconductor ADC it is most natural to use the principle of flux quantization in establishing the quantization levels. Scaling can be achieved by using a flux transformer, picking the right value of L or R and adding or subtracting a constant signal anywhere before the actual quantization is performed. Using these quantization levels the process of analogue-to-digital conversion consists of mapping the analogue input signal values onto a set of discrete output signal values. As the input signal varies in time a sample of it is taken at a certain moment ts . In practice this process requires a certain amount of time called the aperture time ta p or acquisition time for sample-and-hold. The aperture time has to incorporate the aperture times of individual comparators as well as their spread in sampling times. If the input signal changes by more than one LSB during this time, the error in the conversion process exceeds the resolution of the ADC. Bandwidth is directly related to the aperture time. For a harmonic signal with frequency f and amplitude 1 the maximum slew rate is equal to 2π f. The analogue signal needs a minimum time of 1/(2n − 1)π f to slew an amount equal to one LSB. The input signal frequency f must satisfy the relation f ≤ fB = 1/(2n − 1)πta p to ensure that the error does not Copyright © 1998 IOP Publishing Ltd

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exceed ±0.5 LSB, where fB is the input bandwidth of the converter. Here we have assumed a full-scale signal, i.e. twice the amplitude spans the full range of the converter. The influence of nonzero aperture time is depicted in figure I5.2.2.

Figure I5.2.2. The effect of nonzero aperture time. The input signal varies by an amount ∆ during ta p.

However, even in the case of a converter with zero aperture time, serious errors may arise if the input bandwidth is too high. If a signal with frequency f is sampled at a frequency fs mixing products at fs ± f, 2fs ± f, etc, also appear in the output spectrum. Therefore the input bandwidth fB must not exceed fs /2. If a signal is sampled at twice the input bandwidth we say it is sampled at the Nyquist rate fN = 2fB . In this case a very steep low-pass filter is required to ensure that signal frequencies higher than fN /2 are sufficiently reduced in amplitude to disappear in the noise. Such filters are called anti-alias filters and require careful design. However, a signal that is sampled at the Nyquist rate can be reconstructed, i.e. converted from the digital to the analogue domain, with an error not exceeding that of the original quantization error. I5.2.3 Oversampling of converters Fabricating an ADC with a high number of bits may be a costly affair as it requires the use of high-precision components in order to define the transition points in an adequate way. Various clever tricks have been devised to reduce the number of high-precision components and thereby cost (see e.g. Hoeschele 1994 or van de Plassche 1994). Here we will focus on the use of oversampling and noise-shaping techniques. By definition we use the term oversampling if the sampling rate is higher than the Nyquist rate. The so-called oversampling rate (OSR) is given by OSR = fs /fN . A signal with a bandwidth of 20 kHz that is sampled at a rate of 160 kHz is said to be four times oversampled. An obvious advantage of oversampling is that it may significantly reduce the burden on the design and manufacture of the anti-alias filter, leading to a significant cost reduction. The second advantage is that it may reduce the quantizer noise in the input bandwidth. This results from the fact that the quantization error is now randomized over a larger frequency band. The quantization noise power er2 m s is now spread over the bandwidth fs /2 instead of fB . This leads to a reduction of a factor in the quantization noise of a converter. The SNR is increased by 10 log OSR dB, which for a four-times-oversampled converter leads to an increase with 6.02 dB or equivalently we gain an extra bit of dynamic range. It would be nice if, for instance, using oversampling we could use a one-bit converter to arrive at a dynamic range of 16 bits. The advantage of a one-bit ADC is that it is highly linear and hardly requires any costly components. The penalty that has to be paid is an increase in sampling rate, complexity of the digital part and processing speed. Digital filtering is required to remove the noise not lying in the input bandwidth. Furthermore it is customary to convert the digital data stream back to the Nyquist rate with increased word length in a process called decimation. In this

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example this would lead to processing speeds that are a billion times higher than the Nyquist rate which in many cases is prohibitive. A way out is offered by the use of noise-shaping techniques. Noise shaping removes quantization noise from the input bandwidth to out-of-band noise. This is usually at the cost of an increase in total noise power. The out-of-band noise has to be removed by adequate digital filtering. An example of such an operation is delta-sigma analogue-to-digital conversion (for historical reasons it is often called sigma-delta conversion). The basic idea behind it is that the quantization error is fed back to the input so that in a subsequent conversion step a correction can be applied. This is called the delta operation, but it requires a high oversampling rate as the error must not change much between successive samples. A vital improvement came by the introduction of an integrator after the delta operation. Its role is to filter out the high-frequency components leading to a reduction of in-band quantization noise. This constitutes the noise-shaping operation. The process of quantization and error correction is performed by a so-called modulator. Its schematics are presented in figure I5.2.3, where it is assumed that only two quantization levels are available. It can be shown that this first-order feedback correction results in an overall rms

Figure I5.2.3. A schematic diagram of a typical delta-sigma modulator.

noise in the signal band given by nr m s = er m s π/ . Again it has been assumed that input signal and sampling clock are not correlated; it is said that the signal should be sufficiently busy. Each doubling of the oversampling ratio increases the dynamic range by 9 dB, thus providing 1.5 extra bits of resolution. So the use of this first-order feedback already leads to a considerable noise reduction for moderate oversampling rates. This can even be further improved by using higher-order feedback. An extensive analysis can be found in the book by Candy and Temes (1992). If L is the order of the feedback circuit the noise is reduced by 3(2L + 1) dB for every doubling of the sampling rate, giving L + 0.5 extra bits of resolution. However, although not uncommon, feedback loops with order greater than second are difficult to implement as they tend to be less stable. As the modulator only digitizes the signal and reduces the in-band quantization error at the cost of increasing the out-of-band noise, digital filtering is required to remove this extra noise before the output data rate is reduced to the Nyquist rate. This filter also has to remove any remaining alias. In practice it almost always pays to perform the decimation operation in more steps as it requires simpler filters. To give an example, in digitizing 4 kHz telephone signals the modulator produces a one-bit data stream at 1.024 MHz. During the first stage of decimation down to 32 kHz a filter is used that is optimized for removing out-of-band noise as this dominates at high frequencies. In the second stage a low-pass filter is used, optimized to remove alias signals before the data stream is resampled at the Nyquist rate to produce 16-bit words at a rate of 8 kHz. So far it has been assumed that the quantization error may be treated as white noise. This assumes that sample clock and signal are not correlated. If this is not the case, noise patterns appear in the output spectrum that also depend on the amplitude of the input signal. The higher the oversampling ratio the more problematic this becomes. In order to get rid of these patterns and regain white noise a dither signal

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is added to the input. For this dither signal either a small noise signal or a signal at fs /2 is used. This randomizes the noise patterns making them white. The penalty that has to be paid is a reduced dynamic range as the input amplitude must never exceed the limit set by the dynamic range of the converter, otherwise a rapid increase in quantization error occurs. I5.2.4 The comparator A vital element in any ADC is a circuit that compares the analogue input signal to a reference signal. If the input signal is higher than the reference signal it produces a 1, if it is lower a 0. The reference signal level is set at one of the transition points x = 0.5, 1.5, 2.5, etc. In practice there is a certain amount of indeterminacy in this process, i.e. in order to produce a value of 1 the input signal has to be somewhat higher than the reference signal and a 0 is produced if it is somewhat lower. Hence in a small range around the reference signal level the output is undetermined, it becomes either a 1 or a 0 in an unpredictable way. This sets a lower bound to the resolution or the sensitivity of the ADC, i.e. two neighbouring quantization levels can no longer be distinguished if they are taken too close together. In the following we will see that a superconductor comparator, in contrast to its semiconductor counterpart, is periodic in nature. In a semiconductor ADC the comparator is a high-gain amplifier with a clipped output, as depicted in figure I5.2.4.

Figure I5.2.4. A semiconductor comparator.

For an ADC based on the principle of flux quantization the comparator consists of a superconductor loop with a Josephson junction Jq . If an increasing input current is injected into this circuit the current through the junction will increase until its critical current has been reached. Thereafter the junction will switch into the voltage state and the loop inductor will be charged. As a result the current through the junction is reduced and the junction switches back to the superconductor state, whereupon the process will repeat itself. If properly designed at the output of the circuit a single voltage spike is produced with amplitude Ic R n and width τ = Φ 0/Ic R n , where Ic is the junction critical current and Rn its normal-state resistance (see figure I5.2.5). This voltage spike is very narrow: for example if we have a junction with an Ic Rn product of 200 µ V the width τ = 10 ps. The magnetic flux that is trapped in the loop increases by one flux quantum. At the same time the current in the loop reverses sign. If the output pulses are fed

Figure I5.2.5. A one-junction SQUID comparator.

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into a counter a so-called counting ADC has been achieved. This so-called one-junction superconducting quantum interference device (SQUID), also known as a radiofrequency (RF) SQUID, is a rather primitive kind of circuit as it only works with increasing signals. In section 15.2.7 we will see how to deal with this in an appropriate way. Instead of using the output voltage spikes of the circuit, the direction of the current in the loop Il may also be sensed. To that end an extra junction Js is inserted in the loop to probe the direction of the current. Its critical current must be higher than that of the quantizer junction as it should not switch if only the input current Ii n is applied. This so-called quasi-one-junction SQUID (QOS) is depicted in figure I5.2.6.

Figure I5.2.6. A quasi-one-junction SQUID comparator. Jq is the quantizer junction and Js the sense junction.

The direction of the loop current is sensed by applying a current spike Is to the circuit as indicated. If the sum of the loop current and the sense current exceed the critical current of this sense junction, a voltage spike is generated at the output, otherwise no output is generated. If the amplitude of the sense current is equal to the critical current of Js, an output spike is generated if the loop current is clockwise, and no output occurs if it is flowing counterclockwise. In practice the sense current amplitude should be slightly higher as part of it also flows into Jq. The use of a sense junction is a prerequisite for use in a parallel converter scheme. The status of several comparators has to be sensed at one and the same time. The simple one-junction SQUID does not fulfil this requirement as the output pulse appears as soon as the critical current of the quantizer junction is exceeded. In the QOS circuit the quantizer junction will, of course, also switch when its critical current is exceeded. However, this does not produce any output signal of the comparator. One of the major differences between a semiconductor and a superconductor comparator is-that the latter has a periodic characteristic. How this feature may be exploited is discussed in section 15.2.7. Furthermore, no explicit reference signal level is required as the flux quantum itself acts as the reference. However, in some designs this property is not used. I5.2.5 Comparator dynamics As we have already seen in section 15.2.4, Josephson junctions may switch very fast in a properly designed circuit. Taking the example given there, i.e. an output voltage spike of width 10 ps, one may expect an input bandwidth of the order of 100 GHz. In this section it will be demonstrated that other speed-limiting factors may play a more dominant role and a careful design is required to arrive at a very high bandwidth. In the previous section two types of superconductor comparator were introduced, the one-junction SQUID and the QOS. The QOS has been introduced by Ko and Van Duzer (1988) as an alternative for a variety of previously used comparators. The speed-limiting factors they found have been analysed by Fang and Van Duzer (1989), resulting in the conclusion that the QOS comparator offered the highest input bandwidth perspective. If the critical current of the sense junction becomes much higher than that of the Copyright © 1998 IOP Publishing Ltd

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quantizer junction its dynamic properties come close to that of the one-junction SQUID, therefore we will only deal with the dynamic properties of the QOS circuit. Ideally the aperture time of the basic comparators is given by the switching speed of either the quantizer or the sense junction, so it could be 10 ps for the example given in section 15.2.4. For high transition temperature superconductor junctions this might even be ten times faster, giving them a comparative advantage. At present this advantage cannot be exploited due to technological hurdles. In practice the effective aperture time is larger for a variety of reasons. One is that the actual switching time of the junction, embedded in a given circuit, may be larger, but most important is the fact that the pulse does not exactly arrive at the intended moment. This is obvious if several comparators have to act in parallel and not all sampling pulses will arrive at the same time or not all output signals will be available at the same time. However, some of its effects may be reduced by using a Gray encoding scheme for the output. The variation in arrival times of input signal and sampling pulses does not result in an increase of effective aperture time if the input signal before it is divided over several comparators is sampled and its value held for a short period of time. This may, however, have a detrimental effect on the sampling rate and the bandwidth, so a careful analysis of its dynamics is required before one decides to use this solution. For the design of a sample-and-hold circuit we refer to Sage et al (1993). If the signal is of a repetitive nature subsequent samples may be taken at different periods, each having a slight shift in phase. This results in a circuit with a high bandwidth, but as a penalty a relatively long measuring time is required. The purely analogue counterpart of such a circuit is the boxcar detector. Another effect that limits the bandwidth is caused by dynamic hysteresis of the comparator. As a result the instant of time at which a sample of the input signal is taken becomes dependent on the slew rate, including its sign. The actual sample moments may differ substantially between a rising part of the loop current and a declining part of it. This hysteresis effect results from the charging and discharging of capacitors and inductors of the circuit. The former are not explicitly indicated in the circuit drawings, but are always present, either in the form of stray capacitances or in the form of the inevitable junction capacitance. For high-quality tunnel junctions, which are nowadays mostly made of niobium electrodes and an aluminium oxide barrier ( Nb/Alox ), the influence of this capacitance has to be minimized by shunting the junction with a resistor with a value substantially lower than the junction resistance. The so-called capacitance, or Stewart-McCumber, parameter βc = (2πIcR/Φ0 )RC = ωJ R C has to be <1, where R is the shunted resistance of the junction, C its capacitance and ωJ the angular Josephson frequency corresponding to the voltage IcR . If this criterion is met the junction is overdamped and plasma oscillations do not occur. In the following we will neglect capacitance effects. A further requirement for proper operation of the comparator is that the inductance parameter βL = 2π LIc q / Φ0 < 1, where L is the loop inductance and Ic q is the critical current of the quantizer junction Jq. If this condition is met the threshold curve is single valued. This curve is defined as a plot of the amplitude of the sense or sampling current pulses Is that just switch the sense junction into the voltage state versus the input signal Ii n . If the threshold curve is not single valued a dramatic increase in dynamic hysteresis occurs. This is the main reason why several comparator types that have been used in the past do not have a very high bandwidth, although their sampling speed is more than adequate. In figure I5.2.7 dynamic hysteresis effects in the loop current Il are presented for βL = 0.5, an input signal frequency of 1 GHz and a peak-to-peak amplitude of 4 Φ0/L for various ratios of Ic s /Ic q . The quantizer junction product IcR was taken to be 1 mV. The amplitude has been chosen such that four periods of the threshold curve appear, the minimum requirement for a four-bit periodic threshold converter (see section 15.2.7). In the encircled part of the figure eight zero crossings can be seen. The outer two belong to a QOS where the ratio between the critical current of the sense junction and quantizer junction is 1.5: one belongs to an increasing signal, the other to a decreasing signal resulting in a large hysteresis. The inner two zero crossings are from a circuit where the ratio is 10. From these simulations it can be seen that the dynamic hysteresis becomes smaller for larger critical current ratios. The hysteresis effects can be fully explained by the resistive currents

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Figure I5.2.7. A simulation showing dynamic hysteresis in the loop current Il. Currents are given in amps. Each period amounts to exactly a change of one flux quantum in the loop.

produced by the voltages across the junctions. Hence, for a minimum hysteresis the resistance R should be as high as possible. Another way to reduce the hysteresis is to reduce the parameter βL , which can be accomplished by either decreasing the critical current Ic q or by decreasing the loop inductance L. As we will see in section 15.2.6, a decrease in Ic q results in a reduced noise immunity of the circuit. Furthermore, a reduction in L leads to a reduced current sensitivity of the circuit as the flux that has to be quantized is proportional to it. The latter problem can be overcome by using an input coupling transformer. An interesting circuit to reduce the effective critical current without reducing the actual critical currents of the junctions involved has been proposed by Ko et al (1993). It uses a completely symmetric circuit consisting of a SQUID wheel or phase tree with two spokes where the circulating current is sensed by a quantum flux parametron. Due to its inherent symmetry, the dynamic hysteresis does not affect the switching probability: both the 0 and the 1 have equal probability of occurring. However, noise may spoil the behaviour of this circuit at very high frequencies and an edge-triggered sense circuit may be preferred (Bradley and Rylov 1996). The influence of hysteresis effects may also be reduced by the use of redundant coding techniques. In order to avoid too much detail we will leave aside the various methods to reduce dynamic hysteresis effects or their influence on the functioning of an ADC. As high-temperature superconducting junctions have a higher value for the IcR product with respect to dynamic hysteresis, they offer a comparative advantage over classical junctions like that made in Nb/Alox technology. Furthermore, most high-temperature superconductor junctions are nonhysteretic so an external shunt resistor is not needed. A problem with the circuit depicted in figure I5.2.6 is that close to a transition level its response to a sampling pulse is rather slow. Simulations show that delays of several tens of picoseconds may occur. Therefore an edge-triggered version of the QOS circuit as proposed by Bradley and Dang (1991) has to be preferred. In this circuit two sense junctions Js and Js ’ , having the same properties, are used as depicted in figure I5.2.8 and the sampling current pulse has been replaced by a voltage step Vs across both sense junctions. If a sampling voltage signal is applied, at least one of the sense junctions has to switch to the voltage state. If the loop current circulates clockwise the lower junction Js will switch first as it experiences the highest current. Once it is switched the current through the upper junction Js ’ will be reduced to almost zero and hence it does not switch. Therefore the full sampling voltage will appear at the output. In the case of a counterclockwise loop current only the upper sense junction will switch and no output voltage signal appears. When the loop current is close to zero both junctions may switch as can be demonstrated in a computer simulation. This results in an output signal having half the amplitude. If Copyright © 1998 IOP Publishing Ltd

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Figure I5.2.8. The edge-triggered version of a QOS. It is faster than the original QOS.

the SNR is high enough this poses no problem for the read-out circuitry. In this version of the QOS comparator delays of only a few picoseconds occur, making them more suitable for high-speed parallel conversion techniques. I5.2.6 Thermal noise and flux flow Noise in a superconductor circuit, as in any other circuit, may lead to errors. In the case of the comparators that were discussed previously, this may not only result in a variation in the actual moment of sampling the input signal, hence leading to in increase of effective aperture time, but also in switching where it is not appropriate. For a superconductor ADC operating at liquid-helium temperatures this is not so much of a problem, although in some designs it is (Bradley and Rylov 1996). However, when high-temperature superconducting junctions are to be used at elevated operating temperatures care has to be taken that the circuit will not be too noisy. In this section attention will be focused on the requirements to be met by the junctions, especially their critical current density, leaving aside the use of redundant coding techniques, to cope with noise. As is well known, a Josephson junction that is overdamped shows behaviour similar to the Brownian motion of a particle in a tilted washboard potential (Ambegoakar and Halperin 1969). For low tilt angles, i.e. low bias currents, the number of phase slips, resulting in switching spikes, per second is given by f = f Ae−γ, where fA = Ic RΦ0 ≈ 1012 Hz (for IC R = 2 mV ) is the so-called attempt frequency and γ = h Ic /ekBT is the noise-immunity parameter, where h is Dirac’s constant, e the electron charge, kB Boltzmann’s constant and T the absolute temperature. Furthermore the parameter γ has to be large for this approximation to hold. Specifying a minimum acceptable value for the noise-immunity parameter, i.e. γ0 , where the effect of biasing may be included, this leads to a requirement for the critical current Ic ≥ γ0ekBT/h. For a given technology resulting in a given critical current density Jc of the junctions this leads to a minimal junction area. If we now restrict the discussion to edge-type or grain-boundary junctions that are common in high-temperature superconductors and using films of thickness d this leads to a lower bound in the junction width w ≥ γ0ekBT/hdJc . Here we have assumed that the current in the junctions flows along the direction of the planes of the high-temperature superconductor materials being used and that this is parallel to the substrate surface. So-called planar or sandwich junctions, the ones used in classical technology, are not being considered. The technology to fabricate them from high-temperature superconductors and related materials is so immature that at present they cannot be used at all in an electronic circuit and it is unclear whether this will be possible in the near future. Furthermore, the reader

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should be cautioned because dynamic aspects have not been taken into account in this noise analysis. In general this requires the use of an appropriate circuit simulator. Another point of consideration is that it is often required that the junctions should not start to flux flow as the transition from the superconducting state to the normal state becomes less abrupt. This will inevitably lead to reduced operating margins of the circuits built with them. It may also have a detrimental effect on the switching speed. Preventing flux flow sets an upper bound on the junction area and for the types of high-temperature superconductor junction we consider here this leads to a maximum width w ≤ 4λJ , where λJ is the Josephson penetration depth, being a measure of the size of the so-called Josephson vortex. This vortex differs from the ordinary Abrikosov vortex as its core resides in the normal region of the barrier of the junction, resulting in a minimum amount of drag force and hence high speeds. The penetration depth is given by λJ = (4eµ 0 Jcλ L /h)0.5, where µ 0 is the permittivity of vacuum and λL, the London penetration depth. Here it has been assumed that the actual barrier thickness is negligible compared to twice the London penetration depth. Using the London penetration depth of a typical hightemperature superconductor material like YBa2Cu3O7 we arrive at λ J = 300/ µ m, if the critical current density is expressed in A cm−2. So far an upper bound as well as a lower bound has been derived for the width w of the junctions. If we now introduce some numbers, i.e. a film thickness of 100 nm, a London penetration depth of 200 nm and specify the minimum noise-immunity parameter as 40, it is easy to demonstrate that the junction critical current density should exceed 1 kA cm−2. Otherwise both criteria cannot be met simultaneously. In order to arrive at some operating margin, i.e. wm a x /wm i n = 3, a critical current density of about 8 kA cm−2 is required. For comparison in modern low-Tc logic circuit, the technology used has a Jc of 2.5 kA cm−2 and junction areas are a few µm2. For high-temperature superconductors to operate at 40 K with the same noise margins as high a value as 25 kA cm−2 would be required. This is not an easy feat, but several high-temperature superconductor junction technologies can achieve these very high values: at present, however, not with an acceptable parameter spread. Recently the author managed to fabricate a fully functional high-temperature superconductor four-bit ADC having a complexity of 12 junctions with Jc ≈ 8 kA cm−2. For further details on the design, including a noise analysis, refer to the article by Wiegerink et al (1995). Although, as mentioned at the beginning of this section, noise is of less importance at 4 K, even there it may play an important role. For a noise analysis of a so-called quantum flux parametron, an essential element in some ADCs, including dynamic effects the reader is referred to the article by Ko and Lee (1992). In this circuit-shorter exciter rise times lead to an increased switching probability, i.e. noise has more influence at higher frequencies.

I5.2.7 Some analogue-to-digital converters Many types of superconductor ADC have been proposed, simulated and tested. In the examples discussed in the following we will restrict ourselves to three types, namely counting converters, flash converters and oversampling converters. I5.2.7.1 Counting converters As was mentioned before, the characteristics of the basic superconductor comparators that are used are periodic in nature. If the input signal is ramped up, or down, the loop current is purely periodic and the quantizer junction spits out an equidistant train of pulses, one for each flux quantum in a proper single-flux-quantum (SFQ) design. This property can be exploited to full advantage in building a superconductor ADC. In figure I5.2.9 a counting ADC is depicted. The input signal is divided over two one-junction SQUID comparators via a transformer.

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Figure I5.2.9. A counting ADC that tracks the input signal.

The upper comparator produces positive output voltage spikes when the signal decreases, the lower one when it increases. The quantizer junctions also produce negative voltage spikes, the upper one for decreasing signals and the lower one for increasing signals, but it is assumed that they are ignored by the up/down counter. How this counter is being designed is not a matter of concern in this chapter. It suffices to say that SFQ design techniques are required to make it run at high speed. A typical counter would run at a 100 GHz clock speed using Nb/Alox technology. If required, the counter may be preceded by a synchronizer circuit in order to permit synchronous counting as described by Robertazzi and Rylov (1993). I5.2.7.2 Flash converters Another circuit that relies on the periodic nature of the loop current in the comparator is the so-called periodic threshold ADC. It offers the possibility of constructing an η-bit parallel flash converter using only n comparators instead of 2n — 1 as is required for a semiconductor flash converter. As was discussed previously the input bandwidth is maximal if an edge-triggered comparator is used and Gray encoding is employed to minimize errors. A schematic drawing is given in figure I5.2.10.

Figure I5.2.10. An n-bit periodic threshold ADC.

Special care has to be taken to ensure that the input and sampling signals arrive at the comparator stages at the same time. Otherwise the effective aperture time would dramatically increase and hence the input bandwidth decrease. Using Nb/Alox technology a six-bit converter has been demonstrated having an effective resolution of four bits at a 5 GHz input bandwidth (Bradley 1993). This should be compared with a recent semiconductor ADC designed for six bits at a 4 GHz sampling rate having in practice Copyright © 1998 IOP Publishing Ltd

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a resolution of 5.6 effective bits at a 1.8 GHz input bandwidth. In order to arrive at an acceptable sensitivity for the superconductor ADC an input signal coupling transformer was used. It is expected that by improving the design even higher bandwidths and higher resolutions may be achieved. Increasing resolution without sacrificing bandwidth immediately can, for instance, be achieved by a technique called interleaving. Instead of using one comparator for the least significant bit several are employed that operate in parallel. By adding a constant, but different, flux to each of these comparators the full period of the loop current may be subdivided, or interleaved. The added constant flux introduces a phase shift in this current, thereby shifting its zero crossings and hence its internal state, 0 or 1, depending on whether the current flows counterclockwise or clockwise. This results in a subdivision of the levels of the quantizer that produces so-called circular code. An example of such a code is given in table I5.2.2.

For further details the reader is referred to a paper by Ko et al (1993). Dynamic hysteresis effects will, however, become more important if this technique is used, resulting in a reduction in bandwidth. Compared with a fully parallel converter employing 2n — 1 comparators, a penalty has to be paid in using a periodic threshold comparator. The slew rate of the LSB input signal is n times higher than that of a fully parallel converter resulting in an n times lower bandwidth due to dynamic hysteresis effects. Nevertheless it is expected that six effective bits of resolution may be obtained at 10 GHz input bandwidth (Bradley and Rylov 1996). With increasing resolution, placing the switching thresholds becomes quite difficult and this results in a larger error. A way out is to introduce redundancy by using more comparators and to employ a real-time digital error correction scheme (see e.g. Kaplan et al 1996). Using this scheme it is expected that seven effective bits of resolution may be obtained at an input bandwidth of 10 GHz thereby surpassing semiconductor ADCs (cf Poulton et al 1995). I5.2.7.3 Oversampling converters Yet another type of converter in which the periodic nature of a basic superconductor comparator can be used to advantage is the delta-sigma converter. As a result of the very high sampling rates that can be obtained with superconductor circuits it is estimated that 16–22 bits of dynamic range may be obtained with input bandwidths in the range 2–100 MHz. The modulator can easily be made in superconductor technology, as it requires only a few parts. However, the design and layout has to be done very carefully in order to arrive at a high dynamic range and relatively high bandwidth. Its schematic drawing is presented in figure I5.2.11. A high-frequency clock, which may be a sine wave, is fed into a pulse-forming circuit. The pulses are subsequently transferred to a buffer circuit before they are fed as a stream of sampling pulses to the quantizer junction. The input signal is fed into an inductor that acts as an integrator. Each time the quantizer junction switches, a single flux quantum escapes from the integrating inductor resulting Copyright © 1998 IOP Publishing Ltd

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Figure I5.2.11. A schematic diagram of a typical superconductor delta-sigma modulator.

in a subtraction of the digitized signal. The most complicated part, however, is the digital filter that is needed to reconstruct the n-bit word from the digital output of the modulator. It has to be designed using SFQ techniques to arrive at a high bandwidth of the converter. Various other oversampling schemes have been proposed and tested. Examples are a converter based on phase modulation and multichannel timing arbitration (Rylov and Robertazzi 1995) or a combination of oversampling, negative feedback and counting (Lin et al 1995). In the former circuit besides the input signal a clock signal at fs /2 is injected in a one-junction SQUID comparator. The input signal phase modulates the resultant output pulses appearing at an average rate of fs /2. This signal is subsequently demodulated, decimated and filtered. The latter circuit is quite similar to a delta-sigma converter. I5.2.8 Epilogue The ADCs described in the previous section comprise only a fraction of the converters that have ever been designed or tested. Nevertheless, it is felt that the reader has been given an idea about the achievements in this field. To the author’s knowledge at the time of writing no superconductor converters are commercially available. This is primarily due to the fierce competition from semiconductor converters that have the obvious advantage that no cooling is required. Where cooling is already employed, as happens with sensors of extreme sensitivity, superconductor converters can be used to full advantage at little additional complexity and hence cost. An example may be found in the read-out of multichannel gradiometer systems that are used in biomagnetic research. Digital read-out techniques greatly reduce the number of connecting wires to all these channels (over 200 in modern systems), making the system easier to fabricate and reducing the heat leak into its cryogenic part. With the coming of high-temperature superconductor materials and Josephson junctions, cooling requirements are much easier to satisfy. Small cryocoolers (a few hundred cubic centimetres) are available, so systems do not have to be bulky as is the case with classical superconductors. However, at present high-temperature superconductor technology cannot cope with the design requirements, but progress is being made. With respect to fast semiconductor converters their superconductor counterparts have a much lower power consumption. It is expected that steady development will eventually lead to an easier acceptance of superconductor electronics by system designers. References Ambegoakar V and Halperin B I 1969 Voltage due to thermal noise in the dc Josephson effect Phys. Rev. Lett. 22 1364–6

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Bradley P 1993 A 6-bit Josephson flash A/D converter with GHz input bandwidth IEEE Trans. Appl. Supercond. AS-3 2550–7 Bradley P D and Dang H 1991 Design and testing of quasi-one junction SQUID-based comparators at low and high speed for superconductor flash A/D converters IEEE Trans. Appl. Supercond. AS-1 134–9 Bradley P D and Rylov S V 1997 A comparison of two types of single flux quantum comparators for a flash ADC with 10 GHz input bandwidth IEEE Appl. Supercond. to be published Candy J C and Temes G C 1992 Oversampling methods for A/D and D/A conversion Oversampling Delta— Sigma Data Converters—Theory, Design and Simulation ed J C Candy and G C Temes (New York: IEEE) pp 1–29 (This contribution is a review on Oversampling methods used in data conversion. The book further contains a collection of reprints of seminal papers in the field and as such is an important source of references.) Fang E S and Van Duzer T 1989 Speed-limiting factors in flash-type Josephson A/D converters IEEE Trans. Magn. MAG-25 822–5 Hoeschele D F Jr 1994 Analog-to-Digital and Digital-to-Analog Conversion Techniques 2nd edn (New York: Wiley) (A recent textbook on semiconductor conversion techniques.) Kaplan S B, Rylov S V and Bradley P D 1997 Real-time error correction for flash analog-to-digital converter IEEE Appl. Supercond. to be published Ko H and Van Duzer T 1988 A new high-speed periodic threshold comparator for use in a Josephson A/D converter IEEE J. Solid-State Circuits 23 1017–21 Ko H L and Lee G S 1992 Noise analysis of the quantum flux parametron IEEE Trans. Appl. Supercond. AS-2 156–64 Ko H L, Lee G S and Barfknecht A T 1993 Design and experiments on a high-resolution multi-gigahertz sampling rate superconducting A/D converter IEEE Trans. Appl. Supercond. AS-3 3082–94 Likharev K K and Semenov V K 1991 RSFQ Logic/memory family: a new Josephson-junction technology for subterahertz-clock-frequency digital systems IEEE Trans. Appl. Supercond. AS-1 3–28 (An extensive overview of rapid single-flux-quantum logic circuits, basic principles etc by the founders of this field.) Lin J C, Semenov V K and Likharev K K 1995 Design of SFQ-counting analog-to-digital converter IEEE Trans. Appl. Supercond. AS-5 2252–9 Poulton K, Knudsen K L, Corcoran J J, Wang K-C, Nubling R B, Pierson R L, Chang M-C F, Asbeck P M and Huang R T 1995 A 6-b, 4 GSa/s GaAs HBT ADC IEEE J. Solid-State Circuits 30 1109–17 Przybysz J ´ 1993 Josephson analog-to-digital converters The New Superconducting Electronics vol 251, eds H Weinstock and R W Ralston (Dordrecht: Kluwer) pp 329–61 (Lecture from NATO ASI, series E: Applied Sciences, about superconductor converters covering several types of ADC, track-and-hold circuits and a digital SQUID sensor.) Robertazzi R P and Rylov S V 1993 Synchronous flux quantizing analog-to-digital-converter IEEE Trans. Appl. Supercond. AS-1 3114–6 Rylov S V and Robertazzi R P 1995 Superconducting high-resolution A/D converter based on phase modulation and multichannel timing arbitration IEEE Trans. Appl. Supercond. AS-5 2260–3 Sage J P, Green J B and Davidson A 1993 Design, fabrication and testing of a high-speed analog sampler IEEE Trans. Appl. Supercond. AS-3 2562–5 van de Plassche R 1994 Integrated Analog-to-Digital and Digital-to-Analog Converters (Dordrecht: Kluwer) (A recent textbook on semiconductor conversion techniques with an emphasis on integrated circuit versions of converters.) Wiegerink R J, Gerritsma G J, Reuvekamp E M C M, Verhoeven M A J and Rogalla H 1995 An HTS quasi-one junction SQUID-based periodic threshold comparator for a 4-bit superconductor flash A/D converter IEEE Trans. Appl. Supercond. AS-5 3452–8

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I6.1 Thermal detection and antennas P Hartemann

I6.1.1 Introduction This chapter is devoted to the use of superconductors in electromagnetic-wave thermal detection systems and antennas. Applications of low-Tc superconductors to far-infrared or millimetre-wave radiation detection have been investigated for a long time. However, thermal detectors and antennas have benefited from the introduction of high-Tc superconductors operating at about 77 K. In this case the cryogenic refrigeration burden is drastically lightened and practical uses of these devices are expected. Section I6.1.2 deals with the detection of optical or microwave radiation which is based on the abrupt variation of superconductor characteristics around the transition temperature. Section I6.1.3 concerns electromagnetic-wave antennas which take advantage of the very low surface resistance of superconductors at high frequencies. I6.1.2 Thermal detection I6.1.2.1 Introduction This section concerns the main superconducting devices based on thermal effects and designed to be introduced in electromagnetic wave detection systems. It includes thermal detectors and mixers.

Figure I6.1.1. General configuration of a composite thermal electromagnetic radiation detector.

An electromagnetic radiation thermal detector has the general configuration illustrated in figure I6.1.1. A microwave or optical signal power (Pi ) is incident on a layer with a great absorptivity which is equal to its surface emissivity ∈. This coefficient is generally constant over a broad wavelength range. The total dissipated power heats the entire sensor which evacuates power to the surroundings by: Copyright © 1998 IOP Publishing Ltd

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( i ) radiation emitted with emissivity ε by the detector according to the blackbody features; this power is proportional to ∈Aσ T 4, T being the detector temperature, A the useful area and σ the Stefan-Boltzmann constant (σ = 5.67 × 10−8 W m−2 K−4 ). ( ii ) coupling to the temperature-stabilized heat-sink through the thermal conductance G; this power is proportional to the difference T — T0 , T0 being the heat-sink temperature. The expression for the heat balance equation is

where Pb , is the incident radiation from the observed background at temperature Tb (Pb = bAσ Tb4 ), a, b are the geometry dependent constants, C is the global heat capacity of the detector, δ T the elevation of the detector temperature with respect to the heat-sink temperature T0 (δ T = T — T0 ) and t the time. δ T is measured by a thermometer which can be a superconducting resistance or inductance or a Josephson junction (Richards 1994). For a significant incident signal power Pi , the influence of the radiative power emitted by the detector and background radiative power is negligible. However, these radiative powers introduce a minimum noise level. If the incident power Pi , is modulated at the angular frequency ω (ω = 2π f ), the temperature elevation is also variable. From equation (I6.1.1)

where τ = C/G is the thermal time constant. Two types of thermal detector configuration are used according to the dissipative layer dimensions with respect to the incident wavelength: ( i ) for the usual infrared radiation, the dissipative layer is much larger than the wavelength; the incident radiation beam impinges on this layer made of a quasi-blackbody with an absorptivity ε close to one, as for instance the gold black. If the thermometer is a temperature-dependent resistance, the detector is called a bolometer; ( ii ) for far-infrared or submillimetre or millimetre waves, the dissipative layer can be much smaller than the wavelength. The radiation power is collected by a dipole or bow-tie or logarithmic spiral or log-periodic antenna with an efficiency 77, and the radiation-induced electrical power is dissipated in a few micrometre-sized thermally active elements. Any type of thermometer (resistance, Josephson junction and others) may be employed. This kind of thermal detector is called an antenna-coupled microbolometer. The rest of section I6.1.2 deals with bolometers including a superconducting element as thermometer (transition-edge bolometers). Then antenna-coupled microbolometers and detectors with a superconducting inductance or a Josephson junction as thermometer are discussed. I6.1.2.2 Bolometers The temperature-sensitive element is a superconducting thin-film plot or meander line which is cooled at the temperature T0 in the middle of the resistive transition. At T0 , this element shows an electrical resistance R0 and the slope dR/dT is maximum. Its behaviour around the operating temperature is determined by approximating the transition curve by its tangent line at T0 (figure I6.1.2). The resistance change is δ R for a radiation-induced temperature variation of δ T. Then the abrupt change of resistivity around the superconductor transition temperature has to be taken into account. To measure the resistance variation,

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Figure I6.1.2. The transition edge of a superconductor.

the temperature-sensitive element may be biased by a constant voltage and in this case the responsivity is equal to the ratio of the current change amplitude to the modulated incident power. However, generally a constant current I is injected in the superconducting element and introduces a heat dissipation equal to I2(R0 + δ R). The heat balance equation for time-varying terms gives

with δ R = (dR/dt )δ T then

The responsivity (V/W ) of this transition-edge bolometer is S:

The effective thermal conductance is Ge f

f

with

where α is the temperature coefficient of the resistance R0. For superconducting materials α > 0 and Ge f f < G

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with τe f f = C/Ge f f . To maintain the thermal stability by avoiding a value of Ge f f too close to zero, the emission of heat due to the bias current is limited by imposing the following requirement

For the allowed maximum bias current Im a x

In this case

The voltage at the detector output exhibits fluctuations with a root-mean-square value 〈∆V 〉 which may be converted to an incident power Pi n

This noise equivalent power in a 1 Hz bandwidth is denoted by NEP and characterizes the quality of the detection. The overall noise at the sensor output is attributed to various physical origins. ( i ) Photon noise—noise attributed to the fluctuations of radiative powers as a consequence of the quasi-random emission of photons. A detector receives thermal radiation power from the background with a temperature usually close to 300 K through the detector field of view and from cold shields used to reduce the influence of the thermal radiation from the detector surroundings. Other power is radiated by the detector as a blackbody at temperature T0. For a detector cooled at a cryogenic temperature, the fluctuations introduced by this last power are negligible. Then the equivalent noise power corresponding to the total radiative power fluctuations defines the lowest detectable radiation level. It is proportional to the square root of the useful detector area A. The square of this NEP is equal to the sum of two terms due to the contributions of the background and the cold shields. Each term is proportional to T 5, T being the temperature of the considered blackbody source (background or shields). This limit noise equivalent power is designated by NEPB L I P (BLIP = background limited infrared photodetector). ( ii ) Thermal fluctuation noise—noise due to the detector temperature fluctuations as a consequence of the passage of quantized energy carriers (phonons or electrons) between the detector and the heat-sink through the thermal conductance G. The corresponding noise equivalent power is NEPt h

where kB is the Boltzmann constant. ( iii )Johnson noise—noise introduced by current fluctuations in the thermometer resistance. These fluctuations are caused by the random motion of charge carriers

For I « Im a x

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then

For high-Tc superconductors, the Johnson noise is not negligible with respect to the thermal fluctuation noise. ( iv )Sensitive film noise—1/f noise due to the resistance fluctuations δ Rf (in a 1 Hz bandwidth) attributed to phenomena related to the superconducting state (vortex displacements, etc). δ Rf depends on the film quality, the frequency and the biasing current

A noise equivalent temperature of the film (NETf ) is defined

then

( v ) Amplifier noise—this is relatively small and negligible. The total noise equivalent power due to all these uncorrelated noise sources is NEPt

NEP is the noise equivalent power attributed to nonradiative effects

For I « I m a x

This relationship emphasizes the major influence of the geometry-dependent thermal conductance G on the detector performance. By lowering G, the responsivity increases with a lower noise and slower response. Therefore for imaging applications, a bolometer must simultaneously exhibit a small G and a small C to obtain a suitable response speed. Generally, a detector with an area A is characterized by the specific detectivity D*

A superconducting transition-edge bolometer with a very high detectivity D* (D* = 1.1 × 1014 cm Hz1/2 W−1 at a wavelength of 500 µ m and a temperature of 1.27 K) was fabricated using a superconducting aluminium film element on a sapphire substrate suspended by threads (Clarke et al Copyright © 1998 IOP Publishing Ltd

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Figure I6.1.3. A schematic diagram of a superconducting bolometer.

1977). However, detectors are of more interest if they can be used to obtain images using detector arrays. Then focal plane two-dimensional (2D) arrays made of several ten thousands of elementary detectors (pixels) can be mounted in infrared cameras. High-Tc superconductor technology may be compatible with this requirement, but it is not the case for this superconducting aluminium bolometer. The discrepancy between the detector detectivity D* and the limit detectivity D*B L I P due to the thermal radiation powers (D*B L I P = A1/2/NEPB L I P ) must be decreased to a minimum. Then the materials and the structure are selected to thermally isolate the detector and to obtain a low thermal mass. A micromachining technique based on ion milling and wet etching is generally employed to fabricate detector-supporting membranes which are suspended by microlegs (figure I6.1.3). For instance a YBa2Cu3O7 (YBCO) film was deposited on a silicon nitride ( Si3N4 ) coating on a silicon substrate, with a buffer layer of yttria-stabilized zirconia (YSZ) lying between the YBCO and Si3N4. Then micrometre-thick membranes of Si3N4 were fabricated by local chemical etching of the silicon substrate. The measured characteristics of such YBCO films are summarized in table I6.1.1 (Verghese et al 1992). Optics are generally used to form an image on the detectors and the value of the F-number is given by the ratio of the objective focal length, F, to the entrance pupil diameter, D. For an F-number F/D of

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1.2, a detector at 85 K and a background at 300 K, the wavelength-independent limit detectivity is equal to

For a diffraction-limited detector at 10 µ m (diameter ≈ 15.3 µ m) the noise equivalent power with a minimum value of G equal to 3 × 10−7 W K−1 would be

where ∈ = 1, τ = 0.17 ms, ω = 2π × 10 rad s−1 for YBCO/YSZ/Si3N4/Si. The corresponding detectivity would be equal to

In this case, the film noise is dominant (NEPf = 7.2 × 10−13 W Hz−1/2, NEPth = 3.46 × 10−13 W Hz−1/2, NEPJ o h n = 1.68 × 10−13 W Hz−1/2). It can be reduced by improving the YBCO film quality. The responsivity is close to 22 350 V W−1 for a resistance Ro of 3000 Ω. For an F-number F/D of 6 the diffraction-limited detector at a wavelength of 10 µ m (diameter ≈ 76.4 µ m) would show an NEP of 8.39 × 10−13 W Hz−1/2 (D* ≈ 8.1 × 109 cm Hz1/2 W−1) with a time constant of 4.17 ms, the other conditions being identical. In this case D*B L I P ≈ 2.56 × 1011 cm Hz1/2 W−1. Another basic feature for infrared (IR) camera users is the temperature difference of the observed scene equivalent to the noise level. This parameter is called the noise equivalent temperature difference (NETD)

∆f is the pixel read-out bandwidth and M is the variation of power radiated by the observed scene (blackbody) per unit area over the considered wavelength range for a scene temperature change of one degree. For a background temperature of 300 K and an 8–12 µ m atmospheric window, M is close to 2 W m−2 K−1. For a diffraction-limited detector, NETD is independent of the F-number. For a minimum value of G equal to 3 × 10−7 W K−1 (YBCO/YSZ/Si3N4/Si) NETD = 90 mK at 10 µ m (∆f = 50 Hz). If G reaches a minimum value of 10−8 W K−1, NETD = 8.3 mK. The corresponding detectivity would be D* = 1.81 × 1010 cm Hz1/2 W−1 and τ = 5 ms. These results demonstrate the crucial influence of the thermal conductivity G on the detector performance. To fabricate thermal imaging detector arrays with an excellent temperature resolution it is necessary to be skilled in micromachining to reduce G to a minimum. Excellent results were obtained with high-Tc superconducting meander line bolometers (Johnson et al 1994). An epitaxial YBCO film was deposited on a 0.11 µm thick epitaxial YSZ buffer layer on silicon. In this case the coefficient is equal to 0.64 K-1. An Si3N4 film is used to protect the upper YBCO surface against attack by silicon etchant. After micromachining, the thermal conductance is 8.5 × 10−8 W K−1 at 95 K. Then the detectivity D* is equal to (8 ± 2) × 109 cm Hz1/2 W−1 with an NEP of 1.5 × 10−12 W Hz−1/2 at 2 Hz (2 µ A d.c. bias) and a substrate temperature of 80.7 K. These features were measured using filtered blackbody radiation with a spectrum concentrated over the 4-15 µ m wavelength range. The etched pit underneath the sensitive element measures 105 × 140 µ m2 in area and 70 µ m in depth. The resistance R0 is equal to 6100 Ω and the effective thermal time constant is about 105 ms. For a 449 K blackbody radiation chopped at 2 Hz, the maximum responsivity is close to 32 kV W−1, the IR absorptance not being optimized (∈ ≈ 0.58). Copyright © 1998 IOP Publishing Ltd

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Shorter time constants can be obtained as demonstrated by a superconducting bolometer which offers a thermal-fluctuation-noise dominated NEP of 0.63 × 1012 W Hz1/2 at 32 Hz and 71.7 K (D* ≈ 8 × 109 cm Hz1/2 W−1 for wavelengths longer than 12 µ m) (Berkowitz et al 1996). The thermal conductance G is 3 × 107 W K−1 for a responsivity of 3.1 kV W−1. This bolometer is based on a cobalt-doped YBCO film strip on a 0.15µ m thick YSZ membrane supported by four thin legs on an LaA103 substrate. The dimensions are 50 × 50µ m2 for the membrane and 6 × 60 µ m2 for each leg. Undoped YBCO is used for the electrical leads. In these conditions the leads exhibit a transition temperature higher than that of doped YBCO and their resistance is practically zero at the thermometer operating temperature (71.7 K). Then no excess noise is generated in the electrical contacts. For another transition-edge bolometer, a time constant value of 4 ms with a responsivity of 580 V W−1 has been obtained (D* = 3.8 × 109 cm Hz1/2 W−1 at 84.5 K for a wavelength of 6 µ m (Neff et al )). By comparison with other detectors operating at about the same temperature, high-Tc superconducting thin films enable the fabrication of imaging arrays exhibiting competitive characteristics in the usual 8-12 µ m atmospheric window and the performances would be unique for longer wavelengths (beyond 15 µ m). Moreover, the use of silicon as substrate allows the fabrication of bolometer arrays with integration of the read-out and amplification circuitry. I6.1.2.3 Antenna-coupled microbolometers Figure I6.1.4 shows the typical configuration of an antenna-coupled microbolometer (Nahum et al 1991). This planar structure is used to achieve transition-edge microbolometers for detection of far-IR and millimetre-wave radiations or hot-electron microbolometers for detection or mixing of the same radiations.

Figure I6.1.4. A schematic diagram of an antenna-coupled microbolometer.

(a) Transition-edge microbolometers The thermally active element thermalizes the electromagnetic-wave-induced currents and matches the impedance (about 80 Ω) of the broadband normal metal or superconductor planar antenna. A superconducting element can operate either as an antenna load and as a transition-edge thermometer or only as a thermometer. In the latter case the thermometer is electrically isolated from the dissipative element and in intimate thermal contact with it .

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As an example, a high-Tc microbolometer coupled to a log-periodic planar antenna was fabricated. The thermally active element was made of an epitaxial YBCO film deposited on a YSZ buffer layer on silicon (Nahum et al 1991). A YBCO strip (10 µ × 3 µ m) operates as the antenna load and thermometer. It is suspended on a YSZ air-bridge obtained by anisotropic chemical etching of silicon. In this condition the thermal conductance from the air-bridge to the silicon chip is close to 3 × 10−6 W K−1. The incident wave propagates through the silicon substrate before illuminating the antenna, silicon transmitting uniformly over the IR and submillimetre wavelength range. The back of the silicon chip is pressed against a dielectric lens to focus the radiation onto the antenna (780 µ m in outer diameter). At a temperature of 85 K corresponding to the superconducting transition midpoint, the estimated NEP is 3.2 × 10−12 W Hz−1/2 for a modulation frequency of 1000 Hz and a responsitivity of 1070 V W−1. As a consequence of the small size of the superconducting thermometer, the thermal time constant is close to 2 µ s. That is much shorter than that of conventional superconducting bolometers. (b) Hot-electron microbolometers Hot-electron microbolometers were designed to operate either as detectors or as mixers for far-IR and millimetre-wave radiations. Detectors A very small NEP is predicted for microbolometers cooled at very low temperature (0.1 K), the sensitive element being a normal-metal strip. In this condition the thermal isolation of the dissipative element and its temperature rise for a given incident power are greatly enhanced by the behaviour of phonons and electrons. ( i ) the thermal conductivity of materials varies as T03. Moreover, a thermal boundary resistance is introduced by the mismatch of acoustic impedances across the interface between the active region and the substrate. At low temperatures, it is proportional to T0-3 and becomes very pronounced below several kelvin. ( ii ) For a metal at very low temperature (≤1 K) the inelastic collisions between electrons and phonons are not frequent. The inelastic electron-phonon scattering time is proportional to T-3 and reaches about 10 µ s at 0.1 K for copper (Nahum et al 1993) (for comparison at 273 K the electron relaxation time is equal to 2.7 × 10−14 s). The electrons are thermally decoupled from the lattice and an electron-phonon resistance is introduced. Under the radiation-induced current, the ohmic loss and the electron temperature increase. This electron energy enhancement is not quasi-instantaneously transferred to the lattice, the inelastic electron-phonon collisions being infrequent. These electrons are called hot electrons because their energy exceeds the energy at the thermal equilibrium. ( iii )If the antenna is made from a superconductor with a critical temperature higher than the operating temperature, the Andreev reflection of electrons at the superconductor-metal absorber interface traps the thermal energy in the metal and introduces a decrease of the electron energy transfer across this interface (Nahum and Martinis 1993). This reflection occurs when the electron excitation energy above the Fermi energy is smaller than half the binding energy of the Cooper pairs in the superconductor. In this case the electrons are reflected back from the interface as holes with the same particle momentum and an energy lower than the Fermi level. A hot-electron microbolometer consists of a superconducting antenna (in niobium for instance), a thin strip of resistive metal absorber as antenna load and a thermometer to measure the absorber electron temperature. This thermometer is a superconductor-insulator-normal-metal (SIN) junction. A part of the metal strip is included in the junction as the normal electrode. The quasiparticle current through such a junction depends only on the temperature of electrons in the normal electrode (Nahum et al 1993). For this kind of microbolometer, an extremely low NEP of 2.6 × 10−19 W Hz−1/2 at 0.1 K is predicted. The responsivity would be close to 3.5 × 109 V W−1, the metal volume being 2 × 6 × 0.05 µ m3. For

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such an active bolometer volume, the thermal conductance G between the electron and phonon systems would be as low as 1.2 × 10−13 W K−1 at 0.1 K. Experimental results obtained with a device exploiting the Andreev reflection of electrons at lead-copper interfaces approximate to the previously calculated performance (Nahum and Martinis 1993). At 0.1 K, a responsivity of 109 V W−1 was measured for a thermal conductance (G) of 2 × 10−13 W K−1 and a copper volume of 6 × 0.3 × 0.075 µ m3. The amplifier-limited noise equivalent power is close to 3 × 10−18 W Hz−1/2 and can be improved. Mixers An antenna-coupled hot-electron microbolometer may be used as a low-noise mixer for heterodyne detection of terahertz radiation without the frequency limitation introduced by the superconducting gap energy as for Josephson junctions. In this case, the antenna-coupled thermally sensitive element consists of very small and thin (thickness ≤10 nm) parallel bridges made of a low-Tc superconducting film. These strips of superconducting material are biased to operate in the resistive state near the midpoint of the superconductive transition. The radiation-induced current introduces ohmic loss increase and quasiparticle (electron) temperature rise. The heated electrons lose their energy mainly by electron-phonon interaction. The energy is transferred to the lattice of the thin film and the excess heat in the film is carried away by the substrate to reach an equilibrium state. For this hot-electron bolometer, a very short thermal electron relaxation time is required. When two signals with different frequencies are simultaneously received by the antenna, a signal voltage at the difference frequency is observed between the ends of the sensitive element only if the temperature of the bridges can follow the power variation at this intermediate frequency. Then a fast thermal response is necessary. That is obtained by a rapid diffusion of hot electrons out of the ends of the bridges into cooled thick normal-metal leads which serve as a heat-sink. The length of the bridges must be much shorter than the hot-electron diffusion characteristic length, i.e. the bridge length is about 0.2 µ m. Another way to reduce the thermal time constant consists of using bridges made of a superconducting material with a short electron-phonon relaxation time. For instance, niobium nitride (Tc ≈ 11.5 K) offers an electron-phonon relaxation time of about 15 ps at 10 K. Radiations with a wavelength of 119 µ m (2.52 THz) were coupled to such a lattice-cooled mixer with an intermediate frequency of 1.5 GHz and a conversion loss of 25 dB (Semenov et al 1996). Hot-electron microbolometers could be used in space-based telescopes to measure the emission from various radiation sources. I6.1.2.4 Detectors with a superconducting inductance thermometer The internal inductance of a superconducting element is equal to the sum of the usual magnetic inductance LM and the kinetic inductances LK which results from the inertia of the superconducting charge carriers. The expression for the internal inductance per square is Li (see section I5.1)

where d is the film thickness and λL the temperature-dependent penetration depth of the magnetic field.

For a thin film (d/λL ≤ 0.4) the kinetic inductance is dominant

Then

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dLK /dT becomes larger when T approaches Tc . For instance

In this case the kinetic inductance LK is proportional to the reciprocal of the Cooper pair density. Then thermal or magnetic or optical or electronic means can be used to introduce a kinetic inductance change by Cooper pair breaking. The rise in the radiation absorber temperature is transmitted to the inductance and introduces a kinetic inductance change which may be measured by a Wheatstone bridge with four inductances. The sensitive inductance is thermally isolated using a membrane. It is heated by a resistance fed with a current electronically adjusted to balance the bridge. Then a noise equivalent power of 4.4 × 10−11 W Hz−1/2 was measured at 6.6 K (0.8Tc) with a niobium element, the responsivity being 1.9 × 106 V W−1 (Sauvageau et al 1991). This kind of thermal detector avoids the Johnson noise, the resistance being quasi-zero. I6.1.2.5 Detectors with a Josephson junction thermometer The characteristics of Josephson junctions are a function of the temperature. Using various methods this feature can be exploited to measure a temperature change. Generally the thermometer is a superconductor-insulator-superconductor (SIS) junction cooled at liquid-helium temperature (4.2 K). The junction is biased with a constant current to obtain an operating point in the middle of the current-voltage characteristic step corresponding to the abrupt increase in quasiparticle tunnelling current at a voltage of 2∆/e, 2A being the binding energy of the Cooper pairs and e the electron charge (Van Duzer and Turner 1981). Then a temperature elevation induces an energy gap reduction and a voltage decrease of ∆V. The responsivity S of such a thermal detector is ∆V/Pi . Its noise level is attributed mainly to ( i ) thermal fluctuation noise

( ii ) voltage noise 〈∆V 〉 introduced by the junction quasiparticle current fluctuations. For the low frequencies considered here

where V is the voltage, I the bias current and Rd = dV/dI the dynamic resistance. Then the noise equivalent power attributed to nonradiative effects is NEP

At low temperature, the thermal conductance G is very small and (NEPυ)2 proportional to G2 is negligible compared with (NEPt h )2 which is a linear function of G. Thus the thermal fluctuation noise is dominant. In an experimental configuration, the junction is supported by a thin silicon nitride panel which is suspended by four legs (width = 10 µ m, length = 90µ m) above a pit etched in a silicon substrate. An IR absorbing film covers the junction and the entire panel area. In these conditions, the calculated thermal conductance G is 1.1 × 10−7 W K−1 at 4.2 K and NEPt h = 10−14 W Hz−1/2 (∈ = 1). With only two 400 µ m long legs, G would reach 5 × 10−9 W K−1 and NEP would be close to 2 × 10−15 W Hz−1/2 (Osterman et al 1993). Arrays of junction detectors with a superconducting quantum interference device (SQUID) amplifier at each thermometer output were fabricated. Junction detector arrays suitable for imaging systems can be integrated on a single substrate with superconducting processing electronics. Copyright © 1998 IOP Publishing Ltd

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I6.1.3 Antennas I6.1.3.1 Introduction Electromagnetic-wave antennas can benefit from the very low surface resistance of superconductors by reducing the loss of the multi-element antenna feed networks and analogue beam-forming networks or by increasing the efficiency of the electrically small antennas. These advantages concern the receiving or transmitting antennas, but the power-handling capability of superconductors is limited. In the transmitting antennas, the microwave power and the associated magnetic field are often too high and would induce an increase of the surface resistance of superconductors. Then only moderate power can be transmitted by superconducting antennas. Moreover, the constraints imposed by the cryogenic environment, the substrate dimensions and the film deposit processes limit the size of superconductive antennas. I6.1.3.2 Feed networks of array antennas The dimensions of an array antenna depend upon the wavelength, and the spacing between two adjacent elements is usually close to a half-wavelength to avoid grating sidelobes. The first illustration of a superconducting array antenna concerns millimetre waves, the maximum diameter of monolithic high-Tc superconducting films being limited by the available substrate diameter, i.e. 7.5 cm for LaAlO3. A 64-element patch antenna operating at about 30 GHz was printed on a thallium compound film with a diameter of 5 cm (figure I6.1.5) (Lewis et al 1993). The impedance of feed lines is 50 Ω. for a line width of 92 µ m and an LaAlO3 substrate thickness of 254 µ m (∈r ≈ 24). The radiating patches are 1.35 mm wide and 0.9 mm long and resonate at a frequency close to 30.5 GHz at 77 K.

Figure I6.1.5. The pattern of a 30 GHz 64-element antenna array printed on a high-Tc superconducting film (5 cm in diameter). Reproduced from Lewis et al (1993) by permission of IEEE.

The gain Ga of a transmitting antenna is equal to the ratio of An times the maximum power radiated per unit solid angle to the power supplying the antenna. The gain of the superconducting antenna at 77 K is 20.3 dB which is 2 dB higher than that of the antenna with the same pattern in gold cooled at the same temperature. Without any loss, Ga would be equal to 22.2 dB. Then the superconducting antenna losses (1.9 dB) are attributed mainly to spurious Copyright © 1998 IOP Publishing Ltd

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Figure I6.1.6. A schematic diagram of a configuration consisting of an electrostatically coupled normal-conducting patch-array antenna with a superconducting feed network. A 0.5 mm thick vacuum layer provides thermal isolation for a 20 GHz antenna. Reproduced from Herd et al (1996) by permission of IEEE.

electromagnetic surface waves. With a 3 dB beam aperture of 13° the radiation pattern is in excellent agreement with the prediction. The cryostat associated with an array antenna needs a thermally shielding window which introduces a minimal degradation of the radiation pattern. To reach this objective, an architecture based on a proximity coupling between uncooled normal-conducting patches and a high-Tc superconducting feed array was designed (figure I6.1.6). The feasibility of this configuration was demonstrated for a 16-element array operating at 20 GHz with a maximum gain of 13 dB (Herd et al 1996). The feed network loss is proportional to the array geometric dimensions. Then the planar antenna size is limited to avoid an excessive dissipation of microwave power in the feeding normal-conducting microstrip lines. Therefore, the radiation pattern directivity given by the antenna dimensions is restricted by ohmic losses. The use of high-Tc superconductors involves a reduction of line loss per unit length by a factor close to eight at 35 GHz and 77 K, for example. This would allow an increase of the antenna size and directive gain. Millimetre-wave planar antenna sizes up to a few metres would be theoretically possible, but for large array antennas, the cooling environment would be critical. I6.1.3.3 Beam-forming networks The use of superconductors would lead to an improvement of analogue beam-forming system characteristics by a reduction of losses in planar geometry. The radiation pattern of transmitting or receiving antennas must be multibeamed to determine, without scanning, the positions of radar targets or transmitting sources. Moreover multiple beams allow simultaneous communication to information centres at different locations. A discontinuous beam scanning is also possible by switching. In these cases, the antenna far-field radiation pattern at a given frequency shows simultaneous multiple independent beams covering contiguous angular sectors. Such a pattern is obtained by an array matrix such as, for example, a Butler matrix which is made with directional couplers and fixed phase shifters. To generate four beams, a four-element linear antenna array requires a Butler matrix implemented by interconnecting four 3 dB directional couplers and two phase shifters of - π/4 (figure I6.1.7(a)). For eight beams generated by eight elements, the numbers of couplers and phase shifters are 12 and eight respectively. The couplers and phase shifters for a narrow bandwidth can be fabricated according to the microstrip configuration (figure I6.1.7(b)). With normal-

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Figure I6.1.7. (a) A schematic diagram of a half-wavelength-spaced four-element linear antenna array with a Butler matrix for simultaneouly forming four radiation beams. The phase shift through 3 dB couplers is 0° along the straight paths and −90° along the slanted paths. The entire subsystems can be printed on high-Tc superconducting films according to a planar configuration (L = left, R = right), (b) The pattern of this Butler matrix fabricated at 9 GHz using microstrip technology from an epitaxially grown YBCO film on a 20 × 30 × 0.5 mm3 LaAlO3 substrate. The insertion loss is less than 0.1 dB. Reproduced from Chaloupka et al (1993) by permission.

conducting materials, the microwave propagation loss in microstrip waveguides limits the possibilities of Butler matrices. With high-Tc superconductors, miniaturized matrices would be practical. Moreover, they may be connected to superconducting elementary antennas as frequency-selective meander antennas to form a compact front-end microwave subsystem for receivers (Chaloupka et al 1993). I6.1.3.4 Electrically small antennas Electrically small antennas are defined as antennas with dimensions much smaller than the wavelength of operation. To evaluate more easily the impact of superconductors on this domain, it is necessary to remind the reader of some properties of antennas. (a) Radiative efficiency The equivalent circuit of a receiving antenna is shown in figure I6.1.8. The terminating impedance is Rt + j Xt . R I and R r are the ohmic and radiating antenna resistances respectively. Xa is the total antenna reactance and V the radiation-induced voltage. The useful power delivered to the terminating resistance Rt is Pu

To obtain a maximum of power Pu , the antenna must be at least tuned (Xt = −Xa ) or matched if possible (Xt = −Xa and Rt = Rr + Rl ). For a matched antenna

where Ar e f f is the receiving effective aperture area of the antenna and S the incident wave power per unit area (Poynting vector). Copyright © 1998 IOP Publishing Ltd

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Figure I6.1.8. A schematic diagram of a receiving dipole antenna: (a) antenna loaded by the impedance Zt ; (b) equivalent circuit. Rt accounts for conductor and dielectric losses. Rr corresponds to the power scattered by a receiving antenna.

The power re-radiated or scattered by the antenna is Pr

The ohmic loss is Pl

The receiving efficiency ηr is equal to the ratio of the useful power over the incident power. Its general expression is

For a matched antenna ηr = 50% and the relative bandwidth is 2(Rr + Rl )/Xa . For a loss resistance R l negligible compared with the radiation resistance Rr , the useful power Pu is maximum

For a transmitting antenna, the internal resistance Rg of the voltage generator must be taken into account. For an antenna tuned by a circuit with a resistance Rm , the transmitted power is Pr

The radiating efficiency ηt is equal to the ratio of the radiated power over the power supplied to the tuned antenna

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The instantaneous relative bandwidth of the isolated tuned antenna is Bt

The ratio Rr /Xa is the relative radiation bandwidth Br (Br = ∆f /f0 ). This intrinsic feature depends on the antenna geometry only. Then

For a given antenna configuration, the product of the tuned antenna efficiency and the isolated tuned antenna relative bandwidth is constant. The effective relative bandwidth of the fed tuned-antenna is Bt e f f

For the matching condition (Rr = Rl + Rm + Rg )

(b) Directivity An antenna is also characterized by the directivity Da of its radiation pattern in the far-field zone. This dimensionless quantity is equal to the ratio of 4π times the maximum radiated power per unit solid angle to the overall radiated power (Kraus 1988). As a consequence of this definition, the directivity is equal to the ratio of 4π to the solid angle through which the total radiated power would stream with the maximum value of the power per unit solid angle. This solid angle corresponds approximately to the half-power beam aperture of the main radiation lobe. The number Da measures the enhancement in directivity over an isotropic radiator. The gain Ga and the directivity Da are related by the following equation

(c) Characteristics of small antennas A conventional centre-fed resonating thin linear dipole antenna has a total length slightly less than a half-wavelength. Its radiation impedance Zr is real and close to 70 Ω and the directivity is equal to 1.64, i.e. 2.15 dB (Kraus 1988). If the dipole length is exactly equal to a half-wavelength, the radiation impedance is Zr A dipole is electrically small or short when its length (L) is much smaller than the wavelength. Very close to a short dipole, the real part of the Poynting vector deduced from the field expressions is practically zero. Then inside a radian sphere with a radius equal to λ0/2π (λ0 is the wavelength in free space), the power density has an imaginary part (reactive power) larger than the real part (radiated power flux). Therefore the electromagnetic energy is stored in the proximity of a small dipole. Outside this sphere the real part of the Poynting vector is the more dominant as the observation point is remote from the dipole. A short dipole is contained inside the radian sphere. The radiation impedance of such a dipole shows a small resistance Rr (Rr is proportional to (L/λ0)2 ) and a large capacitive reactance Xa (Xa is proportional to (L/λ0 )−1 for a given ratio of the dipole length over its diameter). A large amount of energy is stored in the vicinity of the dipole and the relative radiation Copyright © 1998 IOP Publishing Ltd

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bandwidth Br (Rr /Xa ) proportional to (L/λ0)3 is very small. Moreover, the directivity of any kind of small antenna is close to 1.5 (1.76 dB). For instance, the radiation impedance of a thin dipole with an overall length of λ0/10 (L = 6 cm for 500 MHz) is equal to 2 – j1750 (Ω) (Br ≈ 1.1 × 10−3). To produce significant useful power from this antenna, it is necessary to tune it by an inductance of 0.55 µ H at a frequency of 500 MHz. Such an inductance fabricated with a planar configuration from normal-conducting materials at 300 K exhibits a resistance close to 4.5 Ω. The dipole resistance is about 0.1 Ω and the efficiency ηt is close to 30%. Then this efficiency is limited by the matching network resistance which can be dramatically reduced using superconducting films. At 500 MHz the surface resistance of a cooled normal-conducting material is about 6000 times larger than that of an epitaxial YBCO film at 77 K. Then the resistance of the tuning coil made of a high-Tc superconducting film would be 7.5 × 10−4 Ω. This value is negligible in comparison with the normal-conducting dipole resistance and radiation resistance. Under these conditions the efficiency ηt would be close to 95% and the relative bandwidth Bt of the tuned antenna would be practically equal to the radiation bandwidth Br. The concept of a radian sphere is valid for small loop antennas. In this case the radiation resistance varies as A2/λ40, A being the effective loop area. Then Rr is proportional to (L/λ0)4 (L is the diameter of a circular loop) and rapidly decreases to a very small value when the loop diameter is reduced. Moreover, the radiation reactance Xa is inductive and proportional to L/λ0 for a given ratio of the loop diameter over the wire diameter. It is tuned by a capacitor. The radiation bandwidth Rr /Xa proportional to (L/λ0 )3 is very narrow. To improve the radiating efficiency, the entire loop must be superconducting. Antenna structures where the radiating part is resonant do not require extra tuning networks. Thus H-shaped microstrip patch antennas have been particularly investigated and prototypes have been fabricated (Pischke et al 1991). These antennas operate at 2.4 GHz. The overall width and height of the H-shaped pattern is 6 mm, i.e. λ0/20.8 or λs /4.29, λ0 and λs being the wavelength in air and in the LaAIO3 substrate (∈r = 24) respectively. The width of waveguides forming the H-shape is 0.15 mm for the horizontal bar and 1.5 mm for the vertical bars. This antenna may be viewed as an end-loaded small microstrip dipole. Thus a single-resonant and electrically small structure has been realized by taking advantage of the high substrate permittivity (∈r = 23.5 at 77 K) and the quasi-lumped element geometry. The experimental results reported by Pischke et al (1991) are summarized in table I6.1.2. The dramatic improvement in efficiency due to the introduction of superconductors demonstrates the potentiality of superconductors in this domain. Obviously for a given configuration the instantaneous bandwidth is very narrow using superconductors. This feature involves an accurate matching and imposes some limitations on the use of these antennas.

Another configuration for an electrically small antenna consists of a resonant meander microstrip transmission line printed on a substrate with a high permittivity. At the operating frequency the length of a half-meander lm is equal to a half-wavelength (λg /2) in the microstrip guide. Then the total length of the line is equal to an integer number of half-wavelengths. When the line resonates, the currents in the different meander sections are in phase and lead to a constructive interfering radiation (figure I6.1.9).

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Figure I6.1.9. A schematic diagram of a meander antenna operating at 4.24 GHz (l0 = 70.75 mm). Reproduced from Chaloupka (1992) by permission of Plenum Publishing Corporation.

If the half-meander length is equal to λg /4, the interferences are destructive. Demonstration models of this kind of antenna have been fabricated (Chaloupka 1992). At an operating frequency of 4.24 GHz the wavelengths are 70.75 mm and 14.6 mm in free space and the substrate (LaAIO3) respectively. With a meander length of 8 mm (≈λ0/8.8) and a YBCO line at 77 K the relative 3 dB bandwidth is 4% and the measured radiation efficiency is higher than 50%. Moreover, the frequency response of such an antenna exhibits several secondary resonances for λg /2 equal to (2n + 1)lm and anti-resonances for λg/2 equal to 2nlm (n is an integer). The meander-antenna configuration has been utilized to implement a demonstration of a high-Tc superconducting two-frequency antenna, the number of elementary antennas constituting the arrays being eight at 9.5 GHz and four at 4.5 GHz (Chaloupka et al 1993) (figure I6.1.10). The directivity of electrically small antennas is always very low and many applications require directive transmitting or receiving systems. To overcome in part this drawback, the superdirective antennas are proposed. (d) Superdirective array antennas A superdirective antenna transmits a beam with a directivity substantially greater than that produced by a conventional array formed with a relatively small number of uniformly excited elements. The interelement spacing d of a superdirective array is much smaller than the usual distance of λ/2. Typically d is equal to 0.1λ . In these conditions the elements are strongly coupled. Moreover, the phase shift between currents supplied to the adjacent elements is approximately 180°. This excitation by currents in opposite phase leads to a cancellation of fields in the far zone except in a well-defined narrow beam. To transmit a given power in the far-field zone, the currents must be larger as the spacing d is decreased and tend to be infinite when the interelement spacing is approximately zero. Therefore the radiation resistance of each element decreases when the interelement spacing is reduced. Under these conditions, a great reactive power is stored in the immediate vicinity of the array and the quality factor of the array reaches a very high value. This feature is exacerbated when the elements are electrically small. As an example, a directivity improvement in the end-fire direction is obtained for an array of two parallel λ/2 long lossless dipoles when the interelement spacing is very small (see table I6.1.3). If the currents are in phase, the field maximum is in a direction normal to the array (broadside array) whereas it is in the axis of the array for currents of opposite phase (end-fire array). Using normal-conducting materials, the radiating efficiency of a superdirective antenna is very low Copyright © 1998 IOP Publishing Ltd

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Figure I6.1.10. (a) A schematic diagram of a receiving two-frequency and beam array-antenna, (b) Patterns of the two planar meander-antenna arrays realized from YBCO films on LaAIO3 wafers (5 cm in diameter). The radiating elements show a linear size of λ0/10. Reproduced from Chaloupka et al (1993) by permission.

because of large currents which induce great losses in feed networks and antennas. Under these conditions the utilization of the superdirectivity to improve the performance of small antennas is not practical. This situation changes by introducing high-Tc superconductors mainly in matching networks to reduce drastically the losses as demonstrated by the following preliminary results. A two-element array using normal-conducting electrically short monopoles and superconducting microstrip matching networks was fabricated and tested (Piatnicia et al 1993). The operating frequency is close to 800 MHz ( λ0 = 375 mm). The spacing between 4 cm long (0.1λ0 ) monopoles is equal to 8 cm (0.2λ0 ) and the phase difference between the supplied currents is equal to 140°. Only single-stub microstrip matching networks printed on gold or YBCO films deposited on LaAlO3 substrates were cooled at 77 K. In this demonstration model, a 2 dB loss Copyright © 1998 IOP Publishing Ltd

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Figure I6.1.11. Far-field radiation patterns of two 0.1λ0 long monopole elements 0.2λ0 apart array versus azimuth angle. The plots show the power level normalized to an isotropic radiator. Reproduced from Piatnicia et al (1993) by permission of IEEE, (a) Calculated patterns with lossless matching networks for uniform and superdirective excitations, (b) Pattern measured at zero elevation angle for a superdirective excitation and superconducting matching networks cooled at 77 K. The feed-network loss (≈14 dB) due to phase shifters, power splitter and feedlines is included.

reduction is obtained for a single radiator by using a YBCO microstrip stub. An improvement up to 5 dB is theoretically possible. The measured far-field pattern (figure I6.1.11) exhibits features characteristic of a superdirective antenna. Other superdirective antenna configurations based on dipoles or small helical radiators in high-Tc superconducting wire or loops with superconducting matching networks have been tested in the form of preliminary devices. Superdirective antennas involve narrow bandwidth, and therefore they are very sensitive to detuning due to the lack of accuracy and stability in geometrical and electrical quantities. Consequently the tolerance problem can become too severe. Nevertheless trade-offs can be found and superdirective antennas could be practical using high-Tc superconductors with probably a moderate improvement of the directivity. I6.1.3.5 Antennas for nuclear magnetic resonance instruments The very low resistance of high-Tc superconducting film strips at radiofrequencies is exploited to improve the detection of low-level signals in nuclear magnetic resonance (NMR) instruments. In these pieces of equipment the examined sample is excited by a pulsed magnetic field which is orthogonal to a static magnetic field. Under these conditions, the spin magnetic moments of the atom nuclei in the sample molecules precess around the static field with an angular frequency (Larmor frequency) equal to the product of the d.c. field strength and the gyromagnetic ratio which is a characteristic specific of the nucleus nature. However, this frequency depends slightly on the surroundings in the molecule. If the frequency of the exciting magnetic field is equal to the Larmor frequency of nuclei in the sample, a resonance phenomenon occurs. In this case, the spin magnetic moments are sustained in precession and their phases are brought Copyright © 1998 IOP Publishing Ltd

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to a common value by the effective rotating component of the oscillating magnetic field. When this field is turned off, the spin magnetic moments relax in free precession with two different time constants, i.e. the transverse or spin—spin relaxation time and the longitudinal or spin-lattice relaxation time. The spin magnetic moments precessing in phase create a macroscopic magnetic moment which is driven in the same motion. Hence a magnetic field sensor with a sensitive axis normal to the static field captures an alternating field radiated by the sample. A receiving antenna is set up near the sample to observe this pulsed oscillating magnetic field. The spectrum of the signal collected by the antenna gives the precessional frequencies of all excited nuclei. At radiofrequencies, the antennas consists of a coil which provides an alternating electromotive force induced by the precessing macroscopic magnetic moment. The receiving probe incorporates a coil, a matching network and a preamplifier. The noise at the probe output is mainly generated within the sample under investigation and the coil. When the NMR signal is weak, i.e. for very small or dilute samples or for a low frequency (weak static magnetic field), the noise from a conventional copper coil dominates over the sample noise and gives the lowest level of detectable signals (Withers et al 1994). However, this limit can be overcome by introducing a superconducting coil. In this case the coil resistance is much smaller than that of a conventional coil and consequently the coil loss and thermal noise are greatly lowered. Therefore a superconducting coil involves an improvement of performance in NMR spectroscopy or microscopy (very small sample volume) and in low-field (≤0.5 T) magnetic resonance imaging (MRI) of relatively small human body parts. (a) NMR spectroscopy NMR spectroscopy is a powerful technique used for molecular structure studies in the chemical or pharmaceutical industries and research laboratories. The introduction of superconductors in NMR spectrometer probes involves a gain in sensitivity and data collection speed. Coils have been fabricated for commercialized spectrometers by etching a high-Tc superconducting film in a single loop pattern with a diameter of a few centimetres. For an applied static magnetic field of 9.4 T, the magnetic resonance frequency of hydrogen nuclei (protons) is equal to 400 MHz, the proton resonance frequency being 42.577 MHz per tesla. At zero applied magnetic field, the unloaded quality factor Q of such a coil made of a YBCO film is close to 35 000 at 77 K, but under a field of 9.4 T, the Q factor is very degraded at the same temperature. Thus the operating temperature must be lowered to about 30 K to recover a Q value close to 28 000. These coils are tuned by capacitors formed by interdigital electrodes patterned at the same time that the loop and the useful bandwith are enlarged by the matching network. Then different probes are used to detect the radiation generated by other nuclei such as fluorine-19 (40.054 MHz T−1), phosphorus-31 (17.235 MHz T−1), or carbon-13 (10.705 MHz T−1). The gain in signal-to-noise ratio introduced by superconducting coils allows us to analyse much smaller quantities of rare products or significantly reduce the integration time required to reach a given signal-to-noise ratio. If this gain in signal-to-noise ratio is equal to n, the number of accumulated signal pulses must be divided by n2 to recover the signal-to-noise ratio obtained with a normal-metal coil. Superconducting probes provide a fourfold improvement in sensitivity over conventional probes. Then the data acquisition times can be divided by a factor of 16. (b) Low-field MRI The receiving antennas for NMR imaging can be fabricated with normal metals or superconductors. For magnetic fields above 0.5 T, the noise emitted from the whole human body or body parts such as limbs is greater than the noise from a conventional copper coil. Under these conditions a superconducting coil does not offer any advantage. However, interest in low-field MRI is growing because of the increasing need for cheaper medical systems, but in MRI devices operating with a magnetic field lower than 0.5 T and designed to examine parts of the human body, the noise from a conventional copper coil dominates over the noise from more- or less-conductive biological tissues. In this case superconductors can contribute to increase the capability of the equipment. The very low resistance of a superconducting coil involves a reduction of the coil noise and allows us to improve the quality of low-field images of the human body parts. These

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coils have been fabricated by etching high-Tc superconducting films deposited on substrates. In a hybrid configuration, the pattern consists of a planar spiral loop narrow line and wider circular electrodes on both sides of the spiral loop to form the tuning capacitors (figure I6.1.12). To implement a probe, two spiral lines on different substrates are superposed and isolated by a dielectric spacer to obtain a closed resonant LC circuit formed by a series loop of two coupled inductors and two capacitors. As an example, for a YBCO coil tuned at 4.25 MHz (0.1 T as the static magnetic field strength) the number of turns of each spiral loop is about ten. For a minimum linewidth of 400 µ m the outer diameter is close to 5 cm (Withers et al 1994). Only a weak degradation of the Q value is observed under this low field and the coil operates at a temperature of around 77 K with a Q factor of about 10000. In comparison, copper coils give a typical Q value of several hundreds. To obtain a suitable signal-to-noise ratio, the spacing between the coil and the biological tissue surface must be near 1 mm. For a cooled coil, the thickness of the thermal shield in front of the coil must allow us to carry out this dimensional requirement; this is the case for high-Tc superconducting coils operating at 77 K. Using these coils in low-field MRI devices, an amplitude signal-to-noise ratio improvement by a factor of five to seven has been observed. Monolithic coils with integrated tuning interdigital capacitors are fabricated by patterning double-sided films. Coils based on resonating spiral propagation lines do not need tuning capacitors.

Figure I6.1.12. The pattern of a high-Tc superconducting planar coil (5 cm in outer diameter) for a low-field MRI probe in the hybrid configuration. The black zones correspond to surfaces covered by the superconducting film. Reproduced from Withers et al (1994) by permission of IEEE.

(c) NMR microscopy An NMR microscope is basically a miniature MRI device. A very small sample (about 1 cm in size) of biological tissue is examined under a high magnetic field (typically 7 T). With a YBCO single turn coil cooled at 10 K, the Q value in operation is higher than 50000. By replacing a conventional metal coil with such a superconducting coil, an improvement of about ten in the amplitude signal-to-noise ratio has been obtained. With metal coils, the data acquisition times of images exhibiting the required resolution are impractically long. So the utilization of superconducting coils allows us to reduce the imaging time by a factor of 100 and NMR microscopy could become a practical technique for routine use. I6.1.4 Conclusion The impact of superconductors on electromagnetic-wave thermal detection systems and antennas has been investigated. To conclude with, we review the main contributions of superconductors to the improvement of performance.

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The practicability of superconducting thermal detectors is greatly enhanced by the introduction of high-Tc superconductors such as epitaxial YBCO. The technological process followed to produce these detectors (bolometers) is compatible with the fabrication of large arrays. Silicon can be used as the basic substrate material and the integration of signal read-out and amplification circuits in a monolithic device is feasible. Moreover, the cryogenic environment is similar to that currently utilized for usual semiconductor IR detectors. At present, tested superconducting transition-edge bolometers operating at a temperature of around 80 K exhibit a specific detectivity (D* ) which does not reach the background-limited value. However advances seem possible and the detectivity of these bolometers should become competitive with that of 77 K cooled semiconductor detectors for the 8-12 µ m wavelength range. Above about 15 µ m high-Tc bolometers exceed the performance of any existing detectors (thermal or quantum) operating at the same temperature range or higher. Long-wavelength IR detectors (10–300 µ m) are of great interest for various observations from space platforms. For the same type of application, hot-electron microbolometer mixers based on low-Tc superconductors are excellent candidates as low-noise mixers for terahertz heterodyne receivers. The improvements involved by the introduction of high-Tc superconductors in antennas concern mainly the production of electrically small antennas and receiving antennas for NMR equipment. Small antennas become very efficient when the tuning inductance is superconducting. They have the potential to operate with a poor directivity over a large frequency coverage, but the instantaneous bandwith is restricted and methods to change the operating frequency by electronic tuning need to be developed. In the case of coils used as receiving antennas in NMR instruments, the contribution of superconductors is particularly significant for NMR spectroscopy and low-field MRI as a consequence of the drastic noise reduction due to the very low resistance of high-Tc superconducting coils. The industrial development of these coils is under way.

References Berkowitz S J, Hirahara A S, Char K and Grossman E N 1996 Low noise high-temperature superconducting bolometers for infrared imaging J. Appl. Phys. Lett. 69 2125–7 Chaloupka H 1992 High-temperature superconductor antennas: utilization of low rf losses and of nonlinear effects J. Superconductivity 5 403–16 Chaloupka H, Hein M, Müller G and Piel H 1993 Research and development of analog HTS microwave components for possible space applications Proc. ESA/ESTEC Workshop on Space Applications of High Temperature Superconductors ESA WPP-0052 pp 9–25 Clarke J, Hoffer G I, Richards P L and Yeh N H 1977 Superconductive bolometers for submillimeter wavelengths J. Appl. Phys. 48 4865–79 Herd J S, Poles L D, Kenney J P, Derov J S, Champion M H, Silva J H, Davidovitz M, Herd K G, Bocchi W J, Mittleman S D and Hayes D T 1996 Twenty-GHz broadband microstrip array with electromagnetically coupled high-Tc superconducting feed network IEEE Trans. Microwave Theory Tech. MTT-44 1384–9 Johnson B R, Foote M G, Marsh H A and Hunt B D 1994 Epitaxial YBa2Cu3O7 superconducting infrared microbolometers on silicon SPIE Proc. 2267 24–30 Kraus J D 1988 Antennas (New York: McGraw-Hill) Lewis L L, Koepf G, Bhasin K B and Richard M A 1993 Performance of TICaBaCuO 30 GHz 64 element antenna array IEEE Trans. Appl. Supercond. AS-3 2844–7 Nahum M and Martinis J M 1993 Ultrasensitive-hot-electron microbolometers Appl. Phys. Lett. 63 3075–7 Nahum M, Qing Hu, Richards P L, Sachtjen S A, Newman N and Cole B F 1991 Fabrication and measurements of high-Tc superconducting microbolometers IEEE Trans. Magn. MAG-27 3081–4 Nahum M, Richards P L and Mears C A 1993 Design analysis of a novel hot-electron microbolometer IEEE Trans. Appl. Supercond. AS-3 2124–7

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Neff H, Laukemper J, Khrebtor I A, Tkachenko A D, Steinbeiss E, Michalke W, Burnus M, Heidenblut T, Hefle G and Schwierzi B 1995 Sensitive high-Tc transition edge bolometer on a micromachined silicon membrane Appl. Phys. Lett. 66 2421–3 Osterman D P, Patt R and Madhavrao R 1993 Superconducting niobium infrared thermal detectors and circuits on back-etched substrates IEEE Trans. Appl. Supercond. AS-3 2860–3 Piatnicia A Y, Talisa S H, Gavaler J R, Janocko M A, Talvacchio J, Buckley M J, Leader K M, Moellers J A and Schrote M R 1993 High-temperature superconducting matching networks for electrically short monopole antennas IEEE MTT-S Dig. CH3277–1/93 1417–20 Pischke A, Chaloupka H, Piel H and Splitt G 1991 Electrically small planar HTS antennas 3rd Int. Superconductive Electronics Conf. (ISEC), Extended Abstracts (Glasgow, 1991) (Glasgow: Meeting Makers) pp 340–3 Rice J P, Grossman E N, Borcherdt L J and Rudman D A 1994 High-Tc superconducting antenna-coupled microbolometer on silicon SPIE Proc. 2159 98–109 Richards P L 1994 Bolometers for infrared and millimeter waves J. Appl. Phys. 76 1–24 Sauvageau J E, McDonald D G and Grossman E N 1991 Superconducting kinetic inductance radiometer IEEE Trans. Magn. MAG-27 2757–60 Semenov A D, Gousev Yu P, Nebosis R S, Renk K F, Yagoubov P, Voronov B M, Gol’tsman G N, Syomash V D and Gershenzon E M 1996 Heterodyne detection of THz radiation with a superconducting hot-electron bolometer mixer Appl. Phys. Lett. 69 260–2 Van Duzer T and Turner C W 1981 Principles of Superconductive Devices and Circuits (London: Arnold) Verghese S, Richards P L, Char K, Fork D K and Geballe T H 1992 Feasibility of infrared imaging arrays using high-Tc superconducting bolometers J. Appl. Phys. 71 2491–8 Withers R S, Liang G C, Cole B F and Johansson M 1993 Thin-film HTS probe coils for magnetic-resonance imaging IEEE Trans. Appl. Supercond. AS-3 2450–3 Withers R S, Cole B F, Johansson M E, Liang G C and Zaharchuk G 1994 HTS receiver coils for magnetic-resonance instruments SPIE Proc. 2156 27–35

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I6.2 Superconducting heterodyne receivers Takashi Noguchi and Sheng-Cai Shi I6.2.1 Introduction After the discovery of spectral emission of carbon monoxide (CO) at 115.3 GHz ( 7 = 1– 0) ( J is the rotational state of a molecule and J = 1–0 represents a transition from the lowest excited state ( J = 1) to the ground state ( J = 0)) in 1970 (Wilson et al 1970) millimetre-wave radio astronomy experienced a rapid growth to become one of the most important branches of observational astronomy. The emission spectra of CO have been used extensively in the past to obtain valuable information on the physics and kinematics of interstellar clouds. Higher-J CO transitions, such as the ones at 230 GHz ( J = 2 - 1) and 345 GHz ( J = 3 – 2), also contain spectral and spatial information on the cosmic background, on very distant newly formed galaxies and on the early stages of star formation within molecular clouds in our galaxy. In addition to CO, many molecules have been found by spectroscopy of the interstellar medium using heterodyne receivers. From studies of the interstellar molecular transitions, our understanding of the nature of the interstellar medium has increased enormously. Since the signals received from those molecules are generally extremely weak, the quest for higher sensitivity has led to larger radio telescopes with a good surface accuracy (for reviews on radio telescopes see Baars et al 1994, Ukita and Tsuboi 1994) and the development of low-noise heterodyne receivers to fully exploit the telescopes’ potentiality (for reviews on receiver development see Blundell and Tong 1992, Carlstrom and Zmuidzinas 1996). The key element in heterodyne receivers is the mixer, in which the observed signal (ωs ) is mixed with a local oscillator (ωLO ) to produce a much lower intermediate frequency (ωI F ). Although there are various types of heterodyne mixer receiver, all fulfil the Dicke radiometer equation for their sensitivities (Kraus 1986)

where ∆Tm i n is the minimum detectable temperature, TR X is the receiver noise temperature and ∆v and ∆t represent the pre-detection bandwidth and post-detection integration time respectively. For most heterodyne receiver systems, response is obtained from both sidebands (ωS = ωL O ± ωI F ). Therefore, care must be taken in obtaining the single sideband (SSB) receiver noise temperature from the measured double sideband (DSB). When ωI F « ωS , the receiver response is fairly flat with frequency so that TR X (SSB) 2TR X (DSB). However, in general this is not true and the SSB noise temperature of the receiver system is characterized by where TM , GM and TI F are the mixer noise temperature (SSB) mixer conversion gain (SSB) and noise temperature of the intermediate-frequency (IF) chain respectively. Therefore, a mixer with a low noise temperature and a high conversion efficiency (or low conversion loss) can reach very high sensitivities.

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In principle any type of nonlinear device is suitable for heterodyne mixing. For a long time, Schottky barrier diodes were the most widely used for millimetre- and sub-millimetre-wave mixers, but the recent invention of the superconducting tunnel junction, which is generally called an ‘SIS junction’ for short, has produced major progress in the development of low-noise mixers. Due to their excellent performance as high sensitivity nonlinear elements in heterodyne mixers, SIS mixers have replaced Schottky-barrier-diode mixers in most of the major radio observatories (Blundell and Tong 1992, Carlstrom and Zmuidzinas 1996). I6.2.2 Theory of mixing for SIS mixers SIS mixers make use of the sharp nonlinearity in the quasiparticle tunnelling current that occurs between two superconductors which are separated by a thin barrier formed by an insulating oxide layer. This sharp nonlinearity arises from the ‘energy gap’ in the density of states of a superconductor (Tinkham 1975). Figure I6.2.1 (a) shows a diagram of the density of states as a function of energy for an SIS junction. The tunnelling current is small until the voltage difference across the barrier is so large that filled states below the gap in one of the superconductors are at the same level as the high-density region of empty states above the gap in the other superconductor. There is a discontinuous increase in the current when the voltage reaches the gap voltage VG = 2∆/e , and the current increases nearly linearly with voltage above VG . Here 2∆ is the gap energy. The nonlinearity at the gap voltage is broadened by several effects, such as finite quasiparticle lifetime or various kinds of inhomogeneity which cause a spread in the gap voltage across the area of the junction. When the microwave power is applied to the junction, photon-assisted tunnelling steps are induced in the current-voltage ( I—V ) curve (Dayem and Martin 1962, Tien and Gordon 1963). The absorption of microwave photons by the quasiparticles increases their energy so that tunnelling current occurs at voltages below the gap voltage. The photon-assisted structure in the I—V curve is shown in figure I6.2.1(b). Since the current steps below and above the gap voltage are produced by the absorption and emission of microwave photon(s), respectively, the SIS junctions behave

Figure I6.2.1. Tunnelling current in an SIS junction: (a) energy-band model and (b) d.c. I—V characteristics in the absence of microwave power (broken line) and with microwave power applied (full line). Copyright © 1998 IOP Publishing Ltd

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like photoconductors when they are biased to voltages of less than a photon voltage (h w/e) below the gap, where h is Planck’s constant. The nonlinearity of the SIS junction is so sharp that a quantum-mechanical treatment of the interaction with microwave radiation is required (Werthamer 1966). The quantum-mechanical description of a single junction SIS mixer has been developed by Tucker (1979). Unfortunately, the space allotted here is not enough to include a detailed description of Tucker’s theory, but the reader is referred to the excellent reviews of the quantum theory of mixing for SIS junctions (Phillips and Woody 1982, Tucker and Feldman 1985). Since the normal resistance, RN , of an SIS junction is generally determined by the impedance-matching condition, the junction area must be very small to reduce the capacitance, CJ , especially at high frequencies, while keeping the ωR NC J product constant. Another possible solution to achieve the same ωR NCJ product is to use larger-area junctions in a series array. The latter can be made with less sophisticated equipment and techniques, which are, in practice, crucial factors for most laboratories. For this reason, series arrays of SIS junctions have been employed in many SIS mixers reported. It should be noted, however, that the series-array junctions will have a higher saturation power level per signal but require larger local oscillator (LO) power. Feldman and Rudner (1983) showed theoretically that the series-array SIS mixer behaves like a single-junction SIS mixer, assuming in their analysis that an electrical length of array is short enough compared to the effective wavelength and that all of the junctions of the array are identical. These assumptions are approximately true in most millimetre-wave SIS mixer experiments, since experimental results agree with the theoretical predictions for the array mixers within the errors. In such array mixers, all relevant voltages and impedances are multiplied by the number of junctions N, whereas ωR NCJ is independent of N. Thus Tucker’s mixing theory can be extended to the series-array SIS mixers with no difficulty. Recently, another type of multijunction SIS mixer, the parallel array of SIS junctions, has been proposed (Belitsky et al 1995, Noguchi et al 1995a, b, Zmuidzinas et al 1994). Each junction in the array is separated by a microstrip inductance, or the microstripline is periodically loaded with discrete SIS junctions. Analysis of the parallel array of two SIS junctions, which is the simplest case of the parallel array, was carried out by Zmuidzinas et al (1994) and Noguchi and coworkers (Noguchi et al 1995b, Shi 1996). To analyse the mixing and noise properties of the parallel array, the Tucker quantum theory of mixing for a single SIS junction must be extended. The equivalent circuit of the two-junction system to be analysed is shown in figure I6.2.2, in which the tuning inductance between the two junctions is represented by a chain matrix C. The chain matrix is given by

Figure I6.2.2. Equivalent circuit of the parallel array of two junctions.

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where L is the tuning inductance. Let us assume that the LO voltage developed across junction 2, VL O. 2 , is already known. The LO voltage across junction 1, VL O. 1 , can then be expressed as

where Ci j (i, j = 1,2) are elements of the chain matrix. When an SIS junction is pumped by an LO voltage with nonzero phase of the form

the phase factor W(ω ) (Werthamer 1996, Tucker 1985) becomes

where Jn(x ) is the nth-order Bessel function with argument x and α = eVL O /hωL O Thus the small-signal conversion admittance and noise correlation matrices for the parallel array of two junctions can be obtained as

where Ym n and Hm n represent, respectively, the admittance and noise correlation matrices of the SIS junction pumped by a zero-phase LO voltage. Now we can transform the two-junction circuit into an equivalent single-junction circuit using linear circuit theory and consequently we can calculate the conversion gain and noise temperature of the parallel-array mixer based on the standard Tucker theory. The calculated conversion gain and noise temperature for the parallel array of two junctions are plotted as a function of frequency in figure I6.2.3(a). This figure also shows, for comparison, the same for a single junction with a tuning inductance terminated with an ideal frequency-independent radiofrequency (RF) short. The tuning inductance used in the calculation was determined as L = 2/ω 2 CJ for the two-junction array and L = 1/ω 2 CJ for the single junction, assuming that ωR NCJ = 4 at 1 05 GHz and that RN = 10 Ω. In both calculations we assume that the RF source impedance is independent of frequency and equal to R N /2 for the two-junction array and RN for the single junction. IF load impedance is also assumed to be independent of frequency and equal to 50 Ω. A constant LO voltage of α1 = eV1/hω = 1.3, which optimizes the performance of a single-junction mixer, was assumed to be applied to the SIS junction at each frequency in the single-junction case. The same LO voltage was assumed to be applied to the SIS junction 1 in figure I6.2.2 at each frequency in the case of the two-junction array. The LO voltage developed across the SIS junction 2 was calculated from equation (I6.2.4). Bias voltage was optimized for the mixer performance at each frequency for both cases. The minimum noise temperature in the parallel array is as low as that of the single junction. The noise temperature of the single junction has a rather flat frequency response in the frequency range 80–120 GHz, because there is no limitation on the bandwidth except the ωRNCJ of the junction. On the other hand, the noise temperature in the parallel array is strongly dependent on frequency and increases rapidly below 90 GHz and above 120 GHz. The conversion gain of the parallel array shows a fairly flat frequency dependence in the frequency range 80–120 GHz, while the single junction has a large gain at its resonance frequency. This large gain is usually associated with a negative input resistance of the device. In actual mixers, such a negative

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Figure I6.2.3. Predicted mixing behaviours at the 100 GHz band with a single junction and the parallel array of two junctions: (a) mixer noise temperature (SSB) and conversion gain (SSB) and (b) input and output power coupling coefficients. Full curves represent the parallel array of two junctions and broken curves represent the single junction.

resistance prevents efficient signal power coupling between the junction and the source. In figure I6.2.3(b) we show the calculated input and output coupling coefficients. As can be clearly seen, both the input and output power coupling coefficients for the single junction are negative near the frequency where a large gain is observed. This means that the voltage standing-wave ratios (VSWRs) at the input and output of the device would be so large that a good performance mixer could not be realized under such conditions. In contrast, the parallel array has positive input and output power coupling coefficients over the whole frequency range shown in figure I6.2.3(b). For the parallel array, negative power coupling coefficients or negative resistance at the input and output have never been observed for a wide range of LO voltages. Similar behaviours of the gain, noise and power coupling efficiencies at the input and output have been found up to the gap frequency. Thus, the parallel array has great performance capability as an SIS mixer at millimetre and sub-millimetre wavelengths. The theory for the parallel array of two SIS junctions can be easily extended to that of N junctions or distributed junction arrays in which sets of two junctions are separated by a tuning inductance. In some sense it appears like a lossy transmission line, since a single junction can be approximately regarded as a parallel combination of a resistance and capacitance. It might be also generalized to a very narrow and long SIS junction (Tong et al 1995) which is regarded as a lossy microstripline. In distributed junction arrays, the microwave radiation is coupled in from one end and is gradually absorbed as it propagates along the transmission line due to the photon-assisted tunnelling currents. The bandwidth of such a distributed junction array is not limited by the ωR NC J product of the single junction, which implies that high-quality junctions with a low current density can be used. Hence distributed SIS-junction arrays should be of good use for sub-millimetre-wave SIS mixers. I6.2.3 SIS junctions and integrated tuning circuitry High-sensitivity waveguide SIS mixers have been achieved using adjustable mechanical tuners (Kerr and Pan 1990, Kooi et al 1994, Ogawa et al 1990). However, mechanically tuned mixers are inconvenient for operation and their performance degrades after a period of operation due to the deterioration of the

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mechanical tuners. These problems would become much more serious if mechanical tuners were included in sub-millimetre-wave receivers or complex receiver systems such as focal-plane array receivers, since making high-precision mechanical tuners to operate in the sub-millimetre range is generally difficult and the large number of SIS mixers in focal-plane array receivers would need to be tuned individually. Therefore, for these purposes it is highly desirable to develop SIS mixers without adjustable mechanical tuners. This type of SIS mixer is generally called a ‘tunerless’ or ‘fixed-tuned’ mixer. In order to construct tunerless or fixed-tuned SIS mixers, the effects of design constraints on the performance of SIS mixers must be fully understood. The performance of an SIS mixer depends strongly on the characteristics of the junction itself, which is the primary constraint in the mixer design. A useful parameter in characterizing SIS junctions is the ωR NC J product, which gives a measure of the junction’s tuning bandwidth (∆ω /ω = 1/ωR NC J ). Another design constraint is the embedding impedance of the mixer mount seen by the SIS junction at different frequency ports such as the signal, LO and IF ports. The performance, optimum operating conditions and frequency dependences of single-junction SIS mixers have been analysed systematically by Shi and coworkers (Shi 1996, Shi et al 1993) for various values of the ωR NC J product and different embedding impedances at the signal frequency and IF using the quasi-five-port approximation (Kerr et al 1993). Their most important conclusions are summarized here. ( i ) The optimum value of RS /RN is approximately 0.5 at 100 GHz and is roughly proportional to f 0.5, while the optimum value of RI F /RN is in the range 1–10 independent of frequency. ( ii ) The normalized terminating susceptance B/GN at the signal frequency should be in the range of –0.5 to –1, being independent of the value of RS /RN and frequency. ( iii )The optimum ωR NC J product is around two at frequencies below 300 GHz, but the performance of SIS mixers becomes insensitive to the ωR NC J product above 300 GHz. ( iv )It seems difficult to achieve good input and output matching of SIS mixers simultaneously at frequencies above 200 GHz for the single-junction SIS mixers. These general conclusions provide useful guidelines for the designing of SIS mixers. The tuning circuits presented up to now can be classified into three types. Figures I6.2.4(a) and (b) represent two types of tuning circuit which are traditionally adopted. For the first type, the tuning inductance, which is terminated with an RF short-circuit element such as a quarter-wavelength open stub or a radial stub, is placed in parallel with the SIS junction (Kerr et al 1988, Räisänen 1985b). It has been pointed out that the RF short-circuit element would reduce the tuning bandwidth somewhat if the characteristic impedance of the stub were limited and that fabricating this element with a required performance is generally difficult. For the second type of circuit, the tuning inductance is connected in series with the SIS junction. In this case, since the resulting input impedance is lower than the junction resistance by a factor of 1/[1+(ωR NC J )2], a stepped quarter-wavelength impedance transformer is required to increase the impedance to match the signal source (Büttgenbach et al 1988). Such a small impedance may induce some effects, e.g. a large impedance ratio and the step discontinuity, which generally degrade the matching bandwidth. The third type of circuit is the parallel array of two SIS junctions described above and schematically illustrated in figure I6.2.2. Since neither the RF short-circuit element nor the quarter-wavelength transformer are necessary, this method is free from the parasitic degradation of bandwidth. Additionally it gives a moderate input impedance which may match the source. Thus the parallel array of two junctions turns out to be the most suitable junction-tuning circuit, as far as the tuning bandwidth, input impedance at resonance frequency and ease of fabrication are concerned.

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Figure I6.2.4. Equivalent circuit models of integrated tuning circuits for the SIS junction: (a) parallel inductance and (b) series inductance.

I6.2.4 Designing SIS mixers I6.2.4.1 Junction fabrication Nb/AlOx /Nb tunnel junctions are the most widely used mixing elements in recent SIS mixers, since they have an extremely sharp nonlinearity at the gap voltage even at 4.2 K and a very low sub-gap leakage current and they are physically and chemically stable. The Nb junctions are usually fabricated on a fused-or crystalline-quartz substrate in which the propagation loss of radiation at millimetre and sub-millimetre wavelengths is negligibly small. At first, a base Nb layer is deposited onto the whole substrate, followed by the deposition of an Al layer with a nominal thickness of several tens of ängströms. Subsequently, the Al surface is oxidized in an atmosphere of oxygen introduced into a deposition chamber. Depending on the critical current density of the fabricated junctions, the thickness of the Al oxide layer is controlled by the pressure of the oxygen and oxidation time. After the completion of oxidation, a top Nb layer is deposited onto the oxidized Al. Note here that the Nb/AlOx /Nb tri-layer is completed without breaking the vacuum. Next, the top Nb is removed by reactive ion etching (RIE), except for the junction dots covered with a photoresist. A thick insulator such as SiO2 is deposited successively through the photoresist used in the previous etching process. In some cases, before the deposition of the SiO2 layer an oxide layer is formed by anodization (Kroger et al 1981) in order to reduce the junction sub-gap leakage current and the possibility of shorting junctions. After cleaning the surface of the Nb top layer with an Ar plasma, a thick Nb wiring layer is deposited and is then patterned into the tuner and transformer striplines by RIE. The high-frequency performance of an SIS junction depends upon its normal-state resistance and capacitance, or the ωR NC J product. The ωR NC J product is proportional to the ratio ω/Jc , where Jc is the critical current density of a junction. Since the value of the normal-state resistance is determined by the requirement to match the signal source, the capacitance must be smaller as frequency increases, keeping the ωR NC J product constant. Small-area junctions provide small capacitance but require a high critical current density to achieve the necessary value of normal-state resistance. The ωR NC J product of SIS junctions does not necessarily give an upper limit to the operating frequency for the junction, unlike the case of Schottky diodes, because the capacitance of the junction can be easily resonated out by means of mechanical tuners or integrated tuning structures. A large ωR NC J product requires a large-area junction, which is easy to fabricate with high quality, but results in a limited relative tuning bandwidth. On the other hand, a small ωR NC J product generally makes RF matching and tuning for SIS mixers easy but requires junctions with a small area and high critical current density, which are usually associated with difficult fabrication and degradations of the nonlinearity of the junction. Consequently, it is necessary to look for a trade-off between the junction critical current density and the tuning bandwidth in determining the ωR NC J product for a given frequency. Copyright © 1998 IOP Publishing Ltd

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Once the value of ωR NC J and the normal-state resistance of the junction are given for a frequency of operation, the required junction area can be determined. Here the smallest size of the junction to be used in the SIS mixer will be limited by the fabrication technique. With electron beam lithography, extremely small junctions with an area of ∼0.1 × 0.1 µ m2 can be produced (Bao et al 1995, Dolan 1977), but ultraviolet (UV) photolithography, which has been the most widely adopted technique in the fabrication of junctions to be used in the SIS mixers, can make junctions with a size of ≥1 µ m2 (Dierichs et al 1993, Lichtenberger et al 1991). The performance of Nb/AlOx/Nb SIS mixers deteriorates above the Nb gap frequency of ∼700 GHz and very rapidly deteriorates above twice the gap frequency, due to the onset of absorption of signal photons in the superconducting electrodes (Ke and Feldman 1993). Since the absorption of photons occurs mainly in the superconducting striplines as the impedance transformers or the tuning circuits, striplines made of a normal metal such as Al or Au are sometimes adopted in SIS mixers above the Nb gap frequency, while the Nb/AlOx /Nb junctions are still employed as the mixing element (Bin et al 1996b). In this way, reasonable mixer performance has been obtained (Bin et al 1996a, van de Stadt et al 1996). Another possible way to build SIS mixers above 700 GHz is to use NbN-based tunnel junctions with a gap frequency of ∼1.4 THz. Although the NbN-based junctions with a barrier of MgO attracted much attention at first (LeDuc et al 1991, Shoji et al 1992) and a lot of work has been done to make high-quality NbN-based junctions, it is still difficult to obtain junctions with a high critical current density and sharp nonlinearity. Recently, an NbN/AlN/NbN tunnel junction with a gap voltage 5 mV, a critical current density ≤ 50 kA cm−2 and a sharp nonlinearity has been successfully developed (Wang et al 1996). A quasi-optical SIS mixer using such high-quality NbN junctions has been tested and satisfactory performance has been demonstrated at around 300 GHz (Uzawa 1996). Although NbN-based junctions are expected to give excellent performance as SIS mixers at around 1 THz, few experiments on NbN-based SIS mixers at such high frequencies have yet been reported. Some research is still needed to realize 1 THz SIS mixers employing NbN-based junctions as well as to develop high-critical-current-density NbN-based junctions with very sharp nonlinearity. I6.2.4.2 Mixer mount SIS mixers at millimetre and sub-millimetre wavelengths are generally classified into two categories according to their mounting structure, namely ‘waveguide’ and ‘quasi-optical’ mixers (Blundell and Tong 1992). The difference between these two types of mixer is the signal coupling scheme. While the waveguide mixers employ a combination of feedhorn and waveguide, the quasi-optical mixers utilize a combination of a substrate lens and a thin-film antenna. In general, waveguide mixers have superior performance to quasi-optical ones, because waveguide feedhorns provide a better signal coupling and have an ideal characteristic such as beam-pattern symmetry, sidelobe level and cross-polarization level. Hence, waveguide-mixer receivers are generally better fitted to a radio telescope. It should, however, be noted here that some of the quasi-optical SIS mixers have shown very good performance in real receiver systems at sub-millimetre wavelengths (Gaidis 1996, Zmuidzinas and LeDuc 1992, Zmuidzinas et al 1994). It is of particular importance to know the embedding impedance seen from the junction in SIS mixers. The embedding impedance seen by the junction depends in a complicated way on the waveguide size, RF-filter geometry in the substrate channel, junction lead configuration inside the waveguide and the positions of the tuners. A scaled model measurement is a useful technique in determining the embedding impedance of the mounting circuit at millimetre and sub-millimetre wavelengths (Räisänen et al 1985a). Another method to obtain the embedding impedance of a mixer mount is to use the finite-element method to simulate the mount structure. This method is, however, very time consuming. Once the embedding impedances of the mixer mount at the fundamental as well as at harmonic frequencies are obtained, they can be implemented in the computer program for the optimization and characterization of SIS mixers.

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Figure I6.2.5. A schematic representation of a typical waveguide mixer mount: (a) a cross-section of the mixer mount viewed from the top and (b) a cross-section of the IF channel in which the junction substrate is mounted.

A conventional waveguide mixer mount, in which the SIS junction is located around the centre of the waveguide, is schematically illustrated in figure I6.2.5(a). In traditional SIS mixers, one or two waveguide tuners, a so-called E-plane (or stub) tuner and a back-short tuner, are commonly included to adjust the embedding impedance. Note, however, that only a back-short cavity of a fixed length is adopted in some of the recently developed SIS mixers with an integrated tuning circuit on the junction substrate. The junction substrate is inserted in a long channel milled across the centre of the waveguide. A cross-section of the channel is illustrated in figure I6.2.5(b). The aperture size of the channel is determined so as to have a cut-off frequency higher than the signal frequency, including the effect of the junction substrate. The RF-choke filter is commonly made from (suspended) striplines with alternating low- and high-impedance sections on the junction substrate. While one end of the substrate is grounded, the other end is connected to a bias network. An IF impedance transformer is sometimes adopted, when the output impedance of the junction is much higher than the input impedance of the first IF amplifier. I6.2.5 Experimental results A block diagram of the measurement set-up is illustrated in figure I6.2.6. Before entering the waveguide of the mixer, the signal and LO are combined through a coupler with a coupling efficiency of ∼ −20 dB, which is connected to the feedhorn facing the vacuum window of the dewar and is placed on the 4 K stage of the cryostat or beam splitter with a small reflection (∼1%) which is located in front of the window. Such a simple coupling method is possible because of small LO power requirements (∼10−8 W at 100 GHz). The IF signal at the mixer output port is first amplified by a cooled amplifier, usually attached to the 4 K stage, with a noise temperature ≤ 10 K over ∼1 GHz. Further amplification of the IF signal is accomplished by room-temperature amplifiers. The d.c. bias is supplied through the bias network. The receiver noise temperature referred to its input port can be measured by the Y-factor method using hot (300 K) and cold (80 K) loads placed in front of the window of the cryostat. The receiver noise temperature (DSB), TR X , is given by

where Th o t and Tc o l d are hot- and cold-load temperatures, respectively, and Y is defined as the ratio of the IF responses corresponding to hot- and cold-load inputs. Figure I6.2.7 shows the pumped and unpumped d.c. I—V curves of a parallel array of two Nb/AlOx /Nb junctions as well as the IF responses to hot and cold loads. In the pumped d.c. I—V curve several photonCopyright © 1998 IOP Publishing Ltd

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Figure I6.2.6. A block diagram of the measurement set-up for the SIS mixer.

assisted tunnelling steps are clearly seen. The maximum conversion of the signal occurs usually at the first photon peak below the gap voltage and the maximum Y factor is also obtained at the peak. Note that they are sometimes found at the second or third photon peaks at frequencies lower than 100 GHz. The receiver noise temperature (DSB) of this SIS mixer is approximately 20 K, which is only four times larger than the quantum noise limit of hω/kB , where kB is Boltzmann’s constant.

Figure I6.2.7. D.c. I—V curves pumped and unpumped with LO and IF response curves for the hot- and cold-load input.

The large onset of noise due to the a.c. Josephson current, which is clearly observed in the IF response curves below ∼3 mV in figure I6.2.7, will become a serious problem when the SIS junctions are operating Copyright © 1998 IOP Publishing Ltd

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at higher frequencies (Feldman and Rudner 1983). Usually the optimum bias point of the SIS junction is obtained near the middle of the first step below the gap. The optimum point moves towards the origin, as the operating frequency increases and will encounter the noisy region above ∼300 GHz. The Josephson current and noise can be easily suppressed by applying a magnetic field. Thus SIS mixers operating at frequencies higher than ∼300 GHz should be combined with a permanent magnet or an electromagnet. The performance of recent SIS receivers with Nb/AlOx /Nb junctions, operating at frequencies from 100 GHz to 1 THz, are shown in figure I6.2.8. The receiver noise temperatures (DSB) fall in the range of 3–5(hω/kB ) over 100–600 GHz, but increase rapidly with frequency above 600 GHz due to the onset of the radiation loss in the superconductor or normal metal and reach 10hω/kB at around 1 THz.

Figure I6.2.8. DSB noise temperatures of SIS receivers in the frequency range 0–1100 GHz. Full, dotted and broken curves represent noise temperatures 10, 5 and 3 times the quantum noise limit (hω/kB ), respectively.

I6.2.6 High-Tc superconducting mixers The discovery of high-Tc superconductors produced many expectations about the possibility of developing low-noise superconducting receivers, using them at far-infrared wavelengths as well as sub-millimetre wavelengths, since they would have very large gap voltages. Up to now, great efforts have been made to build high-performance mixers/receivers at centimetre and millimetre wavelengths using high-Tc superconducting materials. However, only a few reports have been published on their noise temperature measurements (Grossman and Vale 1994, Shimakage et al 1997). At present it is already widely known that it is quite difficult to make low-noise mixers using high-Tc superconducting materials. One of the reasons is that it is very difficult to fabricate well-controlled and high-quality high-Tc Josephson or SIS junctions, which are used as mixer elements, mainly due to the extremely short coherence lengths of the order of a few ångströms. It has been recently demonstrated that an SIS junction with fairly good tunnelling characteristics can be made using a high-Tc superconducting film of Ba1−xKxBiO3 grown on a bi-crystal substrate (Takami et al 1997). The Ba1-xKxBiO3 junctions seem quite promising for building SIS mixers at sub-millimetre wavelengths. Measurements of the noise temperature of mixers using the Ba1−xKxBiO3 SIS junction are expected in the near future. The other reason for the difficulty in making these mixers is that high-Tc superconducting films have a large radiation loss especially above millimetre-wave frequencies (Piel and Müller 1991). Since the

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radiation loss is mainly attributed to the granularity of the high-Tc superconductor, it is possible to reduce the loss by employing an epitaxially grown film. Nevertheless, practical applications such as low-noise receivers using these high-Tc superconductors, which are usually difficult to handle, in the lower GHz range have been developed (Barner et al 1995). Further effort is needed to make not only high-quality high-Tc films with low radiation loss but also Josephson or SIS junctions for mixers using high-Tc superconductors in order to reach their full potential. I6.2.7 Summary Great progress in fabricating SIS junctions and developing SIS mixers at millimetre and sub-millimetre wavelengths has been made over the last decade. Nowadays, high-quality Nb/AlOx /Nb junctions with sub-micron sizes can be easily fabricated and several excellent techniques for analysing and designing SIS junctions including their embedding networks have been successfully established. As a result, SIS receivers with nearly quantum-limited noise temperature can be built up to the Nb gap frequency (∼700 GHz). Above the Nb gap frequency the performance of the SIS receivers gradually degrades with increasing frequency due to the onset of radiation loss in superconducting films, but they are still superior to Schottky-diode mixers up to ∼ 1 THz. Beyond 1 THz, quite difficult technological problems have still to be faced in the development of SIS mixers. Waveguide mixers become increasingly difficult at higher frequencies because the critical dimensions scale with wavelength. Quasi-optical SIS mixers are the promising alternative at such high frequencies. While the use of NbN-based junctions for SIS mixers above 1 THz seems promising, losses due to the surface resistance of NbN film at sub-millimetre wavelengths are not well known at present. Great effort must be continuously made to build high-performance SIS mixers above 1 THz. References Baars J M, Greve A, Hein H, Morris D, Penalver J and Thum C 1994 Design parameters and measured performance of the IRAM 30-m millimeter radio telescope Proc. IEEE 82 687 Bao Z, Bhushan M, Han S and Lukens J E 1995 Fabrication of high quality, deep-submicron Nb/AlOx /Nb Josephson junctions using chemical mechanical polishing IEEE Trans. Appl. Supercond. AS-5 2731 Barner J B, Bautista J J, Bowen J G, Chew W, Foote M C, Fujiwara B H, Guern A J, Hunt B J, Javadi HHS, Ortiz G G, Rascoe D L, Vasquez R P, Wamhof P D, Bhasin K B, Leonard R F, Romanofsky R R and Chorey C M 1995 Design and performance of low-noise hybrid superconductor/semiconductor 7.4 GHz receiver downconverter IEEE Trans. Appl. Supercond. AS-5 2075 Belitsky V Yu, Jacobsson S W, Filippenko L V, Holmstedt C, Koshelets V P and Kollberg E L 1995 Fourier transform spectrometer studies (300–1000 GHz) of Nb-based quasi-optical SIS detectors IEEE Trans. Appl. Supercond. AS-5 3445 Bin M, Gaidis M C, Miller D, Zmuidzinas J, Phillips T G and LeDuc H G 1996a Design and characterization of a quasi-optical SIS receiver for the THz band Proc. 7th Int. Symp. on Space Terahertz Technology (Charlottesville, VA, 1996) p 549 Bin M, Gaidis M C, Zmuidzinas J, Phillips T G and LeDuc H G 1996b Low-noise 1 THz superconducting tunnel junction mixer with a normal metal tuning circuit Appl. Phys. Lett. 68 1714 Blundell R and Tong C-Y E 1992 Submillimeter receivers for radio astronomy Proc. IEEE 80 1702 Büttgenbach T H, Miller R E, Wengler M J, Watson D M and Phillips T G 1988 A broad-band low-noise SIS receiver for submillimeter astronomy IEEE Trans. Microwave Theory Technol. MTT-36 1720 Carlstrom J E and Zmuidzinas J 1996 Millimeter and submillimeter techniques Reviews of Radio Science 1993–1995 ed W R Stone (Oxford: Oxford University Press) Dayem A H and Martin R J 1962 Quantum interaction of microwave radiation with tunneling between superconductors Phys. Rev. Lett. 8 246

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1911

Dierichs M M T M, Panhuyzen R A, Honingh C E, de Boer M J and Klapwijk T M 1993 Submicron niobium junctions for submillimeter-wave mixers using optical lithography Appl. Phys. Lett. 62 774 Dolan G J 1977 Offset masks for lift-off photoprocessing Appl. Phys. Lett. 31 337 Feldman M J and Rudner S 1983 Mixing with SIS arrays Reviews of Infrared and Millimeter Waves vol 1, ed K J Button (New York: Plenum) Gaidis M C, LeDuc H G, Bin M, Miller D, Stern J A and Zmuidzinas J 1996 Characterization of low-noise quasi-optical SIS mixers for the submillimeter band IEEE Trans. Microwave Theory Technol. MTT-44 1130 Grossman E N and Vale L R 1994 Heterodyne mixing and direct detection in high temperature Josephson junctions Proc. 5th Int. Symp. on Space Terahertz Technology (Michigan, IL, 1994) p 244 Ke Q and Feldman M J 1993 Quantum source conductance for high frequency superconducting quasiparticle receivers IEEE Trans. Microwave Theory Technol. MTT-41 600 Kerr A R and Pan S-K 1990 Some recent developments in the design of SIS mixers Int. J. Infrared Millimeter Waves 11 1169 Kerr A R, Pan S-K and Feldman M J 1988 Integrated tuning elements for SIS mixers Int. J. Infrared Millimeter Waves 9 203 Kerr A R, Pan S-K and Washington S 1993 Embedding impedance approximation in the analysis of SIS mixers IEEE Trans. Microwave Theory Technol. MTT-41 590 Kooi J W, Chan M S, Bumble B, LeDuc H G, Scaffer P L and Phillips T G 1994 180–425 GHz low-noise SIS waveguide receivers employing tuned Nb/Al-Ox /Nb tunnel junctions Int. J. Infrared Millimeter Waves 15 783 Kraus J D 1986 Radio Astronomy 2nd edn (Powell, OH: Cygnus-Quasar) Kroger H, Smith L N and Jiltie D W 1981 Selective niobium anodization process for fabricating Josephson tunnel junctions Appl. Phys. Lett. 34 280 LeDuc H G, Judas A, Cypher S R, Hunt B D and Stern J A 1991 Submicron area NbN/MgO/NbN tunnel junctions for SIS mixer applications IEEE Trans. Magn. MAG-27 3192 Lichtenberger A W, Lea D M, Li C, Lloyd F L, Feldman M J, Mattauch R J, Pan S-K and Kerr A R 1991 Fabrication of micron size Nb/Al-Al2O3/Nb junctions with a trilevel resist liftoff process IEEE Trans. Magn. MAG-27 3168 Noguchi T, Shi S-C and Inatani J 1995a An SIS mixer using two junctions connected in parallel IEEE Trans. Appl. Supercond. AS-5 2228 Noguchi T, Shi S-C and Inatani J 1995b Parallel connected twin junctions for millimeter and submillimeter wave SIS mixers: analysis and experimental verification IEICE Trans. Electron. E-78 481 Ogawa H, Mizuno A, Hoko H, Ishikawa H and Fukui Y 1990 A 100 GHz SIS receiver for radio astronomy Int. J. Infrared Millimeter Waves 11 717 Phillips T G and Woody D P 1982 Millimeter- and submillimeter-wave receivers Annu. Rev. Astron. Astrophys. 20 285 Piel H and Müller G 1991 The microwave surface impedance of high-Tc superconductors IEEE Trans. Magn. MAG-27 854 Räisänen A R, McGrath W R, Crété D G and Richards P L 1985a Scaled model measurements of embedding impedance for SIS waveguide mixers Int. J. Infrared Millimeter Waves 16 1169 Räisänen A V, McGrath W R, Richards P L and Lloyd F L 1985b Broad-band RF match to a millimeter-wave SIS quasi-particle mixer IEEE Trans. Microwave Theory Technol. MTT-33 1495 Rothe H and Dahlke W 1956 Theory of noisy fourpoles Proc. IRE 44 811 Shi S-C 1996 Quantum-limited broadband mixers with superconducting tunnel junctions at millimeter and submillimeter wavelengths PhD Thesis The Graduate University for Advanced Studies, Japan Shi S-C, Inatani J, Noguchi T and Sunada K 1993 Analytical predictions for the optimum operating conditions of SIS mixers Int. J. Infrared Millimeter Waves 14 1273 Shimakage H, Uzawa Y, Tonouchi M and Wang Z 1997 Noise temperature measurement of YBCO Josephson mixers in millimeter and submillimeter waves IEEE Trans. Appl. Supercond. AS-7 at press Shoji A, Kiryu S, Kashiwaya S, Kohjiro S, Kosaka S and Koyanagi M 1992 Preparation and characteristics of full-epitaxial NbCxN1−x /MgO/NbCx N1−x Josephson junctions Superconducting Devices and Their Applications eds H Koch and H Lübbig (Berlin: Springer) p 208 Takami T, Kuroda K, Wada Y, Hieda M, Tamai Y and Ozeki T 1997 Aiming for SIS mixers using Ba1−xKxBiO3 bicrystal junctions IEICE Trans. Electron. E80-c at press

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Tien P K and Gordon J P 1963 Multiphoton process observed in the interaction of microwave fields with the tunneling between superconductor films Phys. Rev. 129 647 Tinkham M 1975 Introduction to Superconductivity eds B Bayne and M Gardner (New York: McGraw-Hill) Tong C-Y E, Blundell R, Bumble B, Stern J A and LeDuc H G 1995 Quantum-limited heterodyne detection in superconducting nonlinear transmission lines at submillimeter wavelengths Appl. Phys. Lett. 67 1304 Tucker J R 1979 Quantum limited detection in tunnel junction mixers IEEE J. Quantum Electron. 15 1234 Tucker J R and Feldman M J 1985 Quantum detection at millimeter wavelengths Rev. Mod. Phys. 57 1055 Ukita N and Tsuboi M 1994 A 45-m telescope with a surface accuracy of 65 µ m Proc. IEEE 82 725 Uzawa Y, Wang Z and Kawakami A 1996 Quasi-optical submillimeter-wave mixers with NbN/AlN/NbN tunnel junctions Appl. Phys. Lett. 69 2435 van de Stadt H, Baryshev A, Gao J R, Golstein H, de Graauw Th, Hulshoff W, Kovtonyuk S, Schaeffer H and Whyborn N 1996 An improved 1 THz waveguide mixer Proc. 7th Int. Symp. on Space Terahertz Technology (Charlottesville, IL, 1996) p 536 Wang Z, Kawakami A and Uzawa Y 1997 NbN/AlN/NbN tunnel junctions with high current density up to 54 kA/cm2 Appl. Phys. Lett. 70 114 Werthamer N R 1966 Nonlinear self-coupling of Josephson radiation in superconducting tunnel junctions Phys. Rev. 147 255 Wilson R W, Jefferts K B and Penzias A A 1970 Carbon monoxide in the Orion nebula Astrophys. J. 116 L43 Zmuidzinas J and LeDuc H G 1992 Quasi-optical slot antenna SIS mixers IEEE Trans. Microwave Theory Technol. MTT-40 1797 Zmuidzinas J, LeDuc H G, Stern J A and Cypher S R 1994 Two-junction tuning circuits for submillimeter SIS mixers IEEE Trans. Microwave Theory Technol. MTT-42 698

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Glossary

G1

Glossary

A-15 compound: a generic term for intermetallic compounds with a composition of A3B where B atoms form a body-centred cubic lattice and A atoms form a one-dimensional chain in x , y and z directions in the cubic lattice. Superconductive materials such as Nb3Sn, V3Ga and Nb3A1 are of this type of crystal structure. a.c. loss (transient loss): an energy loss arising when a superconductor is used in an alternating magnetic field. According to its origin, it may be classified into three types: hysteresis loss, coupling loss and eddy current loss. adiabatic stabilization: a superconducting magnet design concept to avoid magnetic instability by making superconducting filaments sufficiently thin. alloy superconductor: an alloy which exhibits superconductivity under appropriate conditions. aspect ratio: ratio of the longer to the shorter transverse dimensions of a rectangular composite superconductor. B-1 compound: a generic term for transient metallic carbides, nitrides and oxides with an NaCl type of crystal structure. Superconductive materials such as NbN, NbC and MoN are known to be of this type of crystal structure. BCS theory: a theory on superconductivity stating that the superconducting state is realized by the formation of Cooper pairs of electrons in the vicinity of the Fermi level through lattice distortions (i.e. phonons) Bean-London (critical-state) model: a critical-state model, proposed by Bean and London, for the magnetization process in type II superconductors assuming that the critical current density is constant with respect to the magnetic field. This model may be applied to a narrow range of magnetic fields. braid: a narrow tubular or flat fabric produced by intertwining strands of materials according to a definite pattern. bronze process: a composite diffusion process typically for Nb3Sn and V3Ga composite conductors, where the matrix of Cu-Sn or Cu-Ga bronze containing Nb or V cores is cold-worked into a final shape of the conductor and heat-treated to form a Nb3Sn or V3Ga layer at the interfaces between the matrix and the cores. cable (concentric lay conductor): a conductor constructed with a central core surrounded by one or more layers of helically laid wires. There exist several types of cable including compact round, conventional concentric, equilay, parallel core, rope-lay, unidirectional and unilay. cable-in-conduit conductor: a composite conductor consisting of a cable inside a metal conduit. The conduit is a disturbed mechanical structure that decreases the stress on the cable and also allows forced-flow cooling of the cable to improve the thermal stability of the conductor.

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G2

Glossary

Chevrel phase compound: a generic term for compounds with a typical composition of MMo6Χ8 where M refers to metallic elements from monovalent to quadrivalent ones such as Pb, Sn, Cu and La, while Χ corresponds to chalcogen elements such as S, Se and Te. coherence length, ξ : (1) temperature independent quantity, defined in BCS theory, describing the spatial spread of Cooper-paired electrons; (2) quantity defined in Ginzburg-Landau theory describing the spatial variation of order parameters, and varying with temperature compact round conductor: a conductor constructed with a central core surrounded by one or more layers of helically laid wires and formed into final shape by rolling, drawing or other means. compact stranded conductor: a conductor composed of helically laid monolithic or stranded wires and formed into a final platelike shape by rolling, drawing or other means. composite conductor: a conductor consisting of two or more types of material, each type of material being plain, clad or coated, and assembled together to operate mechanically and electrically as a single conductor. composite diffusion process: a fabrication process for composite conductors, where members of the composite are cold-worked together into a final shape with or without intermediate anneals, and subjected to heat treatment based on solid-state diffusion among the members of the composite to form a desired superconductive phase or an appropriate microstructure containing a superconductive phase with normal-conducting phases. composite superconductor: a conductor incorporating superconductive material. There exist several types of composite superconductor including filamentary, coreless, tape, tubular and hollow conductors. composition: the quantity of each of the components of a mixture: usually expressed in terms of the weight percentage, or the atomic percentage of each of the components in the mixture. compound superconductor: a compound which exhibits superconductivity under appropriate conditions. condensation energy, superconducting, δ G : energy necessary for converting a superconductor from the superconducting to the normal state. conductivity, electric, σ : the ratio of the current density to the potential gradient parallelling the current in a material. This is numerically equal to the conductance between opposite faces of a unit cube of the material. It is the reciprocal of resistivity. conventional concentric conductor: conductor constructed with a central core surrounded by one or more layers of helically laid round wires. The direction of lay is reversed in successive layers, and generally with an increase in length of lay for successive layers. cooling channel: a gap or groove for the flow of liquid helium which is fed there to cool the superconducting magnet or conductor. Cooper pair: pair of electrons, opposite to each other in wave number (momentum) vector and direction of spin. Cooper-paired electrons are free from lattice scattering, thus moving without loss in energy. copper-to-superconductor volume ratio: the volume ratio of the copper to the superconductor in a composite conductor. core: a component of a composite conductor which is surrounded usually by matrix which is another component of the composite.

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Glossary

G3

coreless conductor: a conductor constructed with one or more layers of helically laid wires and formed into final shape by rolling, drawing or other means. coupling current: a current flowing, in an alternating or pulsed magnetic field, between superconducting filaments or strands separated by normal-conducting materials. coupling loss: an a.c. loss caused by the Joule-type heating of coupling current. The coupling loss is often referred to as eddy current loss. coupling time constant: a time interval for which a coupling current caused by an alternating or pulsed magnetic field decays, proportional to the square of the twist pitch and inversely proportional to the matrix resistivity in the direction perpendicular to the filament axis. critical-state model: a model for the magnetization process in type II superconductors stating that the magnetic flux density in a superconductor varies from the surface to the centre with a gradient equal to the critical current density. cryostat: a vessel to keep a material or a device at low temperatures within it. current, critical, Ic : the maximum electrical current below which a superconductor exhibits superconductivity at some given temperature and magnetic field. The critical current is usually defined by means of a resistivity or an electric field criterion. current, persistent: current flowing through a closed, superconducting loop circuit exhibiting no decay with time. current density J : current per unit area. current density, critical, Jc : the critical current divided by the cross-sectional area of the superconductor. demagnetization factor (demagnetizing factor): the ratio of the average demagnetizing field to the average magnetization in a magnetic or superconducting material of finite size. The demagnetizing field may be thought of as arising from surface magnetic poles. diamagnetic material: a material whose susceptibility is negative. diamagnetism: magnetism where the field inside a substance is less than the field in which it is situated. In this case the magnetic susceptibility χ is negative. diamagnetism, perfect: magnetism with a susceptibility equal to −1, exhibited by a type I superconductor below the critical magnetic field, Hc , and a type II superconductor below the lower critical field, Hc 1. The bulk of a superconductor exhibiting perfect diamagnetism is shielded from magnetic fields. dipole coil: a coil to generate a magnetic field with dipolar components in one direction. double pancake coil: a pair of pancake coils so connected with each other as to have their conductor ends appear at the outer circumference of the coil. A magnet is constructed by stacking double pancake coils connected in series at their respective outer circumferences. dynamic stabilization: flux jumps occur when the rate of energy dissipation during a disturbance involving the rearrangement of magnetic flux is greater than the rate of cooling. They can therefore be prevented by slowing down the rate of flux motion or by increasing the rate of cooling. Provided that the superconductive material is sufficiently subdivided, dynamic stabilization may be achieved by embedding the material in a high-conductivity material, such as copper. Screening currents are induced by the flux motion in this material and they decay much more slowly than those induced in superconductive material which has a high resistivity in the normal-conduction state. Thus the flux motion is slowed. In addition the thermal

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G4

Glossary

conductivity of high-conductivity normal materials is much higher than that of superconductive materials in the normal state. Thus both requirements are met by the addition of such material and flux jumps are eliminated. Flux jumps can still occur inside large superconducting filaments but the level of subdivision required to prevent this is less than that required for flux-jump stabilization of isolated filaments. eddy current loss: an a.c. loss caused by the Joule-type heating of eddy current which arises in the normal-conducting component of a superconducting wire, cable, etc in an alternating or pulsed magnetic field. electromagnetic stability: see flux-jump stabilization. energy gap, superconducting, ∆ : one half of the minimum value of energy necessary for destroying a Cooper pair and exciting the two electrons to the normal state. In BCS theory this is given for absolute zero temperature. equilay conductor: conductor constructed with a central core surrounded by more than one layer of helically laid wires, all layers having a common length of lay, direction of lay being reversed in successive layers. external diffusion process: a fabrication process typically for Nb3Sn and V3Ga composite conductors, where a Cu jacket with drilled holes containing Nb or V rods is cold-worked into a final shape, coated on the surface with Sn or Ga and subjected to heat treatment firstly to diffuse the Sn or Ga into the matrix and then to form an Nb3Sn or V3Ga layer at the interfaces between the Cu matrix and the Nb or V cores. filament (elementary filament): a thin, elongated core of superconductive material contained in a composite conductor through which superconducting current flows. filamentary (multifilamentary) conductor: a composite superconductor consisting of more than one superconductive filament embedded in a matrix. film boiling: a phenomenon in which the surface of a material being cooled is completely covered with a film of vaporized coolant. flux-jump stabilization: a superconducting magnet design concept to avoid instability caused by internal magnetic disturbances, that is achieved by subdividing the superconductive material into fine filaments small enough that after an internal magnetic disturbance the energy liberated is so small that the disturbance does not lead to a flux jump. fluxoid (fluxon): quantized magnetic flux lines distributing within a type II superconductor with the unit quantity of magnetic flux quantum, Φ0 , and a radius of coherence length, ξ. forced cooling: a cooling method for superconducting magnets or conductors by forced flow of liquid helium through cooling channels. full stabilization: a design concept in which the amount of high-conductivity material included in a composite superconductor, and the level of cooling provided, are such that, should the superconducting component quench, thus diverting all the current into the normal material, the temperature of the composite will remain below the critical temperature of the superconducting component. In these circumstances the temperature of the superconducting component will always recover to its original level and the current will then transfer back. Ginzburg-Landau (GL) parameter, k : the ratio of penetration depth, λ, to coherence length, ξ : k = λ/ξ. Ginzburg-Landau (GL) theory: a thermodynamic theory interpreting magnetic properties of type II superconductors.

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Glossary

G5

helium, supercritical: 4He which is above a critical temperature, Tc (5.22 K), and a critical pressure, Pc (0.227 MPa). helium, superfluidic: helium exhibiting superfluidity which appears in the temperature-pressure diagram of helium within the region bound by a tie-line connecting the λ. point for saturated helium (2.18 K, 5 kPa) with the upper limit of the λ line of helium (1.76 K, 2.77 MPa). Helmholtz coil: a kind of split-pair coil with a common radius and number of turns, and a separation gap distance which is identical with the common coil radius. hollow conductor (tubular conductor): a conductor in which the individual elements are disposed about one or more hollow passages, the direction of which is along the axial length of the conductor. hybrid magnet: a magnet consisting of different kinds of magnets including normal-conducting and/or superconducting magnets. hysteresis loss (pinning loss): an a.c. loss caused by the movement of fluxoids pinned in a superconductor in an alternating magnetic field. impregnated coil: a coil impregnated with appropriate resin to improve mechanical stability and electrical insulation within the magnet structure. in situ process: a fabrication process typically for Nb3Sn and V3Ga conductors, where a Cu-Nb or Cu-V alloy ingot containing Nb or V fine dendrites dispersed in the Cu matrix is cold-worked into a final shape. It is then coated with Sn or Ga and subjected to heat treatment which first disperses the Sn or Ga throughout the matrix and then forms an Nb3Sn or V?3Ga layer at the interfaces between the Cu matrix and the discrete, elongated Nb or V dendrites. intermediate state: a state of coexistence of superconducting and normal-conducting regions in the bulk of a type I superconductor with finite demagnetizing factor placed in a magnetic field below the critical field. internal-tin process: a fabrication process typically for Nb3Sn composite conductors, where a Cu jacket containing an Sn rod in a hole drilled in its centre and Nb rods in holes drilled in the remaining area is cold-worked into a final shape and subjected to heat treatment firstly to diffuse the Sn into the matrix and then to form an Nb3Sn layer at the surfaces of the Nb cores. isotope effect: a relation, deduced from BCS theory, that the critical temperature is inversely proportional to the square root of the atomic weight of component elements in a superconductor. jelly-roll process: a fabrication process typically for Nb3Sn conductors, where a foil of Cu-Sn bronze and a foil of Nb with slit meshes are lapped and spirally rolled into a cylinder, then cold-worked to a final shape, and subjected to heat treatment to form Nb3Sn at the interfaces between the Cu-Sn and the Nb. Josephson effect: an effect in which superconducting, tunnelling current flows through a thin insulating layer separating two superconducting regions. Josephson junction: a device composed of a thin insulating layer separating two superconducting regions. keystone stranded conductor: a kind of compact stranded conductor with the final shape of a trapezium. Kim (critical-state) model: a critical-state model, proposed by Kim, Hempstead and Strnad for the magnetization process in type II superconductors which assumes that the critical current density is inversely proportional to the internal field.

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G6

Glossary

Kramer’s law: the scaling law for pinning force density proposed by E J Kramer. It fairly well describes the pinning behaviour of Nb3Sn at higher fields. Laves-phase compound: a generic term for intermetallic compounds with a composition of AB2 . Typical superconductive materials of this type of crystal structure include HfV2 and ZrV2 . layered perovskite compound: a generic term for oxide compounds with a typical composition of ABO3 where A and B refer to metallic elements, while O is oxygen. The oxygen ions form an octahedral structure, B ions entering the centre of each octahedron and A ions entering a space built by eight octahedra. Depending on the oxygen content a variety of lattice stacking faults are introduced, resulting in a variety of modifications of the original crystal structure. YBCO, BSCCO and TBCCO are known to be of this group of crystal structure and become superconducting above liquid-nitrogen temperature. Lorentz force, F : electromagnetic force working on electrons moving in a magnetic field, with a direction perpendicular to both electron motion and magnetic field directions. In the case of a superconductor, the Lorentz force acts on the fluxoids in the superconductor in a direction perpendicular to both the transport current and the applied magnetic field. Maddock’s stability criterion: a modification of Stekley’s stability criterion which also takes into account the thermal conduction along the superconductor length, resulting in the reduction in amount of the normal-conducting metal surrounding the superconductor. magnetic field (strength), critical, Hc : the maximum magnetic field below which a superconductor exhibits superconductivity at zero current and temperature. In practice it often means the upper critical field, Hc 2. magnetic field (strength), lower critical, Hc 1: the magnetic field strength above which the Meissner effect is destroyed and magnetic fluxes start to penetrate into the bulk of a type II superconductor. magnetic field (strength), thermodynamic critical, Hc t h : the magnetic field strength which is defined for type II superconductors as δ G = µ 0 H c2t h /2 where δ G is the superconducting condensation energy. magnetic field (strength), upper critical, Hc 2: the magnetic field strength above which the mixed state is destroyed and transition to the normal state occurs in a type II superconductor. magnetic field strength, H : the measured intensity of a magnetic field at a point, can be defined from the magnetic field H = 2π I/R at the centre of a circular loop of current I and radius R. The units are amperes per metre. It can be similarly defined as the field H = I/2π r at a distance r from a very long straight wire carrying a current I. magnetic flux, Φ: the product of the magnetic induction, B, through a surface and the area of the surface, A. When the magnetic induction B is uniformly distributed and directed normal to the surface Φ = B × A. magnetic flux creep: a phenomenon in which fluxoids pinned in a superconductor move due to thermal activation. magnetic flux density, Β : magnetic flux per unit area, identical to magnetic induction. magnetic flux flow: a phenomenon in which fluxoids in a superconductor move when the Lorentz force exceeds the pinning force. magnetic flux jump: the collective, discontinuous motion of fluxoids in a superconductor, produced by mechanical, thermal, magnetic or electrical disturbances. magnetic flux pinning: the trapping of fluxoids at defects in the superconducting material.

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Glossary

G7

magnetic flux quantum, Φ0: the unit or minimum quantity of magnetic flux distributed within a superconductor or its ring. magnetic induction, Β : synonymous with magnetic flux density or magnetic flux per unit area. The magnetic induction, B, is given as B = µ0(H + M) = µ H, where H is magnetic field, M magnetization, µ0 magnetic permeability of vacuum and λ magnetic permeability. B is a magnetic vector quantity which at any point in a magnetic field is measured either by the mechanical force experienced by an element of electric current at the point, or by the electromotive force induced in an elementary loop during any change in flux linkages with the loop at a point. magnetic polarization: the same as magnetization, expressed in teslas or webers per square metre. magnetic shielding: a shielding of magnetic field by using materials of high permeability or superconducting materials. magnetization, Μ : magnetic moment per unit volume in a magnetic material, expressed in amperes per metre. Magnetization in a superconductor occurs due to the distribution of shielding currents. matrix (of composite superconductor): the continuous longitudinal phase of a pure metal, a poly phase alloy or mechanical mixture that is not in the superconducting state at the normal operating conditions of the embedded superconductor. matrix-to-superconductor volume ratio: the volume ratio of the matrix to the superconductor in a composite conductor. Meissner effect: the expulsion of magnetic flux from a superconductor as it enters the superconducting state. minimum propagation zone (MPZ) theory: a superconducting magnet stabilization theory which defines magnet cooling conditions for not propagating the normal zone and not quenching the magnet, by estimating the magnitude of localized disturbance in the superconductor. mixed matrix (of composite superconductor): matrix composed of more than one component. mixed state: a state of coexistence of superconducting and normal-conducting regions in the bulk of a type II superconductor placed in a magnetic field between the lower and the upper critical fields. monolithic conductor: a composite conductor containing superconductor and stabilizer material, and possibly reinforcement and insulating materials, contiguously assembled with one another to form a solid structure that allows no relative motions of the components. normal (-conducting) state: the thermodynamic state in which a superconducting material no longer exhibits any of the characteristics of the superconducting state. normal zone: a region in a conductor or winding in which the superconductor has transformed to normal state. normal zone propagating velocity: the velocity at which the envelope of a normal zone advances along a conductor or through a winding during a quench. This velocity is usually different in the three directions at right angles in a winding so that the normal zone is ellipsoidal until it encounters a winding boundary. nucleate boiling: a phenomenon in which the rate of cooling becomes so large as to cause the formation and detachment of bubbles of vapour at the cooled surface. As the rate of cooling is further increased nucleate boiling gives way to film boiling.

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G8

Glossary

order parameter: thermodynamic quantity defined in GL theory, a measure for the density of Cooper-paired electrons, and roughly proportional to the superconducting energy gap. organic superconductor: an organic material which exhibits superconductivity under appropriate conditions. oxide superconductor: an oxide which exhibits superconductivity under appropriate conditions. pancake coil: a flat whirlpool coil wound in a shape of a thin-sliced pancake. parallel-core conductor: conductor constructed with a central core of parallel-laid wires surrounded by one layer of helically laid wires. peak effect: an effect in which a critical current versus magnetic field curve exhibits a peak near the upper critical field. penetration depth, : the penetration depth of magnetic field in a superconductor that exhibits perfect diamagnetism. persistent-current switch: a thermal, magnetic or mechanical switch to cut off the power supply from a superconducting electric loop circuit which is then operated in a persistent-current mode. pinning centre: a defect in a superconductor at which penetrating fluxoids are pinned. Defects which act as pinning centres include various lattice defects, precipitates, grain boundaries, etc. pinning force: the force that pins fluxoids at pinning centres. pinning force density, Fp : pinning force per unit volume of pinning centres preventing the movement of fluxoids in a superconductor. In the case of a Lorentz force exceeding Fp , the latter is given by Fp = −Jc × B. poloidal magnet: a pulsed field magnet used in a Tokamak fusion reactor to generate a field perpendicular to the toroidal field. The poloidal field is used to heat the plasma and to maintain stability of confinement. The axis of the coils of such a magnet is coincident with that of the torus. pool cooling: a cooling method for superconducting magnets or conductors by directly immersing them in liquid helium. powder metallurgy process: a fabrication process for compound conductors, where powders of component elements of a desired superconductive compound and a matrix material such as Cu are mixed, compacted and cold-worked into a final shape of tape or wire which is then subjected to heat treatment to form the desired compound layer at the surface of the discrete, elongated powders. For example, powders of Nb, Sn and Cu are used for Nb3Sn conductors, and Nb, A1 and Cu for Nb3A1 conductors. pressurized superfluidic helium: superfluidic helium pressurized above 0.1 MPa. proximity effect: penetration of superconducting electrons into normal-conducting regions adjacent to superconducting regions. quadruple coil: a coil consisting of four saddle coils arranged in a quadraxis symmetry to generate a magnetic field with quadrupolar components. quench: the abrupt and uncontrolled loss of superconductivity produced by a disturbance. rapid quench: a design concept in which the normal zone propagation velocity perpendicular to the conductor is artificially enhanced. By this means the winding is protected from burnout because the stored

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Glossary

G9

energy is dissipated more uniformly through the winding but, since the normal zone is growing more rapidly, the quench is also more rapid. react and wind method: a fabrication method for a superconducting coil, where a conductor containing component elements of a required superconductor is wound into a coil after having been heat-treated to form the superconductor. reinforcing member: a structural material incorporated in a composite conductor for mechanical reinforcement. residual resistivity ratio: the ratio of the electric resistivity at 273 K to that at 4.2 K for a normal-conducting metal. For a superconductor this is usually the ratio of the electric resistivity at 273 K to that at a temperature just above the critical temperature, Tc . resistivity, residual: finite electric resistivity remaining at absolute zero temperature. rope-lay conductor: conductor constructed of a bunch-stranded or a concentric-stranded member or members as a central core, around which are laid one or more helical layers of such members. saddle coil: a coil wound with the shape of a saddle to generate a magnetic field in a direction perpendicular to the coil axis. scaling law for pinning force density: an empirical law for describing the dependence of the pinning force density, Fp , on temperature and magnetic field. shim coil: a coil to compensate the inhomogeneity of a magnetic field occurring in the core region of the field. skin effect: an effect in which the intensity of an applied a.c. electric or magnetic field of angular frequency ω falls off exponentially with depth from the surface of a substance. It is characterized by a skin depth δ. solenoid coil: a coil helically wound around an axis using a conductor wire, strand or cable to generate a more or less uniform magnetic field. spacer: a component inserted between two neighbouring components to prevent their contact. split-pair coil: a pair of solenoid coils with a common axis but split by some gap between them. SQUID: acronym for ‘superconducting quantum interference device’, a device used to detect extremely weak magnetic fields using the Josephson effect. stabilization: a design concept in which quenching is prevented. stabilizer: a metal, but not necessarily the matrix, in electrical contact with a superconductor, to act as an electric shunt in the event that the superconductor reverts to the normal state. stable (stability): a superconducting device is stable if it retains its operating characteristics after it has been subjected to a disturbance. Stekly stability criterion: a superconducting magnet stability criterion which states that a superconducting magnet is in a stable condition if the heat evolved in the superconducting metal on quench is less than that taken off from the surface of the metal by the coolant: thermal conduction is taken into consideration essentially only for the direction perpendicular to the superconductor. stranded conductor: a conductor composed of a group of wires, usually twisted together, or of any combination of such groups of wires.

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G10

Glossary

stress effect/strain effect: a change of superconducting properties due to a mechanical or electromagnetic stress/strain imposed on a superconductor. superconducting magnet: a magnet using superconducting wire in its coil(s). superconducting state: the thermodynamic state in which the material exhibits superconductivity. superconducting transition: the combination of values of temperature, T, electric current density, J, and magnetic field, H, at which a transition from the superconducting to the normal state takes place. superconductivity: a property of many elements, alloys and compounds by virtue of which their electrical resistivity vanishes and they become strongly diamagnetic under appropriate conditions. surface diffusion process: a fabrication process for compound superconductors such as V3Ga and Nb3Sn, where a tape of V or Nb is dipped in a Ga or Sn bath, pulled out and heat-treated at a high temperature to form V3Ga or Nb3Sn. A second option is to hold the bath at the reaction temperature so that the V3Ga or Nb3Sn are formed during immersion. surface pinning: a magnetic flux pinning by which magnetic fluxoids are pinned at the interfaces between the superconducting filaments and the matrix in a multifilamentary conductor. surface superconductivity: superconductivity concerning the surface layer region of about the coherence length in depth and persisting beyond the upper critical field, Hc 2 , up to a magnetic field which is termed Hc 3 = 1.69Hc 2 . susceptibility, external: dM/dHa ; where M is the magnetization and Ha is the applied field (not corrected for demagnetizing factor). susceptibility, internal: dM/dHi ; where M is the magnetization and Hi , is the internal field (corrected for demagnetizing factor). susceptibility, magnetic, χ : the ratio of magnetization, M, to the field H producing it. That is χ = M/H. tape conductor: a conductor constructed in the form of flat ribbon or strip. temperature, absolute: (1) temperature measured on the thermodynamic scale, designated as kelvin (K); (2) temperature measured from absolute zero (−273.15 °C). The numerical values are the same for both the Kelvin scale and the ideal-gas scale. temperature, critical, Tc : the maximum temperature below which a superconductor exhibits superconductivity at zero magnetic field and current. temperature, transition: the maximum temperature below which a superconductor exhibits superconductivity at a given magnetic field and current. three-component conductor: a composite superconductor composed of a superconductor and two different matrices. An example of this type of conductor is an NbTi-based conductor with copper and cupro-nickel matrices, the former being for thermal stabilization and the latter for coupling loss reduction. three-stage extrusion process: a fabrication process for NbTi conductors, where a composite made by a two-stage extrusion is again divided into lengths which are again put together side by side in a can and then extruded a third time. toroidal magnet: a closed winding constructed such that the planes of individual equispaced turns or winding sections lie along the radii of a cylinder whose axis is outside the turns or sections. Also a line joining the centres of the turns or sections forms a closed circle with its centre on the axis of the cylinder.

Copyright © 1998 IOP Publishing Ltd

Glossary

G11

The solid shape thus formed is a toroid or torus. The best known examples are the windings used to generate the steady field required for plasma confinement in a Tokamak fusion reactor. training effect: an effect whereby, on first energization of a conductor or winding, a quench occurs before the critical current is reached. Similar quenches occur on subsequent energizations but the current at quench progressively increases until a plateau is reached. transport-current loss: an a.c. loss due to a transport current in combined action with an a.c. magnetic field or due to an a.c. transport current. transposed conductor: a composite conductor in which filaments or strands are plaited together to occupy different relative positions about the conductor axis in a regular manner along its length. transposition length: the distance in which a filament or strand returns to its original relative position in a transposed conductor. trapped (magnetic) flux: the magnetic flux retained in a superconductor when the applied magnetic field is reduced to zero. tube process: a fabrication process typically for Nb3Sn composite conductors, where a Cu tube containing Sn bars is inserted into an Nb tube which is then further inserted in a Cu tube to form a basic composite. The basic composites are inserted into a larger Cu tube, cold-worked into a final shape and subjected to heat treatment firstly to diffuse the Sn into the matrix and then to form an Nb3Sn layer at the interfaces between the Cu and Nb tubes. tubular conductor: a conductor constructed in the form of a tube. twist: the number of turns per unit length made by a filament or strand about a conductor axis. twist pitch: the distance in which a filament or strand returns to its original relative position in a twisted conductor. twisted conductor: a composite conductor in which the filaments or strands spiral about the conductor axis. two-stage extrusion process: a fabrication process for NbTi conductors, where a composite containing NbTi filaments, and made using an extrusion, is divided into lengths, which are put together side by side in a can and then extruded a second time. type I superconductor: a superconductor in which superconductivity with perfect diamagnetism appears below the critical magnetic field, Hc , but disappears above Hc . type II superconductor: a superconductor in which superconductivity appears with perfect diamagnetism up to the lower critical magnetic field, Hc 1 , persists in a mixed state for the magnetic field range between Hc 1 and the upper critical field, Hc 2 , and disappears above Hc 2 . unidirectional conductor: conductor constructed with a central core surrounded by more than one layer of helically laid wires, all layers having a common direction of lay, with increase in length of lay for each successive layer. unilay conductor: conductor constructed with a central core surrounded by more than one layer of helically laid wires, all layers having a common length and direction of lay. vacuum impregnation: a fixing method for a superconducting coil against mechanical or electromagnetic forces, where a coil is impregnated with epoxy resin in vacuum so that the epoxy resin can infiltrate into the smallest recesses of the winding and no air bubbles remain.

Copyright © 1998 IOP Publishing Ltd

G12

Glossary

volume per cent superconductor: that percentage by volume of a composite superconductor which is superconducting under appropriate conditions. wind and react method: a fabrication method for a superconducting coil, where a conductor containing component elements of a required superconductor is wound into a coil and subsequently heat-treated to form the superconductor.

Reprinted, with modifications and rearranged in alphabetical order, with permission from Wada H et al 1995 Terminology for superconducting materials Cryogenics 35 Elsevier Science Ltd, Oxford, UK.

Copyright © 1998 IOP Publishing Ltd