Noise of polyphase electric motors

Author: Jacek F. Gieras | Chong Wang | Joseph Cho Lai

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Noise of Polyphase Electric Motors

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120. Handbook of Electric Motors: Second Edition, Revised and Expanded, edited by Hamid A. Toliyat and Gerald B. Kliman 121. Transformer and Inductor Design Handbook, Colonel Wm T. McLyman 122. Energy Efficient Electric Motors: Selection and Application, Third Edition, Revised and Expanded, Ali Emadi 123. Power-Switching Converters, Second Edition, Simon Ang and Alejandro Oliva 124. Process Imaging For Automatic Control, edited by David M. Scott and Hugh McCann 125. Handbook of Automotive Power Electronics and Motor Drives, Ali Emadi 126. Adaptive Antennas and Receivers, edited by Melvin M. Weiner 127. SPICE for Power Electronics and Electric Power, Second Edition, Muhammad H. Rashid and Hasan M. Rashid 128. Gaseous Electronics: Theory and Practice, Gorur G. Raju 129. Noise of Polyphase Electric Motors, Jacek F. Gieras, Chong Wang, and Joseph Cho Lai

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Noise of Polyphase Electric Motors

Jacek F. Gieras United Technologies Corporation Hamilton Sundstrand, Applied Research Rockford, Illinois, U.S.A.

Chong Wang General Motors Corporation Milford, Michigan, U.S.A.

Joseph Cho Lai The University of New South Wales at the Australian Defence Force Academy Canberra, Australian Capital Territory, Australia

Boca Raton London New York

A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.

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Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 0-8247-2381-3 (Hardcover) International Standard Book Number-13: 978-0-8247-2381-1 (Hardcover) Library of Congress Card Number 2005050213 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe.

Library of Congress Cataloging-in-Publication Data Gieras, Jacek F. Noise of polyphase electric motors / Jacek F. Gieras, Chong Wang, Joseph Cho Lai. p. cm. -- (Electrical and Computer Engineering ; 129) Includes bibliographical references and index. ISBN 0-8247-2381-3 (alk. paper) 1. Electric motors, Polyphase--noise. 2. Electric Motors, Alternating current--Noise. 3. Electric motors, Alternating current--Design and construction. I. Lai, Joseph Cho. II. Wang, Chong. III. Title. IV. Series. TK2785.G535 2005 621.46--dc22

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Preface It has been estimated that more than 65% of the electrical energy produced in developed countries is consumed by electric motors. Electrical motors are the most popular machines of everyday life. They are used either as power machines providing propulsion torque or servo motors operating in a closed loop control with speed or position feedback. Electric motors are embedded in larger systems as their integral part. The noise radiated by electric motors immensely affects the overall noise of the system. Contemporary electric motors are designed with higher magnetic flux density in the air gap than motors manufactured a half century ago. Higher magnetic flux density in the air gap produces higher radial magnetic forces acting on the stator system and, consequently, higher vibration and acoustic noise. With the increased power density of electric motors and more demanding environmental requirements, the prediction of noise at the early stage of design of electrical motors has become a very important issue. Not only electromagnetic, thermal, and economic calculations, but also the level of noise and vibration must be considered, so that the overall performance can be optimized/balanced and specific requirements can be incorporated in the design to avoid large retrofit expenses. However, prediction of noise is more difficult and less accurate than, for example, torque–speed characteristics. This is because only a very small fraction of electrical energy is converted into acoustic energy and correct estimation of some mechanical and acoustic parameters is very difficult. The first book [119] on calculation of noise in electrical motors was published by Jordan in 1950. Details of harmonic field analysis including harmonic torques, noise, and vibration in induction motors are given in the book [87] by Heller and Hamata published in 1977. Analysis of noise in induction machines with emphasis on its reduction is given in monograph [248] by Yang, which was published in 1981. The most comprehensive analysis of noise and vibration in electrical machines is contained in the book [200] by Timar, Fazekas, Kiss, Miklas, and Yang published in 1989. It is also necessary to mention two books on noise and vibration in induction machines published by Russian researchers: Shubov in 1974 [187] and Astakhov, Malishev, and Ovcharenko in 1985 [10], and a book published by Polish researcher Kwasnicki in 1998 [127]. There is no book published so far on noise and vibration of permanent magnet (PM) synchronous

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motors. The demand on these motors is nowadays in second place, after the demand on induction motors. As most books on noise and vibration analysis in electrical machines were published over two decades ago, recent advances in vibro-acoustic theories and technologies are only accessible in learned journals and have not been captured in a single monograph. These advances include the development and application of numerical methods of noise computation such as the finite element method (FEM), boundary element method (BEM), and statistical energy analysis (SEA) [43, 230] to the prediction of noise in electrical machines. With the increase in the importance of noise analysis and synthesis in the modern approach to the design of electrical motors, the authors have made an attempt to prepare a modern monograph on noise calculation in induction and PM synchronous motors addressing electromagnetic, mechanical, and vibro-acoustic issues. The noise and vibration of switched reluctance motors have not been considered here. The authors have devised the book as both an electrical motor noise textbook and a handbook for electrical machine design engineers, research scientists, and graduate students. The book can also be helpful for multidisciplinary research teams working on noise prediction of systems with electrical motors, e.g., electrical vehicles, industrial electromechanical drives, HVAC (heating, ventilating, air conditioning) systems, marine propulsion systems, airborne apparatus, elevators, office equipment, health care equipment, etc. The authors hope that this book will fill the current gap in modern treatment of the analysis and reduction of noise in polyphase electric motors. Jacek F. Gieras Chong Wang Joseph C.S. Lai

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Authors Jacek F. Gieras, Ph.D., graduated with distinction in 1971 (Master’s in Engineering) from the Technical University of Lodz, Poland. He received his Ph.D. in Power Electrical Engineering (Electrical Machines) in 1975 and dr hab. degree (corresponding to D.Sc. in the United Kingdom), also in Electrical Engineering (Electromagnetic Field Theory) in 1980 from the University of Technology, Poznan, Poland. From 1971 to 1998, Dr. Gieras pursued his academic career at several universities worldwide including Poland, Canada, Jordan, and South Africa. He was also a Central Japan Railway Company visiting professor at the University of Tokyo (Endowed Chair in Transportation Systems Engineering), Japan; guest professor at Chungbuk National University, Choengju, South Korea; and visiting professor at the University of Rome La Sapienza, Italy. In 1987, he was promoted to the rank of full professor (life title given by the President of the Republic of Poland). Since 1998 Gieras has been involved in industry-oriented research, cutting-edge technologies, and innovations at United Technologies Corporation, recently in the Department of Applied Research, Hamilton Sundstrand, Rockford, Illinois. Dr. Gieras has authored and coauthored nine books, more than 220 scientific and technical papers and ten patents. His most important books of international standing include: Linear Induction Motors, Oxford University Press, 1994, United Kingdom; Permanent Magnet Motors Technology: Design and Applications, Marcel Dekker, New York, 1996, Second Edition, 2002 (coauthor M. Wing); Linear Synchronous Motors: Transportation and Automation Systems, CRC Press, Boca Raton, Florida, 1999 (coauthor Z. Piech); and Axial Flux Permanent Magnet Machines, Springer-Kluwer, Boston, 2004 (coauthors R. Wang and M. Kamper). He is a Fellow of IEEE, full member of the International Academy of Electrical Sciences, a member of numerous steering committees of international conferences, and cited by all Marquis Who’s Whos.

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Chong Wang, Ph.D., received B.S. and M.S. degrees in Acoustics from Nanjing University, China, in 1964 and 1989, respectively. He received his Ph.D. in Mechanical Engineering from the University of New South Wales, Australia, in 1999. From 1989 to 1995, he was with the Institute of Acoustical Engineering, Northwestern Polytechnical University, China, as an assistant lecturer, lecturer, and an associate professor, where he lectured on several acoustics courses and conducted research work in noise and vibration engineering. In 1999, Dr. Wang joined the Institute for Research in Construction, National Research Council, Canada, as a Canadian Government Laboratory Visiting Fellow. Since 2000, he has been employed as a senior engineer with General Motors Corporation, United States. His research experience and interests include vibro-acoustics, room acoustics, FEA/BEA, SEA, and active noise and vibration control. He has published more than 60 papers in journals and conference proceedings in these areas. Joseph Cho Lai, Ph.D., is a professor of mechanical engineering and associate dean (research) at the Australian Defense Force Academy (ADFA) Campus of the University of New South Wales (UNSW). He was appointed in January 2004. Prior to this appointment, he was the head of the School of Aerospace and Mechanical Engineering from 2001–2003. Dr. Lai obtained a B.Sc. (United Kingdom) in Mechanical Engineering with first-class honors from the University of Hong Kong in 1975. He was awarded a Master of Engineering Science in 1978 and a Ph.D. in Mechanical Engineering in 1981 from the University of Queensland, Australia. He lectured in mechanical engineering at the University of Queensland until June 1985 when he joined the Department of Mechanical Engineering at UNSW/ADFA as a lecturer. Dr. Lai’s research interests are in turbulent shear flows and acoustics and vibration control. He has published 192 papers in journals and conference proceedings in these areas. His current work focuses on the aerodynamic propulsive mechanism of flapping wings as applied to microaerial vehicles, vibro-acoustic communication in termites, and noise and vibration control of machines, including electrical machines. He is a member of four International Standard Organization Acoustics Working Groups. He is a Fellow of the Institution of Engineers, Australia, a Fellow of the Australian Acoustical Society, and an Associate Fellow of the American Institute of Aeronautics and Astronautics.

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Generation and Radiation of Noise in Electrical Machines 1.1 Vibration, sound, and noise . . . . . . . . . . . . . . . . 1.2 Sound waves . . . . . . . . . . . . . . . . . . . . . . . 1.3 Sources of noise in electrical machines . . . . . . . . . . 1.3.1 Electromagnetic sources of noise . . . . . . . . . 1.3.2 Mechanical sources of noise . . . . . . . . . . . 1.3.3 Aerodynamic noise . . . . . . . . . . . . . . . . 1.4 Energy conversion process . . . . . . . . . . . . . . . . 1.5 Noise limits and measurement procedures for electrical machines . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Deterministic and statistical methods of noise prediction 1.7 Economical aspects . . . . . . . . . . . . . . . . . . . . 1.8 Accuracy of noise prediction . . . . . . . . . . . . . . .

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Magnetic Fields and Radial Forces in Polyphase Motors Fed with Sinusoidal Currents 2.1 Construction of induction motors . . . . . . . . . . . . . . . . . 2.2 Construction of permanent magnet synchronous brushless motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A.C. stator windings . . . . . . . . . . . . . . . . . . . . . . . 2.4 Stator winding MMF . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Single-phase stator winding . . . . . . . . . . . . . . . 2.4.2 Three-phase stator winding . . . . . . . . . . . . . . . . 2.4.3 Polyphase stator winding . . . . . . . . . . . . . . . . . 2.5 Rotor magnetic field . . . . . . . . . . . . . . . . . . . . . . . 2.6 Calculation of air gap magnetic field . . . . . . . . . . . . . . . 2.6.1 Effect of slots . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Effect of eccentricity . . . . . . . . . . . . . . . . . . .

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Deformation of the stator core . . . . . . . . Frequencies and orders of magnetic pressure Radial forces in synchronous machines with slotted stator . . . . . . . . . . . . . . . . . 2.7.6 Frequencies of vibration and noise . . . . . . Other sources of electromagnetic vibration and noise 2.8.1 Unbalanced line voltage . . . . . . . . . . . 2.8.2 Magnetostriction . . . . . . . . . . . . . . . 2.8.3 Thermal stress analogy . . . . . . . . . . . . 2.8.4 FEM model . . . . . . . . . . . . . . . . . .

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3 Inverter-Fed Motors 3.1 Generation of higher time harmonics . . . . . . . . . . . . . 3.2 Analysis of radial forces for nonsinusoidal currents . . . . . 3.2.1 Stator and rotor magnetic flux density . . . . . . . . 3.2.2 Stator harmonics of the same number . . . . . . . . 3.2.3 Interaction of stator and rotor harmonics . . . . . . . 3.2.4 Rotor harmonics of the same number . . . . . . . . 3.2.5 Frequencies and orders of magnetic pressure for nonsinusoidal currents . . . . . . . . . . . . . . . . 3.2.6 Interaction of stator harmonics of different numbers . 3.2.7 Interaction of switching frequency and higher time harmonics . . . . . . . . . . . . . . . . . . . . . . . 3.2.8 Interaction of permeance and magnetomotive force (MMF) harmonics . . . . . . . . . . . . . . . . . . 3.2.9 Rectifier harmonics . . . . . . . . . . . . . . . . . . 3.3 Higher time harmonic torques in induction machines . . . . 3.3.1 Asynchronous torques . . . . . . . . . . . . . . . . 3.3.2 Pulsating torques . . . . . . . . . . . . . . . . . . . 3.4 Higher time harmonic torques in permanent magnet (PM) brushless machines . . . . . . . . . . . . . . . . . . . . . . 3.5 Influence of the switching frequency of an inverter . . . . . 3.6 Noise reduction of inverter-fed motors . . . . . . . . . . . . 4 Torque Pulsations 4.1 Analytical methods of instantaneous torque calculation . . . 4.2 Numerical methods of instantaneous torque calculation . . . 4.3 Electromagnetic torque components . . . . . . . . . . . . . 4.4 Sources of torque pulsations . . . . . . . . . . . . . . . . . 4.5 Higher harmonic torques of induction motors . . . . . . . . 4.6 Cogging torque in permanent magnet (PM) brushless motors 4.6.1 Air gap magnetic flux density . . . . . . . . . . . . 4.6.2 Calculation of cogging torque . . . . . . . . . . . . 4.6.3 Simplified cogging torque equation . . . . . . . . . 4.6.4 Influence of eccentricity . . . . . . . . . . . . . . . Copyright © 2006 Taylor & Francis Group, LLC

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xvii 4.6.5 Calculation and comparison with measurements . . . . Torque ripple due to distortion of EMF and current waveforms in permanent magnet (PM) brushless motors . . . . . . . . . . Tangential forces vs. radial forces . . . . . . . . . . . . . . . Minimization of torque ripple in PM brushless motors . . . . . 4.9.1 Slotless windings . . . . . . . . . . . . . . . . . . . . 4.9.2 Skewing stator slots . . . . . . . . . . . . . . . . . . 4.9.3 Shaping stator slots . . . . . . . . . . . . . . . . . . . 4.9.4 Selection of the number of stator slots . . . . . . . . . 4.9.5 Shaping PMs . . . . . . . . . . . . . . . . . . . . . . 4.9.6 Skewing PMs . . . . . . . . . . . . . . . . . . . . . . 4.9.7 Shifting PM segments . . . . . . . . . . . . . . . . . 4.9.8 Selection of PM width . . . . . . . . . . . . . . . . . 4.9.9 Magnetization of PMs . . . . . . . . . . . . . . . . . 4.9.10 Creating magnetic circuit asymmetry . . . . . . . . .

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Stator System Vibration Analysis 5.1 Forced vibration . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Simplified calculation of natural frequencies of the stator system . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Improved analytical method of calculation of natural frequencies . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Natural frequency of the stator core . . . . . . . . . . . 5.3.2 Natural frequency of a frame with end bells . . . . . . . 5.3.3 Natural frequency of a stator core–frame system . . . . 5.3.4 Effect of the stator winding and teeth . . . . . . . . . . 5.3.5 Analytical calculation of natural frequencies for a stator core-winding-frame system . . . . . . . . . . . . . . . 5.4 Numerical verification . . . . . . . . . . . . . . . . . . . . . . 5.4.1 FEM modeling . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Comparison of analytical calculations with the FEM . .

6 Acoustic Calculations 6.1 Sound radiation efficiency . . . . . . . . . . . . . 6.2 Plane radiator . . . . . . . . . . . . . . . . . . . . 6.2.1 Infinite plates . . . . . . . . . . . . . . . . 6.2.2 Finite plates in bending motion . . . . . . 6.3 Infinitely long cylindrical radiator . . . . . . . . . 6.4 Finite length cylindrical radiator . . . . . . . . . . 6.4.1 Acoustically thin shells . . . . . . . . . . . 6.4.2 Acoustically thick shells . . . . . . . . . . 6.4.3 Modal radiation efficiencies of acoustically thick shells . . . . . . . . . . . . . . . . . 6.4.4 Modal averaged radiation efficiency . . . . 6.4.5 Validity of using an infinite length model . Copyright © 2006 Taylor & Francis Group, LLC

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Noise and Vibration of Mechanical and Aerodynamic Origin 7.1 Mechanical noise due to shaft and rotor irregularities . . . 7.2 Bearing noise . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Rolling bearings . . . . . . . . . . . . . . . . . . 7.2.2 Sleeve bearings . . . . . . . . . . . . . . . . . . . 7.3 Noise due to toothed gear trains . . . . . . . . . . . . . . 7.4 Aerodynamic noise . . . . . . . . . . . . . . . . . . . . . 7.5 Mechanical noise generated by the load . . . . . . . . . .

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Effects of boundary conditions on the radiation efficiency . . . . . . . . . . . . . . . . . . . . Calculations of sound power level . . . . . . . . . . . 6.5.1 Sound power radiated from a stator . . . . . . 6.5.2 Total sound power of an induction motor . . . 6.5.3 Permanent magnet synchronous motors . . . .

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Acoustic and Vibration Instrumentation 8.1 Measuring system and transducers . . . . . . . . . . . . . 8.2 Measurement of sound pressure . . . . . . . . . . . . . . 8.2.1 Choice of microphones . . . . . . . . . . . . . . . 8.2.2 The sound pressure sensor–condenser microphone 8.2.3 Sound level meter . . . . . . . . . . . . . . . . . . 8.2.4 Acoustic calibrator . . . . . . . . . . . . . . . . . 8.2.5 Level recorder . . . . . . . . . . . . . . . . . . . 8.3 Acoustic measurement procedure . . . . . . . . . . . . . . 8.3.1 Effect of the operator on measurement results . . . 8.3.2 Measurement position . . . . . . . . . . . . . . . 8.3.3 Standing waves . . . . . . . . . . . . . . . . . . . 8.3.4 Measurements of ambient sound pressure levels . . 8.3.5 Corrections for background sound during source measurements . . . . . . . . . . . . . . . . . . . . 8.3.6 Polar plots . . . . . . . . . . . . . . . . . . . . . 8.4 Vibration measurements . . . . . . . . . . . . . . . . . . 8.4.1 Theory of operation of vibration-measuring transducer . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Characteristics of piezoelectric accelerometers . . 8.4.3 Other vibration transducers . . . . . . . . . . . . . 8.5 Frequency analyzers . . . . . . . . . . . . . . . . . . . . 8.6 Sound power and sound pressure . . . . . . . . . . . . . . 8.7 Indirect methods of sound power measurement . . . . . . 8.7.1 Determination of sound power in an anechoic/ semianechoic room . . . . . . . . . . . . . . . . . 8.7.2 Reverberation room . . . . . . . . . . . . . . . . . 8.7.3 Juxtaposition principle using a reference sound source . . . . . . . . . . . . . . . . . . . .

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xix Direct method of sound power measurement — sound intensity technique . . . . . . . . . . . . . . . . . . . . 8.8.1 Historical perspective . . . . . . . . . . . . . . . 8.8.2 Theoretical background . . . . . . . . . . . . . 8.8.3 Sound intensity probe . . . . . . . . . . . . . . 8.8.4 External noise suppression . . . . . . . . . . . . 8.8.5 Error considerations . . . . . . . . . . . . . . . 8.8.6 Dynamic capability and pressure-intensity index Standard for testing acoustic performance of rotating electrical machines . . . . . . . . . . . . . . . . . . . . 8.9.1 Background . . . . . . . . . . . . . . . . . . . . 8.9.2 Acoustic tests on an induction motor . . . . . . .

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9 Numerical Analysis 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 FEM model for radial magnetic pressure . . . . . . . . . . . . . 9.2.1 Induction motor . . . . . . . . . . . . . . . . . . . . . . 9.2.2 Permanent magnet synchronous motor . . . . . . . . . . 9.3 FEM for structural modeling . . . . . . . . . . . . . . . . . . . 9.4 BEM for acoustic radiation . . . . . . . . . . . . . . . . . . . . 9.4.1 Governing equation and boundary conditions . . . . . . 9.4.2 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Radiating sphere . . . . . . . . . . . . . . . . . . . . . 9.4.5 Application to the prediction of radiated acoustic power from an inverter-fed induction motor . . . . . . . . . . . 9.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Statistical Energy Analysis 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 10.2 Power flow between linearly coupled oscillators . . . . 10.2.1 Two coupled oscillators . . . . . . . . . . . . 10.2.2 Three series coupled oscillators . . . . . . . . 10.2.3 Energy exchange between groups of oscillators 10.3 Coupled multimodal systems . . . . . . . . . . . . . . 10.3.1 General SEA equations . . . . . . . . . . . . . 10.3.2 SEA model establishment . . . . . . . . . . . 10.3.3 SEA parameters . . . . . . . . . . . . . . . . 10.3.4 Limitations of SEA . . . . . . . . . . . . . . . 10.4 Experimental SEA . . . . . . . . . . . . . . . . . . . 10.4.1 General theory . . . . . . . . . . . . . . . . . 10.4.2 Recent developments . . . . . . . . . . . . . . 10.5 Application to electrical motors . . . . . . . . . . . . . 10.5.1 Subsystems of a motor structure . . . . . . . . 10.5.2 Internal and coupling loss factors . . . . . . .

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Contents 10.5.3 Input power to the stator . . . . . . . . . . . . . . . . . . 293 10.5.4 Sound power radiated from the motor structure . . . . . . 295

11 Noise Control 11.1 Mounting . . . . . . . . . . . . . . . . . . . . . 11.1.1 Foundation . . . . . . . . . . . . . . . . 11.1.2 Principles of vibration and shock isolation 11.1.3 Vibration limits . . . . . . . . . . . . . . 11.1.4 Shaft alignment . . . . . . . . . . . . . . 11.2 Standard methods of noise reduction . . . . . . . 11.3 Active noise and vibration control . . . . . . . . 11.3.1 Principles of active noise control . . . . . 11.3.2 Induction motor acoustic noise reduction 11.3.3 Active vibration isolation . . . . . . . . .

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Appendix A Basics of Acoustics A.1 Sound field variables and wave equations A.2 Sound radiation from a point source . . . A.3 Decibel levels and their calculations . . . A.4 Spectrum analysis . . . . . . . . . . . . .

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Appendix B Permeance of Nonuniform Air Gap 327 B.1 Permeance calculation . . . . . . . . . . . . . . . . . . . . . . . 327 B.2 Eccentricity effect . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Appendix C Magnetic Saturation

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Appendix D Basics of Vibration 337 D.1 A mass–spring–damper oscillator . . . . . . . . . . . . . . . . . . 337 D.2 Lumped parameter systems . . . . . . . . . . . . . . . . . . . . . 339 D.3 Continuous systems . . . . . . . . . . . . . . . . . . . . . . . . . 342 Symbols and Abbreviations

347

Bibliography

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1 Generation and Radiation of Noise in Electrical Machines 1.1

Vibration, sound, and noise

Vibration is a limited reciprocating motion of a particle of an elastic body or medium in alternately opposite directions from its position of equilibrium, when that equilibrium has been disturbed. In order to vibrate, the body or system must have two characteristics: elasticity and mass. The amplitude of vibration is the maximum displacement of a vibrating particle or body from its position of rest. Sound is defined as vibrations transmitted through an elastic solid, liquid, or gas with frequencies in the approximate range of 20 to 20,000 Hz, capable of being detected by human ears. Pitch is the perceived tone of a sound which is determined by the sound wave frequency. A sound with a high frequency (short wavelength) has a high pitch, while a sound with low frequency (long wavelength) has a low pitch. Noise is disagreeable or unwanted sound. Distinction is made between airborne noise and noise traveling through solid objects. Airborne noise is the noise caused by the movement of large volumes of air and the use of high pressure. Structure-borne noise is the noise carried by means of vibrations of solid objects.

1.2

Sound waves

A sound wave is generated by a vibrating object and can be defined as a mechanical disturbance advancing with a finite speed through a medium. Sound waves are small-amplitude adiabatic oscillations characterized by wavespeed, wavelength, frequency, and amplitude (Appendix A). In air, sound waves are longitudinal waves, that is, with displacement in the direction of propagation. In other words, the motion of the individual particles of the medium is in the direction that is 1 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors

parallel to the direction of the energy transport. Transverse waves are those with vibrations perpendicular to the direction of travel of the wave and exist in the elastic medium. Examples of transverse waves include waves on a string and electromagnetic waves. Only transverse waves can be polarized, i.e., can have orientation. Polarized waves oscillate in only one direction perpendicular to the line of travel. For example, the polarization of an electromagnetic wave is defined as the orientation of the electric field vector. The electric field vector is perpendicular to both the direction of travel and the magnetic field vector. Polarized waves can be formed from unpolarized waves by passing them through some polarizing process, e.g., a train of unpolarized waves in a rope can be polarized by passing them through a narrow physical gap. Sound waves cannot be polarized. Unpolarized waves can oscillate in any direction in the plane perpendicular to the direction of travel and have no preferred plane of polarization. All sound waves have common behavior under a number of standard situations and exhibit: • reflection, i.e., the phenomenon of a propagating wave being thrown back from a surface between two media with different mechanical properties; • refraction, i.e., the change in direction of a propagating wave when passing from one medium to another; • diffraction, i.e., the process of spreading out of waves, e.g., when they travel through a small slit or go around an obstacle; • scattering, i.e., the change in direction of motion; • interference, i.e., mutual influence of two waves, e.g., the addition of two waves that come in to contact with each other; • absorption, i.e., the incident sound that strikes a material that is not reflected back; • dispersion, i.e., the splitting up of a wave depending on frequency. Sound amplitude can be measured as sound pressure level (SPL), sound intensity level (SIL), sound power level (SWL), and sound energy density (SED) (Appendix A). A human ear can perceive sound waves of sufficient intensity whose frequencies are approximately within the limits from 16 to 20, 000 Hz (audio-frequency range). There is a minimum sound intensity for a given frequency at which the sound can be perceived by the human ear. The minimum sound intensity is different for different frequencies and is called the threshold of audibility. Figure 1.1 shows the audibility zone for the whole audio-frequency range. The range of the sound intensity that can be perceived by the ear is from 10−12 to 1 W/m2 corresponding to 20 µPa sound pressure. The maximum sound intensity at which the ear feels a pain is called the threshold of pain. Sound amplitudes that are extremely loud (at the threshold of pain) have pressure amplitudes of only 100 Pa. Some environmental noise levels are compared in Figure 1.2. Typical sound power levels for common sounds are also given in Table 1.1

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Generation and Radiation of Noise in Electrical Machines 150

Zo ne ty

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Figure 1.1 Sound intensity and audibility zone as a function of frequency. dB 140 Threshold of Pain 130 120 Jet Airliner 110 100 Heavy City Traﬃc 90 Beginning of Hearing Damage 80 70 60 50 40

Average Street Traﬃc Normal Conversation Business Oﬃce Living Room

30 20 10 0

Threshold of Audibility

Figure 1.2 Comparison of some environmental noise levels.

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Noise of Polyphase Electric Motors Table 1.1 Typical sound power levels. Source of noise Quietest audible sound for persons under normal conditions Rustle of leaves Soft whisper, room in a quiet dwelling at midnight Voice, low Mosquito buzzing Department store, clothing department Modern elevator propulsion motor Normal conversation Bird singing Large department store Busy restaurant or canteen Voice, conversation 10-kW, 4-pole cage induction motor Normal street traffic Pneumatic tools Alarm clock ringing Buses, trucks, motorcycles Small air compressors Loud symphonic music Lawn mower Your boss complaining Heavy city traffic Air compressor Heavy diesel vehicle Permanent hearing loss (exposed full-time) Car on highway Steel plate falling Magnetic drill press Vacuum pump Hard rock music Jet passing overhead Jackhammer Jolt squeeze hammer Jet plane taking off Saturn rocket

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Sound power level, dB(A) 10 15 30 40 45 48 50 55 60 70 75 80

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1.3 Sources of noise in electrical machines The frequency of interest for vibration is generally within 0 to 1000 Hz, and for noise is over 1000 Hz. Vibration and noise produced by electrical machines can be divided into three categories: • electromagnetic vibration and noise associated with parasitic effects due to higher space and time harmonics, eccentricity, phase unbalance, slot openings, magnetic saturation, and magnetostrictive expansion of the core laminations; • mechanical vibration and noise associated with the mechanical assembly, in particular bearings; • aerodynamic vibration and noise associated with flow of ventilating air through or over the motor. The load induced sources of noise include: • noise due to coupling of the machine with a load, e.g., shaft misalignment, belt transmission, elevator sheave with ropes, tooth gears, coupling, reciprocating compressor; • noise due to mounting the machine on foundation or other structure. The noise from its source is transmitted through the medium (structure, air) to the recipient (human being, sensor) of the noise. The process of noise generation and transmission in electrical machines is illustrated in Figure 1.3. Basics of acoustics are explained in Appendix A.

1.3.1

Electromagnetic sources of noise

Electromagnetic vibration and noise are caused by generation of electromagnetic fields (Chapter 2). Both stator and rotor excite magnetic flux density waves in the air gap. If the stator produces Bm1 cos(ω1 t + kα + φ1 ) magnetic flux density wave and rotor produces Bm2 cos(ω2 t +lα + φ2 ) magnetic flux densisty wave, then their product is 0.5Bm1 Bm2 cos[ω1 + ω2 )t + (k + l)α + (φ1 + φ2 )] + 0.5Bm1 Bm2 cos[ω1 − ω2 )t + (k − l)α + (φ1 − φ2 )]

(1.1)

where Bm1 and Bm2 are the amplitudes of the stator and rotor magnetic flux density waves, ω1 and ω2 are the angular frequencies of the stator and rotor magnetic fields, φ1 and φ2 are phases of the stator and rotor magentic flux desnity waves, k = 1, 2, 3, . . ., and l = 1, 2, 3, . . .. The product expressed by Equation 1.1 is proportional to magnetic stress wave in the air gap with amplitude Pmr = 0.5Bm1 Bm2 , angular frequency ωr = ω1 ±ω2 , order r = k ±l and phase φ1 ±φ2 . The magnetic stress (or magnetic pressure) wave acts in radial directions on the stator and rotor active surfaces causing the deformation and, hence, the vibration and noise. The slots, distribution of windings in slots, input current waveform distortion, air gap permeance fluctuations, rotor eccentricity, and phase unbalance give

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Figure 1.3 Noise generation and transmission in electrical machines.

rise to mechanical deformations and vibration. Magnetomotive force (MMF) space harmonics, time harmonics, slot harmonics, eccentricity harmonics, and saturation harmonics, produce parasitic higher harmonic forces and torques. Especially, radial force waves in a.c. machines, which act both on the stator and rotor, produce deformation of the magnetic circuit. The stator-frame (or stator-enclosure) structure is the primary radiator of the machine noise. If the frequency of the radial force is close to or equal to any of the natural frequencies of the stator–frame system, resonance occurs, leading to the stator system deformation, vibration, and acoustic noise. Magnetostrictive noise of electrical machines in most cases can be neglected due to low frequency 2 f and high order r = 2 p of radial forces, where f is the fundamental frequency and p is the number of pole pairs. However, radial forces due to the magnetostriction can reach about 50% of radial forces produced by the air gap magnetic field. In inverter fed motors, parasitic oscillating torques are produced due to higher time harmonics in the stator winding currents. These parasitic torques are, in general, greater than oscillating torques produced by space harmonics. Moreover,

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the voltage ripple of the rectifier is transmitted through the intermediate circuit to the inverter and produces another kind of oscillating torque [200].

1.3.2

Mechanical sources of noise

Mechanical vibration and noise (Chapter 7) is mainly due to bearings, their defects, journal ovality, sliding contacts, bent shaft, rotor unbalance, shaft misalignment, couplings, U-joints, gears etc. The rotor should be precisely balanced as it can significantly reduce the vibration. The rotor unbalance causes rotor dynamic vibration and eccentricity which in turn results in noise emission from the stator, rotor, and rotor support structure. Both rolling and sleeve bearings are used in electrical machines. The noise due to rolling bearings depends on the accuracy of bearing parts, mechanical resonance frequency of the outer ring, running speed, lubrication conditions, tolerances, alignment, load, temperature, and presence of foreign materials. The noise due to sleeve bearings is generally lower than that of rolling bearings. The vibration and noise produced by sleeve bearings depends on the roughness of sliding surfaces, lubrication, stability and whirling of the oil film in the bearing, manufacture process, quality, and installation.

1.3.3

Aerodynamic noise

The basic source of noise of an aerodynamic nature (Chapter 7) is the fan. Any obstacle placed in the air stream produces noise. In nonsealed motors, the noise of the internal fan is emitted by the vent holes. In totally enclosed motors, the noise of the external fan predominates. According to the spectral distribution of the fan noise, there is broad-band noise (100 to 10, 000 Hz) and siren noise (tonal noise). Siren noise can be eliminated by increasing the distance between the impeller and the stationary obstacle.

1.4 Energy conversion process Figure 1.4 shows how the electrical energy is converted into acoustic energy in an electrical machine. The input current interacts with the magnetic field producing high-frequency forces that act on the inner stator core surface (Figure 1.5). These forces excite the stator core and frame in the corresponding frequency range and generate mechanical vibration and noise. As a result of vibration, the surface of the stator yoke and frame displaces with frequencies corresponding to the frequencies of forces. The surrounding medium (air) is excited to vibrate, too, and generates acoustic noise. The radiated acoustic power is very small, approximately 10−6 to 10−4 W for an electrical motor rated below 10 kW. It is, therefore, not easy to calculate the acoustic power with reasonable accuracy.

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Electric Power Forces Supply Electromagnetic System

Mechanical System

Displacement

Acoustic Environment

Acoustic Noise

Figure 1.4 Conversion of electric energy into acoustic energy in electrical machines. The stator and frame assembly, as a mechanical system, is characterized by a distributed mass M, damping C, and stiffness K . The electromagnetic force waves excite the mechanical system to generate vibration. The amplitude of vibration is a function of the magnitude and frequency of those forces (Appendix D). The mechanical system can be simply described by a lumped parameter model with N degrees of freedom in the following matrix form ¨ + [C]{q} [M]{q} ˙ + [K ]{q} = {F(t)}

(1.2)

where q is an (N , 1) vector expressing the displacement of N degrees of freedom, {F(t)} is the force vector applying to the degrees of freedom, [M] is the mass matrix, [C] is the damping matrix and [K ] is the stiffness matrix. Equation 1.2

Figure 1.5 Mechanism of generation of vibration and noise in electrical machines.

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can be solved using a structural finite element method (FEM) package. In practice, there are difficulties with predictions of the [C] matrix for laminated materials, physical properties of materials, and errors in calculation of magnetic forces [213].

1.5

Noise limits and measurement procedures for electrical machines

Acoustic quantities (Appendix A) can be expressed in terms of the sound pressure level (SPL) or sound power level (SWL). The sound pressure level is the most common descriptor used to characterize the loudness of an ambient sound level. In general, it is more complicated to measure the sound power level than the sound pressure level. The sound power level measurement is independent of the surface of the machine and environmental conditions. According to the National Electrotechnical Manufacture’s Association (NEMA) [162], the sound pressure level L p A can be related to the sound power level L W A in dB(A), as follows L p A = L W A − 10 log10

2πrd2 S0

(1.3)

where L p A is the average sound pressure level in a free-field over a reflective plane on a hemispherical surface at 1 m distance from the machine, rd = 1.0 + 0.5lm , lm is the maximum linear dimension of the tested machine in meters, and S0 = 1.0 m2 . The noise of electrical machines depends on the type of the machine, its topology, size, design, construction, enclosure, materials, manufacturing, rated power, speed, tolerances, mounting, support, foundation, coupling, bearings, supply, load, etc. Some consequences of noise as, for example, manufacturing, mounting or support are very difficult to predict. In general, the equations for sound pressure level or sound power level as functions of rotational speed n, rated output power Pout , or torque T have the following forms: L p1 = A1 + B1 log10 n

(1.4)

L p2 = A2 + B2 log10 Pout L p3 = A3 + B3 log10 T

(1.5) (1.6)

where A1 , A2 , A3 , B1 , B2 , and B3 are constants. When a motor is tested at no load under conditions specified by [162], the sound power level of the motor shall not exceed values given in Tables 1.2 and 1.3. The enclosures of motors are an open drip proof machine (ODP) type, totally enclosed fan cooled machine (TEFC) type, and weather protected type II machine (WPII) type. The WPII machine is a guarded machine with its ventilating passages at both intake and discharge so arranged that high velocity air and airborne particles blown into the machine by storms or high winds can be discharged without entering

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Noise of Polyphase Electric Motors

Table 1.2 Maximum A-weighted sound power levels L w A in dB(A) at no load for motors with rated speeds 1200 rpm and less according to NEMA [162]. Rated power kW hp 0.37 0.5 0.5 0.75 0.75 1.0 1.1 1.5 1.5 2.0 2.2 3.0 3.0 5.0 5.5 7.5 7.5 10 11 15 15 20 17 25 22 30 30 40 40 50 45 60 55 75 75 100 100 125 110 150 150 200 185 250 220 300 260 350 300 400 350 450 370 500 450 600 520 700 600 800 670 900 750 1000 930 1250 1,100 1500 1,300 1750 1,500 2000

900 rpm and less ODP TEFC WPII 67 67 67 67 69 69 69 69 70 72 70 72 73 76 73 76 76 80 76 80 79 83 79 83 81 86 81 86 84 89 84 89 87 93 87 93 93 96 92 95 97 92 95 97 92 95 97 92 98 100 96 98 100 96 98 100 96 99 102 98 99 102 98 99 102 98 99 102 98 101 105 100 101 105 100 101 105 100 101 105 100 103 107 102 103 107 102 103 107 102

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901 to 1200 rpm ODP TEFC WPII 65 65 67 67 72 72 76 76 81 81 83 83 86 86 88 88 91 91 96 99 99 99 99 102 102 102 102 102 105 105 105 105 107 107 107

64 64 67 67 71 71 75 75 80 80 83 83 86 86 90 90 94 94 98 100 100 100 100 103 103 103 103 103 106 106 106 106 109 109 109

97 97 97 97 99 99 99 99 99 101 101 101 101 103 103 103

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Table 1.3 Maximum A-weighted sound power levels L w A in dB(A) at no load for motors with rated speeds 1201 to 3600 rpm according to NEMA [162]. Rated power kW hp 0.75 1.0 1.1 1.5 1.5 2.0 2.2 3.0 4.0 5.0 5.5 7.5 7.5 10 11 15 15 20 17 25 22 30 30 40 40 50 45 60 55 75 75 100 100 125 110 150 150 200 185 250 220 300 260 350 300 400 335 450 370 500 450 600 520 700 600 800 670 900 750 1000 930 1250 1,100 1500 1,300 1750 1,500 2000 1,700 2250 1,850 2500 2,250 3000

1201 to 1800 rpm ODP TEFC WPII 70 70 70 70 70 70 72 74 73 74 76 79 76 79 80 84 80 84 80 88 80 88 84 89 84 89 86 95 86 95 89 98 89 100 93 100 93 103 103 105 99 103 105 99 103 105 99 103 105 99 106 108 102 106 108 102 106 108 102 106 108 102 108 111 104 108 111 104 108 111 104 108 111 104 109 113 105 109 113 105 109 113 105 109 113 105 110 115 106 110 115 106

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1801 to 3600 rpm ODP TEFC WPII 76 76 76 80 80 82 82 84 84 86 86 89 89 94 94 98 98 101 101 107 107 107 107 110 110 110 110 111 111 111 111 112 112 112 112 114

85 85 88 88 91 91 94 94 94 94 100 100 101 101 102 104 104 107 107 110 110 110 110 113 113 113 113 116 116 116 116 118 118 118 118 120

102 102 102 102 105 105 105 105 106 106 106 106 107 107 107 107 109

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Table 1.4 Expected incremental increase over no-load condition in A-weighted sound power levels L w A , dB(A) for rated load condition for single-speed, threephase, cage induction motors according to NEMA [162] and IEC 60034-9 Standards [93]. Rated power kW hp 1 to 11 1.0 to 15 1 to 37 15 to 50 37 to 110 50 to 150 110 to 400 150 to 500

2p = 2 2 2 2 2

Number of poles 2p = 4 2p = 6 5 7 4 6 3 5 3 4

2p = 8 8 7 6 5

the internal ventilating passages leading directly to the electric parts of the machine itself. The sound power level at rated load should be adjusted according to Table 1.4. The increase in the sound power level under load is mostly due to the change in the air gap magnetic flux density harmonic amplitudes. This effect can be expressed by the following equation [137] Bload 2 L W = 10 log10 (1.7) Bnoload Table 1.5 shows maximum sound pressure level at 1 m from the machine surface according to IEC 60034-9 Standards [93]. Table 1.6 shows maximum sound power level according to IEC 60034-9 Standards [93]. The sound pressure level spectrum is the distribution of effective sound pressures measured as a function of frequency in specified frequency bands. It can also be defined as the resolution of a signal into components, each of different frequency and different amplitude (Figure 1.6). If the sound pressure level spectrum is given in a form of the Fourier series p=

kmax

Pmk sin(ωk t + φk )

(1.8)

k=1

where Pmk is the amplitude of the kth harmonic, ωk = 2π k f is the angular fequency of the kth harmonic, and φk is the phase angle for the kth harmonic, the overall sound pressure level is calculated as a sum of amplitudes squared, i.e., P=

kmax

2 Pmk

W.

(1.9)

k=1

The sound pressure level in dB is then calculated according to Equation A.25. The broad-band noise is the noise in which the acoustic energy is distributed over a relatively wide range of frequencies. The spectrum is generally smooth and continuous. The narrow-band noise is the noise in which the acoustic energy is concentrated in a relatively narrow range of frequencies. The spectrum will generally show a localized “hump” or peak in amplitude. Narrow-band sound may be superimposed on broad-band sound.

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Table 1.5 IEC 60034-9 limits for sound pressure level at 1 m from machine surface, dB(A) [93]. Rated power kW Pout < 1.1 1.1 < Pout < 2.2 2.2 < Pout < 5.5 5.5 < Pout < 11 11 < Pout < 22 22 < Pout < 37 37 < Pout < 55 55 < Pout < 110 110 < Pout < 220 220 < Pout < 400 Rated power kW Pout < 1.1 1.1 < Pout < 2.2 2.2 < Pout < 5.5 5.5 < Pout < 11 11 < Pout < 22 22 < Pout < 37 37 < Pout < 55 55 < Pout < 110 110 < Pout < 220 220 < Pout < 400

1.6

n < 960 rpm ODP TEFC 67 69 72 72 75 75 78 77.5 79.5 78.5 80.5 82 94 85 87 86 88 1900 < n < 2360 ODP TEFC 74 78 82 81 86 83.5 87.5 85.5 89.5 88 94 90.5 93.5 93 96 94 98

960 < n < 1320 ODP TEFC 70 70 74 75 78 78 82 80.5 83.5 82.5 85.5 85 89 87 91 89 92 2360 < n < 3150 ODP TEFC 75 80 83 84 87 86.5 90.5 88.5 92.5 93 96 92.5 93.5 95 98 95 99

1320 < n < 1900 ODP TEFC 71 73 77 81 81 81.5 85.5 83 86 86 88 88.5 91.5 90.5 93.5 92.5 95.5 3150 < n < 3750 ODP TEFC 77 82 85 87 90 90 93 92 95 95.5 98.5 95 98 96 100 98 102

Deterministic and statistical methods of noise prediction

In the efforts to predict the noise emitted from an electric machine, there are two approaches: deterministic and statistical methods. In the deterministic method, shown in Figure 1.7a, the electromagnetic forces acting on a motor structure have to be calculated from the input currents and voltages using an electromagnetic analytical model [254] or the FEM model [226]. The vibration characteristics are then determined using a structural model normally based on the FEM [223, 230]. By using the vibration velocities on the motor structure predicted from the structural model, the radiated sound power level can then be calculated on the basis of an acoustic model.1 The acoustic model may be formulated using either the FEM or boundary-element method (BEM). Generally, for calculating 1 The only commercial FEM software for calculating the electromagnetic field, stator vibration mode shapes, and acoustic noise of electrical machines is JMAG-Studio developed by Japan Research Institute, Ltd., Engineering Technology Division, Tokyo-Osaka-Nagoya, http://www.iri.co.jp/proeng/jmag/e/jmg/.

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Noise of Polyphase Electric Motors Table 1.6 IEC 60034-9 limits for sound power level, dB(A) [93].

Rated power n < 960 rpm 960 < n < 1320 kW ODP TEFC ODP TEFC Pout < 1.1 76 79 1.1 < Pout < 2.2 79 80 2.2 < Pout < 5.5 82 84 5.5 < Pout < 11 82 85 85 88 11 < Pout < 22 86 89 89 93 22 < Pout < 37 89 91 92 95 37 < Pout < 55 90 92 94 97 55 < Pout < 110 94 96 97 101 110 < Pout < 220 98 100 100 104 220 < Pout < 400 100 102 103 106 Rated power 1900 < n < 2360 2360 < n < 3150 kW ODP TEFC ODP TEFC Pout < 1.1 83 84 1.1 < Pout < 2.2 87 89 2.2 < Pout < 5.5 92 93 5.5 < Pout < 11 91 96 94 97 11 < Pout < 22 94 98 97 101 22 < Pout < 37 96 100 99 103 37 < Pout < 55 99 103 101 105 55 < Pout < 110 102 105 104 107 110 < Pout < 220 105 108 107 110 220 < Pout < 400 107 111 108 112

Figure 1.6 Sound pressure level spectrum.

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1320 < n < 1900 ODP TEFC 80 83 87 88 91 92 96 94 97 97 99 99 103 103 106 106 109 3150 < n < 3750 ODP TEFC 86 91 95 97 100 100 103 102 105 104 107 106 109 108 112 110 114

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Generation and Radiation of Noise in Electrical Machines Input Data

Input Data

Electromagnetic Force Model

Electromagnetic Force Model

Force

15

Force Mobility Model Input Power

Structural Model Vibration Velocity Acoustic Model

Sound Power Level (a)

Statistical Energy Model Vibration Power Radiation Eﬃciency Models

Sound Power Level (b)

Figure 1.7 Flowcharts for noise prediction: (a) deterministic method; (b) statistical method. the noise radiated into a space, the BEM is preferred because only the surface of the motor needs to be discretized and the space does not have to be discretized. Although, the analytical and FEM/BEM numerical approaches seem to work well, there are quite a number of limitations for it to be applied in practice (Section 1.8). In the deterministic approach, sometimes, simplified models can be utilized and analytical calculations can be implemented by writing a Mathcad2 or Mathematica3 computer program for fast prediction of the sound power level spectrum generated by magnetic forces. The accuracy due to physical errors may not be high, but the time of computation is very short and it is very easy to introduce and manage the input data set. The main program consists of the input data file, electromagnetic module, structural module (natural frequencies of the stator system), and acoustic module. The following effects can be included: phase current unbalance, higher space 2 Industry

standard technical calculation tool for professionals, educators, and college students. integrated technical computing environment used by scientists, engineers, analysts, educators, and college students. 3 Fully

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Noise of Polyphase Electric Motors

harmonics, higher time harmonics, slot openings, slot skew, rotor static eccentricity, rotor dynamic eccentricity, armature reaction, magnetic saturation. An auxiliary program calculates the torque ripple, converts the tangential magnetic forces into equivalent radial forces, and transfers radial forces due to the torque ripple to the main program. The input data file contains the dimensions of the machine and its stator and rotor magnetic circuit, currents (including unbalanced system and higher time harmonics), winding parameters, material parameters (specific mass, Young modulus, Poisson’s ratio), speed, static and dynamic eccentricity, skew, damping factor as a function of frequency, correction factors, e.g., for the stator systems natural frequencies, maximum force order taken into consideration, minimum magnetic flux density to exclude all magnetic flux density harmonics below the selected margin. The rotor magnetic flux density waveforms are calculated on the basis of MMF waveforms and permeances of the air gap. Magnetic forces are calcualated on the basis of Maxwell stress tensor. The natural frequencies of the stator system are calculated with the aid of equations given in Chapter 5. Those values can be corrected with the aid of correction factors obtained, e.g., from the FEM structural package. Then, using the damping coeffcient as a function of frequency, amplitudes of radial velocities are calculated. The damping factor affects significantly the accuracy of computation. Detailed research has shown that the damping factor is a nonlinear function of natural frequencies. The radiation efficiency factor (Chapter 6), acoustic impedance of the air and amplitudes of radial velocities give the sound power level spectrum (narrow band noise). The overall noise can be found on the basis of Equation 1.9. The overall sound power level calculated in such a way is lower than that obtained from measurements because computations include only the noise of magnetic origin (mechanical noise caused by bearings, shaft misalignment, and fan is not taken into account) and usually, the calculation is done for low number of harmonics of magnetic flux density waves. The FEMs/BEMs, by their nature, are limited to low frequencies. This is because the number of elements required for the model increases by a factor of 8 when the upper frequency of interest is doubled and the number of vibration modes increases significantly with frequency [230]. If the FEMs/BEMs are applied to a large motor for frequencies up to 10,000 Hz, the number of elements and the computing time required will become prohibitive, as discussed in [223]. A method that is particularly suitable for calculations of noise and vibration at high frequencies is the so-called statistical energy analysis (SEA) (Chapter 10), which has been applied with success to a number of mechanical systems such as ship, car, and aircraft structures [140]. This method, however, was applied for the first time to electrical motors in 1999 [43, 223, 230]. The method basically involves dividing a structure (such as a motor) into a number of subsystems and writing the energy balance equations for each subsystem, thus allowing the statistical distribution of energies over various frequency bands to be determined. This method is normally valid for high frequencies where the modal overlap is high [140]. An outline of the calculation procedure using the statistical method is given in Figure 1.7b. The main advantage of the statistical approach is that it does not require all the

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17

details to be modeled. The accurate distribution of the electromagnetic force might not be so important; only the total force in a frequency band is required. Thus, the electromagnetic force needs not be calculated using a FEM model and an approach such as that adopted by Cho and Kim [31] might be suitable. By considering the motor as a simple cylindrical shell, the input power due to this electromagnetic force can be formulated using an analytical ”mobility” model. By invoking a statistical energy model, this input power can then be distributed as vibrational power to different subsystems which make up the motor. If the sound radiation efficiencies of these subsystems are known, then the sound power due to each subsystem can be calculated. Since a motor structure can be decomposed into simple structural elements such as cylindrical shells, plates, and beams, the radiation efficiencies of these simple structural elements can be determined analytically, as depicted in the radiation efficiencies model in Figure 1.7b.

1.7

Economical aspects

Figure 1.8 shows the distribution of the magnetic flux density in the magnetic circuit of a 4-pole permanent magnet (PM) brushless machine. The magnetic flux density in the stator return path (yoke) is proportional to the magnetic flux density in the air gap and can be a measure of both the noise of electromagnetic origin and machine 1.8795e+000 1.8012e+000 1.7229e+000 1.6446e+000 1.5663e+000 1.4879e+000 1.4096e+000 1.3313e+000 1.2530e+000 1.1747e+000 1.0964e+000 1.0181e+000 9.3975e−001 8.6144e−001 7.8313e−001 7.0481e−001 6.2650e−001 5.4819e−001 4.6988e−001 3.9156e−001 3.1325e−001 2.3494e−001 1.5663e−002 7.8313e−002 0.0000e+000

Figure 1.8 (See color insert following page 236.) Distribution of the magnetic flux density in the cross section of a 4-pole brushless machine with surface PMs, as obtained from the 2D FEM.

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Noise of Polyphase Electric Motors

Noise, dB; cost × 200 m.u.

95 90

cost, m.u.

85

noise, dB

80

1 m.u. (monetary unit) = cost of 1 kg of laminations or 4.5 kg of copper

75 70 65 60 55 50 0

0.2

1.2 1 0.8 0.6 0.4 1.4 1.6 Magnetic Flux Density in the Stator Yoke, T

1.8

2

Figure 1.9 Noise level and cost plotted against magnetic flux density (MFD) in the stator yoke for a 200 kW induction motor [2].

cost. Figure 1.9 shows the noise and total cost of an induction machine rated at 200 kW, 50 Hz, 380 V, 1480 rpm [2]. To keep the cost independent of inflation, an arbitrary monetary unit (m.u.) has been used that is equal to the price of 1 kg of steel sheets. Using this unit, the copper wire costs 4.5 m.u./kg and aluminum costs 3 m.u./kg [2]. The minimum of cost is different than the minimum of noise. Therefore, low magnetic flux density (MFD) means low level of noise and, vice versa, increased utilization of the magnetic circuit results in increased noise. The minimum of cost is for the MFD in the stator yoke in the range from 0.6 to 1.0 T. The minimum of noise is for lower MFDs; however, due to increase in dimensions and mass of active materials the cost increases sharply at low MFDs in the stator yoke.

1.8

Accuracy of noise prediction

The results of both analytical and numerical noise prediction may significantly differ from measurements. Forces that generate vibration and noise are only small fraction of the main force produced by the interaction of the fundamental current and the fundamental normal component of the magnetic flux density. The power converted into acoustic noise is only approximately 10−6 to 10−4 of the electrical input power. The accuracy of the predicted, say, sound power level spectrum depends not only on how accurate the model is, but also how accurate are the input data, e.g., level of current unbalance, angle between the stator current and q-axis (in PM brushless machines), influence of magnetic saturation on the equivalent slot opening, damping factor, elasticity modulus of the slot content (conductors, insulation, encapsulation), higher time harmonics of the input current (inverter-fed motor), etc.

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All the above input data are difficult to predict with sufficient accuracy. Listed below are the common problems encountered in the analytical and numerical noise prediction [213, 220]: 1. The most difficult task in analytical calculation of sound power level radiated by electrical machines is the accurate prediction of the natural frequencies of the stator structure. At present, the best method to calculate the natural frequencies of the stator is to use the FEM. This is the only technique that can take into account with reasonable accuracy the end bells, mountings (feets or flanges) and asymmetries due to, e.g., terminal boxes. 2. The calculation of matrices of the mass [M] and stiffness [K ] seems to be obvious in the FEM. However, the physical properties of the materials used in electrical machines design are not known. The anisotropy of laminations, internal stresses caused by manufacturing, and change in stiffness matrix [K ] due to temperature variation (differential thermal expansion of the laminations and housing) are mostly not taken into account. 3. The damping matrix [C] in the FEM is difficult to predict. There are no adequate models available for describing damping in laminated materials and structures composed of different types of materials, e.g. copper, insulation, epoxy, laminations. Practice shows that good values for the damping are absolutely essential for predicting accurate vibrational amplitudes. 4. The force vector {F(t)} has to be found in all points on the inner stator surface. Even the most accurate FEM programs introduce a lot of errors in force calculations [213]. Forces are usually calculated analytically in the preprocessor module on the basis of magnetic flux density harmonics or using a 2D FEM. 5. Because neither the analytical approach nor the FEM/BEM computations guarantee accurate results, the laboratory tests are always very important. 6. The main advantage of the SEA is that it does not require all details to be modeled. 7. The vibration and acoustic noise can be calculated on the basis of modal analysis which is free of calculating the electromagnetic forces [213]. Only flux linkages have to be calculated (Section 9.2.2). 8. The calculated noise level is rather lower than the measured noise level. The calculation is mostly done for low harmonic numbers of the air gap permeance. The measurement gives the total noise level due to all harmonics [220].

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2 Magnetic Fields and Radial Forces in Polyphase Motors Fed with Sinusoidal Currents In this chapter the magnetic field in the air gap of polyphase electrical machines will be discussed. The polyphase winding, usually three-phase, is located in the stator core slots. It has been assumed that the stator is fed with a balanced system of sinusoidal phase currents. For polyphase machines with rectangular and trapezoidal current waveforms, e.g., permanent magnet (PM) brushless machines, this assumption is equivalent to taking into account only the fundamental stator current harmonics.

2.1 Construction of induction motors An induction motor consists of two main parts: stator core and rotor core both with polyphase windings (Figure 2.1a). The stator and rotor ferromagnetic cores are stacked of thin steel laminations. Windings are located in stator and rotor slots or can be made as slotless windings. The stator polyphase winding, in most cases, the three-phase winding has independent phase windings that are star (Y) or delta () connected and fed with a three-phase a.c. current system. Each phase winding consists of multiturn coils connected in series. Series connected coils can also be groupped in two or more parallel paths. The air gap between the stator and rotor core is small, usually less than 1 mm for modern small and medium power induction motors. The rotor winding can be made of (1) three-phase Y-connected multiturn coil groups the terminals of which are connected to three slip rings or (2) in the form of a cage winding. In the first case the induction motor is called the phasewound motor or slip ring motor. Brushes are connected to a three phase variable resistance (rheostat) and slide on slip rings. The variable resistance is necessary 21 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors

(a)

(b)

Figure 2.1 Cross sections and magnetic flux lines of three-phase motors: (a) 4-pole induction motor; (b) 8-pole PM brushless motor. for starting to reduce the high inrush current and also for speed control. The maximum resistance is at the first instant of starting, when the stator electromotive force (EMF) E 1 = 0. As the speed increases, the stator EMF increases too and the stator and rotor currents decrease. When the speed reaches its nominal value, the slip rings can be short-circuited. The rheostat used for the speed control must be rated at larger power than that for starting. The speed decreases as the resistance increases. Speed control rheostat is designed for continuous operation while the starting resistance is smaller because it is designed only for the short time duty. In cage induction motors the rotor winding consists of axial bars short circuited by two end rings forming a “squirrel cage” winding. A cage winding made of aluminum alloy or brass is not insulated from the rotor laminated stack. The polyphase stator winding when fed with polyphase a.c. currents produces rotating magnetic field in the air gap. The stator rotating field induces currents in the rotor winding which in turn interact with the stator field to produce the electromagnetic torque. For motoring operation the rotor spins in the same direction as the stator rotating field. The speed of the stator rotating magnetic field f ns = (2.1) p where f is the stator current frequency and p is the number of pole pairs, is called the synchronous speed. The slip between the rotor and the stator magnetic field for the fundamental harmonic is defined as ns − nm s= (2.2) ns where n m = (1 − s) f / p is the rotor mechanical speed. The slip speed n s − n m expresses the speed of the rotor relative to the rotating magnetic field of the stator. The current in the rotor creates its own magnetic field rotating with a speed that is given by sf sn s = . (2.3) p

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The frequency of the current in the rotor s f is called the slip frequency. Since n s (1 −s )+sn s = n s , the stator and rotor magnetic fields rotate with the same speed. If the stator fundamental field harmonic rotates with the speed given by Equation 2.1, the stator higher space harmonic waves will rotate with the speed nsν = ±

f ns =± . ν νp

(2.4)

The “+” sign relates to the wave rotating in the same direction as the fundamental wave of the magnetomotive force (MMF) (forward-rotating wave) and the “−” sign relates to the wave rotating in the opposite direction (backward-rotating wave). The slip for the νth space harmonic is sν =

±n s ν − n m n s − (n s − n m ) = 1 ∓ ν(1 − s ) =1∓ ±n s ν n s /ν

(2.5)

where s is the slip (Equation 2.2) for the fundamental space harmonic ν = 1 and n s /ν in the denominator is the speed of the rotor νth harmonic that can have either the “+” or “−” sign. The frequency of the current induced in the rotor winding by the magnetic flux of the stator νth harmonic is f 2ν = f sν = f [1 ∓ ν(1 − s )]

(2.6)

The sign “−” in Equations 2.5 and 2.6 is for higher harmonics rotating in the same direction as the fundamental ν = 1 and the “+” sign is for harmonics rotating in the oposite direction.

2.2

Construction of permanent magnet synchronous brushless motors

The stator of a PM synchronous brushless motor is similar to that of an induction motor (Figure 2.1b). The rotor consists of laminated stack with PMs. Rotor laminations are necessary to reduce the rotor core losses due to higher harmonics. PMs can be of surface, bread loaf, interior, or interior double-layer configuration (Figure 2.2). Stators of PM brushless motors can be sinusoidally excited or trapezoidally excited. In the first case the phase windings are fed with sinusoidal waveforms shifted by 360o /m 1 one from another (m 1 is the number of stator phases) and produce rotating magnetic field. The motor works as a synchronous motor and is called sinewave motor. The trapezoidally excited motor also called square wave motor is fed from from waveforms shifted by 360o /m 1 too, but those waveforms are rectangular or trapezoidal (Figure 2.3). Waveforms according to Figure 2.3b are produced when the stator current (MMF) is precisely synchronized with the rotor instantaneous position and frequency (speed). The most direct and popular method of providing the required rotor position information is to use an absolute angular position

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Noise of Polyphase Electric Motors

q

q

d

S

N

S

S

S

S

N

q

N (b)

d

q

S

N

N

S

S

N

N

(a) S

d

S

N

d

N

S

N

S

N

N (d)

(c)

Figure 2.2 Rotors of PM brushless motors with: (a) surface PMs; (b) bread loaf PMs; (c) interior (embedded) PMs; (d) interior double-layer PMs.

i1 Phase A

i1

Phase B

Phase C ωt 0

60

12 0

18 24 0 0 (a)

30 0

360°

0

60

12 0

18 0

24 0

ωt 30 360° 0

(b)

Figure 2.3 Basic stator winding waveforms for three-phase PM brushless motors: (a) sinusoidal waveforms; (b) rectangular waveforms.

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Magnetic Fields and Radial Forces

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encoder mounted on the rotor shaft. Only two phase windings out of three conduct current simultaneously. Such a control scheme or electronic commutation is functionally equivalent to the mechanical commutation in d.c. motors. Therefore, those motors are called d.c. brushless motors.

2.3

A.C. stator windings

In a single-layer winding, only one coil side is located in each slot. The number of all coils is s1 /2 and the number of coils per phase is n c = s1 /(2m 1 ) where s1 is the number of stator slots and m 1 is the number of phases. In a double-layer winding two sides of different coils are accommodated in each slot. The number of all coils is s1 and the number of coils per phase is n c = s1 /m 1 . The number of slots per pole is s1 Q1 = (2.7) 2p where 2 p is the number of poles. The number of slots per pole per phase is q1 =

s1 . 2 pm 1

(2.8)

The number of conductors per coil can be calculated as • for a single-layer winding Nc =

a p aw N 1 a p aw N 1 a p aw N 1 = = nc s1 /(2m 1 ) pq1

(2.9)

• for a double-layer winding Nc =

a p aw N 1 a p aw N 1 a p aw N 1 = = nc s1 /m 1 2 pq1

(2.10)

where N1 is the number of turns in series per phase, a p is the number of parallel current paths, and aw is the number of parallel conductors. The number of conductors per slot is the same for both single-layer and double-layer windings, i.e., Nsl =

a p aw N 1 . pq1

(2.11)

The full coil pitch measured in terms of the number of slots is y1 = Q 1 where Q 1 is according to Equation 2.7. The short coil pitch can be expressed as y1 = β Q 1 =

wc Q1 τ

(2.12)

where w c is the coil pitch (coil span) measured in units of length and τ is the pole pitch. The coil pitch–to–pole pitch ratio is β=

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wc . τ

(2.13)

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The distribution factor of a coil group for the fundamental space harmonic ν = 1 is defined as the ratio of the phasor sum E q —to—arithmetic sum q1 E c of EMFs E c induced in each coil and expressed by the following equation: kd1 =

Eq sin(π/2m 1 ) = q1 E c q1 sin[π/(2m 1 q1 )]

(2.14)

24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

The pitch factor for the fundamental space harmonic ν = 1 is defined as the ratio of the phasor sum—to—arithmetic sum of the EMFs per coil side and expressed

U1

W2

U2

W1

V1

V2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

(a)

V2

U1

W2

V1

U2

W1 (b)

Figure 2.4 Three-phase, 4-pole windings distributed in s1 = 24 slots with full pitch coil groups (q1 = 2): (a) single-layer, (b) double-layer.

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π (2.15) k p1 = sin β 2 The winding factor for fundamental is the product of the distribution factor (Equation 2.14) times pitch factor (Equation 2.15), i.e.

as:

kw1 = kd1 k p1 .

(2.16)

The angle in electrical degrees between the neighboring slots is γ =

360o 2π p≡ p. s1 s1

(2.17)

Figure 2.4 shows simple three-phase, 4-pole windings distributed in s1 = 24 slots with full pitch coil groups. Stator winding factors for higher space harmonics ν = 2m 1 k ± 1 where k = 1, 2, 3, 4, . . . are expressed by similar equations to Equations 2.14, 2.15, and 2.16, i.e., • stator winding distribition factor for the ν-th space harmonic sin[νπ/(2m 1 )] q1 sin[νπ/(2m 1 q1 )] • stator winding pitch factor for the ν-th space harmonic πw νβπ c k p1ν = sin = sin ν 2 2 τ • the resultant winding factor for the ν-th space harmonic kd1ν =

kw1ν = kd1ν k p1ν .

(2.18)

(2.19)

(2.20)

The values of the stator winding distribution factor for higher space harmonics as a function of the number of slots per pole per phase q1 (Equation 2.8) for m 1 = 3 are given in Table 2.1. Table 2.1 shows that for some space harmonics and q1 ≥ 2 the distribution factor is the same as for the fundamental harmonic, i.e., kd1ν = ±kd1 . Those harmonics s1 ν = 2m 1 q1 k ± 1 = k ± 1 (2.21) p where k = 1, 2, 3, . . . , are called stator slot harmonics. The EMF induced by slot harmonics is not reduced. For q1 = 1 all harmonics become slot harmonics. The angle between two neighboring slots (coil sides) for the ν-th space harmonic is ν times larger. For slot harmonics 2π p s1 γν = νγ = k ± 1 = 2π k ± γ (2.22) p s1 where γ for the fundamental space harmonic ν = 1 is according to Equation 2.17. Similarly, the number of rotor slot harmonics is s2 µ=k ±1 (2.23) p where s2 is the number of rotor slots.

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Table 2.1 Winding distribution factor kd1ν according to Equation 2.18 for threephase stator windings. ν 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

2.4 2.4.1

q1 2 0.966 0.707 0.259 -0.259 -0.707 -0.966 -0.966 -0.707 -0.259 0.259 0.707 0.966 0.966 0.707 0.259 -0.259 -0.707 -0.966 -0.966 -0.707 -0.259 0.259 0.707 0.966 0.966 0.707

3 0.960 0.667 0.218 -0.177 -0.333 -0.177 0.218 0.667 0.960 0.960 0.667 0.218 -0.177 -0.333 -0.177 0.218 0.667 0.960 0.960 0.667 0.218 -0.177 -0.333 -0.177 0.218 0.667

4 0.958 0.653 0.205 -0.158 -0.271 -0.126 0.126 0.271 0.158 -0.205 -0.653 -0.958 -0.958 -0.653 -0.205 0.158 0.271 0.126 -0.126 -0.271 -0.158 0.205 0.653 0.958 0.958 0.653

5 0.957 0.647 0.200 -0.149 -0.247 -0.109 0.102 0.200 0.102 -0.109 -0.247 -0.149 0.200 0.647 0.957 0.957 0.647 0.200 -0.149 -0.247 -0.109 0.102 0.200 0.102 -0.109 -0.247

6 0.956 0.644 0.197 -0.145 -0.236 -0.102 0.092 0.173 0.084 -0.084 -0.173 -0.092 0.102 0.236 0.145 -0.197 -0.644 -0.956 -0.956 -0.644 -0.197 0.145 0.236 0.102 -0.092 -0.173

∞ 0.955 0.636 0.191 -0.136 -0.212 -0.087 0.073 0.127 0.056 -0.050 -0.091 -0.041 0.038 0.071 0.033 -0.051 -0.058 -0.027 0.026 0.049 0.023 -0.022 -0.042 -0.02 0.019 0.037

Stator winding MMF Single-phase stator winding

Figure 2.5 shows a flat model of an electrical machine with smooth stator and rotor cores. The stator winding consists of one coil per double pole pitch. The relative magnetic permeability of the stator and rotor cores µr → ∞. Coils are fed with d.c. current I and connected in series to create a single-phase winding. For a single coil with Nc turns and span w c = τ the Ampere’s circuital law can be written as · dl = 2H g = Nc I H (2.24)

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29

t

t

t

Stator

g

Rotor Fc(a )

ν=1

Fc

ν=3 p 2

NcI

-

5 p 2

p

p

3

2

2

p

3p

a

2p

Figure 2.5 Magnetic field and MMF of a single-phase full-pitch winding. where H is the magnetic field intensity in the air gap g. The MMF of a single coil regarded as a magnetic voltage drop across a single air gap is Nc I 2 Introducing the relative permeance (per unit area) of the air gap as µ0 g = , H/m2 g Fcg = H g =

(2.25)

(2.26)

the flat-topped value of a rectangular wave of the magnetic flux density induced by coils in the air gap is B g = µ0 H =

µ0 Nc I = g Fcg . g 2

(2.27)

In Equations 2.26 and 2.27 the air gap is assumed to be smooth (no stator and rotor slots) and the magnetic circuit is assumed to be unsaturated, i.e., the relative magnetic permeability of the stator and rotor cores µr → ∞. Similar to the magnetic flux density wave, the MMF wave is also rectangular (Figure 2.5) and can be resolved into Fourier series as ∞ π Fc (x) = (2.28) Fcmν cos ν x τ ν=1,3,5,...

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Noise of Polyphase Electric Motors

where the higher space odd harmonics ν = 1, 3, 5, . . . and the peak value of the νth harmonic π π 4 4 0.5τ Fcmν = Fcg sin ν . (2.29) Fcg cos ν x d x = τ 0 τ νπ 2 The variation of the MMF Fc (x) with the linear coordinate x can also be expressed as a variation of the MMF with the angular coordinate α. The geometrical angle α=

1π 2π x= x pτ s 1 t1

(2.30)

where 2 pτ = s1 t1 , t1 is the stator slot pitch, and νπ x/τ is replaced by νpα. Considering an a.c. machine with sinusoidal excitation, the d.c. current I should be replaced by a sinusoidal current √ i(t) = 2I cos(ωt) (2.31) where ω = 2π f is the stator current angular frequency, f is the input frequency of the sinusoidal current, and I is the stator r ms current. Putting Equations 2.25 and 2.31 into Equations 2.28 and 2.29 √ π 2 2 Fcmν = Nc I sin ν (2.32) νπ 2 and Fc (x, t) =

π Fcmν cos(ωt) cos ν x . τ ν=1,3,5,... ∞

(2.33)

Equation 2.33 shows that the MMF of a phase winding consists of an infinite number of harmonic MMFs changing in time according to cosinusoidal law cos(ωt) and changing in space also according to cosinusoidal law cos ν πτ x . A doublelayer winding with short coil span w c < τ , i.e., β < 1 can be represented as two windings with full coil spans shifted by (1−β)τ one from another. The magnitude of the ν-th harmonic of the MMF per phase is

π π Fmν = 2Fqν cos ν(1 − β) = 2q1 Fcmν kd1ν cos ν(1 − β) (2.34) 2 2 where the magnitude of the MMF of a group of coils according to Equation 2.14 is Fqν = q1 Fcmν kd1ν (2.35) and the winding distribution factor kd1ν for the νth space harmonic is given by Equation 2.18. Putting Equation 2.32 into Equation 2.34 and regarding that νπ

π νβπ sin cos ν(1 − β) = sin = k p1ν 2 2 2 νπ νπ = 0; sin = ±1 for ν = 1, 3, 5, . . . cos 2 2

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the magnitude of the MMF per phase (Equation 2.34) becomes √

π π 2 2 Nc I kd1ν sin ν cos ν(1 − β) Fmν = 2q1 νπ 2 2 √ 2 2 = 2q1 Nc kd1ν k p1ν I νπ

(2.36)

where the pitch factor k p1ν for the νth space harmonic is given by Equation 2.19. The number of turns Nc per coil of a double-layer stator winding is given by Equation 2.10 and the stator phase current I 1 = a p aw I

(2.37)

where I is the single conductor (aw = 1) coil current. Thus, the amplitude of the ν-th space harmonic of the MMF per phase √ 2 2 N1 kw1ν N1 kw1ν I1 ≈ 0.9 I1 . Fmν = (2.38) π νp νp The resultant winding factor kw1ν for the ν-th harmonic is determined by Equation 2.20. The MMF of a single-phase winding varying in space and time can be resolved into two waves rotating in opposite directions, i.e., F(x, t) =

π Fmν cos(ωt) cos ν x τ ν=1,3,5,... ∞

1 π 1 π Fmν cos ωt − ν x + Fmν cos ωt + ν x . (2.39) 2 ν τ 2 ν τ ∞

=

∞

Equation 2.39 for each harmonic describes two MMF waves with the same amplitudes 0.5Fmν rotating with the same angular speed ω/ p = 2π f / p in opposite directions. Harmonics 0.5Fmν cos(ωt − νπ x/τ ) rotate in the same direction as the fundamental stator magnetic field (and rotor). Those magnetic field waves are called forward-rotating waves. Harmonics 0.5Fmν cos(ωt + νπ x/τ ) rotate in the opposite direction and are called backward-rotating waves. The speed of the MMF waves can be found assuming that the waves are stationary for an observer moving with the angular speed ω, i.e., π cos ωt ∓ ν x = const. τ Thus,

π π x = const; ωdt ∓ ν d x = 0 τ τ and the linear synchronous speed of the νth harmonic wave is ωt ∓ ν

v sν =

1τ 1 f D1in 1 dx =± ω = ±2π = vs dt νπ ν p 2 ν

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(2.40)

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where ω = 2π f is the angular frequency for fundamental harmonic ν = 1, n s = f / p is the rotational speed of the fundamental frequency of the magnetic field called the synchronous speed (Equation 2.1), D1in = 2 pτ/π is the inner diameter of the stator, and v s = ωD1in /(2 p) = 2π( f / p)D1in /2 is the linear synchronous speed of the fundamental harmonic.

2.4.2

Three-phase stator winding

A three-phase symmetrical stator has three the same windings shifted in space by 2π/3 electrical degrees. The stator is fed with three sinusoidal currents i 1A , i 1B , and i 1C . For a balanced input current system, the amplitude of each phase current √ 2I1 is the same and phase shift is 2π/3 electrical degrees, i.e., √ i 1A = 2I1 cos(ωt) √ π (2.41) i 1B = 2I1 cos ωt − 2 3 √ π i 1C = 2I1 cos ωt − 4 . 3 Based on Equation 2.39 the space harmonics of the MMF produced by each phase winding are 1 π 1 π Fmν cos ωt − ν x + Fmν cos ωt + ν x 2 τ 2 τ

1 2π π = Fmν cos ωt − ν x + 0(ν − 1) 2 τ 3

1 2π π + Fmν cos ωt + ν x − 0(ν + 1) 2 τ 3

F1Aν =

2π π x− τ 3 1 2π π 2π + Fmν cos ωt − x− +ν ) 2 3 τ 3

1 2π π = Fmν cos ωt − ν x + 1(ν − 1) 2 τ 3

1 2π π + Fmν cos ωt + ν x − 1(ν + 1) 2 τ 3

F1Bν =

F1Cν =

1 Fmν cos 2

ωt −

2π 3

−ν

4π π x− τ 3 1 4π π 4π + Fmν cos ωt − x− +ν ) 2 3 τ 3 1 Fmν cos 2

Copyright © 2006 Taylor & Francis Group, LLC

ωt −

4π 3

−ν

(2.42)

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33

2π 1 π Fmν cos ωt − ν x + 2(ν − 1) 2 τ 3

1 2π π + Fmν cos ωt + ν x − 2(ν + 1) . 2 τ 3

According to Equation 2.42, harmonic waves of the MMF of each phase are shifted by the angle (ν ∓1)(2π )/3 one from each other. Harmonics ν = 3k = 3, 9, 15, . . . (in general, ν = m 1 k where k = 1, 3, 5, . . . and m 1 = 3) are shifted by (ν ∓ 1)(2π )/3 = (3k ∓ 1)(2π )/3 = 2π k ∓ 2π/3 or 120o . Thus, the sum of those harmonic waves is equal to zero, so that harmonics ν = 3k do not exist in three phase systems. For harmonic waves ν = 6k + 1 (in general, ν = 2m 1 k + 1) the shift angle is (ν − 1)(2π )/3 = 4kπ or 0o . The resultant wave is the arithmetic sum of waves in all three phases. For harmonic waves ν = 6k − 1 (in general, ν = 2m 1 k − 1) the shift angle is (ν − 1)(2π )/3 = 4kπ − 4π/3 or 240o . The resultant wave is 0. The resultant MMF of a three phase symmetrical winding fed with balanced three phase current system does not contain triple harmonic waves. Harmonics ν = 1, 7, 13, 19, . . . rotate forward and harmonics ν = 5, 11, 17, 23, . . . rotate backward. The linear speed of higher space harmonics is expressed by Equation 2.40. The magnitudes of harmonic MMFs according to Equations 2.38 and 2.43 are √ 3 3 2 N1 kw1ν N1 kw1ν Fmν = [Fmν ]m 1 =1 = I1 ≈ 1.35 I1 . (2.43) 2 π νp νp The total MMF of a three-phase symmetrical stator winding fed with a balanced current system ∞ π F1 (x, t) = (2.44) Fmν cos ωt ∓ ν x τ ν=6k±1 or F1 (α, t) =

∞

Fmν cos(ωt ∓ νpα) =

ν=6k±1

∞

Fmν cos(νpα ∓ ωt)

(2.45)

ν=6k±1

where the angle α is determined by Equation 2.30.

2.4.3

Polyphase stator winding

The time-space distribution of the MMF of the k-th phase winding of a symmetrical polyphase winding fed with a balanced current system can be expressed as

∞ 2π 1 π F1k (x, t) = [Fmν ]m 1 =1 {cos ωt − ν x + (k − 1)(ν − 1) 2 ν=1 τ m1

π 2π + cos ωt + ν x − (k − 1)(ν + 1) } (2.46) τ m1

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Noise of Polyphase Electric Motors

where ω = 2π f is the angular frequency for the fundamental harmonic, τ is the pole pitch and ν = 2m 1 k ± 1,

k = 0, 1, 2, 3, 4, 5, . . .

(2.47)

The “−” sign before νπ x/τ is for the forward-rotating field and the “+” sign before νπ x/τ is for the backward-rotating field. The total MMF exicted by all m 1 phases is the sum of MMFs excited by each phase. Using the trigonometric identity cos(α ± β) = cos α cos β ∓ sin α sin β, the following equation can be written on the basis of Equation 2.46 F1ν (x, t) = F11 (x, t) + F12 (x, t) + · · · + F1m 1 (x, t) ∞ 1 π = Fmν cos ωt − ν x 2 ν=1 τ

1 2 × 1 + cos (ν − 1)2π + cos (ν − 1)2π + · · · m1 m1

∞ m1 − 1 1 π cos (ν − 1)2π − Fmν sin ωt − ν x m1 2 ν=1 τ

1 2 × 0 + sin (ν − 1)2π + sin (ν − 1)2π + · · · m1 m1

∞ m1 − 1 1 π sin (ν − 1)2π + Fmν cos ωt + ν x m1 2 ν=1 τ

1 2 × 1 + cos (ν + 1)2π + cos (ν + 1)2π + · · · m1 m1

∞ m1 − 1 1 π cos (ν + 1)2π + Fmν sin ωt + ν x m1 2 ν=1 τ

1 2 × 0 + sin (ν + 1)2π + sin (ν + 1)2π + · · · m1 m1

m1 − 1 sin (ν + 1)2π . (2.48) m1 Series in square brackets can be written as

1 2 m1 − 1 1+cos (ν ∓ 1)2π + cos (ν ∓ 1)2π + · · · + cos (ν ∓ 1)2π m1 m1 m1

sin(ν ∓ 1)π m1 − 1 sin(ν ∓ 1)π = cos (ν ∓ 1)2π = sin[(ν ∓ 1)π/m 1 ] m1 sin[(ν ∓ 1)π/m 1 ] (2.49)

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Table 2.2 Band factor in Equation 2.51 for three-phase stator windings. ν

sin(ν−1)π sin[(ν−1)π/m 1 ]

sin(ν+1)π sin[(ν+1)π/m 1 ]

0 0 0 3 0 0 3 0 0 3 0 0 3 0

0 0 3 0 0 3 0 0 3 0 0 3 0 0

1 3 5 7 9 11 13 15 17 19 21 23 25 27

1 2 m1 − 1 0 + sin (ν ∓ 1)2π + sin (ν ∓ 1)2π + · · · + sin (ν ∓ 1)2π m1 m1 m1

sin(ν ∓ 1)π m1 − 1 = sin (ν ∓ 1)2π = 0. (2.50) sin[(ν ∓ 1)π/m 1 ] m1 Thus, the resultant MMF of a polyphase stator winding is π sin(ν − 1)π 1 cos ωt − ν x Fmν 2 ν=1 sin[(ν − 1)π/m 1 ] τ ∞

F1 (x, t) =

1 π sin(ν + 1)π cos ωt + ν x . Fmν 2 ν=1 sin[(ν + 1)π/m 1 ] τ ∞

+

(2.51)

The so-called band factor in Equation 2.51 is given in Table 2.2 by calculating sin(ν − 1)π = m1 sin[(ν − 1)π/m 1 ]

for

ν = 2km 1 + 1

k = 1, 2, 3, . . .

sin(ν + 1)π = m1 sin[(ν + 1)π/m 1 ]

for

ν = 2km 1 − 1

k = 1, 2, 3, . . . .

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Noise of Polyphase Electric Motors

Otherwise, sin[(ν ∓ 1)π ]/ sin[(ν ∓ 1)π/m 1 ] = 0 including ν = 1. Finally, for higher space harmonics F1 (x, t) =

∞ 1 π [Fmν ]m 1 =1 m 1 cos ωt − ν x 2 ν=2km +1 τ 1

+

∞

1 2 ν=2km

1

π [Fmν ]m 1 =1 m 1 cos ωt + ν x . τ −1

The magnitude of the νth harmonic of the polyphase stator MMF is √ m1 m 1 2 N1 kw1ν Fmν = [Fmν ]m 1 =1 = I1 . 2 π νp

(2.52)

(2.53)

The winding factor kw1ν for the νth space harmonic is given by Equation 2.20. The peak value of the armature line current density or specific electric loading is defined √ as the number of conductors in all phases 2m 1 N1 times the peak stator current 2I1 divided by the armature stack length 2 pτ , i.e., √ m 1 2N1 I1 Am1 = . (2.54) pτ

2.5

Rotor magnetic field

The amplitude of the rotor MMF of an induction machine is √ m 2 2 N2 kw2 I2 Fm2 = π p

(2.55)

where N2 is the number of the rotor turns per phase, kw2 is the rotor winding factor for the fundamental space harmonic, and I2 is the rotor r ms current. The amplitude of the rotor MMF referred to the stator system √ m 1 2 N1 kw1 Fm2 = I2 (2.56) π p where the rotor current referred to the stator system is calculated as I2 =

m 2 N2 kw2 k s I2 . m 1 N1 kw1

(2.57)

The skewing factor for the fundamental harmonic is ks =

sin[π pbs /(t1 s1 )] sin(0.5π bs /τ ) = 0.5π bs /τ π pbs /(t1 s1 )

(2.58)

where bs is the slot skew. In practice bs = 2π τ/s1 = t1 or bs = 2π τ/s2 = t2 .

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The peak value of the rotor line current density of an induction machine is expressed similar to that of the stator (Equation 2.54), i.e., √ m 2 2N2 I2 Am2 = . (2.59) pτ For a synchronous machine, as the rotor rotates at mechanical angular velocity m = , the excitation field and its space harmonics rotate all at the same mechanical angular velocity . The electrical angular frequencies of the rotor higher harmonics are all different and increase with the harmonic number, i.e., ωµ = µp

(2.60)

The magnetic flux density waveform (normal component) of the rotor of an induction machine is given by the equation b2 (α, t) =

∞

B2mµ cos µpα ∓ ωµ t + φµ .

(2.61)

µ=1

For the most important space harmonics the rotor angular frequency of an induction motor is ωµ = ω[(ks2 / p)(1 − s) + 1]. The magnitude of the rotor magnetic flux density can be found as [10] √ m 1 2 N1 kw1ν kw2µ µ0 B2mµ = I1 (2.62) π µp(1 + τ2ν )kw2ν kC g where kw1ν is the stator winding factor for the ν-th space harmonic according to Equation 2.20, kw2µ is the rotor winding factor for the rotor µ-th space harmonic, kw2ν is the rotor winding factor for the stator ν-th space harmonic, τ2ν = X 2ν / X mν is the leakage coefficient of the rotor winding for the ν-th stator harmonic, X 2ν is the rotor winding leakage reactance for the ν-th harmonic, and X mν is the mutual reactance for the ν-th harmonic.

2.6 2.6.1

Calculation of air gap magnetic field Effect of slots

The relationship between the air gap magnetic flux density b(α, t) and the MMF F(α, t) for uniform air gap g is b(α, t) = F(α, t)g (α)

(2.63)

where the relative permeance g (α) of the air gap is a function of the angle α given by Equation 2.30. Assuming a trapezoidal distribution of the magnetic flux density, the amplitude of the νth space harmonic will be equal to the flat-topped value of the flux density multiplied by the following coefficient of slot opening [87] kok =

Copyright © 2006 Taylor & Francis Group, LLC

sin[kρπb14 /(2t1 )] kρπb14 /(2t1 )

(2.64)

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Noise of Polyphase Electric Motors

where t1 is the stator slot pitch, b14 is the stator slot opening and k is the number of harmonics which replace the air gap variation bounded by the active surfaces of ferromagnetic cores of the stator and rotor. The auxiliary function dependent on the relative slot opening is √ κ 2 1 + κ2 √ ρ= . (2.65) 5 + κ 1 + κ2 − 1 The relative slot opening can be estimated as follows • for induction machines

b14 g

(2.66)

b14 g + h M /µ0

(2.67)

κ= • for most PM brushless machines κ≈

where h M is the radial height of the PM per pole. The value of κ may affect the radial magnetic forces, but it is not necessarily a decisive factor in production of noise [220]. The relative permeance of the air gap in Equation 2.63 with slot opening effect being included is ∞ µ0 A0 + Ak cos(ks1 α) 2 gkC k=1,2,3 ∞ = g0 1 + Ak cos(ks1 α) = g0 λg1 (α)

g (α) =

(2.68)

k=1,2,3

where α = 2π x/(s1 t1 ) is according to Equation 2.30, s1 is the number of stator slots and t1 is the slot pitch. The permeances and Fourier coefficients in the above Equation 2.68 are • the constant component of the air gap relative permeance g0 =

A0 µ0 µ0 = = 2 kC g g

(2.69)

• the component of the relative specific permeance of the air gap varying with the angle α when only the stator core is slotted λg1 (α) = 1 +

∞

Ak cos(ks1 α)

(2.70)

k=1,2,3

• the relative value of harmonic permeances of the air gap dependent on the dimensions of the slot openings and air gap [40, 219], i.e., γ1 2 2 π Ak = g (α) cos(ks1 α)dα = −2g kok . (2.71) π 0 t1

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It has been assumed that g (α) above the slot opening is approximated by a linear function. The second term in Equation 2.68 multiplied by g0 yields 2 g0 Ak = −2µ0 γ1 kok /t1 , i.e., as given in classical literature [40]. The angular displacement α according to Equation 2.30 can be expressed by the linear circumferential distance x. For induction machines the constant component (Equation 2.69) corresponds to the relative permeance of physical air gap g increased by Carter’s coefficient kC . For surface magnets g = gkC + h M /µrr ec and for buried magnets totally enclosed by laminations or mild steel g = gkC . The Carter’s coefficient kC =

t1 t1 − γ1 g

(2.72)

where

4

0.5κ arctan(0.5κ) − ln 1 + (0.5κ)2 . (2.73) π The relative permeance due to stator slot openings can also be calculated according to the method proposed by Weber [87, 236]. The coefficient of harmonic permeances in Equation 2.70 is

14 (k)2 Ak = −β(κ) sin(1.6π k) (2.74) 0.5 + kπ 0.78 − 2(k)2 γ1 =

where or

β(κ) = 0.1κ 0.5+1/κ

(2.75)

1 β(κ) = 0.5 1 − √ . 1 + κ2

(2.76)

The relative permeance according to Weber is g0 Ak = µ0 Ak /g where Ak is given by Equation 2.74. The parameter κ is according to Equation 2.66 or Equation 2.67 and b14 = . (2.77) t1 If both the stator and rotor core have slots, the resultant relative air gap permeance is g (α) = g0 λg1 (α)λg2 (α) (2.78) and the resultant Carter’s coefficient is kC = kC1 kC2

(2.79)

where λg1 and λg2 are relative specific permeances of the stator and rotor derived on the assumption that only one core has slots, kC1 is Carter’s coefficient for the stator, and kC2 is Carter’s coefficient for the rotor. The center axes of the stator and rotor teeth of an induction motor are aligned at the time instant t = 0 in the origin of coordinate system, i.e., α = 0 and the rotor angular frequency ω2 = 2π(1 − s) f = 2π(1 − s)n s p = 2π n m p = pm

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Noise of Polyphase Electric Motors

where m is the mechanical angular speed of the rotor and n m is the rotational speed of the rotor in rev/s. Thus, the relative value of harmonic permance for the rotor is ∞ λg2 (α, t) = 1 + Al cos [ls2 (α − ω2 t)] . (2.80) l=1,2,3

Putting Equations 2.69, 2.70, and 2.80 into Equation 2.78 the resultant relative permeance of the air gap of slotted stator and rotor cores is g (α, t) = g0 λg (α, t) =

µ0 λg (α, t) kC g

(2.81)

where λg (α, t) = 1 + +

∞

Ak cos (ks1 α) +

k=1,2,3 ∞

∞

Al cos [ls2 (α − ω2 t)]

l=1,2,3

∞ 1 Ak Al {cos[(ls2 + ks1 )α − ls2 ω2 t] 2 k=1,2,3 l=1,2,3

× cos[(ls2 − ks1 )α − ls2 ω2 t]}.

(2.82)

The first term in Equation 2.82 describes the permeance of the equivalent uniform air gap g = kC g, second term describes harmonics of the stator permeance, third term describes harmonics of the rotor permeance, and the last term represents harmonics of permeance due to reciprocal effect of the stator and rotor. Harmonics of the magnetic flux density in the air gap are obtained by multiplying Equations 2.81 and 2.44 (distribution of the MMF) as given by Equation 2.63.

2.6.2

Effect of eccentricity

The variation of the air gap around the magnetic circuit periphery and with time is g(α, t) = R − r − e cos(α − ω t) = g[1 − cos(α − t)]

(2.83)

where g = R − r , R is the inner stator core radius and r is the outer rotor radius. The relative eccentricity is defined as =

e g

(2.84)

where e is the rotor (shaft) eccentricity and g is an ideal uniform air gap for e = 0. For the static eccentricity the angular frequency = 0. In the case of dynamic eccentricity: • for induction motors = (1 − s) =

Copyright © 2006 Taylor & Francis Group, LLC

f ω (1 − s) = 2π (1 − s) p p

(2.85)

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• for synchronous motors f ω = 2π . (2.86) p p Equation 2.83 has been derived in Appendix B. The constant coefficient (Equation 2.69) in Fourier series (Equation 2.68) for a machine with eccentricity becomes [10]: = =

µ0 1 A0 √ = H/m2 (2.87) 2 gkC 1 − 2 The effect of the static and dynamic eccentricity on the higher harmonics of the air gap performance is difficult to calculate. Most methods are limited only to the influence of eccentricity on the fundamental harmonic of the air gap permeance [87], i.e., • the air gap relative permeance for k = 1 including the static eccentricity and variation of the permeance with slot openings γ1 2 ge,k=1 (α) = g0 Ak=1 λc1 cos(α) = −2µ0 kok λc1 cos(α) (2.88) t1 in which √ 1 − 1 − 2 λc1 = 2 √ (2.89) 1 − 2 • the air gap relative permeance for k = 1 including the dynamic eccentricity and variation of the permeance with slot openings g0 =

ged,k=1 (α, t) = g0 Ak=1 λc1d cos( t − α) γ1 2 = −2µ0 kok λc1d cos( t − α) (2.90) t1 where the angular speed equal to the rotor mechanical angular speed is according to Equation 2.85 or Equation 2.86. The angular displacement α in Equations 2.88 and 2.90 can be replaced by the linear displacement x according to Equation 2.30. To obtain λc1d , the relative static eccentricity in Equation 2.89 is replaced by the relative dynamic eccentricity d defined as ed (2.91) d = g in which ed denotes the dynamic eccentricity (linear radial dimension). To include the eccentricity in calculation of the relative air gap permeance, the permeances (Equation 2.88) and (Equation 2.90) should be added to Equation 2.68 or Equation 2.81. The effect of static and dynamic eccentricity can be taken into account according to Appendix B using the following coefficients √ 1 1 − 1 − 2 λge (α) ≈ √ +2 √ cos(α) (2.92) 1 + 2 1 − 2 √ 1 1 − 1 − 2 λged (α) ≈ + 2 cos ( t − α) . (2.93) 1 + d2 d 1 − d2

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Noise of Polyphase Electric Motors

The resultant relative air gap permeance g (α, t) with slots and eccentricity effects taken into account is given in Appendix B by Equation B.5. Equation B.5 has been obtained by multiplying Equation 2.78 by the product λge (α)λged (α, t) which includes eccentricity. Since the relative eccentricity is small, i.e., ≤ 0.3, the permeance of the air gap due to eccentricity expressed in the form of Fourier series can be limited to the first two terms. For the dynamic eccentricity (neglecting the variation of the permeance with slot openings) ged (α, t) = g0 λged (α, t) 1 1 = µ0 [λoe + λc1 cos(α − t)] ≈ µ0 [1 + cos(α − t)]. gkC gkC (2.94) The wave of the magnetic flux density in the air gap pulsating with the eccentricity can be obtained by multiplying the first harmonic (second term) of Equation 2.94 by the fundamental stator MMF, i.e., cos(α − t)Fm1 cos( pα − ωt − φ) gkC = µ0 Fm1 cos[α(1 + p) − (ω + )t − φ)] + µ0 Fm1 2gkC 2gkC × cos[α(1 − p) + (ω − )t + φ)]

1 = µ0 + 1 − (ω + )t − φ) Fm1 cos pα 2gkC p

1 − 1 + (ω − )t + φ) + µ0 Fm1 cos pα 2gkC p = µ0 Fm1 cos[ν pα ∓ (ω ± )t ∓ φ)]. (2.95) 2gkC

b1 (α, t) ≈ µ0

The stator magnetic flux density higher harmonics due to eccentricity are ν = 1 ±

1 p

(2.96)

and the number of their pole pairs is p ± 1. For p = 1 the eccentricity harmonic numbers are ν = 2 or ν = 0. The second term in Equation 2.95 for ν = 0 represents an axial homopolar flux that can appear in a two-pole machine with eccentricity (Appendix B). The angular frequency for static eccentricity is ω = 2π f . For dynamic eccentricity and induction machines

1 ω ± = ω 1 ± (1 − s) . (2.97) p For synchronous machines s = 0 in Equation 2.97.

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The stator magnetic flux density wave due to eccentricity generate additional magnetic flux density waves in the rotor [64, 200] b2 (α, t ) = B2m µ cos(µ p α − µ t − φµ ) (2.98) µ

where the rotor magnetic flux density higher harmonics due to eccentricity are s2 1 µ = k + 1 ± (2.99) p p and the angular frequency (see also Equation 2.134)

s2 µ = ων + ω k (1 − s ) p

(2.100)

where ων = 2π ν f . The rotor flux density waves originate from the stator flux density waves in the air gap with pole pairs p ± 1. For a synchronous machine, the slip in s = 0, Equation 2.100, and the number of rotor slots s2 = 2 p. The eccentricity fields contribute to the vibration and noise and produce the following effects: • cause unbalance magnetic pull that bends the shaft and magnifies the magnetic pull; • induce voltages in the parallel paths of stator windings and give rise to equalizing currents [123]; • reduce the mechanical stiffness of the shaft and the first critical speed of the rotor due to increased magnetic pull [41, 200].

2.6.3

Effect of magnetic saturation

The slot leakage and residual fields of the stator and rotor MMFs are large [87]. Consequently, the tooth tips become highly saturated (reduction of the relative magnetic permeability) which is equivalent to the increase in the slot opening from b14 to b14sat (Figure 2.6) according to Equation C.10. The maximum slot opening due to the tooth tip saturation corresponds to the region (along the pole pitch) of the maximum slot current. The fictitious slot opening b14sat is a periodic function with the period π/ p, i.e., equal to half the period of the working harmonic. Thus, the first two terms of Fourier series of the fictitious slot opening can be written as b14sat = b140 − b141 cos[2( pα − ωt)]

(2.101)

where b140 is the mean value of the slot opening along the pole pitch and b141 = b140 − b14 is the amplitude of the first harmonic of the slot opening distribution. The air gap permeance with respect to the magnetic saturation can be expressed by the following simplified equation [64, 179] (α, t) ≈ 0 + sat (α, t)

Copyright © 2006 Taylor & Francis Group, LLC

(2.102)

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Noise of Polyphase Electric Motors b14 b14 sat Variation of Slot Opening

b14

t1 b14sat(a)

b14

a=

1 p x pt

t

t (a)

b140

(b)

Figure 2.6 Effect of magnetic saturation on the slot opening: (a) fictitious increase in the slot opening; (b) variation of a fictitious slot opening along the pole pitch due to saturation [87]. where sat (α, t) = −sat cos(2 pα − 2ω1 t − 2φs ).

(2.103)

Only the fundamental harmonic ν = 1 has been included.

2.6.4

Effect of rotor saliency

The effect of the PM rotor saliency can be included with the aid of the rotor magnetic flux density waveform normalized to B0 = 1 T. On the basis of Equation 2.61 the magnetic flux density waveform is b (α, t) = where

∞

b2 (α, t) = bµ cos(µpα ± ωµ t + φµ ) B0 µ=1

(2.104)

τ 4 1 cp sinh(α) − 6 cosh(α) c2p + (µπ/τ )2 µπ bt3 τ 2 2 +3 cosh(α) µπ τ − bp π π bt τ 2 1 τ 4 µπ/τ × sin µ sin µ +6 − + 2 τ τ c2p + (µπ/τ )2 µπ bt2 µπ π π bt × cosh(α) sin µ cos µ ; (2.105) 2 τ

4 bµ = τ

bt =

τ − bp ; 2

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cp = 2

α ; bp

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b p is the width of the pole shoe, and B0 bµ = B2mµ is given by Equation 2.61. The parameter α depends on the distribution of the normal component of the magnetic flux density in the range −0.5b p ≤ x ≤ 0.5b p . For α = 0 the magnetic flux density in the range −0.5b p ≤ x ≤ 0.5b p has constant value (flat curve). For embedded magnets with orifices in pole shoes, α = 0.5 . . . 1 and the magnetic flux density in the range −0.5b p ≤ x ≤ 0.5b p is a concave curve. Thus, the resultant air gap relative permeance with the rotor saliency effect taken into account is gsal (α, t = t0 ) = |b (α, t = t0 )|λg (α, t = t0 )

(2.106)

where b (α, t = t0 ) is according to Equation 2.104, λg (α, t = t0 ) is according to Equation 2.70 or Equation 2.82, and t0 is the arbitrarily chosen time instant. The resultant relative air gap permeance g according to Equation B.5 (Appendix B) with the effect of the rotor saliency taken into account plotted against the circumferential direction x = ατ/π at t = 0 is shown in Figure 2.7.

2.7

Radial forces

2.7.1

Production of radial magnetic forces

On the basis of Equation 2.45 the space and time distribution of the MMF of a polyphase electrical machine fed with sinusoidal and balanced current system can be expressed by the following equations:

3 ⋅ 10−4 2.5 ⋅ 10−4

Λ g (x)

2 ⋅ 10−4

1 ⋅ 10−4 5 ⋅ 10−5 0

0

0.05

0.1

0.15

0.2

0.25 x, m

0.3

0.35

0.4

0.45 0.5

Figure 2.7 Distribution of the air gap relative permeance according to Equation B.5 with the rotor saliency effect taken into account on the basis of Equation 2.106 for a PM brushless motor with s1 = 36, 2 p = 10, τ = 50 mm, b14 = 3 mm, g = 1.0 mm, e = 0.1 mm, ed = 0.1 mm.

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Noise of Polyphase Electric Motors • for the stator F1 (α, t) =

∞

Fmν cos(νpα ∓ ωt)

(2.107)

Fmµ cos(µpα ∓ ωµ t + φ µ )

(2.108)

ν=1

• for the rotor F2 (α, t) =

∞ µ=1

where α is the angular distance from the origin of the coordinate system, p is the number of pole pairs, φµ is the angle between vectors of the stator and rotor harmonics of equal order, ν and µ are the numbers of space harmonics of the stator and rotor, respectively, ω = 2π f is the pulsation of the input current, and Fmν and Fmµ are the peak values of the νth and µth harmonics expressed by Equations 2.43 and 2.55, respectively. The product pα = π x/τ where τ is the pole pitch and x is the linear distance from a given axis results from Equation 2.30. For symmetrical polyphase stator windings and integral number of slots per pole per phase q1 , the stator space harmonics are expressed by Equation 2.47. For induction machines µ is expressed by Equation 2.23. For rotors of synchronous machines µ = 2k − 1

(2.109)

where k = 1, 2, 3, . . .. The instantaneous value of the normal component of the magnetic flux density in the air gap at a point determined by the angle α can be calculated with the aid of Equation 2.63 b(α, t) = [F1 (α, t) + F2 (α, t)]g (α, t) = b1 (α, t) + b2 (α, t)

T

(2.110)

where • for the stator b1 (α, t) = F1 (α, t)g (α, t) =

∞

Bmν cos(νpα ∓ ωt)

(2.111)

ν=1

• for the rotor b2 (α, t) = F2 (α, t)g (α, t) =

∞

Bmµ cos(µpα ∓ ωµ t + φ µ ). (2.112)

µ=1

According to the Maxwell stress tensor, the radial magnetic force per unit area or magnetic pressure waveform at any point of the air gap is pr (α, t) =

1 2 b (α, t) − bt2 (α, t) . 2µ0

(2.113)

Since the magnetic permeability of the ferromagnetic core is much higher than that of the air gap, the magnetic flux lines are practically perpendicular to the stator and

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rotor cores. Thus, the tangential component bt (α, t ) of the magnetic flux density is much smaller than the normal component b(α, t ) and b2 (α, t ) 1 = [F1 (α, t ) + F2 (α, t )]2 2g (α, t ) 2µ0 2µ0 1 = [F 2 (α, t )2g (α, t ) + 2F1 (α, t )F2 (α, t )2g (α, t ) 2µ0 1 + F22 (α, t )2g (α, t )]

pr (α, t ) ≈

=

[b1 (α, t )]2 + 2b1 (α, t )b2 (α, t ) + [b2 (α, t )]2 . 2µ0

(2.114)

There are three groups of the infinite number of radial force waves: • the product [b1 (α, t )]2 of the stator harmonics of the same number ν, i.e., pr ν (α, t ) =

[Bm ν cos(νp α ∓ ωt )]2 B2 = m ν [1 +cos(2νp α∓2ωt )] (2.115) 2µ0 4µ0

• the mixed product 2b1 (α, t )b2 (α, t ) of the stator ν and rotor µ harmonics 2Bm ν cos(νp α ∓ ωt )Bm µ cos(µp α ∓ ωµ t + φµ ) 2µ0 1 = Bm ν Bm µ {cos[(νp α ∓ ωt ) − (µp α ∓ ωµ t + φµ )] 2µ0 + cos[(νp α ∓ ωt ) + (µp α ∓ ωµ t + φµ )]} 1 = Bm ν Bm µ {cos[ p α(ν − µ) ∓ (ω − ωµ )t − φµ ] 2µ0 + cos[ p α(ν + µ) ∓ (ω + ωµ )t + φµ ]} (2.116)

pr νµ (α, t ) =

• the product [b2 (α, t )]2 of the rotor harmonics of the same number µ, i.e., [Bm µ cos(µp α ∓ ωµ,n t + φµ,n )]2 2µ0 1 2 = B [1 + cos(2µp α ∓ 2ωµ t + 2φµ )] 4µ0 m µ

pr µ (α, t ) =

(2.117)

The following trigonometric identities cos2 α = 0.5(1 + cos 2α) and 2 cos α cos β = cos(α − β) + cos(α + β) have been used. The constant term equal to Bm2 ν /(4µ0 ) or Bm2 µ /(4µ0 ) in Equations 2.115 and 2.117 have no significance for noise generation and can be neglected because the static magnetic pressure is uniformly distributed along the air gap. In accordance with Equations 2.115, 2.116, and 2.117, the magnetic forces per unit area (Figure 2.8) can be expressed in the following general form pr (α, t) = Pmr cos(r α − ωr t)

Copyright © 2006 Taylor & Francis Group, LLC

(2.118)

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Noise of Polyphase Electric Motors 3 ⋅ 105 2.5 ⋅ 105

Pr (x)

2 ⋅ 105 1.5 ⋅ 105 1 ⋅ 105 5 ⋅ 104 0

0

0.05

0.1

0.15

0.2

0.25 x, m

0.3

0.35

0.4

0.45

0.5

Figure 2.8 Distribution of the radial magnetic pressure along the air gap periphery at no load for s1 = 36, 2 p = 10, τ = 50 mm, g = 1.0 mm and eccentricity e = 0.1 mm. where Pmr is the amplitude of the magnetic pressure, ωr is the angular frequency and and r = 0, 1, 2, 3, . . . are corresponding orders of radial magnetic forces. The radial forces circulate around the stator bore with the angular speed ωr /r and frequency fr = ωr /(2π ). For small number of the stator pole pairs the radial forces may cause the stator to vibrate and produce acoustic noise. For PM brushless motors the full load armature reaction field is normally less than 20% of the open-circuit magnetic field [254]. Thus, the effect of the first and second term in Equation 2.114 on the acoustic noise at no load is usually lower than that of the third term, i.e., in most cases harmonics in the open-circuit magnetic field have a dominant role in the noise of electromagnetic origin [254].

2.7.2

Amplitude of magnetic pressure

The amplitude Pmr of the radial magnetic pressure of the r -th order in Equation 2.118 depends on the harmonics of the magnetic flux density participating in its production, i.e., • excited by the stator harmonics of the same number ν 2 Bmν N/m2 4µ0 • excited by the stator ν and rotor µ harmonics

Pmr =

Bmν Bmµ N/m2 2µ0 • excited by the rotor harmonics of the same number µ Pmr =

Pmr =

Copyright © 2006 Taylor & Francis Group, LLC

2 Bmµ

4µ0

N/m2 .

(2.119)

(2.120)

(2.121)

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To obtain the amplitude of the radial force, the force pressure amplitude Pmr should be multiplied by π D1in L i where D1in is the stator core inner diameter and L i is the effective length of the stator core.

2.7.3

Deformation of the stator core

By only considering the pure circumferential vibration modes of the stator core, the deflection d of the stator core is an inverse function of the fourth power of the force order r [206] 1 d ∝ 4 (2.122) r Considering the product of the stator harmonics of the same number according to Equation 2.115, the lowest frequency of the magnetic force fr = 2 f varies at twice the line frequency and the order is r = 2 p. Since the force order r = 2 for a two-pole motor and r = 4 for a four-pole motor, the magnetic force in a four-pole motor will be 1/16 of that of the two-pole motor. The twice line frequency noise is generally significant only in two-pole motors because the stator deflection varies with the force order according to Equation 2.122. The largest deformation of the stator yoke ring occurs when the frequency fr is close to the natural mechanical frequency of the stator system. The most important from airborne noise point of view are low circumferential mode numbers, i.e., r = 0, 1, 2, 3 and 4. Vibration mode r = 0. For pulsating vibration mode or “breathing” mode r = 0 the radial magnetic force density p0 = Pmr =0 cos ω0 t (2.123) is distributed uniformly around the stator periphery and changes periodically with time. It causes a radial vibration of the stator core and can be compared to a cylindrical vessel with a variable internal overpressure [87]. Equation 2.123 describes an interference of two magnetic flux density waves of equal lengths (the same number of pole pairs) and different velocity (frequency). According to [154], the “breathing” mode r = 0 can cause acoustically harmful low wave number vibration, even with a high number of poles and sinusoidal stator current. This can happen when nearby low wave numbers are given a high admittance by structural discontinuities such as supports [154]. Vibration mode r = 1. For “beam bending” mode r = 1 [58] the radial pressure p1 = Pmr =1 cos(α − ω1 t)

(2.124)

produces a single-sided magnetic pull on the rotor. The angular velocity of the pull rotation is ω1 . Physically, Equation 2.124 describes an interference of two magnetic flux density waves for which the number of pole pairs differs by one.

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Noise of Polyphase Electric Motors r=0

r=1

P

r=2

P

P

r=3

P

P r=4 P

P

P P P

P P

P P

P

P P

P

P

Figure 2.9 Deformation of the core caused by the space distribution of radial forces. Vibration modes r = 2, 3, 4. For “ovaling mode” r = 2 [58] and r = 3, 4, . . . wave-shaped deflections of the stator core will occur. Figure 2.9 shows space distribution of forces producing vibrations of the order of r = 0, 1, 2, 3, 4.

2.7.4

Frequencies and orders of magnetic pressure

Angular frequencies and orders of radial magnetic pressure result from Equations 2.115, 2.116, and 2.117, i.e.: • excited by the stator harmonics of the same number ν, i.e., ωr = 2ω;

fr = 2 f ;

r = 2νp

(2.125)

• excited by the stator ν and rotor µ harmonics ωr = ω ± ωµ ;

fr = f ± f µ ;

r = (ν ± µ) p (2.126)

• excited by the rotor harmonics of the same number µ, i.e., ωr = 2ωµ ;

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fr = 2 f µ ;

r = 2µp

(2.127)

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For the fundamental space harmonic ν = 1 the frequency of the radial magnetic pressure according to Equation 2.125 is fr = 2 f and the order (circumferential vibrational mode) r = 2 p. The rotating component with 2 p pole pairs and frequency 2 f can cause significant vibration, especially, in large machines, e.g., for Bm ν=1 = 0.8 T the magnitude of the radial magnetic pressure is 127.4 kN/m2 . For higher space harmonics ν > 1 the frequency of the radial force pressure is also fr = 2 f and the order r = 2νp. For salient pole synchronous machines the numbers of higher space harmonics for the stator and rotor are ν = 2km 1 ± 1 and µ = 1 ± 2k, respectively, where k = 0, 1, 2, 3, . . . and m 1 is the number of phases. The vibration order according to Equation 2.126 is r = p[2km 1 ±1 ±(1 ±2k )] or r = 2 p[k (m 1 +1)+1] and r = 2 p[k (m 1 +1)−1], e.g., for k = 0 the vibration order is r = 2 p, for k = 1 and m 1 = 3 the order r = 10 p and r = 6 p, for k = 2 and m 1 = 3 the order r = 18 p and r = 14 p, etc. The frequency of vibration is fr = f ± µf = f [1 ± 1 ± 2k] or fr = 2 f (1 +k ) and fr = 2 f k. The frequency of the rotor space harmonics is f (1 ±2k ). The rotor of a salient pole synchronous machine without any cage winding can be considered as a rotor with the number of slots s0 = 2 p. Thus, the angular frequency of the rotor space harmonics and rotor angular speed = 2 p f / p is f ωµ = ω ± s0 k = ω ± 2 pk 2π = 2π f (1 ± 2k ) (2.128) p The vibrational mode of the rotor harmonics of the same number µ according to Equation 2.127 is r = 2µp = 2(1 ± 2k ) p and frequency fr = 2 f µ . For a salient pole synchronous machine fr = 2 f (1 ± 2k ) and for induction machine fr = 2 f [1 ± k(s2 / p)(1 − s)] where µ = 1 ± 2k, k = 0, 1, 2, 3, . . . and s is the slip (Equation 2.2) for fundamental. Those harmonics of low orders produce significant radial forces, in PM synchronous motors also at zero stator current state. The frequencies of radial forces due to rotor space harmonics of the same number µ for PM synchronous motors are given in Table 2.3.

2.7.5

Radial forces in synchronous machines with slotted stator

Significant vibration and noise can be produced by interaction of the PM rotor field and slotted structure of the stator. The fictitious MMF wave excited by PMs is expressed by Equation 2.108. The PM rotor magnetic flux density waveform modulated by stator slots is a product of the MMF F2 (α, t) and relative air gap permeance λsl (α) due to slot opening varying with the angle α. According to Equations 2.68 and 2.71 sl (α) = −g0

∞

Ak cos(ks1 α)

(2.129)

k=1,2,3 2 where −g0 Ak = 2µ0 γ1 kok /t1 . By multiplying Equations 2.108 and 2.129 the magnetic flux density component produced as a result of the modulation of the

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Noise of Polyphase Electric Motors

Table 2.3 Frequencies fr = 2 f (1 ± 2k ) and orders r = 2(1 ± 2k ) p of forces produced by the rotor magnetic field of synchronous motors. k 0 1 2 3 4 5 6 7 8 9 10

Frequency of radial force fr 2f −2 f and 6 f −6 f and 10 f −10 f and 14 f −14 f and 18 f −18 f and 22 f −22 f and 26 f −26 f and 30 f −30 f and 34 f −34 f and 38 f −38 f and 42 f

Order of radial force r 2p −2 p and 6 p −6 p and 10 p −10 p and 14 p −14 p and 18 p −18 p and 22 p −22 p and 26 p −26 p and 30 p −30 p and 34 p −34 p and 38 p −38 p and 42 p

rotor flux density by the stator slot openings is bλ (α, t ) = −g0

∞

Fm µ cos(µp α ∓ ωµ t + φµ )

µ=1,3,5

=

∞

Ak cos(ks1 α)

k =1,2,3

∞ ∞ 1 g0 Fm µ Ak cos[(µp ± ks1 )α ∓ ωµ t + φµ )]. (2.130) 2 µ=1,3,5 k =1,2,3

In the above Equation 2.130 the trigonometric identity cos α cos β = 0.5[cos(α + β) + cos(α − β)] has been used. The radial force pressure on the stator due to the rotor µth and stator permeance kth harmonic is 1 {0.5g0 Fm µ Ak cos[(µp ± ks1 )α ∓ ωµ t + φµ ]}2 2µ0 1 2 2 2 = F A {1 + cos[2(µp ± ks1 )α ∓ 2ωµ t + 2φµ ]}. (2.131) 8µ0 g0 m µ k

pr λ,µ,k (α, t ) =

Similar to the cogging torque, radial forces according to Equation 2.131 will be present both at zero stator current state and when the motor is fed with current. The order (circumferential mode) of the force pressure pr λ (α, t) is rλ = 2(µp ± ks1 ) and frequency fr λ = 2µf . The most important are low even orders rλ = 2, 4, 6, and 8 for the fundamental stator permeance harmonic k = 1, i.e., µ=

|0.5rλ ∓ s1 | = 1, 3, 5, . . . p

and

fr λ = 2µf.

(2.132)

Table 2.4 shows values of rλ , µ, and fr λ obtained on the basis of Equation 2.132 for selected numbers of pole pairs p and stator slots s1 . The most dangerous case

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Table 2.4 Orders and frequencies of radial forces due to interaction of the rotor field and stator slotted core in three-phase synchronous and PM motors (Equation 2.132). p

rλ

2

4

2

8

3

6

4

8

5 5

2 8

s1

q1

µ

fr λ = 2µf

12 18 24 36 12 18 24 36 18 24 36 24 36 36 36

1 1.5 2 3 1 1.5 2 3 1 1.333 2 1 1.5 1.2 1.2

5 and 7 8 and 10 11 and 13 17 and 19 4 and 8 7 and 11 10 and 14 16 and 20 5 and 7 7 and 9 11 and 13 5 and 7 8 and 10 7 8

10 f and 14 f 16 f and 20 f 22 f and 26 f 34 f and 38 f 8 f and 16 f 14 f and 22 f 20 f and 28 f 32 f and 40 f 10 f and 14 f 14 f and 18 f 22 f and 26 f 10 f and 14 f 16 f and 20 f 14 f 16 f

is for p = 5 and s1 = 36 because the order (circumferential mode) is low, i.e. rλ = 2. This mode should be avoided in the design of PM brushless motors. For example, for 660 rpm, p = 5, s1 = 36 the input frequency is f = 55 Hz and fr λ = 14 × 55 = 770 Hz at rλ = 2. If the number of pole pairs is reduced to p = 4, the input frequency is f = 44 Hz and the frequency of radial forces are fr λ = 16 × 44 = 704 Hz and fr λ = 20 × 44 = 880 Hz at rλ = 8. Reduction of number of poles from 5 to 4 and keeping the same number of slots s1 = 36 increases the order of radial magnetic forces. The higher the order of radial magnetic forces, the lower the acoustic noise of magnetic origin. Similar considerations are given by Walker and Kerruish for large synchronous machines operating at no load [220]. According to [220] the frequency of predominant noise in synchronous machine is fr λ = 2µλ f where µλ is the integer nearest to s1 /(2 p). This frequency is produced by stator harmonic permeance k = 1 and rotor MMF harmonic µ = µλ . The analysis of Equation 2.131 shows that 2µp−2ks1 is least when µ−ks1 / p is least (not in agreement with [220]). For k = 1 the expression µ − s1 / p is least when µ is as close to s1 / p as possible, i.e., µ is the integer nearest to s1 / p. Considering a machine with 2 p = 10, s1 = 36, and f = 55 Hz, the ratio s1 / p = 36/5 = 7.2 and µ = µλ = 7. The frequency of the predominant noise is again fr λ = 2µλ f = 2 × 7 × 55 = 770 Hz and the order rλ = |2(µλ p − s1 )| = |2(7 × 5 − 36)| = 2.

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2.7.6

Noise of Polyphase Electric Motors

Frequencies of vibration and noise

Product of stator space harmonics of the same number According to Equation 2.125, the frequency of the radial magnetic force is equal to the double the frequency of the line fundamental frequency. The order for the stator slot harmonics is r = 2νp = 2(ks1 ± p) (2.133) where ν is according to Equation 2.21. Interaction of stator and rotor harmonics For the fundamental time harmonic the field of each stator winding harmonic at a given point with the coordinate α pulsates with the angular frequency ω = 2π f , where f is the fundamental frequency. For an induction motor the angular frequency of the rotor slot harmonics is [87]

f s2 ωµ = ω ± ks2 m = ω ± ks2 2π (1 − s) = 2π f 1 ± k (1 − s) (2.134) p p where m = 2π f (1 − s)/ p is the mechanical angular speed of the rotor and k = 1, 2, 3, . . .. Thus, the angular frequencies of radial forces due to the stator and rotor slot harmonics result from Equation 2.126

f ωr = ω ± ωµ = ω ± ω ± ks2 2π (1 − s) (2.135) p The frequencies of radial forces s2 fr = f k (1 − s) p

and

s2 fr = f k (1 − s) ± 2 p

(2.136)

The orders are also determined by Equation 2.126 r = (ν ± µ) p = ks1 ± ks2 ± 2 p

(2.137)

where ν is according to Equation 2.21 and µ is according to Equation 2.23. For ν = µ = 1 the orders are r = 0 and r = 2 p. For ν = µ the orders are r = 0 and r = 2νp = 2µp. For a PM synchronous motor with salient poles or embedded magnets and without any cage winding, the higher harmonics of the rotor µ=k

so ± 1 = 2k ± 1 p

(2.138)

are called “pole harmonics” and their “winding coefficients” are equal to “winding coefficients” of the fundamental harmonic. The number of slots for a synchronous

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motor without any cage winding so = 2 p and k = 1, 2, 3, . . .. Thus, the angular frequency of the rotor harmonics (m = = ω/ p) is ωµ = ω ± kso = ω ± 2 pk = ω ± 2 pk(2π

f ) = 2π f (1 ± 2k). p

(2.139)

The noise and vibration frequencies due to the stator winding and rotor harmonics are according to Equation 2.126, i.e., fr = f ± f (1 ± 2k) or

fr = 2(1 + k) f

and

fr = 2k f.

The corresponding orders of radial forces s1 r = (ν ± µ) p = k ± 1 ± 2k ± 1 p = ks1 ± 2 p(k + 1). p

(2.140)

(2.141)

Product of rotor space harmonics of the same number According to Equation 2.127 the angular frequency and frequency of the radial magnetic forces of an induction machine are

s2 ωr = 2ωµ = 2(ω ± ks2 m ) = 4π f 1 ± k (1 − s) (2.142) p

s2 fr = 2 f 1 ± k (1 − s) (2.143) p and the order r = 2µp = 2(ks2 ± p) (2.144) where µ is according to Equation 2.23. For synchronous motors s2 = s0 = 2 p and slip s = 0, so that Equations 2.143 and 2.144 become fr = 2(1 ± 2k) f ;

r = 2 p(1 ± 2k).

(2.145)

Radial force harmonics due to eccentricity The stator and rotor magnetic flux density waves due to eccentricity are expressed by Equations 2.95 and 2.98, harmonic numbers by Equations 2.96 and 2.99, and harmonic angular frequencies by Equations 2.85, 2.86, and 2.100. The angular frequency of the magnetic flux density wave for static eccentricity is ω = 2π f and according to Equation 2.95 for dynamic eccentricity this angular frequency is ω ± . The angular frequency of higher harmonic radial forces in an induction machine due to eccentricity for ν = 1 is calculated with the aid of Equations 2.95, 2.100, and 2.126, i.e.,

s2 r = ω ± ± ων=1 + ω k (1 − s) . p

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Noise of Polyphase Electric Motors

The above equation for static eccentricity ( = 0) gives the following two frequencies

s2 s2 fr = f 2 + k (1 − s) and fr = f k (1 − s). (2.146) p p The basic orders of eccentricity harmonics are r =1

and

r =2

(2.147)

as there is a single-sided magnetic pull due to eccentricity (bending and ovaling modes). However, on the basis of Equations 2.96, 2.99, and 2.126, there are also higher orders of eccentricity force harmonics s2 1 1 r = (ν ± µ ) p = 1 ± ± k + 1 ± p p p p = p ± 1 ± (ks2 + p ± 1) or r = 2( p ± 1) ± ks2

and

r = ks2

(2.148)

that usually can be neglected. Vibration of zero order (r = 0) can also be produced for interference of space harmonics of the same order [10]. The angular frequency of higher harmonic radial forces in an induction machine due to dynamic eccentricity ( = 2π( f / p)(1 − s), according to Equations 2.85, 2.100, and 2.126 is

s2 1 r = ω 1 ± (1 − s) ± 1 ± k (1 − s) . p p The above equation gives the next two frequencies of radial forces due to eccentricity

s2 s2 1 1 fr = f 2 ± (1 − s) + k (1 − s) ; fr = f (1 − s) + k (1 − s) p p p p (2.149) The orders for frequencies (Equation 2.149) are determined by Equations 2.147 and 2.148. For synchronous motors the frequencies of radial forces due to eccentricity are

1 1 fr = 2 f (1 + k); fr = 2 f k; fr = f 2 (1 + k) ± ; fr = f 2k ± . p p (2.150)

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Radial force harmonics due to stator winding parallel paths The effect of eccentricity can produce additional radial forces if the stator winding has parallel current paths. The maximum number of parallel paths is a p = 2 p. The magnitude of the magnetic flux density waveform excited by currents in parallel paths is usually not more than 10 to 15% of the magnitude of the fundamental waveform excited by the stator winding. Interaction of fundamental flux density wave of r = p and frequency f with backward rotating wave (r = p∓1, f = − f ) caused by internal short-circuit currents for > 0 gives radial magnetic force wave of r = 1 and fr = 2 f . The magnitude of this force increases with the load. To avoid noise, Heller and Hamata [87] recommend the following selection of the numbers of rotor slots of induction machines: p (2.151) − p(6c ± 1) = 1 if a p > 2 is odd s2 ± a s2 ± 2 p − p(6c ± 1) = (2.152) if a p > 2 is even 1 a |(s2 ± 2 p) − p(6c ± 1)| = 1

if

ap = 2

(2.153)

where c = 1, 2, 3, . . . is a small integer. Radial force harmonics due to magnetic saturation The saturation magnetic flux density equation is a product of the MMF for ν = 1 according to Equation 2.45 and the second term of the air gap permeance (Equation 2.102), (Equation 2.103), i.e., bsat (α, t) = F1,1 (α, t)sat (α, t) = −Fmν=1 cos( pα − ωt)sat cos(2 pα − 2ωt − 2φs ) 1 1 = − Fmν=1 sat cos(− pα + ωt + 2φs ) − Fmν=1 sat 2 2 × cos(3 pα − 3ωt − 2φs ) = −Bsat [cos( pα − ωt − 2φs ) + cos(3 pα − 3ωt − 2φs )]

(2.154)

where Bsat = 0.5Fmν=1 sat . The above Equation 2.154 describes two stator magnetic flux density saturation waves. One wave has frequency and number of poles the same as the fundamental and second wave has three times higher frequency and number of pole pairs than the fundamental. Equation 2.154 includes both the magnetic saturation of the stator teeth and core (yoke) [64, 179]. Assuming that only the harmonic ν = 3 in Equation 2.154 saturates the teeth, the flux density in the rotor due to saturation is: b2 (α, t) =

∞

Bsat,µ cos(µpα − ωµ − ψµ )

(2.155)

µ=1

where for induction motor µ=k

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s2 + 3. p

(2.156)

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According to Equation 2.134 the angular frequency of the rotor saturation harmonics

s2 ωµ = 3ω + ks2 = 3ω ± k 2π f (1 − s ) . p The angular frequency an order of radial magnetic forces due to saturation of the magnetic circuit can be found with the aid of Equation 2.126

s2 ωr sat = ω ± ωµ = ω ± 3ω ± k 2π (1 − s ) (2.157) p s1 s2 r = (ν ± µ) p = k ± 1 ± k + 3 p . p p The last two equations give the frequencies and orders of radial magnetic forces due to saturation in the following form

s2 s2 fr sat = k (1 − s ) + 4 f and fr sat = k (1 − s ) + 2 f (2.158) p p r = ks1 + ks2 + 4 p

and

r = ks1 + ks2 + 2 p .

(2.159)

Frequencies of radial magnetic forces produced by higher space harmonics in cage induction and PM synchronous motors are given in Tables 2.5 and 2.6.

2.8 2.8.1

Other sources of electromagnetic vibration and noise Unbalanced line voltage

The unbalanced line voltage causes vibration the frequency of which is equal to double the line frequency, i.e., fr unb = 2 f.

(2.160)

The effect of phase unbalance can be included by assigning different values of currents for each phase in Equation 2.42 and then calculating the resultant stator MMF similar to Equation 2.43.

2.8.2

Magnetostriction

Magnetostriction is the change of physical dimensions and crystals of a material in response to changing its magnetization. When a magnetostrictive material is subjected to an alternating magnetic field, it changes its shape and dimensions. Most ferromagnetic materials show some measurable magnetostriction. The

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Table 2.5 Frequencies of radial magnetic forces produced by higher space harmonic ν = 1 (fundamental time harmonic n = 1) in induction motors. Source

Frequency Hz

Product of stator space haromics bν2 of fr = 2 f the same number ν Product of rotor space fr = 2 f [1 ± k(s2 / p)(1 − s)] harmonics bµ2 of the where s is slip for same number µ fundamental space harmonic Product of stator and rotor space harmonics bν bµ — general fr = f ± f µ equations Product of stator and rotor space harmonics bν bµ where fr = [k(s2 / p)(1 − s) ± 2] f ν = ks1 / p ± 1 and fr = [k(s2 / p)(1 − s)] f µ = ks2 / p ± 1 (the so called slot or step harmonics) Product of stator and rotor static eccentricity fr = [2 + k(s2 / p)(1 − s)] f space fr = [k(s2 / p)(1 − s)] f harmonics bν bµ Product of stator and rotor dynamic fr = [2 ± (1 − s)/ p eccentricity +k(s2 / p)(1 − s)] f space fr = [(1 − s)/ p harmonics +k(s2 / p)(1 − s)] f bν bµ Product of stator and rotor magnetic fr = [k(s2 / p)(1 − s) + 4] f saturation space fr = [k(s2 / p)(1 − s) + 2] f harmonics bν bµ

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Order (circumferential mode) r = 2νp r = 2(ks1 ± p) k = 0, 1, 2, 3, . . . r = 2µp r = 2(ks2 ± p) r = (ν ± µ) p

r = ks1 ± ks2 ± 2 p

r =1 r =2

r =1 r =2

r = ks1 + ks2 + 4 p r = ks1 + ks2 + 2 p

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Noise of Polyphase Electric Motors

Table 2.6 Frequencies of radial magnetic forces produced by higher space harmonic ν = 1 (fundamental time harmonic n = 1) in synchronous motors. Source

Frequency Hz

Order (circumferential mode) r = 2νp ν = 2km 1 ± 1 k = 0, 1, 2, 3, . . .

Product of stator space haromics bν2 of fr = 2 f the same number ν Interaction of the rotor fr = 2µλ f magnetic field and slotted where r = 2(µλ p ± s1 ) core of the stator µλ = integer [s1 / p] Product of rotor space haromics bµ2 of the fr = 2(1 ± 2k) f r = 2µp = 2 p(1 ± 2k) same number µ where µ = 1 ± 2k Product of stator and rotor space harmonics bν bµ — general fr = f ± f µ r = (ν ± µ) p equations Product of stator winding and rotor space harmonics fr = 2(1 + k) f bν bµ where fr = 2k f r = ks1 ± 2 p(1 + k) ν = ks1 / p ± 1 and µ = 2k ± 1 Product of stator and rotor static eccentricity fr = 2(1 + k) f r =1 space fr = 2k f r =2 harmonics bν bµ Product of stator and rotor dynamic eccentricity fr = [2(1 + k) ± 1/ p] f r =1 space fr = (2k ± 1/ p) f r =2 harmonics bν bµ Product of stator and rotor magnetic fr = 2(2 + k) f r = ks1 + 2 p(k + 2) saturation space fr = 2(1 + k) f r = ks1 + 2 p(k + 1) harmonics bν bµ

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magnetostrictive coefficient is defined as the change in length l to the length l of a specimen when the magnetization increases from zero to its saturation value, i.e., l = (2.161) l The coefficient can be positive or negative. In the first case, the increase in the magnetic flux density causes expansion of the specimen. In the second case, the specimen shrinks as the magnetic flux density increases. For pure iron the saturation magnetostriction is = −2 × 10−5 to +2 × 10−5 , for laminated steels = −0.1 × 10−5 to +0.5 × 10−5 , for nickel = −5.2 × 10−5 to 0.8 × 10−5 and for polycrystal cobalt = −6 × 10−5 to −2.5 × 10−5 . Some alloys, e.g., iron and dysprosium DyFe2 or iron and terbium TbFe2 , display “giant” magnetostriction under relatively small fields. If the magnetostriction acts to contract a specimen, then this will act against any tensile stress on the material and leads to larger value of the Young’s modulus. The core of an electrical machine or transformer in alternating magnetic fields changes its dimensions cyclically. Figure 2.10 shows a sinusoidal variation of the magnetic flux density and corresponding variation of the magnetostrictive coefficient with time. The curve (t) shown in Figure 2.10 resolved into Fourier series has a constant component and a series of harmonic functions. Those harmonics approximately visualize vibrations relative to the mean position which is determined by the constant component. The fundamental wave of those vibration has a frequency twice the frequency of the flux density f (line frequency), i.e., f ms = 2 f

(a)

(2.162)

b(t)

t

(b)

Λ(t)

t

Figure 2.10 Variation of magnetostrictive coefficient for sinusoidal magnetic flux density: (a) magnetic flux density as a function of time; (b) magnetostrictive coefficient as a function of time.

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Noise of Polyphase Electric Motors

while the order of the fundamental wave of the magnetostriction force in an a.c. rotating machine is rms = 2 p (2.163) For static electromechanical energy converters as transformers, inductors, fluorescent light ballast, etc., magnetostriction is the major cause of acoustic noise. For a.c. rotating machines magnetostriction can contribute considerably to the acoustic noise. Especially, in two pole machines ( p = 1) the magnetostriction force at the twice line frequency (Equation 2.162) can generate undesirable acoustic noise.

2.8.3

Thermal stress analogy

The effect of magnetostriction can be implemented in a similar way as thermal stresses are applied, i.e., the thermal expansion of the free body is calculated based upon the temperature distribution [45, 159]. The elastic strain in the x direction is el,x =

1 (σx − νσ y ) E

(2.164)

where σx is the external stress in the x direction, σ y is the external stress in the y direction, σz = 0, E is Young modulus and ν is Poisson ratio. Including thermal stresses when a material is heated x − αϑ ϑ =

1 (σx − νσ y ) E

(2.165)

where αϑ is the coefficient of thermal expansion, αϑ ϑ is the thermal strain and x = el,x + αϑ ϑ is the total strain (elastic and thermal). For a magnetostrictive material, the magnetostrictive strain λ(B) can also be added to the elastic strain [45], similar to the thermal expansion strain (Equation 2.165), i.e., 1 x − λ(B) = (σx − νσ y ) (2.166) E where the total strain (elastic and magnetostrictive) is x = el,x + λ(B). The magnetostrictive strain λ(B) = 10−6 B 2 (2.167) expressed as a function of the magnetic flux density in the direction of the vector B B is sometiems called the magnetostriction curve [45].

2.8.4

FEM model

The magnetic field is affected by the effect of magnetostriction. Thus, the coupled magnetic and mechanical system must be captured in one magnetomechanical matrix. The effect of magnetostriction can be built into the coupled system using a force distribution that is added to magnetic forces [45, 159, 210, 211].

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The elastic energy of a mechanical finite-element is expressed in terms of the elastic displacement αel and mechanical stiffness matrix K U=

1 T α K αel . 2 el

(2.168)

In general cases the elastic displacement αel = α where α is the total displacement, because the material may show magnetostrictive or thermal expansion. Including the magnetostrictive displacement αms the elastic energy is U=

1 (α − αms )T K (α − αms ). 2

(2.169)

The strains λx , λ y in the x and y directions can be converted into nodal dise (e) placements αms = (αx,i , α ey,i ) where i = 1, 2, 3 means the element nodes with coordinates xi , yi . Considering the midpoint element (xm , ym ) as fixed

(e) αx,i (xi − xm )λx . (2.170) = (yi − ym )λ y α (e) y,i The nodal magnetostriction forces are obtained by multiplying the mechanical (e) stiffness matrix K (e) for one element by the magnetostrictive element αms of the nodes, i.e., (e) (e) Fms = K (e) αms . (2.171) The total magnetostriction force is a sum of magnetostriction forces of all elements, i.e., (e) Fms = Fms = K αms . (2.172) e

The magnetic forces (sum of reluctance forces and Lorentz forces) can be found using virtual works, i.e., A ∂M ∂W Fmag = − AT =− dA (2.173) ∂a ∂a 0 where M is the magnetic “stiffness” matrix, a is the linear displacement, and A is the magnetic vector potential component in the normal direction (z-direction). The magnetic energy 1 W = A T M A. (2.174) 2 The total energy of the magnetomechanical system is the sum of the elastic energy U according to Equation 2.168 and magnetic energy W according to Equation 2.174, i.e., E = U + W. (2.175) The magnetomechanical system is described by the matrix equations [44, 45]: • without magnetostriction

M 0 A T = (2.176) 0 K a R + Fmag

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Noise of Polyphase Electric Motors • with feedback of magnetostriction

M 0 A T = 0 K a R + Fmag + Fms

(2.177)

where R = K a represents external forces and T = M A is the magnetic source term vector. Further procedures for deformation and vibration calculations are given in [44, 45].

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3 Inverter-Fed Motors Most variable speed electric motors are fed from solid state converters (SSCs). The a.c.–a.c. SSC consists of a rectifier, intermediate circuit (filter), and inverter (Figure 3.1). The output voltage and output current of a SSC are nonsinusoidal, that is, contain higher time harmonics as a result of switching. The noise of electric machines with the stator winding nonsinusoidal current increases as compared with an equivalent machine with sinusoidal stator current. The increase in the noise of an induction motor fed from a pulse width modulation (PWM) inverter with switching frequency up to 7 kHz is from 7 to 15 dB(A) [80]. For higher switching frequencies between 7 and 16 kHz the increase in the noise is lower, usually from 2 to 7 dB(A) [80].

3.1

Generation of higher time harmonics

The stator current of an electrical machine fed from or loaded with a SSC is nonsinusoidal, that is, contains higher time harmonics n = 2m 1 k ±1 where k = 0, 1, 2, 3, . . .. The nonsinusoidal current can be expressed in terms of Fourier series as n=∞ √ n=∞ i 1 (t) = I1mn sin(ωn t − φin ) = 2 I1n sin(ωn t − φin ) (3.1) √

n=1

n=1

where I1mn = 2I1n is the peak value of the nth harmonic current. The angular frequency and frequency of higher time harmonic currents are, respectively ωn = nω = 2π n f

f n = n f.

(3.2)

Currents expressed by Equation 3.1 are shifted in each phase of the stator winding of a polyphase machine by the angle 2π/m 1 , where m 1 is the number of phases. For induction motors, the slip for the nth time harmonic is defined as sn =

nn s ∓ n s (1 − s) nn s 65

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(3.3)

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Noise of Polyphase Electric Motors Intermediate Circuit

Rectifier

Inverter VVVF

50 or

d.c.

60 Hz

IM a.c.

a.c.

Figure 3.1 Basic circuitry of a variable voltage variable frequency (VVVF) voltage source SSC for an induction motor. where nn s = n f / p is the synchronous speed for the nth time harmonic, n s (1 − s ) is the rotor speed, and s is the slip for the fundamental harmonic. The “−” sign is for the time harmonics n = 2m 1 k + 1 and the sign “+” is for the time harmonics n = 2m 1 k − 1 where k = 0, 1, 2, 3, . . .. Induction motors operate with small slip 0.02 ≤ s ≤ 0.05, so that 1 sn ≈ 1 ∓ . (3.4) n As the number of time harmonics increases, the slip (Equation 3.4) becomes close to unity. The slip sn , angular frequency ωn of rotor currents, and higher time harmonic slip ssn corresponding to the synchronous speed (Equation 2.1) are: • for forward-rotating magnetic field sn = 1 −

1 (1 − s ); n

ωn = n ωsn = ω[n − (1 − s )];

ssn = 1 − n (3.5)

• for backward-rotating magnetic field sn = 1 +

1 (1 − s ); n

ωn = n ωsn = ω[n + (1 − s )];

ssn = 1 + n (3.6)

where s is the slip for the fundamental harmonic according to Equation 2.2, ω = 2π f is the angular frequency for the fundamental harmonic, f is the input frequency of the fundamental harmonic, and n is the time harmonic number. Higher time harmonics in the stator current of permanent magnet (PM) brushless motors are discussed in Section 4.7.

3.2

Analysis of radial forces for nonsinusoidal currents

The frequencies and orders of most important radial forces produced by SSCs (inverters) are given in Table 3.1. A brief analysis of these forces follows.

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Table 3.1 Frequencies and orders of most important radial magnetic forces produced by inverter harmonics. Frequency Hz

Source Product of stator 2 harmonics b1νn of the same number n Product of rotor harmonics of the same number Product of stator and rotor space harmonics Sums and differences of inverter fundamental and higher harmonics bn bn of different numbers Interaction of the switching frequency f sw and higher time harmonics f n = n f sw ± n f Interaction of the permeance and MMF harmonics associated with time harmonics Rectifier harmonics transmitted to the motor via intermediate circuit and inverter

3.2.1

Order (circumferential mode)

fr,n = 2n f , n = 2km 1 ± 1 where k = 0, 1, 2, 3, . . . fr,n = 2n f µ fr,n = f n ± f µ see also Equation 3.13 fr,n = f ± f n n = 2km 1 ± 1

r = 2νp r = 2 p for ν = 1 r = 2µp r = 2 p for µ = 1 r = (ν ± µ) p r =0 or r = 2p

fr,n = |(± f n ) − f | n = n

r =0

fr,n = |(± f n − f | fr,n = | f n ± µf |

r =0 r =2

fr = 2m 1 k f k = 1, 2, 3, . . .

r = 2p

Stator and rotor magnetic flux density

The space-time variation of the magnetic flux density for the n-th time harmonic can be expressed as a sum of space harmonics • for the stator b1n (α, t) =

∞

b1νn (α, t) =

ν=1

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∞ ν=1

Bmνn cos(νpα − ωn t)

(3.7)

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Noise of Polyphase Electric Motors where b1νn (α, t ) = Bm νn cos(νp α − ωn t ) • for the rotor b2n (α, t ) =

∞

b1µn (α, t ) =

µ=1

∞

Bm µn cos(µp α − ωµ,n t + φµ,n )

(3.8)

µ=1

where b1µn (α, t ) = Bm µn cos(µp α − ωµ,n + φµ,n ). In the above Equations 3.7 and 3.8, ν is the number of the stator space harmonics, µ is the number of the rotor space harmonics, n is the number of the stator time harmonics, Bm νn and Bm µn are the peak values of the magnetic flux density harmonics for the stator and rotor, respectively, α is the angular distance from the origin of the coordinate system, ωn = 2π n f is the angular frequency of the time harmonic current in stator windings, f is the fundamental frequency, ωµ,n is the angular frequency of the space harmonic µ in the rotor system for the given time harmonic n, φµ,n is the angle between vectors of the stator and rotor space harmonics for the given n, and µ0 = 0.4π × 10−6 H/m is the magnetic permeability of free space. Higher space harmonics result from the distribution of stator winding coils in slots, rotor winding slots, or geometry of rotor poles. Higher time harmonics in the stator current are generated by the power supply, that is, utility mains or SSC.

3.2.2

Stator harmonics of the same number

The radial force pressure Figure 2.8 is expressed with the aid of Maxwell stress tensor — Equation 2.114. The first term in Equation 2.114 is the product of the stator harmonics [b1νn (α, t)]2 of the same number ν or n, i.e., [b1νn (α, t)]2 [Bmνn cos(νpα ∓ ωn t]2 = 2µ0 2µ0 2 B = mνn [1 + cos(2νpα ∓ 2ωn t)]. 4µ0

pr,n (α, t) =

(3.9)

For higher time harmonics n > 1 the angular frequency of stator space harmonics is ωn = 2π f n , where f n = n f is the frequency of the n-th higher time harmonic. The constant term for each harmonic ν and n has no significance for noise generation as the magnitudes Bmνn are uniformly distributed along the air gap. For the fundamental harmonic ν = 1 and n = 1 the frequency of the radial force pressure is fr = 2 f and the order (circumferential vibrational mode) r = 2 p. For higher space harmonics ν > 1 and fundamental time harmonic n = 1 the frequency of the radial force pressure is also fr = 2 f and the mode r = 2νp. For ν = 1 and n > 1 the frequency of the radial force pressure is fr,n = 2n f and the order is r = 2 p. Of course, there will be many more harmonics of radial forces as mixed products of stator magnetic flux density harmonics of different numbers, for example, 1 and 7, also produce radial forces.

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69

Interaction of stator and rotor harmonics

The second term in Equation 2.114 is the mixed product of the stator and rotor harmonics 2b1νn (α, t)b2µn (α, t), i.e., pr,n (α, t) = = =

2b1νn (α, t)b1µn (α, t) 2µ0 ∞ 2 ν=1 Bmνn cos(νpα ∓ ωn t) ∞ µ=1 Bmµn cos(µpα ∓ ωµ,n t + φµ,n ) 2µ0 1 2µ0

∞ ∞

Bmνn Bmµn {cos[(νpα ∓ ωn t) − (µpα ∓ ωµ,n t + φµ,n ]

ν=1 µ=1

+ cos[(νpα ∓ ωn t) + (µpα ∓ ωµ,n t + φµ,n ]} ∞ ∞ 1 = Bmνn Bmµn {cos[(ν − µ) pα ∓ (ωn − ωµ,n )t − φµ,n ] 2µ0 ν=1 µ=1 + cos[(ν + µ) pα ∓ (ωn + ωµ,n )t + φµ,n ]}.

(3.10)

The mixed products of the stator and rotor harmonics generate pairs of radial force pressures with pole pairs r = (ν ± µ) p and frequencies fr,n = f n ± f µ,n . For a synchronous machine the angular frequency of the rotor space harmonics for n > 1 is ωµ,n = 2π f n (1 ± k). For an induction machine with the number of rotor slots s2 , the angular frequency of the rotor winding harmonics for time harmonics n ≥ 1 and a “harmonic rotor” angular speed mn = 2π( f n / p)(1 − sn ) is fn ωµ,n = ωn + ks2 mn = ωn ± s2 k 2π (1 − sn ) p s2 = 2π f n 1 ± k (1 − sn ) . (3.11) p According to Equations 3.3 and 3.4, for n = 1 the slip sn=1 = s ≈ 0 and for n > 1 the slip sn ≈ 1. The frequencies of noise and vibration of induction machines due to the mixed products of b1νn (α, t)b2µn (α, t) for µ = ks2 / p ± 1 are s2 fr,n = f n ± f µ,n = f n ± f n 1 ± k (1 − sn ) . p The last equation can also be written as s2 fr,n = f n 1 ± 1 ± k (1 − sn ) p s2 s2 fr,n = f n k (1 − sn ) fr,n = f n k (1 − sn ) ± 2 . p p

(3.12)

(3.13)

The orders of the mixed product harmonics are expressed by Equation 2.137.

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3.2.4

Rotor harmonics of the same number

The third term in Equation 2.114 is the product of the rotor harmonics [b2µn (α, t)2 ] of the same number [b2µn (α, t)]2 [Bmµn cos(µpα ∓ ωµ,n t + φµ,n )]2 = 2µ0 2µ0 1 = Bmµn [1 + cos(2µpα ∓ 2ωµ,n t + 2φµ,n )]. 4µ0

pr,n (α, t) =

3.2.5

(3.14)

Frequencies and orders of magnetic pressure for nonsinusoidal currents

Angular frequencies and orders of the radial magnetic pressure due to higher time and space harmonics result from Equations 3.9, 3.10, and 3.14, i.e.: • excited by the stator harmonics of the same number ν, i.e., ωr,n = 2nω;

fr,n = 2n f ;

r = 2νp

(3.15)

• excited by the stator ν and rotor µ harmonics ωr,n = (nω ±ωµ,n );

fr,n = (n f ± f µ,n );

• excited by the rotor harmonics of the same number µ, i.e., ωr,n = 2nωµ ;

fr,n = 2n f µ ;

r = (ν ±µ) p (3.16)

r = 2µp.

(3.17)

According to Equation 3.15, for the fundamental space harmonic ν = 1 and time harmonics n > 1, the frequency of the radial magnetic pressure is fr,n = 2n f and the order r = 2 p. According to Equation 3.16, for ν = 1 and time harmonics n > 1, the frequency of the radial magnetic pressure is fr,n = n( f ± f µ ) and the order r = (1 ± µ) p. For induction machines the frequencies of magnetic pressures as a function of slip sn are expressed by Equation 3.11. Putting s2 = so = 2 p and sn = 0 for a synchronous machine, the frequency becomes fr,n = f n (1±kso / p) = n f (1 ± 2k). According to Equation 3.17, for space harmonics µ > 1 and time harmonics n > 1, the frequency of the radial magnetic pressure is fr,n = 2n f µ and the order r = 2µp. For a salient pole synchronous machine fr,n = 2n f (1 ± 2k) and for induction machine fr,n = 2n f [1 ± k(s2 / p)(1 − sn )] since according to Equation 2.143 fr = 2 f [1 ± ks2 (1 − s)/ p].

3.2.6

Interaction of stator harmonics of different numbers

Inverters can supply a rich spectrum of higher time harmonics to the stator winding. Higher time harmonics of different numbers can produce significant radial forces, the frequency and order of which can be expressed by the following equations fr,n = (n ± n ) f ;

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r =0

or

r = 2p

(3.18)

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where n = n . Most important are magnetic forces due to sums and differences of the fundamental harmonic f with major higher time harmonics of the stator current [71]. The frequencies of those forces are fr,n = (1 ± n) f where n = 2km 1 ± 1.

3.2.7

Interaction of switching frequency and higher time harmonics

Large amplitude forces are also due to interaction of the switching frequency f sw and higher time harmonics f n = n f sw ± n f [138]. If n is an odd integer, n will be an even integer and vice versa, i.e., f n = f sw ± 2 f , f sw ± 4 f , f sw ± 6 f, . . . and f n = 2 f sw ± f , 2 f sw ± 3 f , 4 f sw ± 5 f, . . .. The most important case is the interaction between the fundamental field harmonic and higher time harmonics for which the frequencies and mode of vibration are [138]: fr,n = |(± f n ) − f |;

3.2.8

r =0

(3.19)

Interaction of permeance and magnetomotive force (MMF) harmonics

Interaction of the permeance field harmonics and MMF harmonics associated with higher time harmonics of the stator current can also result in important vibration, especially at full load and when force orders are low [138]. The frequencies and orders of radial forces are [138] s2 fr,n = |(± f n − f |; r = 0, 2. fr,n = | f n ± f 1 + k (1 − s) |; p (3.20)

3.2.9

Rectifier harmonics

Rectifier harmonics are transmitted via the intermediate circuit and inverter to the stator winding. They can produce radial forces with the following frequency and order fr,n = n f = 2m 1 k f ; r = 2p (3.21) where n = 2m 1 k, k = 1, 2, 3, . . ., and m 1 is the number of phases.

3.3 3.3.1

Higher time harmonic torques in induction machines Asynchronous torques

Asynchronous torques due to higher time harmonics are expressed by the following equations m1 (3.22) Tn = √ pN2 kw2 n I2n cos ψ2n = cT n I2n cos ψ2n 2

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or

2 (I2n ) R2n (I )2 R2n Pw2n = m 1 p 2n =p (3.23) s n n sn ωn sn ωn √ where cT = m 1 pN2 kw2 / 2, N2 is the number of rotor turns per phase, kw2 is the rotor winding factor for fundamental, n is the stator magnetic flux produced by the nth time harmonic, ψ2n is the phase shift between the electromotive force (EMF) (excited by n ) and rotor harmonic current I2n , I2n is the rotor current due to the nth time harmonic referred to the stator system, R2n is the rotor resistance referred to the stator system, Pw2n are the rotor winding losses for the nth time harmonic, and 2π n f ωn n = = (3.24) p p

Tn = m 1

is the angular speed of the stator magnetic field due to the nth time harmonic. The direction of the torque (Equation 3.22) and (Equation 3.23) depends on the number of the higher time harmonic. The torque can be in the direction of rotation (propulsion torque) or in the opposite direction (braking torque). For a three-phase machine (m 1 = 3) higher time harmonics n = 1, 7, 13, . . . generate asynchronous torques in the direction of rotation and higher harmonics n = 5, 11, 17, . . . generate braking torques. For evaluation of torques, the electric losses in the stator winding can be assumed equal to the electric losses in the rotor winding, i.e. [21], Pw2n ≈ 0.5Pwn ≈ 0.5(SC R)2

Pw Pws=1 ≈ 0.5 3 n n3

(3.25)

where SC R = 3 . . . 7 is the starting current ratio (starting–to–rated current), Pwn are stator and rotor winding losses due tho the nth time harmonic, Pw are stator and rotor winding losses at rated load, Pws=1 are winding losses at locked rotor (s = 1), and Pw Pwn = (SC R)2 3 . (3.26) n Assuming that for higher time harmonics sn ≈ 1 and n = n, the asynchronous torque is inversely proportional to n 4 , i.e., Tn = p

Pws=1 Tst Pw2n Pws=1 ≈ ≈ 4 ≈p 3 4 s n ωn 2n × 1 × nω 2n n

(3.27)

where Tst ≈ 0.5Pws=1 / is the starting torque due to the fundamental harmonic. Since higher harmonic asynchronous torques are inversely proportional to n 4 , they are negligible and have little effect on the vibration and noise.

3.3.2

Pulsating torques

Pulsating torques are produced as interaction of currents and fluxes of different frequencies. Their frequencies are significantly higher than the fundamental frequency and their mean value is equal to zero. The number of these torques can

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be extremely large as there is a very large number of interactions among all harmonics. The most important pulsating torques are those produced by the stator fundamental magnetic flux and rotor current higher harmonics expressed by the following equation: 2 pm 1 T p1,n = √ N2 kw2 I2n cos[(n ± 1)]ωt − ψ1,2n ] 2 = 2cT I2n cos[(n ± 1)]ωt − ψ1,2n ]

(3.28)

where is the magnetic flux due to the fundamental harmonic n = 1 and ψ1,2n is the phase shift between the EMF for n = 1 and rotor current I2n . The “−” sign is for harmonics n = 2m 1 k + 1 and the sign “+” is for the harmonics n = 2m 1 k−1. It means that in a three-phase machine the pulsating torques due to the 5th and 7th harmonics have the same frequency 6 f and rotate in opposite directions, i.e., f − (−5 f ) = 6 f and f − 7 f = −6 f . In three-phase induction motors the major torque ripple source is the 6th harmonic component. For the 11th and 13th harmonics the frequency of pulsating torques is 12 f , i.e., f − (−11 f ) = 12 f and f − 13 f = −12 f . The magnitudes of pulsating torques at SC R = 3 . . . 5 are 7 to 12% of the rated torque for the frequency 6 f (due to the 5th and 7th harmonics) and 0.8 to 1.5% for the frequency 12 f (due to the 11th and 13th harmonics [21]. Pulsating torques may have a signifucant influence on the vibration and noise.

3.4

Higher time harmonic torques in permanent magnet (PM) brushless machines

The interaction of the stator current and magnetic flux linkage harmonics produce: • constant torques if harmonics are of the same number; • pulsating torques if harmonics are of different numbers. For trapezoidal EMF and rectangular current the electromagnetic torque is constant (no pulsating torque) [117, 177]. All odd harmonics of the magnetic flux that interact with the stator current harmonics of the same number (except of triple harmonics) produce a constant torque, for example, fundamental harmonic of flux with the fundamental harmonic of current, 5th harmonic of flux and 5th harmonic of current, 7th harmonic of flux and 7th harmonic of current, and so on. However, the contribution of the odd harmonics n > 1 to the constant torque is negligible.

3.5

Influence of the switching frequency of an inverter

There is a significant influence of the switching frequency of a pulse width modulation (PWM) inverter on the overall motor noise. The switching frequency of the inverter is established by the frequency of a triangular waveform. In a PWM

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75

Sound Pressure Level, dB(A)

70

65

60

55 Switching Frequency 1.8 kHz Switching Frequency 3.6 kHz Switching Frequency 7.2 kHz Switching Frequency 14.4 kHz

50

45

40 0

10

20

30

40 50 60 Frequency, Hz

70

80

90

100

Figure 3.2 Influence of the inverter on the sound pressure level of a 9.3 kW (12.5 hp), 60 Hz, 4-pole induction motor for four different switching frequencies [161]. inverter the sinusoidal control signal at the desired frequency is compared with a triangular waveform [156]. At speeds below the rated speed, the noise level of a small motor with a shaft driven fan decreases with the speed (Chapter 7, Equation 7.28). Below the rated speed the magnetic noise is predominant. When the speed exceeds the rated speed, the overall noise increases due to the increased noise of the fan. Figure 3.2 shows the overall noise of a 9.3 kW (12.5 hp), 4-pole induction motor fed from a PWM inverter for four different switching frequencies [161]. For 1.8-kHz switching frequency, the influence of the magnetic noise is considerable and the sound pressure level changes only ±4 dB(A) within the operation frequency range from 10 to 100 Hz. For 3.6, 7.2, and 144 kHz switching frequencies, the magnetic noise and consequently the overall noise is reduced significantly for operation frequencies below the rated frequency of 60 Hz [161]. Table 3.2 compares the sound pressure level radiated by small power induction motors fed with 60 Hz three-phase utility grid and PWM inverters [161]. It is clear that below the rated frequency 60 Hz the sound pressure level decreases as the switching frequency is increased. This is because the noise of magnetic origin

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80

Noise Level (dB)

70

4 kHz Carrier

60 50 40

16 kHz Carrier

30 20 10 0

0

10

20

30

40

50

60

Motor Frequency (Hz)

Figure 3.3 Noise level vs. induction motor operation frequency for 4 and 16 kHz carrier frequencies of X2C series programmable inverters. (Courtesy of TB Wood’s, Chambersburg, PA, U.S.A. [244].) is predominant at low operation frequencies. The overall noise at low speeds is strongly affected by the switching frequency (compare the SPL at 30 and 60 Hz). At operation frequencies above the rated frequency 60 Hz, the influence of the cooling fan on the overall noise is very strong. The increased sound power level at 100 Hz is due to the fan and is practically independent of the switching frequency. The switching frequencies can cause pure tones that are very disturbing to the human ear [147]. The vibration frequncies due to fundamental waves and switching frequencies must not match the natural frequencies of the stator system. Otherwise, the audible noise increases significantly [147]. Modern inverters operate at higher switching frequencies ( f s ≥ 12 kHz), which results in lower harmonic content, especially at low speeds. The inverter losses increase with increasing the switching frequency and motor higher harmonic losses decrease with increasing the switching frequency [147]. Thus, at low speeds the influence of the switching frequency on the efficiency of the motor–inverter system is rather small [147]. Many inverters, for example, the X2C series TB Wood’s inverters have programmable carrier frequencies from 4 to 16 kHz for quiet drive operation [244]. Low carrier frequencies can generate audible motor noise, while at 16 kHz audible noise is virtually eliminated (see Figure 3.3).

3.6

Noise reduction of inverter-fed motors

Inverter-fed induction and PM synchronous motors are noisier than those fed with purely sinusoidal current. The radial magnetic force spectrum is richer and the chance of matching the exciting frequencies with natural frequencies of the stator

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Table 3.2 Influence of inverter on sound pressure level of induction motors, dB(A) [161] Output Utility power kW

0.37 2p = 4 0.75 2p = 2 0.75 2p = 4 2.2 2p = 2 2.2 2p = 4 3.7 2p = 4 5.5 2p = 4 9.3 2p = 4 15.0 2p = 4

grid

Switching frequency of inverter 3.6 kHz

7.2 kHz

14.4 kHz

60 Hz Operation frequency Operation frequency Operation frequency Hz Hz Hz 30 60 100 30 60 100 30 60 100 46.1 52.1 50.5

58.9

42.2 49.3

58.9

37.1 46.8

58.8

61.2 61.2 68.2

73.1

45.9 60.5

72.7

44.7 60.4

72.8

44.7 60.3 52.9

58.8

47.0 57.1

59.5

34.3 46.8

58.5

67.5 60.6 68.5

80.8

54.9 68.4

80.8

53.3 68.2

81.1

53.1 68.7 60.6

63.8

56.6 65.4

64.7

46.6 52.9

63.4

54.4 56.3 58.7

66.8

53.7 65.6

67.0

44.3 54.8

66.5

56.3 61.8 59.6

68.1

52.9 63.6

68.2

43.0 56.5

68.0

61.7 65.1 71.1

73.7

57.2 68.1

73.4

46.7 61.9

73.4

69.2 68.7 73.4

80.9

54.8 69.8

80.8

52.5 69.3

80.7

system is greatly increased. The following strategies can be employed to reduce the acoustic noise of inverter-fed motors; • elimination of higher time harmonics; • operation with low commutation angle (the larger the commutation angle, the higher the magnitude of the torque pulsation); • dynamic variation of the switching frequency [147]; • high switching frequency (above 15 kHz) is a very efficient method but imposes high stress on solid switches and increased switching frequency losses; • random PWM [36, 82, 229]; • active noise control of the motor.

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4 Torque Pulsations In addition to radial forces, torque pulsations and tangential forces also contribute to vibrations and magnetic noise. Normally, tangential forces have less effect on vibrations and noise than radial forces. Tangential forces can either slightly magnify or reduce the amplitude of the first order radial forces [87]. In this chapter, calculations of torque ripple and tangential forces with special attention given to cogging torque of permanent magnet (PM) synchronous motors will be discussed.

4.1

Analytical methods of instantaneous torque calculation

The electromagnetic torque of electrical machines is calculated analytically as T =−

∂W ∂

J

(4.1)

where W is the magnetic field energy in the air gap and is the mechanical angle measured around the air gap periphery. Equation 4.1 can be written as [87] T =−

gL i ∂ 2µ0 ∂

2π

[B(α, t)]2 dα.

(4.2)

0

Neglecting the magnetic saturation B(α, t) =

µ0 µ0 F(α, t) = [F1 (α, t) + F2 (α, t)] g g

(4.3)

where F1 (α, t) is the stator magnetomotive force (MMF) and F2 (α, t) is the rotor MMF. Thus, 2π µ0 L i ∂ T =− [F1 (α, t) + F2 (α, t)]2 dα. (4.4) 2g ∂ 0 77 Copyright © 2006 Taylor & Francis Group, LLC

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For a stationary stator and coordinate system fixed to the stator, only the rotor MMF is dependent on the rotor position [87], i.e., µ0 L i T =− g

2π

[F1 (α, t) + F2 (α, t)]

0

∂F2 (α, t) dα. ∂

(4.5)

Harmonic torques are calculated on the basis of Equation 4.5 putting F1 and F2 in form of the Fourier series. A torque can be produced only by harmonics having an equal number of pole pairs [87].

4.2

Numerical methods of instantaneous torque calculation

The instantaneous torque can be numerically calculated by using • virtual work method

∂w(, θ ) ∂w (i, θ) T =− = ∂θ ∂θ =const i=const

(4.6)

where w, w , , i and θ are the magnetic energy, coenergy, flux linkage vector, current vector and mechanical angle, respectively. • Maxwell stress tensor 1 1 2 F = B( B · n) − B n d S; T = r × F (4.7) µ0 2µ0 T = Li

1 µ0

r Bn Bt dl

(4.8)

l

where n, L i , l, r , Bn and Bt are the normal vector to the surface S, stack length, integration contour, radius, radial (normal) component of the magnetic flux density and tangential component of the magnetic flux density, respectively. • Lorentz force theorem π/(2m 1 p) T = i j (t) 2 pN r L i B(θ, t)dθ j=A,B,C

=

−π/(2m 1 p)

1 [e f A (t)i 1A (t) + e f B (t)i 1B (t) + e f C (t)i 1C (t)] m

(4.9)

where p, θ, m and N are the number of pole pairs, mechanical degree, mechanical angular speed and conductor number in phase belt, respectively. The products e f A (t)i 1A (t), e f B (t)i 1B (t), and e f C (t)i 1C (t) represent electromagnetic power per phase.

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4.3

79

Electromagnetic torque components

The instantaneous torque of an electrical motor T (α) = T0 + Tr (α)

(4.10)

has two components (Figure 4.1), i.e.: • constant or average component T0 ; • periodic component Tr (α), which is a function of time or electrical angle α, superimposed on the constant component. The periodic component causes the torque pulsation also called the torque ripple. There are many definitions of the torque ripple. The torque ripple is caused by both the construction of the machine and power supply. The torque ripple can be defined in one of the following ways: Tmax − Tmin Tmax + Tmin Tmax − Tmin tr = Tav Tmax − Tmin tr = Tr ms [torque ripple]r ms Trr ms tr = = Tav Tav

tr =

(4.11) (4.12) (4.13) (4.14)

where the average torques in Equations 4.12 and 4.14 α+T p Tp 1 1 Tav = T (α)dα = T (α)dα Tp α Tp 0

T Tr T0

Time Tp

Figure 4.1 Constant and periodic components of the torque.

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(4.15)

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and the r ms or effective torque in Equation 4.13 Tp 1 Tr ms = T 2 (α)dα. Tp 0

(4.16)

The r ms torque ripple Trr ms in Equation 4.14 is calculated according to Equation 4.16 in which T (α) is replaced by Tr (α). The time T p in Equations 4.15 and 4.16 is the period of the torque waveform. For sinusoidal waveform the half-cycle √average value is (2/π )Tm , where Tm is the peak torque and the r ms value is Tm / 2. For the waveform containing higher harmonics, the r ms value of the torque ripple is

Trr ms = Trr2 ms1 + Trr2 ms2 + . . . + Trr2 msν . (4.17)

4.4

Sources of torque pulsations

There are three sources of the torque ripple coming from an electrical machine: (a) cogging effect (detent effect), that is, interaction between the rotor magnetic flux and variable permeance of the air gap due to the stator slot opening geometry; (b) distortion of sinusoidal or trapezoidal distribution of the magnetic flux density in the air gap; (c) the difference between permeances of the air gap in the d and q axis. The cogging effect produces the so-called cogging torque, higher harmonics of the magnetic flux density in the air gap produce the field harmonic electromagnetic torque, and the unequal permeance in the d and q axis produces the reluctance torque. The causes of torque pulsation coming from the supply are: (d) current ripple resulting, for example, from PWM; (e) phase current commutation.

4.5

Higher harmonic torques of induction motors

The theory of higher harmonic torques of induction machines due to stator and rotor slots have been developed by G¨orges (1896), Arnold (1909), Punga (1912), Dreyfus (1924), Krondl (1926), Lund (1932), Heller (1947), Alger (1954), Jordan (1962), Oberretel (1965), and others [87]. Harmonic torques of induction motors will not be considered in this book. Any induction motor may be imagined as a series of mechanically coupled asynchronous and synchronous motors having a different number of poles [87]. An asynchronous torque is created if a certain stator MMF harmonic of the order νp produces in the spectrum of the rotor MMFs a harmonic of the same order

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νp = µp [87]. To reduce asynchronous torques due to higher space harmonics, the recommended numbers of slots should be selected according to the following unequalities: s2 ≤ 1.25s1 (4.18) s2 < s1

(4.19)

|s2 − s1 ± p| ≥ 4

(4.20)

where s1 is the number of stator slots and s2 is the number of rotor slots. A synchronous torque is produced if the spectra of the stator and rotor MMF harmonics contain harmonics of the same order νp and if the rotor harmonic of this order is produced by a stator harmonic of another order νp = µp. To reduce synchronous torque the following number of slots should be avoided: • to reduce synchronous torques at standstill s2 = 2m 1 pk

k = 1, 2, 3, . . .

where

(4.21)

s2 = s1 − 2m 1 p

(4.22)

s2 = s1 − 4 p

(4.23)

• to reduce synchronous torques in the motoring region s < 1 s2 = 2m 1 pk + 2 p

(4.24)

• to reduce synchronous torques in the braking region s < 1 s2 = 2m 1 pk − 2 p.

4.6

(4.25)

Cogging torque in permanent magnet (PM) brushless motors

The frequency of the cogging torque is f c = s1 n s = s1

f p

(4.26)

where s1 is the number of stator slots, n s = f / p is the synchronous speed, p is the number of pole pairs and f is the input frequency. There are three sources of the torque ripple coming from the construction of the magnetic and electric circuit of a machine (Section 4.4). The most comprehensive literature review and comparison of methods of the torque ripple reduction is given in [26]. Analytical methods of cogging torque calculation usually neglect the magnetic flux in the stator slots and magnetic saturation of the stator teeth [1, 16, 18, 30, 46, 76, 83, 253, 256]. Cogging torque is derived from the magnetic flux density distribution either by calculating the rate of change of total energy stored in the air gap with respect to the rotor angular

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position [1, 16, 18, 46, 76, 83] or by summing the lateral magnetic forces along the sides of the stator teeth [253]. Many cogging torque calculations are specific to a particular method of reducing the cogging effect [48]. The proposed anlytical method uses the energy of the magnetic field in the air gap and classical equations for the air gap magnetic flux density distribution in a.c. machines with slotted stators [40, 51, 87, 236].

4.6.1

Air gap magnetic flux density

The magnetic flux density distribution in the air gap of a PM machine can be analyzed in a similar way as that of an induction machine with slotted stator and smooth rotor [40, 87]. The normal component of the magnetic flux density in the air gap is a sum of the mean flux density bmean (x ) and periodical component with the period equal to the slot pitch t1 , i.e.: 2π b f (x ) = bmean (x ) + (4.27) Bk cos k t1 k where [40, 87] Bg . (4.28) kC Carter coefficient kC is expressed using the well known Equation 2.72 [87]. Considering the air gap magnetic field distribution only in the interval of one pole pitch τ , the flux density B0 = Bg where Bg is the flat-topped value of the magnetic flux density in the air gap. Fourier series coefficient Bk in Equation 4.27 depends on the approximation of the magnetic flux density distribution over the slot opening. For skewed slots Bk can take the following form (see Equation 2.71) bmean (x ) =

Bk = −2γ1

g 2 2 B0 kok ksk t1

(4.29)

where kok is the coefficient of the slot opening according to Equation 2.64 and ksk is the skew factor. Thus, Equation 4.27 can be brought to the form ∞ 1 2π g 2 2 b f (x ) = B0 − 2γ1 k k cos k x . (4.30) kC t1 k =1 ok sk t1 In order to include the influence of the finite width of PMs or pole shoes on the cogging effect, the flux density B0 must be expressed as a periodic function of the pole pitch τ , i.e., B0 = b P M (x), where b P M (x) is the distribution of the PM magnetic flux density normal component in a machine without slots along the x-axis. Thus, the normal component of the magnetic flux density distribution in the air gap (Figure 4.2) of a PM brushless motor with slotted stator core can simply be expressed as: b P M (x) b f (x) = + bsl (x). (4.31) kC

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bf (x)

83

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6

−1.0

−0.8 −1

0 0

0.1

0.2

0.3

0.4

x

0.5 2⋅p⋅τ

Figure 4.2 Distribution of the normal component b f (x) of the magnetic flux density in the air gap (Tesla vs. meter) of a 10-pole, 36-slot PM brushless machine according to Equations 4.31, 4.32, and 4.33. The x coordinate is in the direction of rotation. The magnetic flux density component excited by the rotor PMs is a sum of higher space harmonics µ, i.e.,

π b P M (x) = (4.32) Bg bµ ksµ cos µ x τ µ and the magnetic flux density component due to stator slots is approximated as

bsl (x) = −2γ1

∞

π 2π g 2 2 kok ksk cos k x Bg bµ ksµ cos µ x . t1 k=1 t1 τ µ

(4.33)

Equation 4.33 describes the distribution of the normal component of the magnetic flux density excited by PMs (slotless machine) for B0 = b P M (x). For periodic variation of the PM flux density ∞

π 1 g 2 2 2π b f (x) = Bg bµ ksµ cos µ x − 2γ k k cos k x . τ kC t1 k=1 ok sk t1 µ (4.34) The coefficient of Fourier series bµ for higher space harmonics µ = 2m 1 k ± 1 in Equation 4.34 depends on the shape of the normal component of the magnetic flux density distribution in the air gap. It can be expressed by Equation 2.105 [73]. The pole width b p –to–pole pitch τ ratio for PM brushless motors αi = b p /τ = 0.65 to 0.92. The parameter α in Equation 2.105 describes the shape of the magnetic flux density waveform under the PM or pole shoe. For a constant (flat-topped) value of the magnetic flux density in the range −0.5b p ≤ x ≤ 0.5b p as shown in

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Figure 4.2 the parameter α = 0. For the magnetic flux density curve changing underneath the pole according to hyperbolic cosine law (concave curve) the parameter 0 < α ≤ 1. Equation 4.34 results from the classical theory of a.c. electrical machines, e.g., [40, 51, 87]. In the above Equations 4.31, 4.32, 4.33, and 4.34, kC is the Carter coefficient of the air gap taking into account slot openings, Bg is the flat-topped value of the periodical magnetic flux density waveform excited in the air gap by the rotor PMs, k = 1, 2, 3, . . ., µ = 2m 1 k ±1 are the rotor higher space harmonics, ksµ is the rotor PM skew factor, τ is the pole pitch, γ1 is the parameter depending on the slot opening b14 and the air gap g, t1 is the stator slot pitch, k0k is the stator slot opening factor and ksk is the stator slot skew factor. In general, both PMs and stator slots can be skewed. The skew factors are defined by the following equations: • the rotor PM skew factor ksµ

sin µπ b f s /(2τ ) = µπ b f s /(2τ )

(4.35)

• the stator skew factor for k = 1, 2, 3, . . . ksk =

sin [kπ pbs /(s1 t1 )] sin [kπ bs /(2τ )] = . kπ bs /(2τ ) kπ pbs /(s1 t1 )

(4.36)

The stator slot opening factor is defined by Equation 2.64 [87]. The skew of PMs is b f s and the skew of stator slots is bs . For most PM configurations the air gap g is to be replaced by an equivalent air gap g ≈ g + h M /µrr ec where g is the mechanical clearance, h M is the radial height of the PM (one pole), and µrr ec = 1.02 to 1.1 is the relative recoil permeability of the PM material [75].

4.6.2

Calculation of cogging torque

If the magnetic saturation and armature reaction are negligible, the cogging torque is independent of the stator current. The frequency of the fundamental component of the cogging torque is f c = s1 n s , where s1 is the number of the stator slots and n s is the rotor speed in rev/s. Assuming that the total energy W of the magnetic field is stored in the air gap, the cogging torque is expressed as Tc (x) = −

D2out d W dW =− d 2 dx

(4.37)

where D2out ≈ D1in is the rotor outer diameter, D1in is the stator inner diameter and the x axis is in the direction of rotation. The mechanical angle =

Copyright © 2006 Taylor & Francis Group, LLC

2τ . D2out

(4.38)

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The rate of change of the air gap coenergy is Li g W (x ) = 2µ0

b2f (x )d x

(4.39)

where L i is the effective length of the stator stack, µ0 is the magnetic permeability of free space, g is the air gap, and b f (x ) is the air gap magnetic flux density distribution according to Equation 4.31. Substituting Equations 4.31 and 4.39 into Equation 4.37 and assuming that the maximum energy change is in the interval X + a ≤ x ≤ X + b, the cogging torque equation becomes L i g D2out ∂ Tc (X ) = − 2µ0 2 ∂ x

X +b X +a

2

b P M (x ) + bsl (x ) kC

d x.

(4.40)

According to [46], the maximum energy change occurs for a = 0.5t1 and b = 0.5b14 + ct where ct = t1 − b14 . Taking into account only fundamental space harmonics µ = 1 and k = 1, the expression in the square bracket under the integral will have the following form

2

b P M (x ) + bsl (x ) kC

1 = B cos(βx ) − AB cos(αx ) cos(βx ) kC

2

where g 2 A = 2γ1 kok k2 ; t1 =1 sk=1 α=

2π ; t1

B = Bg bµ=1 ks µ=1

(4.41)

π . τ

(4.42)

β=

With a stationary stator, only the magnetic flux density excited by the rotor PMs depends on the rotor position with respect to the coordinate system fixed to the stator [87]. Magnetic flux density waveforms are visualized in Figure 4.3. It is easier to take the integral over b f (x)2 assuming that the rotor is stationary and the stator moves with synchronous speed, that is, only stator slots expressed by the term A cos αx change their position. Thus, L i g D2out X +b 1 Tc (X ) = − 2 B cos(βx) − AB cos(αx) cos(βx) . 2µ0 2 kC X +a ×(−ABα sin αx cos βx)d x (4.43) After performing integration with respect to x, the cogging torque equation takes the following form: Tc (X ) = −

L i g D2out [I1 (X ) + I2 (X )] 2µ0 2

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(4.44)

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bf (x) N

bsl(x) x

S

ns

Rotor bPM(x)

bPM (x)

bf (x)

x

S

N

Rotor

ns

Figure 4.3 Magnetic flux density waveforms in the air gap: b P M (x) excited by the rotor PMs, bsl (x) due to stator slots and b f (x) resultant. Only b P M (x) moves with the rotor.

where 2α − AB 2 sin(αx) cos2 (βx) d x kC X +a 2 a+b AB a−b 2 sin α X + = sin α kC 2 2 α a+b + sin (α + 2β) X + α + 2β 2 a+b × sin (α + 2β) 2 α a+b + sin (α − 2β) X + α − 2β 2 a−b × sin (α − 2β) 2

I1 (X ) =

X +b

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(4.45)

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2α A2 B 2 sin(αx ) cos(αx ) cos2 (bx ) d x X +a a+b 2 1 = AB sin 2α X + sin[α(a − b)] 2 2 α a+b + sin 2(α + β) X + × sin[(α + β)(a − b)] 4(α + β) 2 α a+b + sin 2(α − β) X + × sin[(α − β)(a − b)] . 4(α − β) 2 (4.46)

I2 (X ) =

X +b

Since I1 (X ) >> I2 (X ), the following shorter form of cogging torque equation can be used L i g D2out Tc (X ) ≈ − I1 (X ). (4.47) 2µ0 2 Figure 4.4 clearly shows that the fundamental frequency of the cogging torque (first integral) is f c = s1 n s , i.e., there are s1 pulses per one revolution. The fundamental waveform is modulated by another waveform with frequency f cp = 2 pn s , i.e., 2 p pulses per revolution. The second integral indicates that there is negligible frequency f c2 = 2s1 n s , i.e., 2s1 pulses per revolution.

4.6.3

Simplified cogging torque equation

Putting b P M (x) = Bg in Equation 4.40, the cogging torque will become a waveform with predominant frequency f c = s1 n s and constant amplitude. Such a simplified equation, which does not take into account the finite width of rotor poles, has been derived in earlier publications by the first author [75,76]. For b P M (x) = Bg , µ = 1 and k = 1 the simplified cogging torque equation has the following form 2 X +b Bg L i g D2out d − ABg cos(αx) d x 2µ0 2 d X X +a kC X +b 2 d Bg 2 2 2 2 2 − ABg cos(αx) + A Bg cos (αx) d x d X X +a kC kC

Tc (X ) = − =−

L i g D2out 2µ0 2

=−

L i g D2out [K 1 (X ) + K 2 (X )] 2µ0 2

(4.48)

where 4 a+b a−b 2 K 1 (X ) = − ABg sin α X + sin α kC 2 2 a+b K 2 (X ) = A2 Bg2 sin 2α X + sin[α(a − b)]. 2

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(4.49)

(4.50)

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Noise of Polyphase Electric Motors 0.2 0.16 0.12 0.08 I1(x) 0.04 0 I2(x) −0.04 −0.08 −0.12 −0.16 −0.2 −0.2 0.2

0 0

0.1

0.2

0.3

0.4

0.3

0.4

x

0.5 2⋅p⋅τ

(a) 0.005 0.005 0.004 0.003 0.002 0.001 0 I2(x) −0.001 −0.002 −0.003 −0.004 −0.005 −0.005

0 0

0.1

0.2 x

0.5 2⋅p⋅τ

(b)

Figure 4.4 Comparison of the first and second term in Equation 4.44: (a) integrals I1 (X ) and I2 (X ) according to Equation 4.45 and 4.46; (b) integral I2 (X ). Calculation results for a PM brushless motor with embedded magnets, s1 = 36 stator slots, 2 p = 10 poles, g = 1 mm, L i = 198 mm, t1 = 14 mm, τ = 50.3 mm, b14 = 3 mm, b f s = 0, bs = 14 mm, Bg = 0.76 T, n s = 11 rev/s. Since K 1 (X ) >> K 2 (X ) (Figure 4.5), Equation 4.48 can also be written in a shorter form: L i g D2out Tc (X ) ≈ − K 1 (X ) (4.51) 2µ0 2

4.6.4

Influence of eccentricity

The variation of the air gap for static eccentricity can be expressed analytically as [87] π g(x) = g 1 − cos x (4.52) pτ where g is the mean air gap, = e/g is the relative static eccentricity and e is the displacement between the stator and rotor axis. See also Equation 2.83 in which

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Torque Pulsations 0.1 0.08 0.06 0.04 K 1(x) 0.02 0 K 2(x) −0.02 −0.04 −0.06 −0.08 −0.1 −0.1

89

0.1

0 0

0.1

0.2

0.3

0.4

0.5 2⋅p⋅τ

0.3

0.4

0.5 2⋅p⋅τ

x (a)

0.005 0.004 0.003 0.002 0.001 0 K 2 (x) −0.001 −0.002 −0.003 −0.004 −0.005 −0.005 0.005

0 0

0.1

0.2 x (b)

Figure 4.5 Comparison of the first and second term in Equation 4.48: (a) integrals K 1 (X ) and K 2 (X ) according to Equations 4.49 and 4.50; (b) integral K 2 (X ). Simulation has been done for a PM brushless motor with the same parameters as specified in Figure 4.4 caption. α is according to Equation 2.30. Since the air gap is now the function of the x coordinate, the magnetic energy change depends on the variation of the air gap with the x coordinate. It is difficult to analyze Equation 4.40 with the air gap g(x) in the square bracket (second term) dependent on the x coordinate. It is more convenient to use the simplified Equation 4.48 in which A = 2γ1

g(x) 2 k ksk=1 . t1 0k=1

(4.53)

Thus, Tc (X ) = −

L i g D2out [K 1 (x) + K 2 (X ) + K 3 (X ) + K 4 (X ) + K 5 (X )] 2µ0 2 ≈−

L i g D2out [K 1 (X ) + K 3 (X )] 2µ0 2

Copyright © 2006 Taylor & Francis Group, LLC

(4.54)

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where K 1 (X ) and K 2 (X ) are according to Equations 4.49 and 4.50, respectively, and the remaining terms are K 3 (X ) = 2

1 a+b a+b sin ( + α) ABg2 sin ( + α) x + kC 2 2

a+b a−b + sin ( − α) x + sin ( − α) 2 2

(4.55)

a−b a+b K 4 (X ) = −A2 Bg2 2 sin x + sin 2 2 a+b a−b + sin ( + 2α) x + sin ( + 2α) 2 2

(4.56)

a+b a−b + sin ( − 2α) x + sin ( − 2α) 2 2 K 5 (X ) = −A2 Bg2 2

a+b 1 sin 2( + α) x + sin[( + α)(a − b)] 4 2

1 a+b + sin 2( − α) x + sin[( − α)(a − b)] 4 2 1 a+b + sin 2γ1 x + sin[γ1 (a − b)] 2 2

(4.57)

1 a+b + sin 2α x + sin[α(a − b)] . 2 2 Figure 4.6 shows that K 3 (X ) >> K 4 (X ) and K 3 (X ) >> K 5 (X ). Including the finite width of poles the cogging torque equation eventually be Tc (X ) = −

L i g D2out [I1 (x) + I2 (X ) + K 3 (X ) + K 4 (X ) + K 5 (X )] 2µ0 2 ≈−

L i g D2out [I1 (X ) + K 3 (X )]. 2µ0 2

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(4.58)

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0.002 2 ⋅ 10−3 K 3 (x) 0.001 K 4 (x) K 5 (x)

0 −0.001

−2 ⋅ 10−3 −0.002

0 0

0.1

0.2

0.3

0.4

x

0.5 2⋅p⋅τ

(a) 1 ⋅10−4 1 ⋅10−4

5 ⋅10−5

K 4 (x)

0 −5 ⋅10−5

−1 ⋅10−4

−1 ⋅10−4

0 0

0.1

0.2

0.3

0.4

0.3

0.4

x

0.5 2⋅p⋅τ

(b) 1 ⋅10−6 1 ⋅10−6

K 5 (x)

5 ⋅10−7 0 −5 ⋅10−7

−1 ⋅10−6 −1 ⋅10−6

0 0

0.1

0.2 x

0.5 2⋅p⋅τ

(c)

Figure 4.6 Comparison of the third, fourth, and fifth terms in Equation 4.54: (a) integrals K 3 (X ), K 4 (X ) and K 5 (X ) according to Equations 4.55, 4.56, and 4.57; (b) integral K 4 (X ); (c) integral K 5 (X ). Simulation has been done for a PM brushless motor with the same parameters as specified in Figure 4.4 caption and relative eccentricity = 0.2.

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4.6.5

Calculation and comparison with measurements

The results of analytical calculation of the cogging torque according to Equation 4.44 which includes the finite width of the rotor magnet are shown in Figure 4.7a while the results of calculations according to the simplified Equation 4.48 are shown in Figure 4.7b. The calculated and measured cogging torque produced by the same motor 2 p = 10, s1 = 36 and the same air gap magnetic flux density Bg = 0.76 T is shown in Figure 4.8. To capture the cogging torque pulsations, a torque transducer, oscilloscope and data acquisition system have been used. It can be seen that the measured cogging torque waveform is modulated by a sinusoid

1

1 0.8 0.6 0.4 0.2

T c (x)

0 −0.2 −0.4 −0.6 −0.8 −1 −1

0 0

0.1

0.2

0.3

0.4

0.5 2 ⋅p ⋅ τ

0.3

0.4

0.5 2 ⋅p ⋅ τ

x (a)

1.0

T c (x)

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4

−0.6 −0.8 −1.0 −1

0

0.1

0

0.2 x (b)

Figure 4.7 Cogging torque calculation results (Nm vs. meter): (a) according to Equation 4.44; (b) according to Equation 4.48. Design data of the PM brushless motor are the same as specified in Figure 4.4 caption, relative eccentricity ≈ 0.2.

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1.2 Calculations Measurements

1 0.8 0.6

Torque, Nm

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2

0

0.1

0.3

0.2

0.4

0.5

x. m

Figure 4.8 Comparison of calculated cogging torque according to Equation 4.58 and measured cogging torque for one full rotor revolution. Design data of the PM brushless motor are the same as specified in Figure 4.4 caption, relative eccentricity ≈ 0.2.

with the period equal to one revolution 2 pτ , ie.: π K (x) = −M sin x pτ

(4.59)

This effect is due to the misalignment of the rotor and transducer shafts. The small shaft misalignment is difficult to avoid. The constant M in Equation 4.59 is proportional to the degree of misalignment. In comparison with the finite element method (FEM), the presented analytical method gives immediate results and allows for fast analysis of different magnetic circuit geometries of PM brushless motors. It can easily be implemented in the design procedure of commercial PM brushless motors and new products. Equation 4.44 can capture the effect of the finite width of PMs while the simplified Equation 4.48 can only predict the fundamental frequency of the cogging torque and its approximate amplitude. Equation 4.48 can only be recommended for 2-pole machines with large pole shoe–to– pole pitch ratio αi ≥ 0.9. The effect of eccentricity is included by K 3 (x), K 4 (x) and K 5 (x) and can be predicted with the aid of Equation 4.54. For the detailed study of the cogging torque the best results can be obtained using Equation 4.58.

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There is a certain discrepancy between the analytical prediction and measurements caused by the misalignment of the rotor and torque transducer shafts. On the other hand, cogging torque measurements are difficult and the accuracy of measurements is always questionable.

4.7

Torque ripple due to distortion of EMF and current waveforms in PM brushless motors

Torque ripple due to shape and distortion of the electromotive force (EMF) and current waveforms is called the commutation torque. Assuming no rotor currents (no damper, very high resistivity of magnets and pole faces) and the same stator phase resistances, the Kirchhoff’s voltage equation for a three-phase machine can be expressed in the following matrix form: 

    v 1A i 1A R1 0 0  v 1B  =  0 R1 0   i 1B  0 0 R1 v 1C i 1C      i L L L e d  A B A C A   1A   f A  L B A L B LC B i 1B + e f B . + dt L i 1C efC C A LC B LC

(4.60)

For inductances independent of the rotor angular position the self inductances L A = L B = L C = L and mutual inductances between phases L AB = L C A = L C B = M are equal. For no neutral wire i a A + i a B + i aC = 0 and Mi a A = −Mi a B − Mi aC . Hence,     i 1A v 1A R1 0 0  v 1B  =  0 R1 0   i 1B  v 1C i 1C 0 0 R1 



     L1 − M 0 0 i 1A ef A  d  i 1B  +  e f B  . 0 L1 − M 0 + efC 0 0 L 1 − M dt i 1C

(4.61)

The instantaneous electromagnetic torque is the same as that obtained from Lorentz Equation 4.9, i.e., Td =

1 [e f A i 1A + e f B i 1B + e f C i 1C ]. 2π n

(4.62)

) At any time instant, only two phases conduct. For example, if e f A = E (tr f , (sq) (sq) ) , i 1B = −I1 and i 1C = 0, the instantaneous e f B = −E (tr f , e f C = 0, i 1A = I1

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electromagnetic torque according to Equation 4.62 is [28, 177] (sq)

Td =

) 2E (tr f I1

(4.63)

2π n

(sq)

) where E (tr are flat-top values of trapezoidal EMF and square wave current f and I1 (Figure 4.9). For constant values of EMF and currents, the torque (Equation 4.63) does not contain any pulsation [177]. Since e f = ωψ f = (2π n/ p)ψ f where ψ f is the flux linkage per phase produced by the excitation system, the instantaneous torque (Equation 4.62) becomes

Td = p(ψ f A i 1A + ψ f B i 1B + ψ f C i 1C )

(4.64)

To obtain torque for periodic waves of the EMF and current, it is necessary to express those waveforms in forms of the Fourier series. The magnetic flux on the basis of Equation 4.32 wc f (t) = L i b P M (x) cos(µωt)d x 0

=

4 τ Li Bg 2 π S

∞

1 π sin(µS)ksµ [1 − cos(µ w c )] cos(µωt) 3 µ τ µ=1,3,5,...

(4.65)

where w c is the coil pitch, S is according to Figure 4.9a and ksµ is according to Equation 4.35. The EMF induced in the phase A is calculated on the basis of the

Ef (tr)

e fA S

el. degrees 0

60

120 180

240

300

360

(a) i aA S1

I1(sq) el. degrees

0 60 S2

120

180 240

300

360

(b)

Figure 4.9 Idealized (solid lines) and practical (dash lines) waveforms of: (a) phase EMF; (b) phase current.

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electromagnetic induction law e f A (t ) = −N1

=

d [ f (t )] dt

∞

π 8 τ Li 1 Bg f N1 1 − cos µ w c sin(µωt ). (4.66) sin(µS )k s µ π S µ2 τ µ=1,3,5,...

The EMFs e f B (t ) and e f C (t ) induced in the remaining phase windings are shifted by 2π/3 and −2π/3, respectively. For phase A, the current rectangular waveform expressed as Fourier series is i 1A (t ) =

∞ 4 (sq ) 1 I1 cos(ν S1 ) sin(νωt ). π ν ν=1,3,5,...

(4.67)

Harmonics of the magnetic flux and current of the same order produce constant torque. In other words, the constant torque is produced by all harmonics 2m 1 k ± 1 where k = 0, 1, 2, 3, . . . of the magnetic flux and armature currents. Harmonics of the magnetic flux and current of different order produce pulsating torque. However, when the 1200 trapezoidal flux density waveform interacts with 1200 rectangular current, only a steady torque is produced with no torque pulsations [177, 178]. In practice, the stator current waveform is distorted and differs from the rectangular shape. It can be approximated by the following trapezoidal function i 1A (t ) =

∞ 1 4Ia(sq ) [sin(ν S1 ) − sin(ν S2 )] sin(νωt ) π(S1 − S2 ) ν=1,3,5,... ν 2

(4.68)

where S1 − S2 is the commutation angle in radians. The EMF wave also differs from 1200 trapezoidal functions. In practical motors, the conduction angle is from 1000 to 1500 , depending on the construction. Deviations of both current and EMF waveforms from ideal functions result in producing torque pulsations [28, 177]. Peak values of individual harmonics can be calculated on the basis of Equations 4.66, 4.67, and 4.68. Using the realistic shapes of waveforms of EMFs and currents as those in practical motors, Equation 4.68 takes into account all components of the torque ripple except the cogging (detent) component. Figure 4.10 shows trapezoidal EMF, trapezoidal current and electromagnetic torque waveforms, while Figure 4.11 shows the same waveforms in the case of sinusoidal EMFs and currents. The torque component due to phase commutation currents Tcom = Td − Tav where Tav is the average electromagnetic torque. The frequency of torque ripple due to phase commutation is f com = 2lm 1 f where l = 1, 2, 3, . . .. For sinusoidal EMF and current waveforms the electromagnetic torque is constant and does not contain any ripple. Torques shown in Figure 4.11 are not ideally smooth due to a slight current unbalance.

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97

300 200 efA(ωt) ef B(ωt)

0

efC(ωt) −200 −300

0 0

5

10

15

20

25

30 2⋅π⋅p

20.94

26.18

31.42 2⋅π ⋅p

4

5

6 2⋅π

ωt (a)

25 25 12.5 i1A(ωt) i1B(ωt)

0

i1C(ωt) −12.5 −25

−25

0 0

5.24

10.47

15.71 ωt (b)

150 150 100 Td(ωt) Tcom(ωt)

50 0 −50

−100 −100

0 0

1

2

3 ωt (c)

Figure 4.10 Electromagnetic torque produced by a medium power PM brushless motor (m 1 = 3, 2 p = 10, z 1 = 36): (a) trapezoidal EMF waveforms; (b) trapezoidal current waveforms; (c) resultant electromagnetic torque Td and its commutation component Tcom . The current commutation angle is 50 .

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Noise of Polyphase Electric Motors 300 300 200 efA(ωt) 100 ef B(ωt) efC(ωt)

0 −100 −200

−300

−300

0 0

5

10

15 ωt

20

25

30 2 ⋅ π ⋅p

(a) 25

i1A(ωt) i1B(ωt)

25 20 15 10 5 0

i1C(ωt) −5 −10 −15 −20 −25 −25

0 0

5

10

15 ωt

20

25

4

5

30 2⋅π⋅p

(b) 150 150 100

Telm(ωt) Tcom(ωt)

50 0 −50

−100 −100

0 0

1

2

3 ωt

6 2⋅π

(c)

Figure 4.11 Electromagnetic torque produced by a medium power PM brushless motor (m 1 = 3, 2 p = 10, z 1 = 36): (a) sinusoidal EMF waveforms; (b) sinusoidal current waveforms; (c) resultant electromagnetic torque Td and its commutation component Tcom .

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4.8 Tangential forces vs. radial forces For an accurate analysis of magnetic forces in electrical machines it is necessary to include both radial and tangential forces. The radial magnetic pressure obtained on the basis of Maxwell stress tensor, is expressed by Equation 2.114. The tangential component of the magnetic pressure can also be obtained with the aid of Maxwell stress tensor, i.e., pt (α, t ) =

1 b(α, t )bt (α, t ) = b(α, t )a (α, t ) µ0

N/m2

(4.69)

where b(α, t ) is the normal component of the magnetic flux density waveform according to Equation 2.110 or Equation B.6, bt (α, t ) is the tangential componet of the magnetic flux density distribution and a (α, t ) is the line current density. The radial and tangential components of the magnetic pressure for an 8-pole PM brushless motor are plotted in Figures 4.12 and 4.13. Every harmonic of the magnetic flux density in the air gap corresponds to a certain harmonic of the line current density. As a result of interaction of rotating harmonics of the magnetic flux density in the air gap and harmonics of the line current density, rotating tangential forces varying in the space and time are produced. Those forces can also be found on the basis of Biot-Savart law. The following boundary equalities for tangential components of the magnetic field can be written • at the stator core–air gap boundary bt1 (α, t ) = ±µ0 a1 (α, t )

(4.70)

Radial Magnetic Pressure, Pa

600000 500000 400000 300000 200000 100000 0 −100000

0

100

200 300 400 500 Circumference, mm

600

700

Figure 4.12 Distribution of the radial magnetic pressure along the air gap circumference for a PM brushless motor with 2 p = 8 and s1 = 36 as shown in Figure 2.1b as obtained from the 2D FEM simulation.

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Tangential Magnetic Pressure, Pa

300000 200000 100000 0 −100000 −200000 −300000

0

100

200 300 400 500 Circumference, mm

600

700

Figure 4.13 Distribution of the tangential magnetic pressure along the air gap circumference for a PM brushless motor with 2 p = 8 and s1 = 36 as shown in Figure 2.1b as obtained from the 2D FEM simulation. • at the rotor core–air gap boundary bt2 (α, t) = ±µ0 a2 (α, t)

(4.71)

where the line current density waveforms can be obtained by differentiating the MMF waveforms • for the stator

or

√

m 1 2 N1 kw1ν dF1ν (x, t) π a1ν (x, t) = =− I1 sin ωt ∓ ν x dx π pτ τ

π = Am1ν sin ωt ∓ ν x (4.72) τ a1 (α, t) =

∞

Am1ν sin(νpα ± ωt)

(4.73)

ν=1

• for the rotor of an induction motor √

m 2 2 N2 kw2µ dF2µ (x, t) π a2µ (x, t) = =− I2 sin ωµ t ∓ µ x + φµ dx π pτ τ

π (4.74) = Am2µ sin ωt ∓ ν x τ or ∞ a2 (α, t) = Am2µ sin(µpα ∓ ωµ t + φµ ). (4.75) µ=1

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For the rotor of a PM synchronous motor it is recommended to use the tangential component bt2 (α, t) of the rotor magnetic flux density instead of the line current density a2 (α, t). Every harmonic of the tangential magnetic force, similar to the radial force, has a certain frequency fr and an order r which describes its distribution around the stator periphery. The tangential magnetic pressure due to the stator and rotor magnetic flux density components is obtained by putting Equations 4.70 and 4.71 into Equation 4.69. Three groups of the tangential pressure can be identified as a result of interaction: • harmonics of the stator magnetic flux density and stator line current density ptν (α, t) = b1ν (α, t)a1ν (α, t)

(4.76)

• harmonics of the stator line current density and rotor magnetic flux density ptνµ (α, t) = b2µ (α, t)a1ν (α, t)

(4.77)

• harmonics of the rotor magnetic flux density and rotor line current density ptµ (α, t) = b2µ (α, t)a2µ (α, t).

(4.78)

Frequencies and orders of tangential magnetic forces are expressed by the same Equations 2.125, 2.126, and 2.127 as for radial magnetic forces. The stator and rotor MMF waveforms are expressed by Equations 2.107 and 2.108 in which ∞ F1 (α, t) = ∞ F (α, t) and F (α, t) = F (α, t). The peak values of 1ν 2 2µ ν=1 µ=1 the stator and rotor line current densities Am1ν and Am2µ are expressed by Equations 2.54 and 2.59. Note that the peak values of line current densities obtained from the MMF waveforms contain stator and rotor winding factors kw1ν and kw2ν for higher space harmonics. Thus, the following relationships between the MMF and line current densities amplitudes for the νth space harmonics can be written • for the stator F1mν =

1τ Am1ν ; νπ

Am1ν =

1 τ Am2µ µπ

Am2µ =

νπF1mν τ

(4.79)

µπ F1mµ . τ

(4.80)

• for the rotor F2mµ =

Assuming the magnetic saturation of the stator and rotor core to be negligible, on the basis of Ampere’s circuital law the peak values of the magnetic flux densities as functions of MMFs are • for the stator B1mν = µ0

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F1mν kC g

(4.81)

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Noise of Polyphase Electric Motors • for the rotor

F2m µ (4.82) kC g The ratio of the radial–to–tangential magnetic pressures created by the stator and rotor magnetic flux density components are: B2m µ = µ0

• for the product of the stator harmonics of the same number pr ν (α, t ) b1ν (α, t ) τ = = cot(νp α ∓ ωt ) pt ν (α, t ) 2µ0 a1ν (α, t ) 2νπkC g

(4.83)

• for the product of the stator ν harmonics and rotor µ harmonics pr νµ (α, t ) 2b1ν (α, t )b2µ (α, t ) τ = = cot(νp α ∓ ωt ) pt νµ (α, t ) 2µ0 b2µ (α, t )a1ν (α, t ) νπkC g

(4.84)

• for the product of the rotor harmonics of the same number pr µ (α, t ) b2µ (α, t ) τ = = cot(µp α ∓ ωµ t + φµ ) pt µ (α, t ) 2µ0 a2µ (α, t ) 2µπ kC g

(4.85)

Since τ >> g, the magnitude of the radial magnetic pressure is much higher than that of the tangential magnetic pressure. Consequently, the tangential forces producing electromagnetic torques are much smaller than radial forces. Peaks in Figure 4.13 have been obtained due to inaccuracy of the finite element method (FEM) model. Equations 4.83, 4.84, and 4.85 can be used for calculation of the tangential magnetic pressure on the basis of radial magnetic pressure.

4.9

Minimization of torque ripple in PM brushless motors

The torque ripple can be minimized both by the proper motor design and motor control. Measures taken to minimize the torque ripple by motor design include elimination of slots, skewed slots, special shape slots and stator laminations, selection of the number of stator slots with respect to the number of poles, decentered magnets, skewed magnets, shifted magnet segments, selection of magnet width, direction-dependent magnetization of PMs. Control techniques use modulation of the stator current or EMF waveforms [20, 59].

4.9.1

Slotless windings

Since the cogging torque is produced by the PM field and stator teeth, a slotless winding can totally eliminate the cogging torque. A slotless winding requires increased air gap, which in turn reduces the PM excitation field. To keep the same air gap magnetic flux density, the height h M of PMs must be increased. Slotless PM brushless motors use more PM material than slotted motors.

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4.9.2

103

Skewing stator slots

Normally, the stator slot skew bs equal to one slot pitch t1 can reduce the cogging torque practically to zero value — Equations 4.35 and 4.36. Sometimes, the optimal slot skew is less than one slot pitch [136]. On the other hand, the stator slot skew reduces the EMF which results in deterioration of the motor performance. Skewed slots are less effective in the case of rotor eccentricity.

4.9.3

Shaping stator slots

Figure 4.14 shows methods of reducing the cogging torque by shaping the stator slots, i.e., (a) (b) (c) (d)

bifurcated slots (Figure 4.14a), empty (dummy) slots (Figure 4.14b), closed slots (Figure 4.14c), teeth with different width of the active surface (Figure 4.14d).

Bifurcated slots (Figure 4.14a) can be split into more than two segments. The cogging torque increases with the increase of the slot opening. When designing

(a)

(b)

(c)

(d)

Figure 4.14 Minimization of the cogging torque by shaping the stator slots: (a) bifurcated slots; (b) empty slots; (c) closed slots; (d) teeth with different width of the active surface.

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Noise of Polyphase Electric Motors

closed slots (Figure 4.14c), the bridge between the neighboring teeth must be properly designed. Too thick a bridge (thickness measured in radial direction) increases the stator slot leakage to an unacceptable level. Too narrow a bridge can be ineffective due to high saturation. Because closed slots can only accept “sewed” coils, it is better to close the slots by inserting an internal sintered powder cylinder or to separate the stator yoke and tooth-slot section. In the second case, stator slots are open externally from the frame side.

4.9.4

Selection of the number of stator slots

The smallest common multiple Ncm between the slot number s1 and the pole number 2 p has a significant effect on the cogging torque. As this number increases the cogging torque decreases [83]. Similarly, the cogging torque increases as the largest common factor between the slot number and the pole number increases [30, 83].

4.9.5

Shaping PMs

PMs thinner at the edges than in the center (Figure 2.2b) can reduce both the cogging and commutation torque ripple [75]. Magnets may require a polygonal cross section of the rotor core. Decentered PMs together with bifurcated stator slots can suppress the cogging torque as effectively as skewed slots with much less reduction of the EMF [175].

4.9.6

Skewing PMs

The effect of skewed PMs on the cogging torque suppression is similar to that of skewing the stator slots. Fabrication of twisted magnets for small rotor diameters and small number of poles is rather difficult. Bread loaf shaped PMs with edges cut at a slant are equivalent to skewed PMs.

4.9.7

Shifting PM segments

Instead of designing one long magnet per pole, it is sometimes more convenient to divide the magnet axially into K s = 3 to 6 shorter segments. Those segments are then shifted one from each other by equal distances t1 /K s or unequal distances the sum of which is t1 . Fabrication of short, straight PM segments is much easier than long, twisted magnets.

4.9.8

Selection of PM width

Properly selected PM width with respect to the stator slot pitch t1 is also a good method to minimize the cogging torque. The magnet width (pole shoe width) is to be b p = (k + 0.14)t1 where k is an integer [136] or including the pole curvature b p = (k + 0.17)t1 [92]. Cogging torque reduction requires wider pole shoes than the multiple of slot pitch.

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4.9.9

105

Magnetization of PMs

There is a choice between parallel, radial, and direction-dependent magnetization. For example, if a ring-shaped magnet of a small motor is placed around the magnetizer poles without an external magnetic circuit, the magnetization vectors will be arranged similar to the Halbach array. This method also minimizes the torque ripple.

4.9.10

Creating magnetic circuit asymmetry

Magnetic circuit asymmetry can be created by shifting each pole by a fraction of the pole pitch with respect to the symmetrical position, or designing different sizes of North and South magnets of the same pole pair [18].

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5 Stator System Vibration Analysis In previous chapters the magnitudes, frequencies, and orders of radial and tangential magnetic forces have been analyzed. Analytical and numerical methods of computations of these forces have been discussed. The noise and vibration of the motor structure are the direct responses of the excitation by these forces. For example, if the frequency of the radial magnetic force is close to one of the natural frequencies of the stator system and the force order r is the same as the circumferential vibrational mode m of the stator system, significant vibration and acoustic noise can be produced. In this chapter, the modal vibration behavior of the stator system is discussed. Several analytical approaches for the calculation of natural frequencies and vibrational modes of cylindrical stators with frames (enclosures) are given.

5.1

Forced vibration

The stator and frame of a motor structure are basically cylindrical shells. A complete analytical analysis of the vibration of the system may be difficult. However, if only the stator is the concern, or if the overall vibration behavior of the motor is attributed to the stator, a circular cylindrical shell can be employed as a simplified model of the stator for an analytical approach. The vibration of a continuous structure due to the force excitation is discussed in Appendix D. For a cylindrical shell, although the equation of motion is of the same form as Equation D.32, the differential operator is more complicated than that for the beam and the plate [190]. In fact, due to the curvature of the shell, the vibration in the three orthogonal directions, radial, axial, and tangential, are coupled to each other. A result of this coupling is that any excitation in one direction would cause vibrations in all three directions. Such effects are more prominent at low frequencies. This makes it difficult to solve the equations directly, and various 107 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors

approximations/simplifications have to be made in the analytical solution [190]. A detailed and accurate analysis may have to reply on the numerical approach such as the finite element method (FEM). The stator vibration is primarily induced by the electromagnetic force acting on the inner surface of the stator core. As described in previous chapters, the electromagnetic force has radial and tangential components. Both components, strictly speaking, may excite the vibration in the radial direction which is the primary concern in acoustic radiation. But since the radial component is approximately an order of magnitude larger than the tangential component in general, it might be reasonable to neglect the contribution from the tangential component in the electromagnetic force, which means that the coupling in the vibration between the radial and tangential directions could be neglected. The forced vibration response of a cylindrical shell under the excitation of the electromagnetic force in the radial direction thus should be of the form as given in Equation D.37. As an example, a stator could be simplified as a circular cylindrical shell with both ends free of constraints. It is subject to the electromagnetic force as expressed in Equation 2.118. If only the circumferential modes of the stator are of interest, it can be shown that the force component of the order r only excites the mode of the same order for the stator, i.e. r = m. The amplitude of vibration displacements of mode m can be derived as Fm /M Am = 2 (ωm − ωr2 )2 + 4ζm2 ωr2 ωm2

(5.1)

where M is the mass (kg) of the cylindrical shell, ωm is the angular natural frequency of the mode m, ωr is the angular frequency of the force component of the order r , and ζm and is the modal damping ratio. The amplitude of force Fm = π D1in L i Pmr , in which D1in is the inner diameter of the stator core, L i is the effective length of the stator core, and Pmr is the magnitude of the magnetic pressure of the order r according to Equations 2.115, 2.116, or 2.117. It can be seen that when the frequency of the excitation force is close to the natural frequency of the stator; the vibration reaches a maximum and excessive noise may be generated. This result indicates that among all the electromagnetic force components, only those of which the frequencies are close to the natural frequencies of the stator modes are important to the stator vibration response. By rearranging Equation 5.1 a magnification factor can be defined as hm =

Am 1 = . 2 2 Fm /Mωm [1 − ( fr / f m ) ]2 + [2ζm ( fr / f m )]2

(5.2)

The h m value calculated by Equation 5.2 is plotted against ( fr / f m ) and different damping ratios ζm in Figure 5.1. The magnification factor increases as f = | fr − f m | decreases and the damping factor ζm decreases. Particularly at fr = f m , the vibration amplitude is controlled by the mechanical damping in the structure. Analytical determinations of the mechanical damping are generally difficult. Experimental approaches are usually adopted. The mechanical damping

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109

10 xm = 0.05

Magnification factor hm

8 6

xm = 0.1

4

xm = 0.2

2 0

xm = 0.3 0 0

1

0.5

2

1.5

2.5

3 3

fr/fm

Figure 5.1 Magnification factor h m as a function of ( fr / f m ) for different damping factors ζm . of the stator is frequency dependent and very much affected by winding and the lamination of the stator core. An empirical expression for the small and medium sized electrical machines is suggested as [248] ζm =

1 (2.76 × 10−5 f m + 0.062). 2π

(5.3)

This relationship is plotted in Figure 5.2. For a more accurate result, the modal damping ratio ζm can be extracted directly from modal testing [223]. The conversion between various damping parameters, such as the damping ratio, logarithmic decrement, and the loss factor, and so forth, can be found in Table 10.1. 0.06 0.06 0.05 0.04 ξ m(f ) 0.03 0.02 0.01 0

0

0 0

2000

4000

6000 f

8000

1 ⋅ 104 10000

Figure 5.2 Variation of damping ratio ζ with frequency [248].

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Noise of Polyphase Electric Motors

It should be noted that Equation 5.1 is derived for a single circumferential mode. The total response of the stator subject to the electromagnetic force excitation should be the superposition of the contributions from all the modes, as discussed in Appendix D. However, if the vibration modes are well separated over the frequency domain, Equation 5.1 can be employed around the natural frequency of each mode. Expressed as a function of the magnetic pressure magnitude Pmr , Equation 5.1 becomes, Fm π D1in L i Am = hm = Pmr h m . (5.4) M ωm2 M ωm2 The amplitude of the vibration velocity for mode m is, Vm = ωr Am = 2π fr

5.2

π D1in L i Pmr h m . M ωm2

(5.5)

Simplified calculation of natural frequencies of the stator system

Calcultation of natural frequencies of the stator system is essential in the vibration analysis of electric machines. In classical approaches [87, 119, 200, 248], the stator system, that is, the stator core, winding, and frame (enclosure) is considered as a single thick ring loaded with teeth and winding. Circumferential vibrational modes of a thin cylinder can be visualized in a similar way to force orders as shown in Figure 2.9. The natural frequency of the stator system of the mth circumferential vibrational mode can be expressed as: Km 1 fm = (5.6) 2π Mm where K m is the lumped stiffness (N/m) and Mm is the lumped mass (kg) of the stator system. For a stator core with the thickness h c , mass Mc , and the mean diameter Dc , the lumped stiffness and lumped mass in the breathing mode, that is, the circumferential vibrational mode m = 0 is [119, 248] K 0 = 4π

Ec hc L i ; Dc

M0 = Mc kmd = π Dc h c L i ρc ki kmd

(5.7)

where ρc is the mass density of the stator core, ki is the stacking factor, and kmd is the mass addition factor for displacement defined as [248] kmd = 1 +

M t + M w + Mi . Mc

(5.8)

In the above equation, Mt is the mass of all stator teeth, Mw is the mass of stator windings, Mi is the mass of insulation, and Mc is the mass of the stator core cylinder (yoke).

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111

Equations 5.6 and 5.7 give the following formula for the natural frequency of m = 0 circumferential mode Ec 1 f0 = . (5.9) π Dc ρc ki kmd Typically, for laminations, the elasticity modulus E c = 200 × 109 Pa = 200 GPa, mass density ρc = 7700 kg/m3 , and stacking factor ki = 0.96. This will give the phase speed cc = 200 × 109 /7700 = 6096 m/s. For the circumferential vibrational mode m = 1 (bending mode) K 1 = 4π F1 =

Ec hc L i ; Dc

M1 =

2 ; 1 + κ 2 kmr ot /kmd

Mc kmd M0 = 2 F12 F1

(5.10)

hc κ=√ . 3Dc

(5.11)

In Equation 5.11, kmr ot is the mass addition factor for rotation [248] hc M w + Mi 1 s1 ct L i h 2t hc 2 kmr ot = 1 + + 1+ + π Dc I c Mt 3 2h t 2h t =1+

s1 ct L i h 2t π Dc I c

M w + Mi Mt

1+

(4h 2t + 6h c h t + 3h 2c )

(5.12)

where

h 3c L i (5.13) 12 is the area moment of inertia about the neutral axis parallel to the cylinder axis, s1 is the number of stator teeth (slots), ct is the tooth width, and h t is the tooth height. Ic =

Then, the natural frequency for the circumferential mode m = 1 is [119, 248] Ec 2 1 f1 = = f 0 F1 . (5.14) π Dc ρc ki kmd 1 + κ 2 kmr ot /kmd Generally, for circumferential vibrational modes m ≥ 2 [119, 248] K m = 16π Mm = Mc

kmd m 2 + 1 kmd m 2 + 1 = π D h L ρ k c c i c i Fm2 m 2 Fm2 m 2

Fm =

E c Ic 2 (m − 1)2 ka2 Dc3

κ 2 (m 2 − 1)[m 2 (4 + kmr ot /kmd ) + 3] 1+ m2 + 1

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−1/2

(5.15)

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where the coefficient ka > 1 accounts for end bells and frame support (foot or flange mounting). The natural frequency of the stator system for circumferential modes m ≥ 2 can be derived as Ec Ic m(m 2 − 1) 2 1 m(m 2 − 1) √ fm = ka Fm = f 0 κ √ ka Fm . 2 π Dc ρc ki kmd h c L i m2 + 1 m2 + 1 (5.16) Equation 5.16 is in fact a modified equation for circumferential vibration of a ring derived by Hoppe [90]. Most monographs on noise analysis of electrical machines, for example, [87, 119, 200, 248], recommend Equations 5.9, 5.14, and 5.16 for calculating natural frequencies of stator systems. However, Equations 5.9 to 5.16 do not guarantee a good accuracy of the calculation since the stator system is a complex structure which consists of the laminated stack with yoke and teeth, winding distributed in slots, encapsulation (potting), and frame (enclosure or casing). Equations 5.9 to 5.16 can only be used to estimate the natural frequency of the stator core alone, without any frame and end bells and with only partial influence of the winding and teeth.

5.3

Improved analytical method of calculation of natural frequencies

The analytical approach presented in this section is an attempt to consider the effects of winding, teeth, and particularly the outer frame on the natural frequencies of the stator system. In this method, the stator core, winding, and teeth, and the frame are modeled separately. Then a formula is suggested for the system.

5.3.1

Natural frequency of the stator core

If the stator length–to–mean diameter ratio is L i /Dc ≤ 1, the stator core can be considered as a ring and the following classical formula for m ≥ 2, first published by Hoppe in 1871 [90], can be used 1 m(m 2 − 1) √ fm = 2π m2 + 1

E c Ic 2 m(m 2 − 1) √ = ρl (0.5Dc )4 π Dc2 m 2 + 1

E c Ic ρc L i h c

(5.17)

where the area moment of inertia Ic about the neutral axis parallel to the ring axis is given by Equation 5.13, h c is the stator yoke thickness, and the mass per unit circumference ρl =

1 Mc = ρc (π Dc )L i h c = ρc L i h c . π Dc π DC

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(5.18)

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113

According to Equations 5.6 and 5.17 the lumped stiffness and mass of the finite length cylinder are, respectively, Km =

16π E c Ic m 2 (m 2 − 1)2 Dc3 m2 + 1

Mm = Mc = πρc Dc L i h c .

(5.19) (5.20)

However, Hoppe’s equation does not give good results if L i /Dc ≥ 1. Better results are obtained if the stator of an electrical machine is regarded as a cylindrical shell of infinite length. In most mechanical engineering textbooks, the natural frequency of an infinitely long shell for circumferential vibrational modes m ≥ 0 is expressed as [135] Ec

m fm = (5.21) π Dc ρc (1 − νc2 ) where νc is the Poisson ratio for the stator core. On the basis of Donnel-Mushtari theory [135], the parameter m , that is, the roots of the second order characteristic equation of motion are: • for circumferential mode m = 0

0 = 1 • for circumferential modes m ≥ 1 1

m = (1 + m 2 + κ 2 m 4 ) ± (1 + m 2 + κ 2 m 4 )2 − 4κ 2 m 6 ; 2

(5.22)

(5.23)

• where the nondimensional thickness parameter κ2 =

h 2c 3Dc2

(5.24)

To express Equation 5.21 in the form of Equation 5.6, the lumped stiffness can be written as Km =

4 2m π L i h c E c . Dc 1 − νc2

(5.25)

Meanwhile, Mm = πρc Dc L i h c , which is actually the stator core mass Mc . It is recommended to use Equation 5.21 even if L i /Dc < 1. The stator core (yoke) behind the teeth of 2-pole machines (2 p = 2) is thicker in the radial direction than that of machines with 2 p > 2. Therefore, the stiffness (Equation 5.25) of 2-pole machines is higher and deflection of the stator core is smaller than in machines with a larger number of poles. Two-pole machines can often be less noisy than multipole machines.

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5.3.2

Natural frequency of a frame with end bells

A frame (enclosure) with end bells behaves as a circular cylindrical shell with both ends constrained mechanically. Mathematically, both ends can be approximately described as being either simply supported or clamped. With the circumferential modal patterns being illustrated in Figure 2.9, typical axial modal patterns for circular cylindrical shells with simply supported ends are shown in Figure 5.3. The axial vibrational modes are n = 1, 2, 3 . . .. For a closed circular cylindrical shell of finite length L f with simply supported condition at both ends, the characteristic equation of motion has the following form [135]

6mn − (C2 + κ C2 ) 4mn + (C1 + κ C1 ) 2mn − (C0 + κ C0 ) = 0.

(5.26)

This is actually a third order equation with respect to the nondimensional frequency parameter 2mn . Three sets of roots correspond to the vibrations in three orthogonal directions, in which the smallest real root is associated with the natural frequency of the flexural vibration of the frame. According to Donnell-Mushtari theory, the constants in Equation 5.26 are [135] 1 C2 = 1 + (3 − ν f )(m 2 + λ2 ) + κ 2 (m 2 + λ2 )2 (5.27) 2

1 2 2 2 2 23 − νf 2 2 2 2 C1 = (1−ν f ) (3 + 2ν f )λ + m + (m + λ ) κ (m + λ ) (5.28) 2 1 − νf C0 =

λ = nπ

Rf ; Lf

1 (1 − ν f ) 1 − ν 2f λ4 + κ 2 (m 2 + λ2 )4 2 C2 = C1 = C0 = 0 R f = 0.5(D f − h f );

κ2 =

(5.29) (5.30) h 2f

12R 2f

(5.31)

where ν f is the Poisson ratio, R f is the mean frame radius, D f is the external frame diameter, h f is the frame thickness, and L f is the frame length. When both ends of the frame are clamped, the characteristic Equation 5.26, is still valid, and the constants in the equation are the same as shown in Equations 5.27 n=1

n=2

n=3

Figure 5.3 Nodal patterns for a circular cylindrical shell supported at both ends by shear diaphragms.

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to 5.30. However, Equation 5.26 should be solved with the parameter λ in Equation 5.31 replaced by λe , which according to Arnold and Warburton is [135] λe = nπ

Rf ; L f − L0

L0 = L f

0.3 n + 0.3

(5.32)

where L 0 is the correction to the length of the “clamped-clamped” shell, which depends on the axial modal number n. For either a “clamped-clamped” shell or a shell simply supported at both ends, the circumferential vibrational modes m = 0, 1, 2, 3, . . . should be calculated for each of the axial vibrational mode n = 1, 2, 3, . . .. The axial mode n = 0 does not exist. Thus, the natural frequency of the frame is Ef K mn 1 1 mn f mn = = (5.33) 2π R f 2π Mf ρ f (1 − ν 2f ) where E f is the elasticity modulus of the frame material, and ρ f is the material density. The lumped stiffness for the frame therefore is K mn =

2mn E f V f 2 2mn π L f h f E f = Rf R 2f 1 − ν 2f 1 − ν 2f

(5.34)

and M f is the mass of the frame, i.e., M f = ρ f V f = ρ f (2π R f )L f h f ;

5.3.3

V f = 2π R f L f h f .

(5.35)

Natural frequency of a stator core–frame system

Most electrical machines have stator cores pressed into frames (enclosures). A frame increases both the mass and stiffness of the stator. If the stator core is short, an external structure produces a pressure which is transmitted uniformly throughout the stator core. Since the resultant moment of inertia of coaxial cylinders is equal to the sum of moments of inertia of each cylinder, the natural frequency of the statorframe system is to be calculated under the assumption that the lumped stiffness of (f) the stator core K m(c) and frame K mn are in parallel, that is, the equivalent stiffness (c) (f) is K m + K mn . The same assumption has been made for equivalent masses of the stator core and frame, that is, the resultant mass of the stator-frame system is Mc + M f . On the basis of these assumptions, the natural frequency of the stator-frame system can be expressed as1 : (f) K m(c) + K mn 1 f mn ≈ (5.36) 2π Mc + M f (f) where K m(c) can be determined based on Equation 5.19 or 5.25, and K mn is according to Equation 5.34. Lumped masses Mc and M f are calculated from Equations 5.20 and 5.35.

1 See also S. Huang, M. Aydin, and T. A. Lipo, Electromagnetic vibration and noise assessment for surface mounted PM machines, IEEE PES Summer Meeting, Vol. 3, pp. 1417–1426, Vancouver, BC, Canada, 2001.

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In large machines the deflection of the stator core which produces noise is about ±0.25 × 10−4 mm [220]. On the other hand, the manufacturing tolerances between the outer diameter of the core and inner diameter of the frame are from ±0.02 to ±0.1 mm. This means that the mechanical coupling between the core and frame is small for low amplitude deflections [220]. Small machines with stamped ring laminations are more likely to produce some mechanical coupling than segmental laminations of large machines. The mechanical coupling between core and frame is increased due to radial expansion after heating. In the calculation of noise and vibration for most constructions of electrical machines, the • frame may be neglected only in large machines; • frame and its moment of inertia cannot be neglected in small machines.

5.3.4

Effect of the stator winding and teeth

The winding is located in slots which are separated by steel teeth. The toothslot zone with the winding can be regarded as an additional ring internal to the stator core (yoke). Thus, the natural frequency of the stator-frame system with the winding will be f mn ≈

(f)

K m(c) + K mn + K m(w) Mc + M f + Mw

1 2π

(5.37)

where K m(w) is the lumped stiffness of the tooth-slot zone including the winding and Mw is the mass of the teeth, winding, and insulation. The lumped stiffness of the tooth-slot zone can be determined based on the method given in Section 5.3.1. • for short length winding (finite length cylinder) K m(w) =

m 2 (m 2 − 1) 2π E w Iw m2 + 1 Rw3

(5.38)

• for long length winding (infinitely long cylinder) K m(w) =

2m E w Vw Rw2 1 − νw2

(5.39)

The nondimensional frequency parameter m is according to Equations 5.22 and 5.23 for Rw = Rc − h sl , where h sl is the slot depth, Vw = 2π Rw h t (L i + 2h ov ) is the volume of the winding considered as a cylinder, L i is the effective length of the stator stack, h ov is the axial length of a one-sided end connection, E w is the equivalent elasticity modulus for winding and insulation, and νw is the Poisson ratio for the winding with insulation. The elasticity modulus for copper winding is 110 to 120 GPa and for polymer insulation 3 GPa. The equivalent elasticity modulus for copper and insulation assuming the slot filling factor 0.5 is approximately 9.4 GPa [241]. The Poisson ratio for copper is 0.33 to 0.36 and for polymer insulation exceeds 0.36. The mass of winding Mw = m 1 ρw N1ltur n aw a p sw

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(5.40)

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where m 1 is the number of phases, ρw is the mass density of conductors (8890 kg/m3 for copper, 2700 kg/m3 for aluminum), N1 is the number of turns per phase, ltur n is the average length per turn, a p is the number of parallel current paths, aw is the number of parallel wires, and sw is the conductor cross section area. Similar to the stator core, Equation 5.39 gives better results than Equation 5.38 even if (L i + 2h ov )/(2Rw ) < 1.

5.3.5

Analytical calculation of natural frequencies for a stator core-winding-frame system

A cylindrical stator system (core, winding, frame, end bells) of a 10 kW a.c. machine with s1 = 36 slots will be considered. The mass of the stator core (yoke) is 8.58 kg, mass of teeth is 10.27 kg, mass of the winding is 11.45 kg, mass of the insulation is 0.57 kg, and mass of the aluminum frame is 6.24 kg. The dimensions of the stator system are: • • • • • • • • •

stator core inner diameter D1in = 0.16 m stator core outer diameter D1out = 0.233 m effective length of the stator core L i = 0.1975 m stator tooth width ct = 5.87 mm stator slot opening b14 = 3 mm thickness of stator yoke (core) h c = 8.42 mm length of winding overhang h ov = 48 mm diameter of frame D f = 0.246 m length of frame L f = 0.359 m

Material parameters are specified below: • • • • • • • • •

specific mass density of laminations ρc = 7700 kg/m3 specific mass density of copper ρw = 8890 kg/m3 specific mass density of frame (aluminum) ρ f = 2700 kg/m3 modulus of elasticity of laminations E c = 200 × 109 Pa modulus of elasticity of winding with insulation E w = 9.4 × 109 Pa modulus of elasticity of frame (aluminum) E f = 71 × 109 Pa Poisson ratio of laminations νc = 0.3 Poisson ratio of copper νw = 0.35 Poisson ratio of frame (aluminum) ν f = 0.33

Natural frequencies of the stator core The mean radius of the stator core (yoke) is Rc = 0.5(D1out − h c ) = 0.5(0.233 − 0.00842) = 0.112 m and the parameter κ according to Equation 5.24 is κ 2 = h 2c / (12Rc2 ) = 0.00842/(12 × 0.1122 ) = 0.000469. Since the stator core is regarded as an infinite shell, the parameter m and natural frequencies f m of the core are only a function of circumferential modes m = 0, 1, 2, 3, . . .. The parameter m is calculated with the aid of Equations 5.22 and 5.23, e.g., m = 0.08 for m = 2. Natural frequencies of the stator core calculated on the basis of Equation 5.21 for circumferential modes 0 ≤ m ≤ 8 are given in Table 5.1.

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Table 5.1 Natural frequencies of the stator core for 0 ≤ m ≤ 8 calculated on the basis of Equation 5.21. 0 7572

1 116

2 464

circumferential mode m 3 4 5 6 1313 2517 4070 5971

7 8218

8 10,811

Natural frequencies of the stator winding The stator winding distributed in slots is regarded as a homogenous cylinder with an equivalent modulus of elasticity E w = 9.4 × 109 Pa (copper, insulation, encapsulation) and specific mass density equal to that of copper, that is, ρw = 8890 kg/m3 . The mean radius of the stator winding is Rw = 0.5(D1in +h t ) = 0.5(0.16 + 0.028) = 0.094 m and the parameter κ according to Equation 5.24 is κ 2 = h 2c /(12Rw2 ) = 0.0028/(12 × 0.0942 ) = 0.0074. Since the stator winding is regarded as an infinite shell, the corresponding parameter m and natural frequencies of the winding are only a function of the circumferential modes m = 0, 1, 2, 3, . . .. The parameter m is calculated with the aid of Equations 5.22 and 5.23, e.g., m = 0.307 for m = 2. Natural frequencies of the stator winding calculated on the basis of Equation 5.21 with E c , ρc , νc , and Rc replaced by E w , ρw , νw , and Rw , respectively, for circumferential modes 0 ≤ m ≤ 8 are given in Table 5.2.

Natural frequencies of the frame The mean radius of the frame is R f = 0.25(D f + D1out ) = 0.25(0.246+0.233) = 0.12 m and the paremeter κ according to Equation 5.24 is κ 2 = h 2f /(12R 2f ) = 0.000246 where the thickness of the frame h f = 0.5(D f − D1out ) = 0.5(0.246 − 0.233) = 0.0065 m. A cylindrical frame with end bells can be regarded as a “clamped-clamped” shell for which the equivalent wavelength λe as a function of axial vibrational modes n = 1, 2, 3, . . . is expressed by Equation 5.32. Putting λe into Equations 5.27, 5.28, 5.29, and 5.30, the roots of the third order characteristic Equation 5.26

Table 5.2 Natural frequencies of the winding for 0 ≤ m ≤ 8 calculated on the basis of Equation 5.21 in which E c = E w , ρc = ρw , νc = νw , and Rc = Rw . 0 1857

1 113

2 570

circumferential mode m 3 4 5 6 1360 2471 3901 5647

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7 7708

8 10,080

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Table 5.3 Smallest polyroots 2mn of Equation 5.26 for 0 ≤ m ≤ 8 and n = 1, 2, 3. axial mode n 0 1 2 1 0.65 0.251 0.079 2 0.902 0.599 0.3 3 0.937 0.787 0.538

circumferential mode m 3 4 5 6 7 8 0.049 0.082 0.174 0.344 0.623 1.048 0.176 0.17 0.254 0.433 0.731 1.182 0.379 0.34 0.414 0.603 0.926 1.414

can be easily found with the aid of, for example, Mathcad technical calculation tool. For m = 0 and n = 1 the polyroots are 201 = 0.65, 0.837, and 2.02; for m = 2 and n = 1 the polyroots are 221 = 0.079, 2.318, and 6.516; for m = 4 and n = 1 the polyroots are 241 = 0.082, 6.372, and 18.729, and so on. Table 5.3 contains the smallest polyroots 2mn for 0 ≤ m ≤ 8 and 1 ≤ n ≤ 3. The natural frequencies f mn of the aluminum frame with end bells found on the basis of Equation 5.33 are given in Table 5.4.

Natural frequencies of the stator system Natural frequencies of the stator system (core with teeth, winding with insulation, and frame) have been calculated using simplified Equation 5.37. The stiffness of the core K m(c) is given by Equation 5.25, the stiffness of the winding K m(w ) is given (f) by Equation 5.39, and the stiffness of the frame K mn is given by Equation 5.34. The mass of the core with teeth is Mc = 8.58 + 10.27 = 18.85 kg, mass of the winding with insulation is Mw = 11.45 + 0.57 = 12.02 kg, and mass of the frame is M f = 6.24 kg. The natural frequencies of the stator system for 0 ≤ m ≤ 8 and 1 ≤ n ≤ 3 are given in Table 5.5. It is always recommended to verify the natural frequencies of the stator system obtained from analytical Equations 5.17, 5.21, 5.33, 5.36, and 5.37 with the 3D FEM structural modeling.

Table 5.4 Natural frequencies of the frame with end bells for 0 ≤ m ≤ 8 and 1 ≤ n ≤ 3 calculated on the basis of Equation 5.33. axial mode n 1 2 3

circumferential mode m 0 5819 6856 6990

1 3616 5589 6406

2 2033 3952 5294

3 1604 3027 4444

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4 2072 2976 4210

5 3010 3638 4644

6 4233 4751 5606

7 5697 6173 6947

8 7392 7850 8584

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Table 5.5 Natural frequencies of the stator system for 0 ≤ m ≤ 8 and 1 ≤ n ≤ 3 calculated on the basis of Equation 5.37. axial mode n 1 2 3

5.4 5.4.1

circumferential mode m 0 4722 4897 4921

1 1299 2002 2293

2 995 1568 2010

3 1720 1950 2270

4 3039 3133 3309

5 4775 4831 4940

6 6904 6947 7028

7 9419 9458 9526

8 12,316 12,352 12,414

Numerical verification FEM modeling

Numerical approach in engineering computations is typically used when a system is far too complex to analyze analytically. One of the most popular generalpurpose 3D FEM analysis software packages is Ansys. In the FEM, a complex system is divided into very small pieces called elements, the shape and size of which are decided by the user. The accuracy of the FEM results generally depends on the number of elements in the model. The forces or loads on each element are calculated one by one using differential equations. The results then can be presented in numerical or graphical form. The following generic steps are taken to solve a problem in Ansys or other FEM software packages: • Building the geometry. 2D or 3D representation of the object or system is constructed. • Defining material properties. A library of the necessary material properties that compose the object or system is created. • Mesh generation. Most software packages can generate the mesh automatically. However, it must be defined how the modeled system should be broken down into finite elements. • Application of loads. Constraints such as physical loadings or boundary conditions are assigned. • Solution. Solver module solves numerically differential equations. The FEM software needs to understand what kind of solution is expected, that is, steady state, transient, and so forth. • Postprocessing. The results can be presented in form of graphs, contour plots, or tables. Structural analysis of machine housings, machine parts, and machine tools is the most common application of the FEM. The term “structural” implies not only civil engineering structures but also naval, aeronautical, and mechanical structures. The following types of structural analyses can be performed with Ansys: • Static analysis. It is used to determine displacements, forces, and stresses under static loading conditions.

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• Modal analysis. It is used to calculate the natural frequencies and mode shapes of a structure. • Harmonic analysis. It is used to determine the response of a structure to harmonically time-varying loads. • Transient dynamic analysis. It is used to determine the response of a structure to arbitrarily time-varying loads. • Spectrum analysis. This is an extension of the modal analysis to calculate stresses and strains due to a response spectrum or a power spectral density (PSD) input (random vibration). • Buckling analysis. It is used to calculate the buckling loads and determine the buckling mode shape. Nonlinearities in static, dynamic, and buckling analyses can include plasticity, stress stiffening, large deflection, large strain, hyper elasticity, contact surfaces, and creep. The modal analysis is necessary to determine the vibration characteristics, that is, natural frequencies and mode shapes of a structure or machine component. The natural frequencies and mode shapes are important parameters in the design of a structure for dynamic loading conditions. They are also necessary in a spectrum analysis, mode superposition harmonic analysis, and transient analysis. In vibrational and acoustic analysis of electrical rotating machinery, it is necessary to compare the frequencies and orders of radial magnetic forces with natural frequencies and vibrational modes of the stator system. All frequencies of magnetic forces that are close to natural frequencies of the stator system for the same force order as vibrational mode can produce dangerous vibration and excessive noise. Figure 5.4a shows the Ansys element plot of the stator laminated core with 36 skewed axial slots. The Ansys model consists of 3D solid elements and surface to surface contact elements taken from the Ansys element library. The laminated stator is modeled as a continuous solid. The material properties assigned to the stator laminations are orthotropic in nature. The orthotropic material properties enable a difference in stiffness between the radial and axial directions. The stator

(a)

(b)

Figure 5.4 Element plots: (a) stator laminated stack; (b) stator winding.

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n=2

n=3

Figure 5.5 Circumferential modes m = 2 for axial modes n = 1, 2, and 3 for an aluminum frame with end bells (outer diameter 0.246 m, inner diameter 0.233 m, and length 0.359 m) obtained from the 3D FEM analysis. stack is connected to the frame (enclosure) using a bonded contact pair. Since the stator is pressed into the frame, the contact pair is also used to calculate the prestress effect of the interference fit on the natural frequencies of the system. The volume of material representing the stator winding is modeled also using solid elements. The stator axial slots are considered fully filled with the winding material. The winding parameters (specific mass density, elasticity modulus, Poisson ratio) are an equivalent representation of copper, insulation, and epoxy resin. Winding elements are shown in Figure 5.4b. The Ansys model of the aluminum frame is shown in Figure 5.5. For the circumferential mode m = 2 three axial modes, i.e., for n = 1, 2, and 3 has been plotted. It has been assumed that the frame is furnished with end bells (“clampedclamped” shell).

5.4.2

Comparison of analytical calculations with the FEM

First, a simple thin circular square solid ring with its diameter 0.508 m, thickness h c = 0.00127 m, axial length L i = 0.00127 m, modulus of elasticity E c = 206.8 × 109 Pa, and specific mass density ρc = 7955 kg/m3 will be considered. The natural frequencies f 0 for m = 0 and f 2 for m = 2 of radial vibration will be determined. The frequency f 0 can be calculated with the aid of Equation 5.9 for ki = 1 (solid homogeneous ring) and kmd = 1 (no mass addition), i.e., 1 206.8 × 109 f0 = = 3194.6Hz. π 0.508 7955

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Table 5.6 Comparison of natural frequencies for 0 ≤ m ≤ 4 and n = 1, 2, 3 obtained from the 3D FEM and analytical calculations for an aluminum frame with end bells. Values calculated analytically are in parentheses. axial mode

m=0

m=1

n=1

-

2927 2927 2919 2924 (3616) (3563) 5184 5095 5134 5093 5139 (5589) (5516) 6311 6031 6176 6045 6201 (6406) (6331)

n=2

n=3

(5819) (5693) 6175 6220 6157 6218 (6856) (6772) 6865 6612 6794 6637 6818 (6990) (6914)

circumferential mode m=2 m=3 m=4 1884 1841 1833 1827 1829 (2033) (1985) 3626 3519 3524 3506 3526 (3952) (3889) 5182 4918 4980 4906 4996 (5294) (5222)

1687 1533 1520 1513 1517 (1604) (1518) 2986 2772 2764 2743 2764 (3027) (2950) 4491 4129 4148 4087 4157 (4444) (4364)

2298 1966 1941 1924 1933 (2072) (1963) 3202 2782 2773 2727 2767 (2976) (2876) 4504 3923 3939 3843 3937 (4210) (4114)

method ANS N1 N2 N3 N4 A(5.33) AWL ANS N1 N2 N3 N4 A(5.33) AWL ANS N1 N2 N3 N4 A(5.33) AWL

The frequency for m = 2 can be calculated according to Hoppe’s formula (Equation 5.17), i.e., 2(22 − 1) 206.8 × 109 × 2.168 × 10−13 2 √ f2 = = 12.373Hz π × 0.5082 22 + 1 0.013 where ρc L i h c = 7955 × 0.00127 × 0.00127 = 0.013 kg/m and the moment of inertia according to Equation 5.13 is Ic = 0.001273 × 0.00127/12 = 2.168 × 10−13 m4 . The same frequencies obtained from Ansys are f 0 = 3226.4 Hz and f 2 = 12.496 Hz. The convergence is good. For thin square solid ring analytical equations give slightly smaller values of natural frequencies. In the next example natural frequencies of a frame of an electrical machine will be calculated. Tables 5.6 and 5.7 show the comparison of natural frequencies of an aluminum frame with end bells (outer diameter 0.246 m, inner diameter 0.233 m, length 0.359 m) obtained from the 3D FEM and analytical calculations. Two different FEM packages, namely Ansys and Nastran have

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Table 5.7 Comparison of natural frequencies for 5 ≤ m ≤ 8 and n = 1, 2, 3 obtained from the 3D FEM and analytical calculations for an aluminum frame with end bells. Values calculated analytically are in parentheses. axial mode n=1

n=2

n=3

m=5

circumferential mode m=6 m=7 m=8

method

3404 2891 2836 2804 2819 (3010) (2897) 4080 3425 3410 3323 3392 (3638) (3526) 5177 4309 4343 4178 4326 (4644) (4535)

4826 4136 4023 3968 3990 (4233) (4118) 5395 4516 4470 4335 4434 (4751) (4636) 6351 5186 5231 4979 5193 (5606) (5491)

ANS N1 N2 N3 N4 A(5.33) AWL ANS N1 N2 N3 N4 A(5.33) AWL ANS N1 N2 N3 N4 A(5.33) AWL

6510 5656 5447 5355 5387 (5697) (5582) 7025 5939 5825 5631 5762 (6173) (6055) 7893 6446 6474 6123 6405 (6947) (6829)

8436 7446 7087 6937 6988 (7392) (7275) 8920 7654 7421 7149 7317 (7850) (7731) 9731 8027 7991 7527 7881 (8584) (8465)

been used. In order to show the convergence of the results with the number of elements, two mesh sizes have been examined in the Nastran analysis. Results obtained from different element types are also listed in Tables 5.6 and 5.7. The following abbreviations have been used: ANS = Ansys (cube elements), N1 = Nastran 2003 14 × 40 HEX8 elements, N2 = Nastran 2003 30 × 60 HEX8 elements, N3 = Nastran 2003 14 × 40 CQUARD4 elements, N4 = Nastran 2003 30 × 60 CQUARD4 elements, A(5.33) = analytical method according to Equation 5.33, AWL = analytical method according to Wang and Lai [226]. Both analytical methods give reasonable estimations on the natural frequencies except for the low order modes, such as m = 1, n = 1. Figure 5.6 shows circumferential modes m = 2 and 3 and axial mode n = 1 obtained from the 3D FEM Ansys for the stator with aluminum frame equipped with end bells (outer diameter 0.246 m, inner diameter 0.233 m, length 0.359 m), laminated core, and copper winding. The diameters are as follows: stator inner

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Stator System Vibration Analysis m=2

125 m=3

Figure 5.6 (See color insert following page 236.) Circumferential modes m = 2 and 3 and axial mode n = 1 for the stator (aluminum frame with end bells, laminated core, and copper winding) obtained from the 3D FEM Ansys. (teeth) diameter 0.1601 m, stator core (yoke) inner diameter 0.1769 m, stator outer diameter 0.233 m, frame outer diameter 0.246 m. The stator core length L i = 0.1975 m, stator winding length with end connections 0.2935 m, stator frame length L f = 0.359 m. The natural frequencies obtained from analytical calculations (Equation 5.37) and Ansys 3D FEM software package for the stator system (aluminum frame with end bells, laminated core, copper winding) have been compared in Table 5.8. Although, the accuracy of Equation 5.37 is reasonable, it can be found that as the system becomes complicated, the accuracy of analytical solutions deteriorates. For a complete electrical machine system, it is always recommended to use the FEM analysis for accurate results. Table 5.8 Comparison of natural frequencies obtained from analytical calculations and Ansoft 3D FEM software package for the stator system (aluminum frame with end bells, laminated core, copper winding). Values calculated analytically are in parentheses. axial circumferential mode m mode n 0 1 2 3 4 5 6 7 8 1 4289 − 1009 2089 3332 4541 5785 − − (4722) (1299) (995) (1720) (3039) (4775) (6904) (9419) (12,316) 2 − − 1789 2388 3401 4553 − − − (4897) (2002) (1568) (1950) (3133) (4831) (6947) (9458) (12,352) 3 4934 − − − − − − − − (4921) (2293) (2010) (2270) (3309) (4940) (7028) (9526) (12,414)

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6 Acoustic Calculations The sound radiated from structures is an impact of the structural vibration on the environment. From the vibro-acoustic point of view, it is a result of interactions between the structure and the ambient medium. Generally speaking, more vibrations may produce more sound. However, the relation between the two is not simple, especially for machinery structures. The sound radiation generally depends on the structure dimensions, boundary conditions, material properties, external excitations, and ambient medium conditions, and so on. In acoustics, in order to facilitate the discussions on the mechanism of sound generations from various structures, simple structural elements, such as plates, spheres, and cylinders, are treated as basic sound radiators because of their frequent occurrences in practical applications [160]. For more than a century, numerous papers and books have been published on almost every aspect of the topic. It is, therefore, not wise and impossible to cover all the materials in a document of limited size. The intention of this chapter is to summarize the outcome that may be related to the sound radiation from electric motor structures, and to present readers with the most recently updated results and techniques to benefit their relevant studies and practices. For this purpose, plane and cylindrical radiators are discussed directly without much emphasis on the basic acoustics theories. Readers might need to refer to Appendix A and [124, 176] for prerequisite knowledge for the contents of this chapter.

6.1

Sound radiation efficiency

As it is known, a vibrating structure produces sound waves in the ambient medium. The sound pressures at any specific location in the medium space, and the sound power radiated by the structure are the two parameters used to quantify the local and global acoustic effects in the field. They are generally functions of vibration levels on the structure surface, which are further determined by the boundary conditions and the external excitations. For a specific structure, higher vibration levels mean higher sound pressures in the field and higher radiated sound power. 127 Copyright © 2006 Taylor & Francis Group, LLC

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Noise of Polyphase Electric Motors

In many cases, however, it is not the absolute value, but the transfer function from the structural vibration to the acoustic response that is of greatest interest. A properly defined vibro-acoustic transfer function could fundamentally reveal the energy transfer mechanisms and efficiently eliminate the duplications of the tedious derivation/calculation in accounting for the interactions between the structure and the ambient medium, thus simplifying the vibro-acoustic analysis. One of the most common definitions used in vibro-acoustic analysis is the sound radiation efficiency, as given by, σ =

ρ 0 c0 S v 2

(6.1)

where ρ0 , c0 are the density and the sound speed in the ambient medium (air) respectively, S is the area ofthe radiating surface, is the sound power radiated from the structure, and v 2 is the spatial averaged mean square velocity over the structure radiating surface. This expression relates the radiated sound power to the spatial averaged vibration level. Physically, it indicates the capability of a structure in sound radiation in comparison with a piston with the same surface area and same vibration levels under the condition that the piston circumference greatly exceeds the acoustic wavelength [176]. However, since the vibration distribution is generally affected by the excitation, the spatial averaged mean square velocity could be quite different for different excitation configurations. For example, a point force excitation might result in different sound radiation efficiencies from a distributed force or point moment excitations for the same structure. As a result, the modal radiation efficiency (σm ) associated with each vibration mode (m) is normally introduced in theoretical analyses [37], m σm = (6.2) ρ0 c0 S v m2 where m is the sound power radiated by the mth mode of the structure, and v m2 is the spatial averaged mean square velocity of the mth mode. The modal radiation efficiency is generally independent of the external excitation and, hence, reflects the inherent characteristics of the structure in the energy transfer. Since the total sound power is the sum of the power radiated from each vibration mode m , i.e., = m , the relationship between Equations 6.1 and 6.2 is [37], m

σm v m2 m σ = . v m2

(6.3)

m

As Equation 6.3 takes into account the contribution from each vibration mode, the result is also called the modal averaged radiation efficiency. In practice, the modal averaged radiation efficiency is what can be directly measured.

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On the other hand, since the radiated sound energy is a loss to the structural vibrational energy, the radiation loss factor can be defined to describe the extent to which the vibration is damped due to the radiation of power from the structure to the ambient medium, i.e., ηrad = (6.4) ωE where ω is the angular frequency, E is the total vibrational energy of the structure. For a uniform structure with a spatially uniform vibration distribution over the structure surface E = M v 2 , where M is the mass of the structure [37]. The relationship between the radiation efficiency and the radiation loss factor is, σ =

M k0 ηrad ρ0 S

(6.5)

where k0 = ω/c0 is the acoustic wavenumber. Compared to the radiation efficiency, the radiation loss factor is a small number, especially for the radiation into the air. Since the radiation loss factor is also related to the spatial averaged mean square velocity over the structure surface, the modal averaged radiation loss factor (ηrad ) can be expected to have the same relationship to the corresponding modal averaged radiation efficiency (σ ) as that shown in Equation 6.5. Another parameter often used in the sound radiation analysis is the acoustic radiation impedance, defined on the radiating surface of the structure as, p( r )S ZR = (6.6) v r=r 0 where p( r ) is the sound pressure in the field, v is the vibration distribution over the structure, S is the radiating surface area, and the subscript r = r0 denotes the location on the surface of the structure. The real part of the impedance represents the energy radiated into the ambient medium, while the imaginary part of the impedance represents the energy stored in the near field of the source [124]. This expression is convenient to use when the vibration amplitude is uniform over the structure; examples are, a point source, a piston radiator, or a plate vibrating in one of its modes. The relation between the acoustic radiation impedance and the radiation efficiency can be found as, σ =

e[Z R ] ρ 0 c0 S

(6.7)

where e denotes the real part of a complex number.

6.2

Plane radiator

Plane structures are very common in industries. Substantial research has addressed the importance of the sound radiation from plane structures. For electric machine structures, plates are often found as accessory components that are necessary for the assembly and the installations; for example, the end-shields and the support

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which in some cases are found important in the total sound power contributions. In this section, the sound radiation from an infinite plate is briefly reviewed to benefit the analysis of finite plates. Then the finite-sized plates are discussed in regard to their modal radiation and modal averaged radiation efficiencies.

6.2.1

Infinite plates

In order to have a better understanding of the sound radiation phenomena of practical plane radiators, it is helpful to have a simple review on the sound radiation from an infinite plate (Figure 6.1). For simplicity, it will be assumed an infinitely extended plate in the y = 0 plane is vibrating transversely with a velocity distribution given as (omitting the time component e jωt , without loss of generality), v(x) = v 0 e− jk p x

(6.8)

where k p is the wavenumber of the plate in bending motion. This equation actually represents a one-dimensional approaching wave propagating in the +x direction with the wavelength and the wave speed being λ p and c p , respectively. Since the vibration is uniform in the z direction (−∞, +∞), and so will be the acoustic field, the acoustic wave motion equation can be simplified as ∂ 2 p(x, y) ∂ 2 p(x, y) + + k02 p(x, y) = 0 ∂x2 ∂ y2

(6.9)

where p(x, y) is the radiated sound pressure in the field. The solution of Equation 6.9 may be generally written as p(x, y) = p0 e− jkx x e− jk y y

(6.10)

y

l0 ky

k0 kx

kp

q

x lp

Figure 6.1 Radiation from an infinite plate.

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where k x and k y are the two components of the acoustic wavenumber k0 in the x and the y directions, respectively, and k02 = k x2 + k 2y . This expression must satisfy the acoustic boundary condition at y = 0, which is 1 ∂ p(x, y) v(x) = − jωρ0 ∂ y y=0 then,

kx = k p

p0 = v 0 ρ0 c0 k0 /k y = v 0 ρ0 c0 /

(6.11)

1 − k 2p /k02 .

The radiated sound pressure in the field is therefore given as √2 2 v 0 ρ0 c0 p(x, y) = e− jk p x e− j k0 −k p y 1 − k 2p /k02

(6.12)

From Equations 6.6 and 6.12, the radiation impedance per unit area of the plate can be obtained as p(x, y) ρ 0 c0 ρ 0 c0 ZR = = = (6.13) v(x) y=0 2 2 1 − k /k 1 − λ2 /λ2 p

0

0

p

Equation 6.13 indicates that for an infinite plate, the sound radiation has two fundamentally different behaviors. If the wavelength in the plate λ p is greater than the acoustic wavelength λ0 in the air (or equivalently, the wave speed in the plate c p is faster than the speed of sound in the air c0 , namely the supersonic effect), the radiation impedance of the plate is a real number, indicating that the energy is transmitted into the ambient medium. The sound radiation efficiency per unit area of the plate can be written as σ =

e(Z R ) 1 1 = = , or simply ρ0 c0 1 − k 2p /k02 1 − λ20 /λ2p

σ =

1 . (6.14) cos θ

If the structural wavelength is smaller than the wavelength of the sound wave in the air (or equivalently, the structural wave speed is slower than the speed of sound in the air, namely the subsonic effect), the radiation impedance is a pure imaginary number, which means that the sound pressure is 90◦ out of phase with the plate velocity. In this case, the energy will be exchanged between the plate and the acoustic medium with no sound energy radiated. The sound radiation efficiency of the plate is therefore zero, and the sound pressure simply decays along the y direction in the form as √2 2 jv 0 ρ0 c0 p(x, y) = (6.15) e− jk p x e− k p −k0 y . k 2p /k02 − 1 An important phenomenon that must be addressed here is, when the acoustic wavelength is matching the structural wavelength, i.e., λ p = λ0 , Equation 6.12

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Radiation Eﬃciency, σ

100

10

1

0.1 1

0.1

10

f/fc

Figure 6.2 Radiation efficiency as a function of frequency for an infinite plate.

gives an infinity solution for the sound pressure in the field. This is because the radiated sound wave, under this condition, is propagating in the direction parallel to the structural wave in the plate, both waves having the same wavelengths and speeds. As a result, the sound wave will be enhanced continuously along the propagation path and the energy will be accumulated to infinity. In practice, due to the finite size and the damping in the plate, the radiated sound pressure always remains finite. However, large radiation still can be observed. Note that the wavelengths of both structural and acoustic waves are frequency dependent. A frequency, namely the critical frequency of the plate f c , at which the structural wavelength is equal to the acoustic wavelength (or equivalently, the structural wave speed is equal to the speed of sound in the air) exists, i.e., [55], cp = and therefore,

1/4 √ ω Eh 2 = ω· kp 12ρ(1 − ν 2 ) √

3c0 fc = πh

ρ(1 − ν 2 ) . E

(6.16)

In Equation 6.16 h is the thickness of the plate. E, ρ, and ν are the Young’s modulus, the density and the Poisson’s ratio of the plate material respectively. Figure 6.2 shows the variation of the sound radiation efficiency vs. the frequency. It is seen that for an infinite plate, sound radiation only occurs in the frequency range above the critical frequency f c . The radiation efficiency approaches to unity quickly as the frequency increases, indicating that the plate is an efficient radiator at high frequencies.

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6.2.2

133

Finite plates in bending motion

When a plate is finite in size, its vibration will have modal behaviors. The modal parameters, such as natural frequencies, mode shapes, and so on, are inherent dynamic characteristics of a structure, depending on the geometry, material properties, and boundary conditions of the structure. Theoretically, a vibration of a structure can be described by a superposition of all the vibration modes with different participation amplitudes that are determined by the external excitations (see Appendix D). Consequently, the sound radiation due to the vibration of the structure can and will be related to the structural modal behavior. In order to discuss the effect of the finite size on the sound radiation from a plate, it is instructive to consider a plate strip of width l in the x direction embedded in an infinite baffle with simply supported boundary conditions at both edges, as shown in Figure 6.3. With the baffle, the sound radiated from each side of the plate does not interfere with each other, such that only the sound field on one side of the plate is considered here. Again, assume that the vibration in the z direction is uniform (−∞, +∞), the velocity distribution for a particular mode m over the y = 0 plane is,

v m (x) =

v 0 sin(mπ x/l) 0 < x < l 0 0 > x > l.

(6.17)

By transforming v m (x) to the wavenumber domain, Vm (k x ) =

l

v m (x)e jkx x d x = v 0 ·

0

mπ [(−1)m e− jkx l − 1] · l k x2 − (mπ/l)2

(6.18)

where Vm (k x ) and v m (x) satisfy, v m (x) =

1 2π

∞

−∞

Vm (k x )e− jkx x dk x .

(6.19)

Equation 6.19 indicates that the modal vibration distribution in the x direction can be decomposed into a series of waves propagating along an infinite plate in the x direction with different wavenumbers and magnitudes. The sound radiation from each of these waves thus can be treated as the radiation from an infinite plate with the velocity distribution as Vm (k x )e− jkx x . Based on Equation 6.12, the sound

y

lpm = 21/m x l

Figure 6.3 Modal behavior of a plate strip in a baffle.

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pressure produced on the plate surface associated with each wave component is ρ 0 c0 k 0 Pm (k x )| y=0 = Vm (k x ) . k02 − k x2

(6.20)

Correspondingly, the total sound pressure due to all the wave components can be written as, ∞ 1 ρ0 c0 k0 − jkx x pm (x)| y=0 = Vm (k x ) e dk x . (6.21) 2π −∞ k02 − k x2 With the known vibration velocity distribution and its induced sound pressure on the plate surface, the time-averaged sound power radiated per unit length (in the z direction) of the plate is,   l ∞ 2 c k |V (k )| 1 ρ 0 0 0 m x m (ω) = e e  pm (x)| y=0 v m∗ (x)d x = dk x  2 4π 0 −∞ k2 − k2 0

x

(6.22) where “∗ ” denotes the complex conjugate. Note that the term within the integral is real only for −k0 < k x < k0 . By using Equation 6.18, Equation 6.22 can be rewritten as k0 m 2l 2 k x l − mπ 2 m (ω) = πρ0 c0 k0 v 0 dk x . (6.23) sin2 2 −k0 (k 2 l 2 − m 2 π 2 )2 k02 − k x2 x Although Equation 6.23 cannot be evaluated in a closed form, approximate solutions corresponding to the two distinguished regions discussed for the infinite plate can be obtained. • If k0 mπ/l, i.e., the acoustic wavelength is greater than the structural wavelength λ pm (or equivalently, in the subsonic region where the structural wave speed is slower than the speed of sound), Equation 6.23 becomes [37] 2 1 ρ0 c0 k0 v 02l 2 k0 ρ0 c0 k0 v 02 l m (ω) ≈ dk x = (6.24) 2m 2 π 3 2 mπ −k0 k02 − k x2 then the corresponding modal radiation efficiency is, 2 m 2k0 l 4m = σm = = . l mπ ρ0 c0lv 02 ρ0 c0l v m2

(6.25)

• If k0 mπ/l, i.e. the acoustic wavelength is shorter than the structural wavelength λ pm (or equivalently, in the supersonic region where the structural wave speed is faster than the speed of sound), one gets m (ω) ≈

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ρ0 c0 k0lv 02 /[4 k02 − (m π/l )2 ] ≈ ρ0 c0lv 02 /4 . The modal radiation efficiency is, m k0 ≈ σm = ≈ 1. (6.26) 2 2 k0 − (m π/l )2 ρ0 c0 l v m Compared to the infinite plate, the most prominent feature of a finite plate in its sound radiation is that the modal radiation is not zero in its subsonic region. Given an explanation, a finite plate with simply supported edges in a particular mode is considered in Figure 6.4. For a higher order mode of which structural wavelengths λ px , λ py along the two directions are smaller than the acoustic wavelength λ0 in the air, as shown in Figure 6.4a, the adjacent subsections of which the vibration is 180◦ out of phase are separated by less than a wavelength in the surrounding air. The air displaced outward by one subsection moves to occupy the space left by the adjacent subsection, without being disturbed much in the far field, and thus very little power is radiated. In other words, the sound radiated from adjacent subsections cancel each other. At the corners, however, such “cancelation” is not complete with four monopole sources remaining and making primary contributions to the total sound power radiated from the plate. If only one of the structural wavelengths is smaller than the acoustic wavelength, for example λ px < λ0 as shown in Figure 6.4b, the adjacent subsections are separated by less than an acoustic wavelength in the l x direction, while the adjacent subsections in the l y direction are beyond an acoustic wavelength. In this case, the cancelations only happen along one direction as shown in the graph with two edges remaining as effective radiators. Along the l y direction, the two

Uncanceled Corners

+ ly l0 l py

Uncanceled Edges

− + +

−

+ Typical Region of Cancelations

+

−

+

l0

lpy

+

ly − +

l0

(a)

−

Typical Region of Cancelations − + −

lx

lx lpx

−+

−

lpx

l0 (b)

Figure 6.4 Sound radiation from: (a) uncanceled corners; (b) uncanceled edges of a finite plate.

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subsections radiate sound independently. Thus, edges normally radiate more power than corners in the frequency range where they both occur. In the supersonic region, the structural wavelengths in both directions are greater than the acoustic wavelength, i.e. λ px , λ py > λ0 . All subareas radiate sound independently and make their contributions in the total sound power. If the plate is large compared to the acoustic wavelength, the boundary effect is negligible and the acoustic behavior of a finite plate approaches to that of an infinite plate. Figure 6.5 shows the modal radiation efficiencies of a series of modes associated with a baffled finite plate with simply-supported boundary conditions. As expected, all the modal radiation efficiencies generally increase as the acoustic wavenumber k0 in the subsonic region, and reach maximums around k0 = k p (k p = (nπ/l x )2 + (mπ/l y )2 for a simply-supported plate, where n, m are the mode numbers along the l x and l y directions, respectively). In the supersonic region, the radiation efficiencies are unity as that of the infinite plate. An interesting and important feature that may be concluded from Figure 6.5 is that, for each mode (n, m), the frequency at which the acoustic wavenumber k0 matches the structural wavenumber k p is different. In the frequency domain, lower order modes reach unity in their sound radiation efficiencies at lower frequencies, thus being more efficient in sound radiation than higher order modes. Apparently, an accurate calculation of the sound power radiated from a finite plate depends on a detailed analysis of the plate vibration from which the participation amplitudes of each mode in the total response can be determined.

10

Modal Radiation Eﬃciency

1 0.1 Mode (1, 1)

0.01

Mode (1, 3) Mode (3, 3)

0.001

Mode (1, 2) Mode (2, 3)

0.0001

Mode (2, 2) 0.00001 0.01

0.1

1

10

k0/kp(m, n)

Figure 6.5 Modal radiation efficiencies of a finite plate in a baffle [221].

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However, if the plate is mechanically excited by a random noise, the vibration response within a frequency band f is primarily dominated by the modes of which the natural frequencies are within this band. Assume that the modal energies within this band are equally distributed, general expressions for the modal averaged radiation efficiency of the plate might be derived. Since each mode generally takes effect around their own natural frequencies because of a relatively large participation amplitude in the total vibration response, it has been shown that statistically a maximum that could exceed unity can still be observed in the modal averaged sound radiation efficiency around f c for a finite plate [55]. Two specific plate configurations that were considered in the analysis are that the plate is embedded in an infinite baffle and that the plate is unbaffled with the edges being acoustically free. Baffled plates Specifically, at low frequencies, the behavior of the plate is dominated by the first mode, normally the piston mode, of the plate. In this case, the radiation efficiency of the plate embedded in an infinite baffle can be calculated using the following equation [176, 212] σ =1−

2J1 (4k0 S/π ) 4S f 2 ∼ 2 8k0 S/π c0

for

λ0

√

2S

(6.27)

where J1 is the Bessel function of the first order. Ver and Holmer [212] suggested this equation generally applies below the first natural frequency of the plate, that is, f < [c02 /(2S f c )][P 2 /(8S) − 1] , where S is the area of one side of a plate, and P is the plate perimeter. When the baffled plate is in the multimodal region, a formula for the modal averaged radiation efficiency based on the simply supported boundary conditions is given in [146], i.e.,  σ cor ner + σ edge f < fc     √ σ = (6.28) l x /λc + l y /λc f = f c      √ 1/ 1 − f c / f f > fc where σ cor ner , and σ edge are the modal averaged radiation efficiencies associated with the radiation regions around corners and along the edges of the plate respectively. They were given as, σ cor ner

8 λ2 = 4· c × π S

σ edge =

√ (1 − 2α)/(α 1 − α 2 ) f < f c /2 0 f > f c /2

1 Pλ2c (1 − α 2 ) ln[(1 + α)/(1 − α)] + 2α · · 2 4π S (1 − α 2 )3

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(6.29)

(6.30)

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√ where α = f / f c . It was suggested that for the clamped boundary conditions, 3 dB should be added to the radiation efficiency in the range f < f c [212]. Figure 6.6 compares the modal averaged radiation efficiencies of two plates calculated analytically and by the boundary element method (BEM). In BEM calculations, the baffled plate was excited mechanically by a point force. To simulate the equally distributed modal energies, the BEM results were averaged over three excitation locations randomly chosen on the plate. It is seen that for small plates, the piston mode could dominate the acoustic behavior even higher than the critical frequency. As the plate becomes bigger, and thinner, the piston mode may only work at very low frequencies. However, Figure 6.6b shows that the piston mode region is wider than that suggested for simply supported and clamped boundary conditions. This is simply because the first mode of the plate (piston mode) is more efficient in sound radiation than higher order modes. Equation 6.28 works well in the multimodal region, which is above the frequency corresponding to the lower one of the natural frequencies f 21 and f 12 , i.e., f > [c02 /(2S f c )][P 2 /(8S) + 3l y /(2l x ) − 1] for a rectangular plate with l x and l y being the long and short edges of the plate, respectively. For baffled plates with completely free boundary conditions, it was shown that rigid motions, one piston and two rotation motions, may predominately contribute to the low frequency sound radiation. As a result, it seems difficult to derive a general expression for the modal averaged radiation efficiency because, depending on the excitations, the sound radiation efficiency may behave quite differently due to the contribution from one or some of these rigid motions [15]. However, studies have shown that if there is any localized constraints along which the two rotation motions are restricted, Equations 6.27 and 6.28, though developed based on simply supported boundary conditions, might give reasonable results in the sound radiation efficiency. Unbaffled plates As mentioned above, the baffle around the plate separates the space into halves, in which the sound radiation considered is only from one side of the plate. In practice, a typical example of this is a plate backed with a large cavity enclosed by rigid walls. The sound radiation from the two sides are isolated from each other or their interactions can be ignored. However, in many places, plane structures are found unbaffled. In this case, the sound radiation from both sides of the plate and their interactions should be considered. An instant effect of removing the baffle in the sound radiation discussions is that the radiating area in the definition of the sound radiation efficiency (Equation 6.1) is twice that of the baffled plate. For an unbaffled plate, the radiation from both sides may exchange the energy due to the oscillating inertial flow around the edge. Since the vibration velocities on both sides are 180◦ out of phase relative to their normal directions (into the ambient medium), the overall sound pressure field is of different characteristics from that of a baffled plate. Specifically, at low frequencies where the piston mode dominates the plate behavior, the unbaffled plate becomes a dipole source

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10 BEM, Simply-Supported Plate

Radiation Eﬃciency, σ

BEM, Clamped Plate 1

Equation 6.27

0.1

0.01

0.001 1000

100

10

10000

Frequency (Hz) (a) 10

Radiation Eﬃciency, σ

BEM, Simply-Supported Plate BEM, Clamped Plate Equation 6.28

1

Equation 6.27

0.1

0.01

0.001 1000

100

10

10000

Frequency (Hz) (b)

Figure 6.6 Modal averaged radiation efficiencies of a baffled plate: (a) 0.17 m×0.19 m×0.01 m aluminum plate, 9 × 10 elements in the BEM model; (b) 0.6 m×0.7 m×0.005 m steel plate, 40 × 46 elements in the BEM model. acoustically. Asymptotic solutions for the radiation efficiency of an unbaffled circular plate with radius a were derived by Silbiger [188], i.e.,

σ =

8(k0 a)4 /(27π 2 ) k0 a 1 1 k0 a 1.

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(6.31)

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On the basis of Equation 6.31, a general expression for an arbitrarily shaped plate with the unbaffled acoustic boundary condition can be obtained as,

σ =

√ 4S 2 f 4 /c04 λ0 > √2S 1 λ0 ≤ 2S

(6.32)

where S is the area of one side of a plate. In the multimodal region, below the critical frequency, instead of pulsating sphere and cylinder sources radiating into the half space, corners and edges of an unbaffled plate will behave more like oscillating sphere and cylinder sources, which are less efficient in sound radiation [124]. As a result, the modal averaged radiation efficiency of an unbaffled plate with simply supported boundary conditions can be written as [172] σ = F plate (Fcor ner σ cor ner + Fedge σ edge )

for

f < fc

(6.33)

where F plate , Fcor ner , Fedge are the correction factors due to the unbaffled condition. Particularly, F plate is for the effect of the inertial flows that surround the plate when the acoustic wavelength is greater than the plate dimensions, while Fcor ner , Fedge account for the localized inertial flow effects associated with corner and edge radiations. Semiempirical expressions for these correction factors were given as, F plate =

53 f 4 S 2 /c04 ; 1 + 53 f 4 S 2 /c04

Fcor ner =

1 13α 2 ; 2 1 + 13α 2

Fedge =

1 49α 2 . 2 1 + 49α 2 (6.34)

It can be seen that these factors take effects in different frequency ranges corresponding to different multimodal radiation behavior of the plate. Above the critical frequency, since the acoustic wavelength is smaller than the structural wavelength, the radiation from each small subsection on the plate will not cancel each other and is making contributions independently in the total radiated sound energy. Although the corners and edges may still be less efficient due to inertial flow effects if the acoustic wavelength is greater than the thickness of the plate, the effect on the total radiation is limited and therefore may be ignored. As a result, the modal averaged radiation efficiency of an unbaffled plate is still unity above the critical frequency. Figure 6.7 compares the modal averaged radiation efficiencies of two plates calculated analytically and by the BEM. The BEM results were also averaged over three excitation locations. It can be seen that Equation 6.32 gives a good estimation for the sound radiation from the piston mode. In the multimodal region, there is about 3 dB difference in the sound radiation efficiency between simply supported and clamped boundary conditions below the critical frequency. Again, Equation 6.33 is valid in the range f > [c02 /(2S f c )][P 2 /(8S) + 3l y /(2l x ) − 1] for a rectangular plate.

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10

Radiation Eﬃciency, σ

1

0.1

0.01 BEM, Simply-Supported Plate BEM, Clamped Plate

0.001

Equation 6.32 0.0001 100

1000 Frequency (Hz)

10000

(a) 10 BEM, Simply-Supported Plate BEM, Clamped Plate

Radiation Eﬃciency, σ

1

Equation 6.33 Equation 6.32

0.1 0.01 0.001 0.0001 0.00001 10

1000

100

10000

Frequency (Hz) (b)

Figure 6.7 Modal averaged radiation efficiencies of an unbaffled plate: (a) 0.17 m×0.19 m×0.01 m aluminum plate, 9 × 10 elements in the BEM model; (b) 0.6 m×0.7 m×0.005 m steel plate, 40 × 46 elements in the BEM model.

6.3

Infinitely long cylindrical radiator

Cylindrical shells are one of the most widely used structures in industries, for example, the casing of electric machines, and pipes used in various constructions. Due to the geometrical curvature, the vibration and the sound radiation is

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of different behaviors from that of plane structures. Acoustically, an elementary cylindrical radiator is an infinitely long circular cylindrical shell. By employing the cylindrical coordinates (Figure 6.8), the acoustic wave equation can be written as (omitting the time component e jωt , without loss of generality), ∂ 2 p(r, θ, z) 1 ∂ p(r, θ, z) 1 ∂ 2 p(r, θ, z) ∂ 2 p(r, θ, z) + + + + k02 p(r, θ, z) = 0. ∂r 2 r ∂r r2 ∂θ 2 ∂z 2 (6.35) The general solution of this equation is [160] p(r, θ, z) = p0 (Ae jkz z + Be− jkz z ) cos mθ Hm(2) (kr r )

(6.36)

where p0 , A, and B are constants that are determined by the acoustic boundary conditions on the cylindrical shell surface, m is the circumferential modal number of the cylindrical shell, Hm(2) (·) is the second kind of Hankel function of the m th order, kr and k z are the two components of the acoustic wavenumber k0 in the r and the z directions respectively, i.e. k02 = kr2 + k z2 . Assume that the vibration velocity distribution on the surface of the infinite length shell (r = a) is taking the form as, v(θ, z) = v 0 e− jksz z cos mθ (6.37) where v 0 is the amplitude, ksz is the z component of the structural wavenumber ks , as shown in Figure 6.8. This equation represents a structural wave traveling along the +z direction of the cylindrical shell. With the acoustic boundary condition on the surface of the shell, i.e., 1 ∂ p(r, θ, z) v(θ, z) = − jωρ0 ∂r r =a

ksz ks ksq = n/a

rr z

aa

q

kz k0 kr

Figure 6.8 Cylindrical shell coordinates, structural and acoustic wavenumbers.

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one obtains, A = 0;

B = 1;

k z = ksz ;

p0 = −

jv 0 ρ0 c0 k0 (2)

m (kr a) kr d Hd(k r a)

.

(6.38)

The circumferential distribution taking the same form in Equations 6.36 and 6.37 is also a result of complying with the acoustic boundary condition. Thus, Equation 6.36 can be rewritten as p(r, θ, z) = −

jv 0 ρ0 c0 k0 (2)

m (kr a) kr d Hd(k r a)

cos mθ Hm(2) (kr r )e− jkz z

(6.39)

where kr = k02 − k z2 . In the far field, i.e., kr r >> 1, an asymptotic solution of the radiated sound pressure can be obtained as jv 0 ρ0 c0 k0 2 j 2 (1+ 1 ) p(r, θ, z) = − e π 2 cos mθe− jkr r e− jkz z . (6.40) (2) d Hm (kr a) π k r r kr d(kr a)

The above Equation 6.40 indicates that for a particular axial structural wavenumber ksz (= k z ), the sound radiation from an infinite length cylindrical shell only happens under the condition k0 > k z . In the range k0 < k z , the pressure is 90◦ out of phase with shell surface velocity. No sound wave is propagating into the far field and the sound pressure near the shell surface decays along the radial direction. Based on Equation 6.40, the sound power radiated into the far field per unit axial length is 2π 2π l 1 v 02 ρ0 c0 k02 = | p(r, θ, z)|2r dθdz = cos mθdθ. (2) 2 m (kr a) 0 2ρ0 c0l 0 0 πlkr3 d Hd(k r a) (6.41) The mean square velocity on the shell surface is given by, 2π l v 02 2π 1 2 2 v = |v(θ, z)| adθdz = cos mθdθ. (6.42) 4πal 0 0 4π 0 The radiation efficiency per unit length thus has a simple and closed form as,  k0 ≤ k z 0 (2) 2 σ = (6.43) m (kr a) k0 > k z .  2k02 / πakr3 d Hd(k r a) Equation 6.43 clearly indicates that an infinite length cylindrical shell does not radiate sound below a certain frequency where the acoustic wavenumber matches the axial component of the structural wavenumber. In this frequency range, the acoustic wavelength is greater than structural wavelength in the axial direction, and the sound generated from the two adjacent subsections with 180◦ out of the phase movement will interfere with each other, resulting in a complete cancelation along the axial direction of the shell. Beyond this frequency range

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when the acoustic wavelength is smaller than the structural wavelength in the axial direction, the sound generated from adjacent subsections is independent of each other and the total sound radiation from the shell becomes more efficient. As frequency increases, an asymptotic value of unity can be expected as for a plane radiator. For simplicity, Equation 6.43 is sometimes employed to calculate modal radiation efficiencies of finite length cylindrical shells by setting k z equal to n π/l, which are the modal wavenumbers in the axial direction of a cylindrical shell of length l and with simply-supported boundary conditions at both ends, where n is the axial mode number. According to Equation 6.43, for each vibration mode, there exists a cut-off frequency, at which the corresponding acoustic wavenumber k0 is equal to k z = n π/l. Below this cut-off frequency, the modal radiation efficiency is zero. Above this frequency, the modal radiation efficiency increases as frequency until it reaches unity where the acoustic wavenumber k0 is equal to the structural wavenumber (n π/l )2 + (m /a )2 . This result is not obvious because if one uses infinite flat plates as an analogy, the modal radiation efficiencies of infinite length cylindrical shells would be expected to be zero when k0 < (n π/l )2 + (m /a )2 due to the intercell cancelation. However, for cylindrical shells, the structural wave speeds in the axial and the circumferential directions have dispersion effects, the wave in the axial direction increasing faster with frequency than that in the circumferential direction due to curvature effects below the ring frequency. Therefore, the axial wave would catch up the acoustic wave much faster than the circumferential wave as the frequency increases. Hence, there is a frequency range within which the axial component of a vibration mode is supersonic while the circumferential component is still subsonic. The modal radiation efficiency would continue to increase with frequency until both components are supersonic. The frequency at which the axial component becomes supersonic is the cut-off frequency of this mode. Generally, the cut-off frequency increases with number n. Figure 6.9 shows the radiation efficiencies for a series mode associated with a simply supported shell calculated by Equation 6.43. Particularly, if the axial vibration distribution is uniform, i.e. k z = 0, Equation 6.43 gets a simpler form as [200] σ (m , k0 a ) =

=

2 2 (2) π k0 a d Hdm(k0(ka0)a )

(6.44)

(k0 a )2 [Ym (k0 a )Jm +1 (k0 a ) − Jm (k0 a )Ym +1 (k0 a )] [m Jm (k0 a ) − (k0 a )Jm +1 (k0 a )]2 + [mYm (k0 a ) − (k0 a )Ym +1 (k0 a )]2

where Jm (k0 a ) and Jm +1 (k0 a ) are the first kind Bessel functions of the m th , (m + 1)th order respectively, Ym (k0 a ) and Ym +1 (k0 a ) are the second kind of Bessel functions (Neumann functions) of the m th , (m + 1)th order, respectively. The sound radiation efficiencies of various mode number m vs. k0 a are plotted in Figure 6.10. Equation 6.44 allows for a simple calculation of the sound radiation efficiency of the pure circumferential modes m = 0, 1, 2, 3, . . ..

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145

Modal Radiation Eﬃciency

10

1

0.1 Mode (1, 2) 0.01

Mode (2, 2) Mode (2, 1)

0.001

Mode (3, 2) Mode (2, 3)

0.0001 0

0.5

1

1.5

2

k0 /ks(m, n)

Figure 6.9 Modal radiation efficiencies of a circular cylindrical shell calculated by the infinite length model l/a = 3.

6.4

Finite length cylindrical radiator

In practice most circular cylindrical structures are finite in length. As the length becomes shorter, it can be expected that the end effects would become more important, and the results obtained by using an infinite length model could be in severe error. By reviewing the literature, the techniques used in the analysis could be categorized into two groups, namely the statistical approach and the deterministic analysis. A statistical approach [150, 198] is employed when the resonant frequencies are densely packed in a frequency band, and the acoustic properties are lumped together so that individual frequency analysis becomes impractical. Obviously, this method is only valid at high frequencies or for relatively thin shells with high modal densities. The most successful work about a cylindrical source was carried out by Szechenyi [198] who developed a set of general, but approximate equations for predicting the radiation efficiencies of cylindrical shells having large length to thickness and radius to thickness ratios within a given frequency bandwidth. These results have been verified by experiments except for low frequencies [198]. In deterministic analysis, specific vibration velocity distributions and their acoustic response, which could be affected by the geometry and the boundary conditions of a cylindrical shell at each frequency, are considered. Since at low frequencies where the modal density is low, the behavior of each individual mode becomes important, the corresponding modal averaged radiation efficiency may have different characteristics from that at high frequencies. Various studies using deterministic analysis have been reported. The early contribution to the acoustic

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Modal Radiation Eﬃciency

2

1.5

1

0.5 m=1

m=2

m=4

m=5

m=3

0 0

1

2

3

4

5

6

7

8

9

10

k0a

Figure 6.10 Radiation efficiency factor curves according to Equation 6.44 for a cylindrical radiator and m = 0, 1, 2, . . . 5.

radiation from finite length cylindrical shells was made by Williams [238] who analyzed the radiation of finite length cylinders with a uniform radial vibration velocity profile and compared the sound field with that of the infinite length model. It was found that infinite length models may overestimate the acoustic radiation from finite length cylinders, and the error would increase as the length of the cylinder becomes shorter. Then, Schenck (1968) presented a surface Helmholtz integral formulation for obtaining approximate solutions of acoustic radiation problems for an arbitrary surface, which improved the accuracy of Williams, results in the axial direction. In order to take the reaction of the fluid inside the cylindrical shell into account, Sandman [182] examined the model of a baffled finite length cylindrical shell. It was found that the model is a reasonable approximation in determining the acoustic radiation from the surface of finite length cylindrical shells, and is applicable to arbitrary radial vibration distribution. Based on this model, Stepanishen [192] discussed the radiation impedance of different vibration modes of a finite length cylindrical shell, and Zhu [255] examined the “relative sound intensities” for different vibration modes. Also, Stepanishen [193] and Laulagnet et al. [134] investigated the effects of fluid on the acoustic radiation of finite length cylindrical shells. In this section, the sound radiation from acoustically thin shells is first introduced briefly. Then the baffled finite length cylindrical shell model is employed to calculate the modal radiation efficiencies and modal averaged radiation efficiencies for different boundary conditions. The limitation of the infinite length model and the effects of boundary conditions on the modal radiation efficiencies of finite length cylindrical shells are discussed last.

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6.4.1

147

Acoustically thin shells

The acoustically thick and thin shells are distinguished in terms of shell curvature effects on the acoustic radiation behavior. For cylindrical shells, the ring frequency is an important parameter used to indicate the frequency range in which curvature effects are important. The ring frequency fring is defined for the condition when the wavelength of extensional waves in the shell is equal to the shell circumference (compare Equation 5.21), E 1 fring = (6.45) 2πa ρ(1 − ν 2 ) where a is the radius of the cylindrical shell, ρ is the density of the material, E is the Young’s modulus, and ν is the Poisson ratio. Above the ring frequency, the structural wavelength is so short that the structural wave propagation is very much controlled by the local stiffness. The curvature effects of the shell disappear, and the structure will behave like a plate. The sound radiation, as a result, will be governed by the compliances between the total structural wavenumber ks and the acoustic wavenumber k0 , as discussed in Section 6.2. In this frequency range, the critical frequency defined for the plate f c (Equation 6.14) is of physical significance because of no dispersion effects for the structural wavenumbers. Below the ring frequency, the curvature changes the stiffness in the axial direction, making the structural wave speed in this direction faster than that in the circumferential direction. Then the sound radiation is primarily determined by the behavior of each individual mode in the total response. The physical significance of the critical frequency defined for the plate f c becomes questionable, as will be seen in Section 6.4.2. Given these two different acoustical features, the cylindrical shells are consequently defined as acoustically thin for fring / f c < 1, and thick for fring / f c > 1 [89]. However, it should be emphasized that a shell can be geometrically thin (thickness h >>radius a) but still acoustically thick. For example, for an aluminum (Young’s modulus, E = 71 × 109 Pa, density ρ = 2700kg/m3 and the Poisson ratio, ν = 0.33) or steel (E = 210 × 109 Pa, ρ = 7850 kg/m3 and ν = 0.3) circular cylindrical shell, by using Equations 6.42 and 6.14, one obtains, fring h ≈ 67 . fc a

(6.46)

If a / h is smaller than 67, the shell would be considered to be acoustically thick but still geometrically thin. In practice, a typical example of acoustically thin shells is an airplane fuselage. For acoustically thin cylindrical shells, because of the high modal density, statistical analysis can be employed [150, 198]. By measuring the area ratio of the supersonic and subsonic regions that vibration modes possibly fall into in the wavenumber contour map, Szechenyi [198] generated a family of curves of the sound radiation efficiency for various cylindrical shell configurations, as shown in Figure 6.1