High Performance AC Drives: Modelling Analysis and Control

Power Systems Mukhtar Ahmad High Performance AC Drives Modelling Analysis and Control ABC Prof. Dr. Mukhtar Ahmad ...

Author: Mukhtar Ahmad

142 downloads 1907 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Power Systems

Mukhtar Ahmad

High Performance AC Drives Modelling Analysis and Control

ABC

Prof. Dr. Mukhtar Ahmad Department of Electrical Engineering Aligarh Muslim University Aligarh, 202002 India E-mail: [email protected]

ISSN 1612-1287 e-ISSN 1860-4676 ISBN 978-3-642-13149-3 e-ISBN 978-3-642-13150-9 DOI 10.1007/978-3-642-13150-9 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2010926860

c Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover Design: deblik, Berlin, Germany Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

In memory of: My father, Mr. Mushtaq Ahmad And My mother Mrs. Zeenat

Preface

Variable speed is one of the important requirements in most of the electric drives. Earlier dc motors were the only drives that were used in industries requiring operation over a wide range of speed with step less variation, or requiring fine accuracy of speed control. Such drives are known as high performance drives. AC motors because of being highly coupled non-linear devices can not provide fast dynamic response with normal controls. However, recently, because of ready availability of power electronic devices, and digital signal processors ac motors are beginning to be used for high performance drives. Field oriented control or vector control has made a fundamental change with regard to dynamic performance of ac machines. Vector control makes it possible to control induction or synchronous motor in a manner similar to control scheme used for the separately excited dc motor. Recent advances in artificial intelligence techniques have also contributed in the improvement in performance of electric drives. This book presents a comprehensive view of high performance ac drives. It may be considered as both a text book for graduate students and as an up-to-date monograph. It may also be used by R & D professionals involved in the improvement of performance of drives in the industries. The book will also be beneficial to the researchers pursuing work on sensorless and direct torque control of electric drives as up-to date references in these topics are provided. It will also provide few examples of modeling, analysis and control of electric drives using MATLAB/SIMULINK. An approach applying first principles that will give reader understanding of the basic concepts of high performance ac drives has been used. The main emphasis of the book is on sensorless control of ac drives, as these controlled drives provide high performance at low cost with high reliability. The book written mainly with the above objectives is divided into seven chapters. The first chapter deals with the basics of electric drives, their requirement in industries and load dynamics. Since the induction machine is the most commonly used ac motor in the drives, its modeling is taken in the chapter 2.Here dynamic model of induction motor in different reference frames have been described. Also state space model of the induction machine used in the simulation is also discussed. In chapter 3, vector control, or field oriented control which transforms the dynamic structure of ac machine into that of separately excited compensated dc motor for both induction and synchronous motor is discussed. Chapter4 deals

VIII

Preface

with sensorless control and direct torque control of induction machine. Chapter 5 is dedicated to vector and direct torque control of permanent magnet synchronous motor and brushless dc motor. These motors are now very widely used in industries. In this chapter the model of the PMSM and brushless DC motor are presented along with the control schemes. Chapter 6 is devoted to Switched Reluctance Motor (SRM) drives. The SRM had its origin in 1850 but has received considerable interest since 1980s. These motors are now finding use in many variable speed drives. The switched reluctance motors have many advantages, e.g. High efficiency, can be designed for ratings from few watts to M watts and can be employed in harsh working environments. In this chapter principle of operation of SR motor, various types of its configurations, equivalent circuit, and design procedure are discussed. Also the modeling, simulation and control of these drives is presented in simple manner. Sensorless operation and control of these drives is also discussed. In chapter 7 multi-phase drives which are now being considered for many applications have been described. Chapter 8 deals with fuzzy logic and application of neural network in the control of high performance drives. The content of this book and the material has been developed by this author while teaching graduate students in AMU Aligarh and UPM Malaysia. I am also involved in research in the area of high performance drives and multiphase drives. I am thankful to the chairman Department of electrical engineering and the vicechancellor Aligarh MuslimUniversity for awarding me leave for academic pursuits to write this book. Finally, I am very grateful to my wife Maimoona for the patience and support to carry out this work.

Mukhtar Ahmad Aligarh

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction to Electric Drives . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Electric Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Power Electronic Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Inverters for Adjustable Speed Drives . . . . . . . . . . . . . . 1.4 Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Typical Load Torque/Speed Curves . . . . . . . . . . . . . . . . 1.6 Load Dynamics and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Multi-quadrant Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Duty Cycle and Motor Rating . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 2 2 4 4 4 5 8 9 10 11

2

Modeling of Induction and Synchronous Machines . . . . . . . 2.1 Induction Machine Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Equivalent Circuit of Induction Motor . . . . . . . . . . . . . . . . . . . . 2.3 Dynamic Model of a Two-Phase Induction Machine . . . . . . . . 2.3.1 Transformation to Obtain Constant Inductances . . . . . 2.3.2 Dynamic Model of Three-Phase Machine . . . . . . . . . . . 2.4 Selection of Reference Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Modeling in Arbitrary Reference Frame . . . . . . . . . . . . 2.5 Models in Other Reference Frames . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Stator Reference Frame Model . . . . . . . . . . . . . . . . . . . . 2.5.2 Rotor Reference Frame Model . . . . . . . . . . . . . . . . . . . . . 2.5.3 Synchronously Rotating Reference Frame Model . . . . . 2.6 Space Phasor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Speed Control of Induction Motor . . . . . . . . . . . . . . . . . . . . . . . 2.8 State Space Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 16 18 20 22 23 25 25 26 26 27 34 34

X

Contents

2.9 Modeling of Synchronous Machine . . . . . . . . . . . . . . . . . . . . . . . 2.9.1 Production of Torque in Cylindrical Rotor Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9.2 Salient Pole Synchronous Machine . . . . . . . . . . . . . . . . . 2.10 Dynamic Modeling of Synchronous Machine . . . . . . . . . . . . . . 2.11 Space Phasor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

Vector Control of Induction Motor Drives . . . . . . . . . . . . . . . 3.1 Speed Control of Induction Motor . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Volts/Hz Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Flux and Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Introduction to Vector Control . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Space Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Transformation of Space Vector from One Reference Frame to Other . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Principle of Vector Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Direct Vector Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Direct Vector Control Sensing Line Voltages and Currents (Rotor Flux) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Direct Vector Control Stator Flux Model . . . . . . . . . . . 3.6.3 Direct Vector Control Sensing Induced EMF and Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Direct Vector Control with VSI Using Space Vector Modulation (SVM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Indirect Vector Control or Feed Forward Control . . . . . . . . . . 3.9 Case Study 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Direct Torque Control and Sensor-Less Control of Induction Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Sensorless Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Direct Torque Control Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Torque and Flux Control . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 DTC Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Switching Table Based DTC Scheme . . . . . . . . . . . . . . . . . . . . . 4.4.1 Direct Self Control Scheme . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Main Features of DTC . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Sensorless Control of Induction Motor . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36 37 39 41 43 44 47 47 47 49 50 50 52 55 58 59 61 61 63 67 68 74 75 77 77 78 78 79 81 85 86 87 87 95

Contents

5

6

XI

Control of Permanent Magnet Machine (PM) . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Design Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Modeling of PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Modeling of Brushless DC Motor . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Drive Operation with Inverter . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 120 Degree Angle Switch-On Mode . . . . . . . . . . . . . . . . 5.5.2 Voltage and Current Control PWM Mode . . . . . . . . . . 5.5.3 Current Control with Half Wave Converter Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Speed Control Using PWM Inverter . . . . . . . . . . . . . . . . . . . . . 5.7 Vector Control of PMSM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Operating Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Constant Torque Angle Control . . . . . . . . . . . . . . . . . . . 5.8.2 Unity Power Factor Control . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Maximum Torque Per Ampere Control . . . . . . . . . . . . 5.8.4 Flux Weakening Control . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 Direct Torque Control of PM Motor . . . . . . . . . . . . . . . . . . . . . 5.10 Sensorless Control of PM Motor . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.1 Position Information from Measurement of Voltages and Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10.2 Position Information from Measurement of Inductance Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Sensorless Control of BLDC Motor . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97 97 100 101 102 107 108 108

124 125 128

Switched Reluctance Motor Drives (SRM) . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Linear SR Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Basic Principle of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Design of SR Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Selection of Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Stator and Rotor Pole Angle Selection . . . . . . . . . . . . . 6.4.3 Selection of Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Converters for SR Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Asymmetric Bridge Converter . . . . . . . . . . . . . . . . . . . . . 6.5.2 Six Switch Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Variable dc Link Buck Converter . . . . . . . . . . . . . . . . . 6.6 Buck-Boost Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Analytical Model of SR Machine . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Control of SR Motor Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 General Purpose SRM Drive with Speed/Position Sensor . . . 6.9.1 Design of Current Controllers . . . . . . . . . . . . . . . . . . . . . 6.9.2 Torque Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

129 129 129 131 133 134 135 135 138 138 139 141 142 145 146 148 150 151 151

109 110 110 112 112 113 115 116 116 119 123

XII

Contents

6.10 Direct Torque Control of SRM Drive . . . . . . . . . . . . . . . . . . . . 6.11 Sensorless Control of SRM Drives . . . . . . . . . . . . . . . . . . . . . . . 6.11.1 Position Information from Inductance Variation . . . . . 6.11.2 Estimation Based on Inductance Measurement with External Signal Injection . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

152 156 156

7

Control of Multiphase AC Motor Drives . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Modeling of a Five Phase Induction Motor . . . . . . . . . . . . . . . . 7.3 Machine Model in Arbitrary Reference Frame . . . . . . . . . . . . . 7.4 Vector Control of Five-Phase Induction Motor . . . . . . . . . . . . 7.5 Five-Phase Inverters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 SVPWM Five-Phase Voltage Source Inverter . . . . . . . . 7.6 Five-Phase Permanent-Magnet Motor Drives . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

161 161 161 164 166 167 169 171 172

8

Fuzzy Logic and Neural Network Applications in AC Drives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Basic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Fuzzy System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Fuzzification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Fuzzy Rule Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Fuzzy Inference Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Mandani Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Defuzzification Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Centre of Area (COA) Method . . . . . . . . . . . . . . . . . . . . 8.6.2 Mean of Maxima Method . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 Centre of Maxima Method . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Speed Control of Induction Motor Using Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Formation of Fuzzy Set and Fuzzy Rules for the System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Neural Network Based Control . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Artificial Neuron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Artificial Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.3 Feed Forward Neural Networks . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

157 159

175 175 176 178 178 178 178 178 179 179 180 180 180 182 183 183 185 185 188

Chapter 1

Introduction

1.1

Introduction to Electric Drives

For almost three quarters of the twentieth century, most of the ac drives for industrial and domestic uses have been designed to operate at constant speed. Constant speed drives are highly ineﬃcient for variable speed operations. Now, with the availability of economical variable frequency electric supplies, resulting from advances in power electronic switching devices, and the low cost microprocessor based control, electric drives are increasingly being used for operating industrial loads at any one of a wide range of speeds. In many modern industries adjustable speed drives are required for precise and continuous control of speed, torque, or position, with long term stability, good transient performance and high eﬃciency. The industrial loads may be of constant torque type, constant power type, or may have torque as a function of speed. The electric drive makes use of electric motors as prime mover due to the following advantages. They can be brought very close to the working machine (load), can be operated at any desired speed through power electronic control, and can be started and reversed in very short periods of time. The electric motors are also available in the market in a wide range of power ratings; from few watts to few thousand of kilowatts capacity. Another advantage of electric drives is that speedtorque characteristic of electric motors can be easily modified to suit the load characteristic. A modern electric drive has four components: (i) Electric motor (ii) Power electronic converter (iii) Controller (iv) Actual apparatus or equipment (Load) These components are described in the following sections. M. Ahmad: High Performance AC Drives, Power Systems, pp. 1–11. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

2

1.2

1 Introduction

Electric Motors

Motors used in electric drives are:(i) DC motors- shunt, series, compound, separately excited, and switched reluctance motors (ii) AC motors- Induction, synchronous, Permanent magnet, and switched reluctance motors (iii) Special machines- stepper motor etc. The selection of a type of motor to be used in a particular application depends on many factors such as:(i) (ii) (iii) (iv) (v)

1.3

Speed – torque characteristics of the load Initial and running cost Availability of spare parts and trained personnel Peak torque capability Thermal capacity

Power Electronic Converters

Adjustable speed drives are very common these days, due to their higher eﬃciency, better performance, and higher productivity. The type of power electronic converter topology and its control depends on the type of motor drive selected. The power converters employed in the adjustable speed drives (ASD) are: (i) Controlled rectifiers- to provide controlled dc output voltage from fixed 1-phase or 3-phase ac supply (ii) DC to AC converters- to provide controlled ac voltage or current at desired frequency. The dc supply is obtained either from the battery or through uncontrolled ac to dc conversion using diodes. The matching between Power Electronic Converter (PEC) and the motor is important. The current rating of the PEC is selected on the basis of steady state and peak torque requirement of the motor. The voltage of PEC must be selected based on maximum voltage appearing at motor terminal. Since this book deals with ac machines only, the PEC considered here is basically a dc to ac converter or Inverter.

1.3.1

Inverters for Adjustable Speed Drives

In large number of industries the ac drives are required to be operated at diﬀerent speeds. In order to obtain variable speed, these ac machines are fed from inverters with variable voltage and variable frequency supply. The inverters can be broadly classified into two types: (i) voltage source inverters,

1.3 Power Electronic Converters

3

+ 1

3

5

Lo ad

4

6

2

Fig. 1.1 Three phase bridge inverter

and (ii) current source inverters. The voltage source inverter or VSI is more common and it produces well defined switched voltage waveforms at the terminal of the motor. The voltage control of these inverters is usually obtained using pulse width modulation (PWM). The current source inverter or CSI provides switched current waveforms at the motor terminals. A stiﬀ dc bus current is maintained by use of a large inductor in the dc link. 1.3.1.1

Three –Phase Bridge Inverter

A three – phase voltage source bridge inverter is shown in Figure 1.1. The dc supply for the inverter is obtained by rectifying ac voltage. In addition, a relatively large electrolytic capacitor is connected in parallel to “stiﬀen” the link voltage, and provide a path for the rapidly charging currents drawn by the inverter. The size of the capacitor required is quite large in the range of 2000 to 10000 micro Farads. Also a series inductance is connected to limit the fault current. An inverter used to supply an induction motor also requires a switching device capable of being turned oﬀ and on through the gate. The devices that can be used are BJT, MOSFET, IGBT, GTO or Thyristor with external commutation circuit. If thyristors are used, the external commutation circuit will require at least one extra thyristor and one capacitor per phase. Also additional inductors are required to change the polarity of the capacitor for commutation. Thus thyristors with external commutation are rarely used. Transistors have nearly replaced the Thyristor completely in inverter circuits except for large power applications where GTOs are used. The basic operation of six step voltage inverter can be explained for the circuit shown in Figure 1.1. A three-phase output can be obtained from six transistors and six diodes. Two types of operation are possible with this circuit; 180 degree conduction for each device, or 120 degree conduction. The 180-degree

4

1 Introduction

conduction has better utilization of switches, produces a higher output voltage under any operating condition and is more commonly used method. In 1800 conduction method, three transistors are conducting at any instant of time. There are six modes of operation in a cycle of 360 degrees. Each mode is therefore for 60 degrees. The transistors are numbered in the sequence of their switching as 1-2-3, 2-3-4, 3-4-5, 4-5-6, and 6-1-2. That means transistors 1,2 and 3 are turned on for sixty degrees time and then 2-3-4 transistors are turned on for next 60 degrees and so on.

1.4

Controllers

Controllers are required to match the motor characteristic with that of load. A number of control strategies have been developed for various motor drives. The electric controller in general controls the current and voltage or flux linkage and torque within the PEC. The electric sensors (observers) refer to voltage, current, flux as measured or calculated state variables. The electric sensor gets its input both from the power source and the PEC output. The output of this controller is commands for improving the performance of power converter (like improving the power factor, reducing harmonics etc.). Motion sensors (observers) refer to mean position, and/or speed and torque as measured or calculated state variables. The motion controller gets its input from motion sensors and delivers output in the form commands relating to motions (speed, position, torque). The electric and motion controllers are combined as single controller and are realized with analog or digital circuits. The present trend is to use microcontrollers, and/ or Digital signal processors (DSPs) as controllers, specially for high performance drives.

1.5

Load

An electric drive is required to match the characteristic of the motor with that of load. The characteristic of the load is described by the torque vs speed or torque vs time and speed vs time or position vs time relationships.

1.5.1

Typical Load Torque/Speed Curves

The torque/speed characteristics of most of the industrial loads can be classified into the following four general categories (see Figure 1.2): I. II. III. IV.

Constant torque type Torque linearly proportional to speed Torque proportional to square of the speed Torque inversely proportional to speed.

1.6 Load Dynamics and Stability

5

Speed

Speed

Torque

Figure 1.2 (a) Constant torque

Torque

Figure 1.2 (b) Torque proportional to speed Speed

Speed

Torque

Figure 1.2 (a) Constant torque

Torque

Figure 1.2 (b) Torque proportional to speed

Fig. 1.2 Speed/torque characteristics of common type loads

The nature of load torque- speed characteristics depends on particular application. A low speed hoist, piston compressors, conveyor machines etc. exhibit constant torque (independent of speed) type of characteristics. Mixers, stirrers, and separately excited dc generators connected to a constant resistance load, eddy current brakes, and calendaring machines have torque proportional to speed. In fans, ship propellers and compressors the torque is proportional to the square of the speed. The lift machines, winding machines, lathes, wire drawers etc. have torque that is inversely proportional to speed. Most of the loads require extra eﬀort at the time of starting to overcome static friction. In power application it is known as breakaway torque and the control engineers call it stiction. An electric drive has to match the speed/torque (and power) characteristics of load with that of motor. The motor and power electronic converter is, therefore chosen or designed for this purpose. However, the motor to mechanical load matching should be for steady state as well as for transient conditions.

1.6

Load Dynamics and Stability

A motor driving a load machine attains equilibrium speed when motor torque is equal to the load torque. The basic equation governing the dynamics of a

6

1 Introduction

motor load system with rigid mechanical coupling between motor and load is given by dω = Tm − TL − Tf (1.1) J dt Where T m is the motor electromagnetic torque, Tl is the actual load torque, J is the moment of inertia of motor and load combined reduced to motor shaft, ω is the angular speed, and Tf is friction torque. The friction torque is a combination of static friction torque, Coulomb friction torque, viscous friction torque, and windage friction torque. It can be approximately taken as proportional to speed. Then dω = Tm − TL − B ω (1.2) dt The steady state stability of the motor load system can be obtained by linearizing Eq. (1.2). To evaluate the stability of the motor load system, speed/torque curves for the motor and load are drawn as shown in Fig. 1.3. The points of intersection are the equilibrium points. To check whether an equilibrium point is also stable, a small change in speed is made. If the net torque acting on the motor is such as to bring the drive back to its original condition, the equilibrium point is a stable point; otherwise not. As an example the speed/torque curves for motor load systems are drawn and the condition of stability evaluated. The point A in Fig. 1.3 (a) is a stable point since for the small decrease in speed the motor torque is more than the load torque and therefore, the motor will accelerate the load and bring it to point A. The point B is also stable point, but the point C is unstable. Here a decrease in speed will make the motor torque less than the load torque, consequently the motor will further decelerate. Mathematically, the steady state stability can be obtained as follows: at point A J

Tm = T

Speed

L

+ Bω and

dω =0 dt

(1.3)

Speed

Torque

Torque

Fig. 1.3 Speed/torque curves of motor load system

1.6 Load Dynamics and Stability

7

Assuming B to be negligible, a disturbance in supply, load, or any part of drive will change these values by small amounts ∆TM , ∆TL and ∆ω. The Eq.(1.1) can therefore, be written as (TM + ∆TM ) − (TL + ∆TL ) = J

d(ω + ∆ω) dt

(1.4)

d∆ω = ∆TM − ∆TL (1.5) dt If the deviations are small, the speed/torque curves of the motor and load can be assumed to be linear. Then dTL dTM ∆TM = ∆ω and ∆TL = ∆ω (1.6) dω dω Or

J

L where dTdωM and dT dω are derivatives of motor and load torque at the equilibrium point. The Eq.(1.5) therefore, can be written as dTL dTM d − J (∆ω) + ∆ω = 0 (1.7) dt dω dω

This is first order linear diﬀerential equation. If initial deviation in speed at t=0 is ∆ω0 then the solution of Eq. (1.7) results in −1 dT L dTM − ∆ω = ∆ω0 exp t (1.8) J dωM dω An operating point will be stable when ∆ω approaches zero as t approaches infinity. For this condition the exponent in Eq.(1.8) must be negative. Or dTM dTL > dω dω

(1.9)

Example 1.1 A dc shunt motor with the following speed–torque characteristics ω=200-0.1T, where T is the electromagnetic torque in N-m drives a load having speed torque relation as ω=2TL. Calculate the speed and torque at which the motor will operate and check its stability. Solution Here B=0, hence the equilibrium occurs at T = TL Or Or

ω 2 ω =190.476 rad/sec ω = 200 − 0.1

Now the torque at which the motor will operate is given by T=ω/2 Or T=190.476/2 = 95.238 Nm.

(1.10)

8

1 Introduction

Z

torque Fig. 1.4 Speed -torque curves of motor and load

dT L To check for stability of this point dT dω = 2 and dω = −10 , the point is therefore a stable point. The graphical representation of the system is shown in Figure 1.4.

1.7

Multi-quadrant Operation

An electric drive may be operated in one direction of rotation or both directions of rotations depending upon requirements. Similarly the motors generally operate in two modes -motoring and braking. In motoring mode it converts electrical energy into mechanical energy, and the torque developed is in the direction of motion. In braking mode it converts mechanical energy into electrical energy and the torque developed by the motor opposes the motion. The power developed by the motor is the product of torque and the speed. If the torque and speed are both positive, or negative, it is motoring operation. If the torque and the speed have opposite signs, the motor works in braking mode. The multi-quadrant operation of electric drives is shown in Figure 1.5. The I quadrant operation where torque and speed are both positive is called forward motoring. Similarly III quadrant operation is called reverse motoring as both torque and speed are negative. In II and IV

Forward braking

Z

Forward motoring

II

I T

Reverse motoring

III

IV

Reverse braking

Fig. 1.5 Multi-quadrant operation of Drives

1.8 Duty Cycle and Motor Rating

9

quadrant the torque and speed have diﬀerent directions and therefore the motor is working in braking mode. During motoring operation the power flows from the mains to the motor. In braking mode, if regenerative braking is used the power flows to the mains. In such cases the power electronic converter must be designed to handle power in both directions. In case of dynamic braking the kinetic energy of motor –load system is dissipated as heat in the rotor and braking resistor.

1.8

Duty Cycle and Motor Rating

The power rating of a motor to drive a particular load is selected on the basis of thermal loading of motor. The capacity of the motor must match the requirement of load and depending on the conditions of loading, the steady state temperature rise of the must be within permissible limits. The limits on temperature rise are decided by the type of insulation used in the motor. Selection of proper power rating of motor is important from the point of view of both economy and maintenance. The power of the motor specified by the manufacturer is for continuous duty loads. However there are large number of drives used for duties that are not continuous. There are diﬀerent patterns of load variation with time and these are categorized into eight major types .These are(i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Continuous duty. Short time duty. Intermittent periodic duty. Intermittent periodic duty with starting. Intermittent periodic duty with starting and braking. Continuous operation with periodic duty. Continuous operation with periodic duty with starting and braking. Continuous duty with periodic speed changes.

The load vs time curve of continuous duty load is shown in Fig. 1.6. The power rating of the motor is .selected on the basis of thermal loading of the motor. The thermal loading of the motor depends on the type of duty cycle as defined by the load/time curve. The current drawn by the motor depends on the magnitude of the load. The heat produced by the current depends on the magnitude of the current and the duration for which it is applied. Due to the heat generated inside the motor, its temperature rises . When the temperature of the motor is above the surrounding air, it starts dissipating heat to the atmosphere. Steady state temperature is reached when the heat produced inside the motor is equal to the heat dissipated to the atmosphere. This temperature rise must not be more than the permissible value for the class of insulating material used. For selecting the power rating of motor for continuous duty loads, the maximum continuous power demand of the load is obtained. A motor with next higher power rating from commercially available ratings is selected.

1 Introduction

load

load

10

Time Time

(i)

load

load

(ii)

Time

(iii) Time

load

(iv)

(v)

Time

Fig. 1.6 Load Versus Time

For short time duty loads, the motor is loaded for a small time compared to the heating time constant of the motor and is allowed to cool down before restarting. In such cases motor can be overloaded by a factor K (K>1) such that the maximum temperature rise is up to permissible limit. For intermittent periodic duty loads shown in Fig. 1.6 (iii), (iv) and (vi) the motor rating is calculated based on mean current or mean torque and is further derated to take into account the energy loss at starting and braking.

1.9

Problems

1. A motor is driving a load whose torque- speed relationship is given by TL = 0.10ω. The motor develops a constant torque T = 10 N-m. calculate the time taken for the speed to change from zero to steady state value. The moment of inertia of drive is J = 0.10 Kg-m2 . 2. A motor torque and load torque of a drive is given by: T = 150-0.628ω N-m, load torque TL = 100 N-m. Initially the drive is operating in steady state. If the load torque is changed to TL = -100 N-m. Calculate the final equilibrium speed.

References

11

3. Obtain the equilibrium points, when motor and load torques are specified by the following equations; √ T = -1-2ω and TL = −3 ω 4. A motor operating with a suitable control system develops a torque given by the relation T = aω -b . The motor drives a load for which the torque is given by TL = cω 2 , where a, b and c are positive real constants. Find the equilibrium speeds and the stability of equilibrium point.

References 1. Dubey, G.K.: Fundamentals of Electrical Drives. Narosa Publishing House, New Delhi (2001) 2. Rashid, M.H. (ed.): Power Electronics Handbook, ch. 26. Academic Press, London (2001) 3. Leonard, W.: Control of Electric Drives. Springer, New York (1985) 4. Slemon, G.R.: Electric Machines and Drives. Addison–Wesley, Reading (1992) 5. Van Wyk, J.D.: Power electronic converters for motion control. Proc. IEEE 82(8), 1164–1193 (1994) 6. Bose, B.K.: Power electronics – a technology review. Proc. IEEE 80, 1303–1334 (1992) 7. Bose, B.K.: Recent advances in power electronics. IEEE Trans. Power Electron. 7, 2–16 (1992) 8. Mohan, N.T., Undeland, T.M., Robbins, W.P.: Power Electronics, 3rd edn. Wiley, New York (1995) 9. Subramanyam, V.: Electric Drives: Concept and Applications. Tata McGraw Hill, New Delhi (1994) 10. Ahmad, M.: Industrial Drives. Macmillan India (1996)

Chapter 2

Modeling of Induction and Synchronous Machines

2.1

Induction Machine Theory

Induction motor is the most widely used ac motor in the industry. An induction motor like any other rotating machine consists of a stator (the fixed part) and a rotor (the moving part) separated by air gap. The stator contains electrical windings housed in axial slots. The induction machines used in industries are mainly three-phase, except for small power where single phase machines are common. Each phase on the stator has distributed winding, consisting of several coils distributed in number of slots. The distributed winding results in MMF due to the current in the winding to be stepped waveform similar to a sine wave. The MMF wave has its maximum value at the center of the winding. In three-phase machine the three windings have spatial displacement of 120 degrees between them. When balanced three phase currents are applied to these windings, the resultant MMF in the air gap has constant magnitude equal to 23 times the magnitude of one phase, and rotates at an angular speed of ω electrical radians per second. Here ω is the angular frequency of the supply current. The actual speed of rotation of magnetic field depends on the number of poles in the motor. This speed is known as synchronous speed of the motor and is given by sec ωs = 2πf/prad/

(2.1)

Where p is number of pole pairs. It can also be expressed as f/p rev/sec or 60f/p rev/min. If the rotor of an induction motor has a winding similar to the stator it is known as wound rotor machine. These windings are connected to slip rings mounted on the rotor. There are stationary brushes touching the slip rings through which external electrical connected. The wound rotor machines are used with external resistances connected to their rotor circuit at the time of starting to get higher starting torque. After the motor is started the slip are short circuited. M. Ahmad: High Performance AC Drives, Power Systems, pp. 13–45. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

14

2 Modeling of Induction and Synchronous Machines

Another type of rotor construction is known as squirrel cage type rotor. In this construction the rotor slots have bars of copper or Aluminium shorted together at each end of rotor by end rings. In normal running there is no difference between a cage type or wound rotor machine as for as there electrical characteristics are concerned. When the stator is energized from a three phase supply a rotating magnetic field is produced in the air gap. The magnetic flux from this field induces voltages in both the stator and rotor windings. The electromagnetic torque resulting from the interaction of the currents in the rotor circuit (since it is shorted) and the air gap flux, results in rotation of rotor. Since emf in the rotor can be induced only when there is a relative motion between air gap field and rotor, the rotor rotates in the same direction as the magnetic field, but it will not run at synchronous speed. An induction motor therefore always runs at a speed less than synchronous speed. The diﬀerence between rotor speed and synchronous speed is known as slip. The slip s is given by(2.2) s = (ωs − ωr )/ωs

2.2

Equivalent Circuit of Induction Motor

The steady state characteristics of induction machines can be derived from its equivalent circuit, which is similar to that of a transformer. In order to develop a per phase equivalent circuit of a three-phase machine, a wound rotor motor as shown in Fig. 2.1 is considered here. In case of a squirrel cage motor, the rotor circuit can be replaced by an equivalent three-phase winding. When three-phase balanced voltages are applied to the stator, the currents flow in them. These currents produce a rotating magnetic field in the air gap. Similarly the currents induced in the rotor winding also produce rotating magnetic field. These fields rotate at the same speed in the air gap, and a resultant air gap field is produced that rotates at synchronous speed.

Fig. 2.1 3-phase induction motor

2.2 Equivalent Circuit of Induction Motor

15

Fig. 2.2 Equivalent circuit

The resultant field induces voltages in both stator (at supply frequency f), and rotor windings (at slip frequency f2). The equivalent circuit, therefore is identical to that of a transformer, and is shown in Figure 2.2. Here Rs is the stator winding resistance, Ls is self inductance of stator, Lr is self inductance of rotor winding referred to stator, Rr is rotor resistance referred to stator Xm is magnetizing reactance and Rm is resistance corresponding to core loss, and s = slip. The steady state performance equations are the air gap power, mechanical and shaft output power, and electromagnetic torque. The real power transmitted to the air gap Pa of the motor is the diﬀerence between total power input Pi and the stator copper loss and is given as Pa = Pi − 3I2s Rs

(2.3)

If core losses are neglected the air gap power is Pa = 3I2r

Rr s

(2.4)

It can be bifurcated as Pa = 3I2r Rr + 3I2r Rr (1−S S)

(2.5)

Thus the air gap power is the sum of rotor copper losses and the power converted as output in mechanical form. The mechanical power output Pm is therefore given as Pm = 3I2r Rr (1−S S)

Watts

(2.6)

The electromagnetic torque Te is related to the mechanical power as Pm = Te ωm OR torque Te = 3I2r Rr

(1 − s) 1 = 3I2r Rr sωm sωs

Where ωm is the rotor speed in radians/second.

(2.7)

16

2 Modeling of Induction and Synchronous Machines

Fig. 2.3

If the shunt branch (without the resistance Rm ) is shifted to the input, an approximate equivalent circuit as shown figure 2.3 is obtained. This approximation is justified and the performance of the machine predicted from approximate equivalent circuit is within 5 percent from the actual performance. From this circuit the magnitude of the rotor current is obtained as Vs Ir = (Rs + Rr /s)2 + ω2s(Ls + Lr )2

(2.8)

and substituting the value in equation 2.7, the torque equation is obtained Te =

2.3

V2s 3Rr 2 sωs (Rs + Rr /s) + ω2s (Ls + Lr )2

(2.9)

Dynamic Model of a Two-Phase Induction Machine

The steady state characteristics of an induction machine can be derived from its equivalent circuit. In adjustable speed drives feed back is generally used, and for fast response the study of dynamic behavior involving transients in voltages, currents or torque is required. A dynamic model of the machine is essential for high performance drives and is developed here. The dynamic model of an induction machine for a two phase motor is considered first, and then the equivalence between three phase and two phase machine models is derived. A two-phase induction machine has two windings on the stator displaced from each other by 90 electrical degrees in space. The rotor is also considered to have equivalent two phase windings as shown in Figure 2.4. The stator windings are placed along the d and q axis, and rotor windings are placed on α, and β axes. At any instant the rotor winding α, is at an angle θ from the stator d axis. The terminal voltages of the stator and rotor windings can be expressed as the sum of voltage drops in resistances and rate of change of flux linkages. The flux linkages are the products of currents and inductances (self and mutual). In a two-phase machine there are four windings (circuits) and each circuit has a self and mutual inductance. The self inductance of phases d, q, α and β consists of inductance due to

2.3 Dynamic Model of a Two-Phase Induction Machine

17

Fig. 2.4

main flux and the leakage flux, and does not depend on rotor position. These inductances can be written asLαα = Lββ = Lrr

(2.10)

Ldd = Lqq = Ls

(2.11)

The mutual inductance between the stator windings d and q, and between rotor windings α and β is zero as these windings are displaced from each other by 90 degrees. Therefore, Lαβ = Lβα = 0

(2.12)

Ldq = Lqd = 0

(2.13)

The mutual inductances between the stator and rotor windings are function of the rotor position θr . Since the distribution of MMF in the air gap is sinusoidal, the mutual inductances vary as the cosine function of θr . Here θr is the electrical angle between rotor α axis and stator d axis. Therefore Lαd = Ldα = Lsr cosθr

(2.14)

Lβd = Ldβ = Lsr sinθr

(2.15)

Lαq = Lqα = Lsr sinθr

(2.16)

Lβq = Lqβ = −Lsr cosθr

(2.17)

Where Lsr is the peak value of the mutual inductance between stator and rotor winding. Vds , Vqs , Vα , and Vβ are the terminal voltages of stator d and q-axis windings, and rotor α and β windings respectively. The currents in

18

2 Modeling of Induction and Synchronous Machines

these windings are, ids , iqs , iα and iβ respectively. The voltage equations in terms of voltage drops and flux linkages can be written asVds = (Rs + Ls p)ids + Lsr p(iαCosθr ) + Lsr p(iβ Sinϑr )

(2.18)

Vqs = (Rs + Ls p)iqs + Lsr p(iα Sinθr ) − Lsr p(iβCosθr )

(2.19)

Vα = (Rrr + Lrr p)iα + Lsr p(ids Cosθr ) + Lsr p(iqs Sinθr )

(2.20)

Vβ = (Rrr + Lrr p)iβ + Lsr p(ids Sinθr ) − Lsr p(iqs Cosθr )

(2.21)

Here the stator winding resistances Rd = Rq = Rs

(2.22)

Rα = Rβ = Rrr

(2.23)

and rotor resistances The solution of these equations (Eq. 2.18–Eq. 2.21) is complicated by the fact that inductances are functions of position of rotor. It is possible to simplify these equations using transformation to replace time dependent inductances by constant values.

2.3.1

Transformation to Obtain Constant Inductances

In order to replace time varying inductances with constant inductances mathematical transformation to remove the terms cos θr and sin θr can be used. H.C. Stanley in 1930s showed that time varying inductances in the voltage equations of an induction machine due to electric circuits in relative motion can be eliminated by transforming the rotor variables to variables associated with fictitious stationary winding. For this purpose the rotor is assumed to have a fictitious winding on d and q axis which are fixed to the stator (Fig 2.5). These windings have the same number of turns as the actual rotor windings and produce the same mmf. If the fictitious rotor currents are iqrr and irdr , then the transformation as given below can be used. idrr iα cosθr sinθr = (2.24) iqrr iβ sinθr −cosθr Substituting these values, the voltage equations (2.18 to 2.21) can be written as – ⎤ ⎡ ⎡ ⎤ ⎡ ⎤ 0 Lsr p 0 Rs + Ls p Vqs iqs ⎢ ⎢ Vds ⎥ ⎥ ⎢ 0 Rs + Ls p 0 Lsr p ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ ids ⎥ (2.25) ⎣ Vqrr ⎦ = ⎣ Lsr p −Lsr ω Rrr + Lrr p −Lrr ω ⎦ ⎣ iqrr ⎦ Vdrr Lsr p Lrr ω Rrr + Lrr p Lsr ω idrr

2.3 Dynamic Model of a Two-Phase Induction Machine

19

Fig. 2.5

r Where ω = dθ dt . The Equation 2.25 has only constant inductance terms. However the rotor quantities can be referred to the stator side as is normally done to obtain steady state equivalent circuit. If rotor voltage is made equal to stator voltage, the rotor and stator windings can be physically connected. Since the performance of this winding has to be same as actual winding the following parameter changes are required.

If a =

Stator turns per phase Rotor turns per phase

(2.26)

Then the referred quantities are as followsRr = a2 Rrr

(2.27)

Lr = a2 Lrr

(2.28)

iqr =

iqrr a

(2.29)

idr =

idrr a

(2.30)

vqr = avqrr

(2.31)

vdr = avdrrr

(2.32)

The magnetizing inductance of the stator and the mutual inductance now have the following relation. (2.33) Lm = aLsr Substituting new values from equations (2.27 - 2.33) into equation (2.25) the simplified dynamic equations for two-phase machine are obtained as

20

2 Modeling of Induction and Synchronous Machines

⎡

⎤ ⎡ ⎤⎡ ⎤ Vqs 0 Lm p 0 Rs + Ls p iqs ⎢ Vds ⎥ ⎢ ⎢ ⎥ 0 Rs + Ls p 0 Lm p ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ids ⎥ ⎣ Vqr ⎦ ⎣ Lm p −Lm ω Rrr + Lrr p −Lrr ω ⎦ ⎣ iqr ⎦ Vdr Lm p Lrr ω Rrr + Lrr p Lm ω idr

(2.34)

This equation has some impedance elements which depend on rotor speed. For constant rotor speed the machine is in steady state and equation 2.34 is linear.

2.3.2

Dynamic Model of Three-Phase Machine

The equation 2.34 gives the dynamic model of a two-phase motor. However, three-phase induction machines are more common: two-phase machines are rarely used in industries. A three-phase machine can be represented by an equivalent two-phase machine using axis transformation. A three-phase machine and its equivalent two-phase machine is shown in figure 2.6. Here ds and qs are the direct and quadrature axis of the stator and dr and qr are

Zr

(a)

(b)

Fig. 2.6

Zm

2.3 Dynamic Model of a Two-Phase Induction Machine

21

direct and quadrature axis of the rotor. The axis transformation relates current or voltages on a-b-c axes with the currents or voltages on d-q-0 axes. This transformation, that relates the voltages, currents, and flux linkages associated with the stator winding, with variables associated with fictitious windings on d and q axis on the rotor, was first proposed by R. H. Park in 1920s. With such a transformation called Park’s transformation time varying inductances can be eliminated. If the qs axis is lagging the ‘a ’ axis by θc , then the currents in d-q-0 axes are related to the currents in a-b-c axes by the Park’s transformation given below: The superscript ‘s’ is used to indicate that the d-q-0 axis is on the stator. ⎡ ⎤⎡ s ⎤ ⎡ s ⎤ 2π cosθc cos(θc − 2π iqs ias 3 ) cos(θc + 3 ) ⎣ is ⎦ = 2 ⎣ sinθc sin(θc − 2π ) sin(θc + 2π ) ⎦ ⎣ is ⎦ (2.35) bs ds 3 3 3 s 1 1 1 i is0s cs 2 2 2 The current i0 is present only if a-b-c currents are unbalanced, and it does not produce a resultant magnetic field. The equation 2.33 can be written in compact form as(2.36) isdq0 = [Tsabc ] isabc Similarly for obtaining iabc from iqd0 inverse Park’s transformation may be used Or (2.37) isabc = [Tsabc ]−1 isdq0 where [Tsabc ]−1

⎡

⎤ sinθc 1 cosθc 2π ⎦ = ⎣ cos(θc − 2π 3 ) sin(θc − 3 ) 1 2π ) 1 cos(θc + 3 ) sin(θc + 2π 3

(2.38)

The transformation as given above for currents is also applicable for voltages and flux linkages. If θc is made equal to 0, so that qs axis is aligned with the phase ‘a’ axis, and the system is balanced (without neutral connection), then only d and q components are present. The transformation can be written as isa = isqs

(2.39)

⎤ ⎡ ⎤ 1⎤⎡ 1 −√12 − isqs ias √2 ⎣ is ⎦ = 2 ⎣ 0 − 3 3 ⎦ ⎣ ibs ⎦ ds 3 2 2 1 1 1 ics 0 2 2 2

(2.40)

⎡

For balanced three-phase system ias + ibs + ics = 0, or is0s = 0;

(2.41)

22

2 Modeling of Induction and Synchronous Machines

And

1 1 isds = − √ isbs + √ iscs 3 3

(2.42)

The voltage equation in terms of flux linkages can be written as in equation (2.34). From these equations, for balanced system the equivalence between three-phase induction machine and two-phase induction machine is established. However, if the system is unbalanced, is0s = 0 then equation 2.34 will have two more terms as followsv0s = (Rs + Llsp)i0s

(2.43)

v0r = (Rr + Llr p)i0r

(2.44)

where, Lls and Llr are leakage inductances of stator and rotor.

2.4

Selection of Reference Frame

The dynamic performance of AC machines is complex due to the fact that the rotor windings and stator windings have relative motion between them. To simplify the representation of a three phase machine it was replaced by an equivalent two-phase machine in section 2.3.2. However, the time varying inductances still make the model complex. It will be shown here and in later sections that a much simplification in the model of ac machines results by selecting a reference frame and represent all the variables on that reference frame. The selection of reference frame is based on the requirement of motor control strategy. The basic approach in obtaining simplified model involves the transformation of stator and rotor equations to a common reference frame represented by d-q axis. The reference frame can be revolving or stationary, can be attached to the stator or rotor. The possible choices for d-q axes frames in ac machines are, The d-q axis fixed to • the stator or stationary frame (stator reference frame) • the rotor and rotates at the speed of rotor (rotor reference frame) • rotor flux vector and rotates at the speed of rotor flux (rotor field reference frame) • a frame rotating at synchronous speed. (synchronously rotating reference frame) or field frame It is also possible to attach the d-q axis to a reference frame rotating at an arbitrary speed. If the reference frame is considered to be rotating at an arbitrary speed, such a reference frame is known as arbitrary reference frame. The models based on a particular frame of reference provide simplification of the system equations and are very commonly used in high performance control of ac drives.

2.4 Selection of Reference Frame

qg

23

qs

dg

Zg ds

v ds

i ds

Fig. 2.7 Position of stationary and general reference frames

2.4.1

Modeling in Arbitrary Reference Frame

The modeling of an induction motor in arbitrary reference frame is the most general. Models in other reference frames can be obtained as a particular case of this general model. For obtaining the model in arbitrary reference frame (general reference frame), the relationship between ds and qs axes on the stator and the general reference frame denoted by dg and qg can be obtained as follows. Figure 2.7 shows the relative position of ds and qs axes and dg and qg axes. Here dg and qg axes are assumed to be rotating at an arbitrary speed ωg . It is assumed that the windings on both the reference frames have equal number of turns. As shown in figure, qg is leading the qs axis by an angle θg . The general reference frame currents can be resolved on the d and q axes, and relationship between the currents is given by g

isqds = [Tg ] iqds

(2.45)

Here the system is assumed to be balanced, and has only d and q components. cosθg sinθg g (2.46) [T ] = −sinθg cosθg The reference frame is rotating with the speed given by θ˙ g = ωg

(2.47)

Likewise, the stator voltages can also be transformed as g

vqds = [Tg ] vqds

(2.48)

For rotor quantities again the same assumptions are made. The fictitious rotor winding on d and q axis on rotor has same number of turns as the winding on d –q axis on the arbitrary reference frame. The angle between q

24

2 Modeling of Induction and Synchronous Machines

axis of arbitrary reference and the rotor q axis is, θg . Since the rotor speed is ωr , the relative speed between d-q axes on rotor and d-q axes on arbitrary reference frame is (ωg − ωr ). The relationship between the rotor currents transformed from arbitrary reference frame is given by g irqdr = [Tg ] iqdr (2.49) and voltages can be written as g

vrqdr = [Tg ] vqdr

(2.50)

Similar equation can be written using flux linkages as variables. Since the flux linkages are continuous even if the currents and voltages are discontinuous, the equations in flux linkages result in great reduction in number of variables. The stator and rotor flux linkages in arbitrary reference frame are defined as λgqs = Ls igqs + Lm igqr g

g

g

λds = Ls ids + Lm idr λgqr = Lr igqr + Lmigqs

(2.51)

g

λgdr = Lr idr + Lmigds The q axis stator voltage in arbitrary reference frame can be written as g

g

vgqs = Rs igqs + ωg (Ls ids + Lmidr ) + Lm

d d g iqr + Ls igqs dt dt

(2.52)

In simplified form it can be written as d g λ dt qs

g

vgqs = Rs igqs + ωg λds +

(2.53)

Similarly direct axis voltage vds can be written as g

g

vds = Rs ids −ωg λgqs +

d g λ dt ds

(2.54)

For rotor the q axis and d axis voltages can be written as – g

vgqr = Rr igqr + (ωg − ωr )λdr + and

d g λ dt qr

(2.55)

d g λ (2.56) dt dr Substituting the values of flux linkages from equations 2.51, the induction motor model in arbitrary reference frame is obtained. The model is very g

vgdrr = Rr idr − (ωg − ωr )λgqr +

2.5 Models in Other Reference Frames

25

useful for vector control and direct torque control applications and is given below. ⎡ g ⎤ ⎡ ⎤⎡ g ⎤ vqs ωg L s Lm p ωg L m Rs + Ls p iqs ⎢ vg ⎥ ⎢ ⎥ ⎢ ig ⎥ −ω L R + L p ω L L p g s s s g m m ⎢ ds ⎥ ⎢ ⎥ ⎢ ds g ⎥ ⎣ vgqr ⎦ = ⎣ Lm p (ωg −ωr )Lm Rr + Lr p (ωg −ωr )p ⎦ ⎣ iqr ⎦ g Lm p −(ωg −ωr )Lr Rr + Lr p −(ωg − ωr )Lm vdr igdr (2.57) g For single fed wound rotor machine and for squirrel cage machines vdr = g vqr = 0. The torque developed by the motor in d-q component can be obtained from Eq.(2.57), written in compact form as [v] = [R][i] + [Lp][i] + [Gωr ][i] + [Fωg ][i]

(2.58)

The instantaneous power input to the motor is obtained by multiplying voltage equation with the current as [i]t [v] = power input. The Eq.(2.58) indicates that the term associated with [R] is power loss, while [Lp} is associated with rate of change of stored magnetic energy, and the term [Fωg ] is fictitious power associated with mathematical manipulation of equations. Thus the only term associated with real power is [i]t [Gωr ][i] which is the power input minus losses or is the electrical power output of the motor. The electromagnetic torque Te = P2 [i]t [Gωr ][i]. The term P2 relates the mechanical radians with electrical radians. Or 3 P g g (2.59) (λds igqs − λgqs ids ) Te = 2 2 Here the term 32 is introduced to make the power equivalence of three-phase and two-phase machine. The model in steady state can be obtained by putting p=jωg in equation (2.57).

2.5

Models in Other Reference Frames

The most common reference frame models used are [1, 2, 3, 4]: 1. Stator reference frame model; 2. Rotor reference frame model; 3. Synchronously rotating reference frame model. These models are derived here.

2.5.1

Stator Reference Frame Model

The dynamic model of induction motor in stator reference frame is also known as Stanley model. This model is useful for finding the performance of stator

26

2 Modeling of Induction and Synchronous Machines

controlled drives. For deriving stator reference frame model the speed of reference frame is taken as 0, i.e. ωg = 0. Substituting the value of ωg = 0 in equation (2.57) the equations for voltages can be written as: ⎤ ⎡ ⎤⎡ s ⎤ iqs vsqs Rs + Ls p 0 Lm p 0 ⎢ vs ⎥ ⎢ ⎥ ⎢ is ⎥ + L p 0 L p 0 R s s m ⎥ ⎢ ds ⎥ ⎢ ds ⎥ = ⎢ ⎣ 0 ⎦ ⎣ Lm p −ωr Lm Rr + Lr p −ωr Lr ⎦ ⎣ isqr ⎦ ωr L m Lm p ωr Lr Rr + Lr p 0 isdr ⎡

(2.60)

As can be seen from this equation, in stationary frame the variables appear as sine waves in steady state with sinusoidal inputs. The torque equation is 3 Te = 2

2.5.2

P (λsds isqs − λsqs isds ) 2

(2.61)

Rotor Reference Frame Model

The dynamic model in rotor reference frame can be obtained by substituting the value of ωr = ωg in equation (2.57) models are used when the induction motor is controlled from the rotor side. Here the speed of rotor reference frame is (2.62) ωr = ωg Then the voltage equation can be written as : ⎡ r ⎤ ⎡ ⎤⎡ r ⎤ iqs vqs Lm p ωr L m Rs + Ls p ωr Ls ⎢ vr ⎥ ⎢ −ωr Ls Rs + Ls p −ωr Lm Lm p ⎥ ⎢ ir ⎥ ⎥ ⎥ ⎢ ds ⎢ ds ⎥ = ⎢ ⎦ ⎣ irqr ⎦ . ⎣ 0 ⎦ ⎣ Lm p 0 Rr + Lr p 0 0 Rr + Lr p 0 Lm p 0 irdr

(2.63)

The torque is given by – Te =

2.5.3

3 2

P (λrdr irqr − λrqr irdr ) 2

(2.64)

Synchronously Rotating Reference Frame Model

If the reference frame de −qe is assumed to be rotating at synchronous speed equal to the stator supply frequency ωg = ωs

(2.65)

2.6 Space Phasor Model

27

The model in synchronous reference frame is obtained by substituting equation 2.65, in the general equation and using superscript ‘e’ as given below: ⎤⎡ e ⎤ ⎡ e ⎤ ⎡ iqs vqs ωs L s Lm p ωs L m Rs + Lsp e ⎥ ⎥ ⎢ ⎢ ve ⎥ ⎢ i −ω L R + L p −ω L L p s s s s s m m ⎥ ⎥ ⎢ ds ⎥ =⎢ ⎢ ds e ⎣ vqr ⎦ ⎣ Lm p (ωs −ωr )Lm Rr + Lr p (ωs −ωr )Lr ⎦ ⎣ ieqr ⎦ −(ωs −ωr )Lm Lm p −(ωs −ωr )PLr Rr + Lr p iedr vedr (2.66) The model obtained in synchronous reference frame transforms the sinusoidal quantities in a-b-c frame to dc quantities in d-q-0 frame. These models are very useful for vector control schemes of induction motor. All the models given above are in d-q-0 frame. To obtain the model in a-b-c variables the Park’s transformation as given in equation (2.31) may be used. In induction motor control the most important equation is the electromagnetic torque equation, which can be obtained in any reference frame using the quantities of that frame: 3P Lm (iqs idr −ids iqr )(N.m) (2.67) 22 Similarly the equation for electromagnetic torque in terms of flux linkages in any reference frame can be obtained using the quantities in that frame and is given by: 3P (iqs λds −ids λqs ) Te = (2.68) 22 Te =

2.6

Space Phasor Model

The space phasor model of ac machine can be developed [5, 6, 7, 8] using the concept of “space vectors”. In ac machines the stator has a distributed winding with several coils distributed around the periphery. The MMF distribution in space therefore, has a stepped waveform, which can be approximated to a sine wave. For dc current flowing in the ‘a’ phase of the stator winding, there is sinusoidal distribution of the MMF and the flux density in space. The peak value of this flux density is along the axis of the coil which is considered here as reference axis (α = 0). However, if the MMF wave of phase ‘a’ of stator is described by an equivalent current phasor ¯ias ; it will be assumed to have a magnitude of Ias and direction along the axis of the winding (α = 0). Since the distribution of MMF is sinusoidal the eﬀect of this current at an angle α will be Ias cosα. In three phase induction motor the three phase windings are identical with 120 degree phase displacement between them. Thus if axis of phase ‘a’ current is taken as reference, the current space phasor for phase ‘b’ and c will have +120 degree and +240 degrees (–120 degree) from phase ‘a’. If dc current is flowing in all the three windings, the current space vectors will have their

28

2 Modeling of Induction and Synchronous Machines

I bs

I as

I cs Fig. 2.8 Current space vectors with dc currents in a, b and c windings

positions as shown in Figure 2.8. The combined stator currents as given by Equation (2.69) is also a current vector, called resultant stator current space vector ¯is , which can be obtained as ¯is = Ias ∠0 + Ibs∠ 2π + Ics ∠ − 2π = {Ias + aIbs + a2 Ics } 3 3

(2.69)

where, 2π 4π = Re [a]; cos = Re [a2 ] 3 3 This current vector can be resolved along d-q axes as 2π

a = ej 3 ; cos

¯is = ¯ids + j¯iqs

(2.70)

(2.71)

The current space vector of a three-phase machine has a fixed direction in space for each phase that is along the axis of the magnetic flux density produced by the MMF of respective winding. The magnitude of each phase current space vector is the magnitude of the current, and the angle is the angle of the axis of the phase winding with the reference axis. If instead of dc, ac current is applied to the three phase windings of the stator, the magnitude of the current space vector will be varying sinusoidally with time. In order to obtain the resultant current space vector, the time variation of the current is also considered. Suppose

ias =

√ 2Is cosωs t;

ibs =

√ √ 2π 4π 2Is cos(ωs t− ); and ics = 2Is cos(ωs t − ) 3 3 (2.72)

2.6 Space Phasor Model

29

Then the resultant current space vector is given by: ¯is = ias ej0 + ibs ej( 2π3 ) + ics ej( 4π3 )

=

¯isd + j¯isq

(2.73)

√ ¯is = 3 2Is ejωs t (2.74) 2 This means√that the stator current space vector has a constant magnitude equal to 23 2Is and it rotates with a constant angular speed equal to ωs rad/sec. The current, voltage or flux space phasors are the resultant stator or rotor current, voltage or flux quantities obtained by taking vector sum of these quantities in appropriate axes frame. Similarly the complex rotor current phasor ¯ir in a-b-c frame with phase ‘a’ as reference can be written as: Or

¯ir = 2{ iar + aibr + a2 icr} 3

(2.75)

In a similar way sinusoidal flux density wave can be described by a space vector. It is however, preferred to choose the corresponding distribution of the flux linkage with a particular three-phase winding as the characterizing quantity. For example the flux linkage space vector λ¯ s of the stator winding is written as ¯ s = 2 (λas + λbs + λ λ cs) 3 ¯ s = ls¯is + lsr¯ir λ

(2.76) (2.77)

Where ls is the three-phase stator winding inductance, and lsr is the threephase mutual inductance between stator and rotor windings. The three phase inductance of a winding is defined here as inductance of a winding corresponding to flux linkage due to currents in all the three phase windings. For example ls = λiasas , where λas = flux linkage of phase ‘a’ winding due to currents in phase a, b and c and can be written as λas = (Ls ias + Lab ibs + Lac ics )

(2.78)

Where Lab = Lac is the mutual inductance between stator phase windings. ¯ s induces a voltage in the stator The rotating stator flux linkage vector λ windings given by ¯s dλ (2.79) v¯ s = dt It is related to the stator winding phase voltages by the equation v¯ s =

2 vas + avbs + a2 vcs 3

(2.80)

30

2 Modeling of Induction and Synchronous Machines

Similarly rotor voltage space vector v¯ r can be written as: v¯ r =

2 var + avbr + a2 vcr 3

(2.81)

The three-phase stator voltages in equivalent d-q-0 stationary frame are obtained as ⎡ ⎤ ⎡ ⎤ 1⎤ ⎡ 1 −√12 − vds vas √2 ⎣ vqs ⎦ = ⎣ 0 − 3 3 ⎦ ⎣ vbs ⎦ (2.82) 2 2 v0s v 1 1 1 cs Again for balanced system vas + vbs + vcs = 0

(2.83)

The resultant voltage space vector v¯ s can be expressed in rectangular coordinates as v¯ s = Vs cosθ + jVs sinθ where, Vs is the magnitude of the resultant voltage space vector and θ is the angle with a reference axis. Suppose we take d axis as the reference, then v¯ s = vds + jvqs

(2.84)

The space vector voltages, currents, and flux linkages have so far been represented in their respective coordinates (stator for stator, rotor for rotor). For proper representation of an ac machine in space phasor form all these quantities have to be represented on common frame. For example, if rotor reference frame and stator reference frame are in alignment, then ¯ir will be same in stator reference frame also. However, if at any instant of time, rotor reference frame makes an angle θr with the stationary stator reference frame, then the rotor current space vector ¯ir in stator reference frame can be written as isr = (irdr + jirqr )(cosθr + jsinθr )

Or

isr

(2.85)

= (isdr + jisqr ) = irdr cosθr −irqr sinθr + j(irdr sinθr + irqr cosθr ) OR

isdr isqr

=

cosθr −sinθr sinθr cosθr

irdr irqr

(2.86)

(2.87)

r Where ωr = dθ dt If all the quantities are transferred to a general reference frame rotating at the speed of ωg , then the quantities on stator frame are related to general frame by¯ g ejθg , ¯is = ¯ig ejθg ¯s =λ (2.88) λ s s s s

2.6 Space Phasor Model

31

For quantities in rotor frame transferred to general frame the equations are ¯ g ej(θg −θr ) ¯r =λ λ r r

(2.89)

The voltage equations on general reference frame therefore can be written using space vectors as : v¯ s = rs¯is +

Here

¯s dλ ¯s + jωg λ dt

(2.90)

¯r dλ ¯r + j(ωg −ωr )λ v¯ r = rr¯ir + dt ¯ r = lr¯ir + lsr¯is ¯ s = ls¯is + lsr¯ir and λ λ

(2.91) (2.92)

Now if d-q frame is used instead of a-b-c, then the space phasors can be written as : v¯ s = vds + jvqs

¯is = ids + jqs

;

v¯ r = vdr + jvqr

;

¯ir = idr + jiqr ;

¯ s = λds + jλqs λ

;

¯ r = λdr + jλqr λ

(2.93)

In dq0 frame the equations can be obtained as : vds = rs .ids +

dλds −ωg .λqs dt

vqs = rs .iqs +

dλqs −ωg .λds dt

dλdr −(ωg −ω.r ).λqr = 0 dt dλqr + (ωg −ω.r ).λdr = 0 vqr = rr .iqr + dt vdr = rr .idr +

Te =

(2.94) (2.95) (2.96)

3P 2 2 (λd iqs −λq ids )

3P Lm (iqs idr −ids iqr ) (2.97) 22 Using Park’s transformation the equations in a-b-c frame can be obtained. For unbalance system v0s andv0r may also be required. The d-q-0 model of the induction machine operates with real (not complex) variables. The complex variable or space phasor model and d-q-0 models are equivalent as they are based on the same assumptions. =

Example 2.1 Develop a dynamic model of an induction motor in the synchronous reference frame and rotor reference frame. The motor has the following parameters.

32

2 Modeling of Induction and Synchronous Machines

5hp, 200V, 3-phase, 50 Hz, 4 –pole, star connected, 1400 rpm. Rs = 0.277Ω

Rr = 0.183Ω, Lm = 0.0538HLs = 0.0553HLr = 0.056H

Eﬀective stator to rotor turn ratio a=3. Solution: The applied voltages are 200 √ 2sinωt = 163.3sinωt = vm sinωt vas = √ × 3 Here ω = ωs = 2πfs

2π ) 3 2π vcs = 163.3sin(ωs t + ) 3 The transformation from abc to dq0 is obtained by substituting θc = θs = ωs t ⎡ ⎤ ⎡ e ⎤ ⎤⎡ 2π cosθs cos(θs − 2π vqs vas 3 ) cos(θs + 3 ) 2 ⎣ ve ⎦ = ⎣ sinθs sin(θs − 2π ) sin(θs + 2π ) ⎦ ⎣ vbs ⎦ (2.98) ds 3 3 3 1 1 1 v v0 cs 2 2 2 vbs = 163.3sin(ωs t−

Substituting these values in the equation we get, veqs = 0volts and veds = 163.3volts and v0 = 21 (vas + vbs + vcs ) = 0 Since the stator voltages in synchronous frame are dc, the currents will also be dc. The motor model in synchronous reference frame therefore is given by ⎡ ⎤ ⎡ ⎤⎡ e ⎤ iqs ωs L s 0 ωs L m 0 Rs ⎢ vm ⎥ ⎢ ⎥ ⎢ ie ⎥ −ω L R −ω L 0 s s s s m ⎢ ⎥ =⎢ ⎥ ⎢ ds ⎥ ⎣ 0 ⎦ ⎣ Rr (ωs −ωr )Lr ⎦ ⎣ ieqr ⎦ 0 (ωs −ωr )Lm 0 −(ωs − ωr )Lm 0 −(ωs −ωr )Lm Rr iedr (2.99) Substituting the values of impedance elements and ωr = 1400rpm in the above equation the currents in synchronous frame are obtained as: ⎡ ⎤ ⎡ e ⎤ iqs −11.13A ⎢ 17.22A ⎥ ⎢ ie ⎥ −1 ⎥ ⎥ ⎢ ds = ⎢ ⎣ −2.04A ⎦ ⎣ ieqr ⎦ = [V] [Z] −17.52A iedr The electromagnetic torque is given by Te =

s s 3P s s 2 2 Lm (iqs idr −ids iqr )

= 26.9 N-m

2.6 Space Phasor Model

33

The actual machine currents are obtained by taking the inverse Park’s transformation as: iabc = [Tabc ]−1 iqd0 Or ias = 17.1sin(ωst−0.57) ibs = 17.1sin(ωs t − 2.67) ics = 17.1sin(ωs t + 1.52) As can be seen from the result the three-phase currents are balanced. (b) Rotor reference frame model Here θs = (ωs − ωr )t = ωsl t=2π(6.66t) ⎤ ⎤ ⎡ ⎡ r ⎤ ⎡ 2π cosθs cos(θs − 2π vqs vas 3 ) cos(θs + 3 ) ⎣ vr ⎦ = ⎣ sinθs sin(θs − 2π ) sin(θs + 2π ) ⎦ ⎣ vbs ⎦ ds 3 3 1 1 1 vcs v0 2 2 2 or

⎤ ⎤ ⎡ ⎤ ⎡ vrqs 0 − jvm vm sin(ωs −ωr )t ⎣ vr ⎦ = ⎣ vm cos(ωs −ωr )t ⎦ = ⎣ vm + j0 ⎦ ds 0 0 v0 ⎡

Since in rotor reference frame the stator voltages appear the slip frequency, p=jωsl ⎡ r ⎤ ⎡ ⎤⎡ r ⎤ iqs vqs Rs + Ls p ωr Ls Lm p ωr L m ⎢ vr ⎥ ⎢ −ωr Ls Rs + Ls p −ωr Lm Lm p ⎥ ⎢ ir ⎥ ds ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ds ⎣ 0 ⎦ = ⎣ Lm p ⎦ ⎣ irqr ⎦ 0 Rr + Lr p 0 0 Lm p 0 Rr + Lr p 0 irdr substituting the values obtained for voltages and the system parameters the currents are obtained as ⎡ r ⎤ ⎡ ⎤ iqs 17.1sin(ωsl t − 0.58) ⎢ ir ⎥ ⎢ 17.1cos(ωsl t − 0.58 ⎥ ⎢ ds ⎥ ⎢ ⎥ ⎣ irqr ⎦ = ⎣ 14.7sin(ωsl t − 3.04 ⎦ 14.7cos(ωsl t − 3.04 irdr The actual phase currents can be obtained by inverse transformation iabc = [Trabc ]−1 iqd0 ⎤ ⎡ cosθs sinθs ias ⎣ ibs ⎦ = ⎣ cos(θs − 2π ) sin(θs − 2π ) 3 3 2π ics cos(θs + 2π 3 ) sin(θs + 3 ) ⎡

⎤⎡ r ⎤ 1 iqs 1 ⎦ ⎣ irds ⎦ 0 1

34

2.7

2 Modeling of Induction and Synchronous Machines

Speed Control of Induction Motor

In order to obtain high performance, and fast dynamic response in induction motors, it is important to develop appropriate control schemes. In separately excited dc machine, fast transient response is obtained by maintaining the flux constant, and controlling the torque by controlling the armature current. In order to achieve independent control of flux and torque in induction machines, the stator (or rotor) flux linkages phasor is maintained constant in its magnitude and its phase is stationary with respect to current phasor [6, 7]. As discussed earlier in this chapter, there are three distinct flux space phasors in the induction machine: λm -air gap flux, λs -stator flux, and λr - rotor flux. Their relationship with current is given by

¯ m = Lm . ¯is + ¯ir λ

;

¯ s = Ls¯is + Lm¯ir and λ ¯ r = Lr¯ir + Lm¯is λ

(2.100)

Vector control can be performed with respect to any of these. flux space phasors by attaching the reference system d -axis to the direction of respective flux phasor. These schemes are described in the chapter 3.

2.8

State Space Model

The state-space model of induction machine is important for study of transient analysis using simulation on computer. In general rotating reference frame is selected and electrical variables are either current or flux, or both. If flux linkages are selected as main variables, the state space equations in rotating frame can be obtained as follows. If equations (2.90) are considered for voltages in d-q axis, then these equations are repeated here. vds = rs .ids +

dλds −ωg .λqs dt

vqs = rs .iqs +

dλqs −ωg .λds dt

(2.101)

dλdr −(ωg −ω.r ).λqr = 0 dt dλqr + (ωg −ωr )..λdr = 0 (2.102) vqr = rr .iqr + dt The flux linkage expressions in terms of the currents can be written as vdr = rr .idr +

λqs = Ls iqs + Lm (iqs + iqr ) λds = Ls ids + Lm (ids + idr )

2.8 State Space Model

35

λqr = Lr iqr + Lm (iqs + iqr )

(2.103)

λdr = Lr idr + Lm (ids + idr ) These equations (2.103) can be rearranged in form of state equations ˙ = [A] [X] + [B] [U] X

(2.104)

T [X] = λds λqs λdr λqr

(2.105)

T [U] = vds vqds vdr vqr

(2.106)

where [X] is the state vector, which for flux linkages as states, is given as

[U] is input vector and is given by

[A] and [B] are coeﬃcient matrices of known quantities. Arranging the equations (2.101) and (2.102) in state space form as given by Eq. (2.104). The following equations are obtained⎡˙ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ λds 0 ωg 0 0 λds rs ids vds ˙ ⎥ ⎢ ⎢λ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0 0 ⎥ ⎢ qs ⎥ = ⎢ ωg 0 ⎥ ⎢ λqs ⎥ + ⎢ rs iqs ⎥ + ⎢ vqs ⎥ ˙ dr ⎦ ⎣ 0 0 ⎣λ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ 0 ωg −ωr λdr rr idr vdr ⎦ ˙λqr 0 0 −(ωg − ωr ) 0 λqr rr iqr vqr

(2.107)

In equation (2.107) the currents are unknown quantities which can be replaced by flux linkages and inductances from equations (2.103) as given belowiqs = (Lr λqs −Lm λqr )/(Lr Ls −L2m ) iqr = (Lm λqs −Ls λqr )/(L2m − Lr Ls )

(2.108)

ids = (Lr λds −Lm λdr )/(Lr Ls −L2m )

idr = (Lm λds −Ls λdr )/(L2m − Lr Ls )

If L0 = L2m −Lr Ls and substituting the values of the currents and rearranging the equation ⎤ ⎡ ⎡˙ ⎤ ⎡ ⎤⎡ ⎤ λds ωg rs Lm /L0 0 vds λds −rs Lr /L0 ˙ ⎥ ⎢ ⎥ ⎢ ⎢λ ⎢ ⎥ ωg rs Lm /L0 0 −rs Ls /L0 ⎥ ⎢ qs ⎥ = ⎢ ⎥ ⎢ λqs ⎥ + ⎢ vqs ⎥ ˙ dr ⎦ ⎣ rr Lm /L0 ⎣λ 0 −rr Ls /L0 ωg −ωr ⎦ ⎣ λdr ⎦ ⎣ 0 ⎦ ˙ qr 0 λqr 0 rr Lm /L0 −(ωg − ωr ) −rr Ls /L0 λ (2.109) This equation gives state representation of induction machine. The output equation in state variable form is represented as [Y] = [C] [X] + [D [U]]

36

2 Modeling of Induction and Synchronous Machines

Here [Y] is output. The torque Te is given by 3P 3P (λds iqs −λqs ids ) = Lm (iqs idr −ids iqr ) (2.110) 22 22 Now if the currents instead of fluxes are selected as the state variables, the state voltage equations can be written as Te =

⎡

⎤ ⎡ ⎤⎡ ⎤ vqs r s + Ls p ωs L s Lm p ωs L m iqs ⎢ vds ⎥ ⎢ ⎥ ⎢ ids ⎥ −ωs Ls r s + Ls p −ωs Lm Lm p ⎢ ⎥ =⎢ ⎥⎢ ⎥ ⎣ vqr ⎦ ⎣ Lm p (ωs −ωr )Lm rr + Lr p (ωs −ωr )Lr ⎦ ⎣ iqr ⎦ vdr −(ωs −ωr )Lm Lm p −(ωs −ωr )Lr rr + Lr p idr (2.111) Here T [U] = vds vqds vdr vqr , T [X] = ids iqs idr iqr

Rearranging the equation (2.111) in state variable form ⎡

⎤ ⎡ ⎤ ⎡ ⎤ vqs rs ωs L s 0 ωs L m iqs ⎢ vds ⎥ ⎢ ⎥ ⎢ ids ⎥ −ωs Ls rs −ωs Lm 0 ⎢ ⎥ −⎢ ⎥ =⎢ ⎥ ⎣ vqr ⎦ ⎣ 0 (ωs −ωr )Lm rr (ωs −ωr )Lr ⎦ ⎣ iqr ⎦ vdr −(ωs −ωr )Lm 0 −(ωs −ωr )Lr rr idr (2.112) ⎤ ⎡ Ls 0 Lm 0 ⎢ 0 Ls 0 Lm ⎥ ⎥ If ⎢ (2.113) ⎣ Lm 0 Lr 0 ⎦ = [L] 0 Lm 0 Lr then ⎤ ⎡ ⎡ ⎤⎡ ⎤ ⎤ ˙iqs iqs vqs rs ωs L s 0 ωs L m ⎢ ˙ids ⎥ vds ⎥ −ωs Ls rs −ωs Lm 0 ⎥ ⎢ ids ⎥ −1 ⎢ ⎢ ⎥ = [L] −1 ⎢ ⎣ ⎣ v ⎦ − [L] ⎣ ⎦ ⎣ ˙iqr ⎦ iqr ⎦ 0 (ωs −ωr )Lm rr (ωs −ωr )Lr qr ˙idr idr −(ωs −ωr )Lm 0 −(ωs −ωr )Lr rr vdr ⎡

2.9

(2.114)

Modeling of Synchronous Machine

In synchronous machine the stator has a winding identical to that of an induction machine. The rotor carries dc current to produce flux in the air gap. The construction of rotor may be salient type with non uniform air gap, or non salient pole type with uniform air gap. The non salient pole

2.9 Modeling of Synchronous Machine

37

Fig. 2.9 Non salient pole synchronous machine

rotor has winding in slots along the rotor periphery as shown in Figure 2.9. The high speed machines are generally non salient pole type whereas low speed machines have salient poles. The dc current to the rotor winding is supplied through slip ring and brushes. Since the stator winding is three phase like induction motor it produces rotating magnetic field when three phase balanced supply is connected. The speed of the MMF is given by ωs = 2πf radians per sec., where f= frequency of stator supply. As the rotor carries only dc current the field produced by rotor is stationary. In order to produce torque the two fields must run at the same speed. The synchronous motor therefore, runs always at synchronous speed which is the speed of stator field. OR ωr = ωs = 2πns p where ns = rotor speed in rev.per sec. Also, there is no stator induced induction in the rotor; therefore, the rotor MMF is exclusively supplied by the field winding. Thus a synchronous machine can have a power factor that can be leading, lagging or unity.

2.9.1

Production of Torque in Cylindrical Rotor Machine

The mechanism of torque production in synchronous machine is similar to an induction machine [11, 12]. For sinusoidal M.M.F. and constant air gap the motor produces a constant torque. The expression for steady state torque can be obtained from the equivalent circuit shown in Figure 2.10. The current If in the field winding produces a flux λf in the air gap. The current is in the

38

2 Modeling of Induction and Synchronous Machines

Fig. 2.10 Equivalent circuit of non-salient pole machine

stator winding produces a flux λa . The flux λa consists of two parts; leakage flux λal , which links with the stator winding only, and armature reaction flux λar that links the field winding. The resultant flux in the air gap λr is therefore a combination of these two fluxes. The fluxes λf and λar are rotating in the air gap and induce voltages in the stator winding shown as Ef andEar respectively. The Voltage Ear depends on λar , and therefore on is , and lags is by 90 degrees. The voltage Ear therefore, can be represented as a voltage drop across a reactance Xar . If the leakage reactance Xal due to λal is combined into one reactance Xs , then Xs is known as synchronous reactance of the machine. The per phase equivalent circuit and the phasor diagram for non- salient pole synchronous motor is shown in figure 2.9. Here Rs is the per phase resistance of the stator winding and Vs is the applied voltage per phase. The angle δ between Vs and Ef is known as power or torque angle. The steady state torque developed by the motor can be obtained from the equivalent circuit as follows Taking Vs as reference is =

=

Vs ∠0− Ef ∠δ Zs ∠θs

Vs ∠ − θs Ef ∠ − θs + δ − Zs Zs

(2.115)

The per phase complex power S=Vs i∗s . And real 3- phase power input to the motor Pi is given as 3Vs is cosφ. If the resistance of the stator winding is neglected Zs = Xs ∠ π2 and Pi =

3Vs Ef sinδ Xs

(2.116)

The torque Te neglecting losses in the machine is given by Te =

Pi ωs

=

3 Vs E f sinδ ωs Xs

(2.117)

2.9 Modeling of Synchronous Machine

39

= Tmax sin δ where, and ωs =

Tmax =

3 Vs E f , ωs Xs

(2.118) (2.119)

2πns 60

ns = synchronous speed of the motor in r p.m. As these equations for power and torque show, both power and torque vary sinusoidally with angle δ. The maximum torque Tmax is also known as pull out torque. That means if the motor is loaded with more torque than this value it will stall.

2.9.2

Salient Pole Synchronous Machine

Rotors of synchronous motors in general are of salient pole types and have non-uniform air gaps. The magnetic reluctance is low along the poles and high between poles. The flux produced by armature current is more along the pole axis, called the d-axis, and less along the interpolar axis, called the q-axis. The d axis flux and q axis flux can be accounted for by the d axis armature reactance Xad and q axis armature reactance Xaq . If the leakage armature reactance due to the leakage flux is included the d axis and q axis synchronous reactances can be obtained as – Xd = Xad + Xal d axis synchronous reactance Xq = Xaq + Xal q-axis synchronous reactance Here Xd > Xq . The phasor diagram of a salient pole motor is shown in Figure 2.11. Again here the stator resistance is neglected. The excitation or the emf due to the field current Ef is aligned with the q axis, because the field current and therefore the field flux is aligned with d axis. The armature current Is can be resolved along d axis as Id and along q axis as Iq , then the applied voltage Vs can be represented as Vs = Ef + +IdjXd + Iq jXq

(2.120)

Is cosφ = Iq cosδ − Id sinδ

(2.121)

from the phasor diagram

The input power Pi can be written as Pi = 3Vs (Iq cos δ−Id sinδ)

(2.122)

40

2 Modeling of Induction and Synchronous Machines

Vqs

Vs

I qs X qs

δ I qs

φ

Vf + I ds X ds

Is

Fig. 2.11 Phasor diagram of salient pole machine

From the phasor diagram the values of Id and Iq can be obtained as Id =

Vs cosδ−Ef Xd

and Iq =

Vs sinδ Xq

(2.123) (2.124)

substituting these values in equation yields Pi = 3 and torque Te =

Pi ωs ,

(Xd −Xq ) E f Vs sin2δ sinδ + 3V2s Xd 2Xd Xq

since ωs =

Te = 3(

(2.125)

2πns 60

(Xd −Xq ) E f Vs P ){ sin2δ} sinδ + 3V2s 2ωe Xd 2Xd Xq

(2.126)

2.10 Dynamic Modeling of Synchronous Machine

2.10

41

Dynamic Modeling of Synchronous Machine

In salient pole synchronous machine there are 3 windings on the stator, one field winding on the rotor and two damper windings also on rotor. The damper windings consist of short circuited squirrel cage windings placed in slots on the rotor surface. Each of these six windings is characterized by resistance, self inductance, and mutual inductance (except the two damper windings which do not have mutual inductance between them). Since the air gap is not uniform, the inductances will vary with the position of rotor, except the self inductances of field and damper windings. The pole surfaces are so shaped as to obtain variation in reactance as sinusoidal as possible. If at any instant of time α is the electrical angle between the axis of phase ‘a’ of stator winding with rotor pole axis, then all the inductances that depend on the rotor position will be functions of the angle α. The self inductances Laa , Lbb , and Lcc vary periodically with the angle α, and are given as Laa = L1 + L2cos2α

(2.127)

Lbb = L1 + L2cos2(α −

2π ) 3

(2.128)

Lcc = L1 + L2 cos2(α +

2π ) 3

(2.129)

The mutual inductances satisfy the equations Lij = Lji Also a positive current in stator winding i will give a negative flux linkage component in the other two stator windings. The double frequency component for mutual inductance has the same amplitude as the self inductance component. The mutual inductance component therefore can be expressed as π Lab = −L3 − L2 cos2(α + ) 6

(2.130)

π Lbc = −L3 − L2 cos2(α − ) 2

(2.131)

π (2.132) Lac = −L3 − L2 cos2(α − ) 6 The rotor self inductance is constant and is expressed as Lff , the mutual inductance between rotor and stator windings varies between a positive maximum value to negative maximum. It is expressed as

42

2 Modeling of Induction and Synchronous Machines

Laf = L4 cosα Lbf = L4 cos(α−

(2.133)

2π ) 3

(2.134)

4π ) (2.135) 3 similarly the mutual inductances between the stator windings and the damper windings can be expressed as (the damper windings are put on the d and q axis on the rotor) ⎫ LaD = L5 cosα LaQ = L6 sinα ⎪ ⎪ ⎪ ⎬ 2π 2π LbD = L5 cos(α− 3 ) LbQ = L6 sin(α− 3 ) (2.136) ⎪ ⎪ ⎪ 4π ⎭ LcD = L5 cos(α− 4π 3 ) LcQ = L6 sin(α− 3 ) Lcf = L4 cos(α−

The self inductances of the damper windings are constant and there is no mutual inductance between them. LDD = L7 and LQQ = L8 The inductances can be written in matrix form as ⎡ a b c f D ⎢ ⎢ ⎢ a Laa Lab Lac Laf LaD ⎢ ⎢ ⎢ b Lba Lbb Lbc Lbf LbD ⎢ ⎢ [L] = ⎢ c Lca Lcb Lcc Lcf LcD ⎢ ⎢ ⎢ f Laf Lbf Lcf Lff LfD ⎢ ⎢ ⎢ D LaD LbD LcD LfD LDD ⎣ Q LaQ LbQ LcQ 0 0

(2.137)

Q

⎤

⎥ ⎥ LaQ ⎥ ⎥ ⎥ LbQ ⎥ ⎥ ⎥ LcQ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎦ LQQ

(2.138)

The voltage current relationship can be expressed in the matrix form for stator in stator reference frame and for rotor in rotor reference frame as [V] = [R] [I] +

dλ dt

T where, [V] = Va Vb Vc in stator reference

T [I] = Ia Ib Ic

[r] = diag Rs Rs Rs ; where Ra = Rb = Rc = Rs

(2.139)

2.11 Space Phasor Model

43

The flux linkages λ in stator reference ⎡ ⎡ ⎤ a b c λa ⎢ a Laa Lab Lac ⎢ λb ⎥ ⎢ ⎢ ⎥ ⎢ b Lba Lbb Lbc ⎢ λc ⎥ ⎢ ⎢ ⎥ = ⎢ c Lca Lcb Lcc ⎢ λf ⎥ ⎢ ⎢ ⎥ ⎢ f Laf Lbf Lcf ⎣ λD ⎦ ⎢ ⎣ D Lad Lbd Lcd λQ Q Laq Lbq Lcq

frame can be written as ⎤ f D Q ⎡ ⎤ ia Laf Lad Laq ⎥ ⎥ ⎢ ib ⎥ ⎢ ⎥ Lbf Lbd Lbq ⎥ ⎥ ⎢ ic ⎥ ⎢ ⎥ Lcf Lcd Lcq ⎥ ⎥ ⎢ if ⎥ ⎢ ⎥ Lff Lfd 0 ⎥ ⎥⎣ i ⎦ Lfd Ldd 0 ⎦ D iQ 0 0 Lqq

(2.140)

The d-q-0 model of the synchronous machine can be obtained in the same way as has been done for induction machine using Park’s transformation. If damper winding is included it can be modeled as cage type winding along d and q axis. The field winding is active on d axis only. L L λabc iabc (2.141) = TS SR i f DQ λfDQ LSR LR Or ⎡

⎤ ⎡ λ0 L0 0 0 ⎢ λ d ⎥ ⎢ 0 Ld 0 ⎢ ⎥ ⎢ ⎢ λq ⎥ ⎢ 0 0 Lq ⎢ ⎥=⎢ ⎢ λ f ⎥ ⎢ 0 kMf 0 ⎢ ⎥ ⎢ ⎣ λD ⎦ ⎣ 0 kMD 0 λQ 0 0 kMQ

⎤⎡ ⎤ i0 0 0 0 ⎢ ⎥ kMf kMD 0 ⎥ ⎥ ⎢ id ⎥ ⎢ ⎥ 0 0 kMQ ⎥ ⎥ ⎢ iq ⎥ ⎥ ⎥ Lf LfD 0 ⎥ ⎢ ⎢ if ⎥ ⎣ ⎦ LfD LD 0 iD ⎦ 0 0 LQ iQ

(2.142)

Where LS, LSR and LR matrices have been transformed using Park’s transformation.

2.11

Space Phasor Model

Using the same procedure as developed for the induction motor the space phasor model of the salient pole synchronous machine can be obtained as follows. The stator current space phasor ¯is , in stator coordinates is ¯iss = 2 {ias + aibs + a2 ics } (2.143) 3 The stator flux linkage space phasor in stator coordinates can similarly be written as

¯ s = 2 λa + aλb + a2 λc λ s 3

(2.144)

where λa is the flux linkage of phase ‘a’ winding due to currents in all other windings.

44

2 Modeling of Induction and Synchronous Machines

λa = Laa ia + Labib + Lac ic + Laf i+ f Ladr idr + Laqr iqr

(2.145)

The stator voltage equations in stator reference frame can be written as ¯s ¯ ss = Rs¯iss + dλs V dt

(2.146)

The stator equation (neglecting the eﬀect of damper windings) in d-q frame can be written in the same form as for induction motor as Vd = Rs id +

dλd − ωr λ q dt

(2.147)

dλq + ωr λ d (2.148) dt However, here the flux linkages are having diﬀerent values from the case of induction machine due to field winding on d axis Vq = Rs iq +

λq = Lq iq , and λd = Ld id + L4 ir

(2.149)

and dλf dt

(2.150)

3 λf = Lff If − L4 id 2

(2.151)

Vf = Rf If + where

References [1] Krause, P.C.: Analysis of Electrical Machinery. McGraw Hill Book Company, New York (1986) [2] Krishnan, R.: Electric Motor Drives, Modeling Analysis and Control. Prentice Hall, Englewood Cliﬀs (2001) [3] Boldea, I., Nasar, S.A.: Electric Drive. CRC Press, Boca Raton (1999) [4] Bose, B.K.: Modern Power Electronics and AC Drives. Pearson Education Inc., London (2002) [5] Vithyathil, J.: Power Electronics. McGraw Hill Inc., New York (1995) [6] Vas, P.: Sensorless Vector Control and Direct Torque Control. Oxford University Press, Oxford (1998) [7] Ong, C.M.: Dynamic Simulation of Electric Machinery. Prentice Hall, New Jersey (1998) [8] Sen Gupta, D.P., Lynn, J.W.: Electric Machine Dynamics. Macmillan Press, London (1980)

References

45

[9] Trzynadlowski, A.M.: Control of Induction Motors. Elsevier, Amsterdam (2001) [10] Slemon, G.R.: Electrical Machines for variable Frequency Drives. Proceedings of IEEE 82, 1123–1129 (1994) [11] Sen, P.C.: Principles of electric machines and power electronics. John Wiley & Sons, Chichester (1997) [12] Leonhard, W.: Control of Electric Drives. Springer, Berlin (1985)

Chapter 3

Vector Control of Induction Motor Drives

3.1

Speed Control of Induction Motor

For applications where high degree of accuracy in speed control is not required simple methods based on steady state equivalent circuit have been employed. Since the speed of an induction motor in rev. per minute is given by N=

120f (1 − s) P

(3.1)

Where f = frequency of supply, P = number of poles, and s = slip. Thus the speed of the motor can be changed by controlling the frequency, or number of poles or the slip. Since, number of poles of a motor is fixed at the time of construction, special motors are required with provision of pole changing windings. These motors are not used now. With the availability of semiconductor devices for static frequency conversion, most of the adjustable speed drives are now inverter fed drives. Here only inverter fed drives are discussed briefly.

3.1.1

Volts/Hz Control

Equation (3.1) indicates that the speed of an induction motor can be controlled by varying the supply frequency f. Earlier, when high power semiconductor devices were not available, it was a diﬃcult task to obtain variable frequency supply. However, now PWM inverters are available that can easily provide variable frequency supply with good quality output wave shape. The open loop volts/ Hz control is therefore quite popular method of speed control for induction motor drives where high accuracy in control is not required. The frequency control also requires proportional control in applied voltage, because then the stas tor flux λs = V ωs (neglecting the resistance drop) remains constant. Otherwise, if frequency alone is controlled, then the flux will change. When frequency is increased, the flux will decrease, and the torque developed by the motor will M. Ahmad: High Performance AC Drives, Power Systems, pp. 47–75. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

48

3 Vector Control of Induction Motor Drives

+ 1

3

5

M 4

6

2

Fig. 3.1 Voltage inverter fed induction motor

decrease. When frequency is decreased, the flux will increase and may lead to the saturation of magnetic circuit. Since in PWM inverters the voltage and frequency can be controlled independently, these drives are fed from a PWM inverter. The control scheme is simple as shown in Fig. 3.1with motor being supplied by three-phase supply through diode rectifier, filter and PWM inverter. The drive does not require any feed back and is used in low performance applications where precise speed control is not required. Depending on the desired speed the frequency command is applied to the inverter, and phase voltage command is directly generated from the frequency command by a gain factor, and input dc voltage of inverter is controlled. The speed of the motor is not precisely controlled by this method as the frequency control only controls the synchronous speed [1-4]. There will be a small variation in speed of the motor under load conditions. This variation is not much when the speed is high. When working at low speeds, the frequency is low, and if the voltage is also reduced then the performance of the motor deteriorates due to large value of stator resistance drop. For low speed operation the relationship between voltage and frequency is given by v = v0 + kf

(3.2)

Where v0 = voltage drop in the stator resistance. In order to get better performance than one can obtain in open loop control, an outer speed loop is connected in the induction motor drive control. The speed of the motor is then controlled in closed loop operation by regulation of slip. The scheme of speed control with slip regulation is shown in Figure 3.2. A speed encoder measures the speed of the motor ωr . This value of speed is compared with the speed command ω∗r . The speed loop error is used to generate the command value of slip ω∗sl through a PI controller and limiter. The slip command is added to the measured speed to generate command frequency signal ω∗e . The voltage command signal for the inverter is also generated using frequency command and V/f function generator.

3.1 Speed Control of Induction Motor

49

Vdc

1 2S

Zr*

7e* Zr

Fig. 3.2 Speed control with slip regulation

In closed loop operation, the limits on the slip speed and reference speed are externally adjustable. However, since there is no closed loop flux control, the line voltage variations will cause drift in flux. This may cause the change in value of torque from the desired value.

3.1.2

Flux and Torque Control

The disadvantages of V/f control are mainly due to drift in flux, which can be minimized using flux control. The closed loop control of flux and torque means addition of two more loops in the drive. The flux and torque feedback signals are not measured directly, but estimated from the terminal voltages and currents. The complete scheme is shown in Figure 3.3. The flux vector can be estimated from the voltage equations derived in chapter 2. The drive as shown in the Figure 3.3 has three closed loops. There is an inner torque loop within the speed loop which improves the speed loop’s response. The flux control loop controls the inverter voltage. With constant rotor flux command,

Fig. 3.3

50

3 Vector Control of Induction Motor Drives

as the speed increases, the voltage increases proportionally until square wave mode is reached and field weakening mode starts. However, if PWM operation is desired in field weakening mode, the flux command must be decreased. inversely with speed signal. The flux control loop is generally slower than torque control loop. As the speed command ω∗r is increased by the torque loop, the flux is decreased temporarily till flux control compensates it. The coupling between the torque and flux loop slows down the torque response.

3.2

Introduction to Vector Control

The steady state performance of an inverter fed induction motor drives is comparable to a separately excited dc motor. However, the dynamic performance of an induction motor with simple controls as described above is much slow in comparison to dc machines. This drawback has restricted the application of induction motor drives in industries requiring high performance, such as, in reversing sheet rolling mills, and many machine tool drives. The vector control or field oriented control of ac machines has been evolved which makes it possible to control ac motor in a manner similar to the control of a separately excited dc motor. The main objective of vector control is to achieve superior performance under torque and speed change. In separately excited dc motor the MMF produced by the field current and the MMF produced by the armature current are spatially in quadrature. There is therefore, no magnetic coupling between the field circuit and armature circuit. Thus, the armature current can be changed independently, and torque can be controlled faster, keeping field flux constant. In ac machines also, the torque is produced by the interaction of current and flux. But in induction motor the power is fed to the stator only, the current responsible for producing flux, and the current responsible for producing torque are not easily separable. The basic principle of vector control is to separate the components of stator current responsible for production of flux, and the torque. The vector control in ac machines is obtained by controlling the magnitude, frequency, and phase of stator current, by inverter control. Since, the control of the motor is obtained by controlling both magnitude and phase angle of the current, this method of control is given the name vector control.

3.3

Space Vectors

The principle of vector control can be easily understood if the concept of “space vectors” is used. The space phasor model of an induction motor is described in 2.6. The stator current space vector ¯is is given as ¯is = ias ej0 + ibs ej( 2π3 ) + ics ej( 4π3 ) .

(3.3)

3.3 Space Vectors

51

This space vector has constant amplitude and rotates at synchronous speed. The d-q-0 model as obtained in chapter 2 can be written as. ⎡ ⎤ ⎡ ⎤⎡ ⎤ 1 − 21 − 21 ids ias ⎣ iqs ⎦ = ⎣ 0 − 3 3 ⎦ ⎣ ibs ⎦ (3.4) 2 2 ics i0 1 1 1

Here the currents in a, b and c phases are assumed to be balanced. Thus i0 = ias + ibs + ics = 0

(3.5)

Similarly, stator phase voltage space vector is obtained as2π

4π

v¯ s = vas + vbsej 3 + vcs ej 3

The voltages for equivalent two –phase machine are obtained as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 1 − 12 − 21 vds vas ⎣ vqs ⎦ = ⎣ 0 − 3 3 ⎦ ⎣ vbs ⎦ 2 2 v0s vcs 1 1 1

(3.6)

(3.7)

Again for balanced system v0s = vas + vbs + vcs = 0

(3.8)

The resultant voltage space vector v¯ s can be expressed in rectangular coordinates as v¯ s = vs cosθ + jvs sinθ where, vs is the magnitude of the resultant voltage space vector and θ is the angle with a reference axis. Suppose we take d axis as the reference, then v¯ s = vds + jvqs

(3.9)

Similar to the definition of stator current and stator voltage space vectors, it is possible to define the space vector stator flux linkage as – 4π 2π ¯ s = 2 (λas + λbs ej 3 + λcs ej 3 ) λ 3

(3.10)

It can be written in terms of inductances as – ¯ s = Ls¯is + Lmm¯ir λ

(3.11)

Where Ls = total three-phase stator inductance and, Lmm is the three-phase magnetizing inductance equal to 32 (Lsr ). In d-q axis it can be written as – ¯ s = λds + jλqs λ

(3.12)

52

3 Vector Control of Induction Motor Drives

Equation (3.12) gives the flux linkage for each phase of the equivalent twophase machine in stator reference frame.

3.4

Transformation of Space Vector from One Reference Frame to Other

The equations (3.3) to (3.12) have been derived for stator currents, voltages and flux linkages in stator reference frame. In an induction motor the currents flow in both stator and rotor windings. Since the rotor windings of an induction motor are short circuited, the voltages in these windings are zero. For a two phase machine if rotor quantities in rotor reference frame are considered, the rotor current space vector can be written as ¯irr = irrd + jirrq

(3.13)

Where the r suﬃx indicates the rotor reference frame. If rotor reference frame and stator reference frame are in alignment, then ¯ir will be same in stator reference frame also. However, if rotor reference frame makes an angle θr with the stationary stator reference frame, as shown in Fig. 3.4 then the rotor current space vector ¯ir , in stator reference frame can be written as ¯isr = (ir + jirrq )(cosθr + jsinθr ) rd Or OR

(3.14)

¯isr = (is + jisrq ) = irrd cosθr −irrq sinθr + j(ir sinθr + irrq cosθr ) rd rd

isrd isrq

cosθr −sinθr = sinθr cosθr

irrd irrq

(3.15)

Equation (3.15) relates rotor current space vector in stator reference frame, to the rotor current space vector in rotor frame of reference. In case the rotor is rotating, the angle θr will be changing with time. The selection of reference frame therefore, has eﬀect on the nature of the space vectors. For example, the stator current space vector when seen from stator frame of reference equation (3.3) has constant amplitude and rotates at constant synchronous speed. However, the same vector with respect to a synchronously rotating reference frame will have constant amplitude and will be stationary ( DC). In case of flux linkage, equation (3.12) shows the resultant space vector λs which is split into λd and λq components along d and q axis respectively. These components can also be considered as the flux linkages of two phase machine having stator windings along d and q axis. However, each stator winding will have a flux linkage from all the windings carrying current. Since the d axis and q axis are at 90-degrees, there will be no flux linkage in d axis winding from the current in q axis winding. Similarly, q axis winding will

3.4 Transformation of Space Vector from One Reference Frame to Other

53

qs

qr Ir

dr

Tr ds

(a)

(b) Fig. 3.4 Space vectors in stator and rotor reference frames

not have any flux linkage from current in d winding. If rotor circuit is open circuited, then λs can be written in terms of self inductance Ls of the two windings. (3.16) λs = Ls isds + jLs isqs Or λs = Ls¯is

(3.17)

54

3 Vector Control of Induction Motor Drives

Similarly, if 3-phase rotor winding is replaced by equivalent two-phase winding, the rotor flux linkage vector due to rotor currents only and in rotor reference frame will be given by ¯ r = Lr¯ir λ

(3.18)

However, when stator and rotor windings both carry current which is the normal case, there is mutual flux linkages between the stator and rotor windings. The power transfer from the stator to rotor takes place through the mutual flux linkage, and is responsible for production of torque. If rotor is assumed to be stationary and the rotor reference axis coincides with the stator reference axis as shown in Figure 3.4(b), then the mutual inductance between the stator direct axis winding and rotor direct axis winding will be maximum equal to Lˆ m Similarly the quadrature axis will also have the mutual inductance equal to Lˆ m . Now, if the rotor is turned by an angle θr , then the mutual inductance will be Lˆ m cosθr . However, under this position the quadrature winding of rotor will also have mutual inductance with the direct axis winding of the stator equal to Lˆ m cos (θr + π2 ). Thus the direct axis flux linkage in the stator winding due to the currents in the rotor windings only will be (3.19) λsds = Lˆ m irdr cosθr −Lˆ m irqr sinθr Similarly, the flux linkage of quadrature axis winding of the stator due to current in the rotor windings only can be written as λsqs = Lˆ m irqr cosθr −Lˆ m irdr sinθr

(3.20)

The total flux linkage of the stator winding due to currents in the rotor windings only and substituting the values from Eq.(3.17) is λss = Lˆ m isr

=

Lˆ m irr ejθr

(3.21)

If the eﬀect of stator current is also considered, the total stator flux linkage space vector in the stator reference frame can be expressed as λss = Ls¯iss + Lˆ m¯isr

(3.22)

= Ls¯iss + Lˆ m¯irr ejθr

(3.23)

Now, if the rotor is rotating, the angle θr will be time varying. Similarly, the total flux linkage space vector of the rotor in rotor reference frame can be expressed as λrr = Lr¯irr + Lm¯iss e−jθ (3.24) Or λsr = Lr¯isr ejθr + Lm¯iss

(3.25)

3.5 Principle of Vector Control

55

The torque Te = 23 P2 LLmr λsdr isqs . If ¯imr is defined as a rotor magnetizing current space vector in stator coordinates, then total rotor flux linkage can be obtained by multiplying this current with the mutual inductance Lm . Or Lm ismr = λsr = Lr¯irr ejθr + Lm¯iss (3.26) Equations (3.24) and (3.26) show that total flux linkage space vector in one reference frame can be expressed as flux linkage due to self inductance in that reference frame plus the flux linkage due to mutual inductance in other reference frame transformed to this reference frame. The stator voltage equations in terms of space vectors and reference frame fixed to the stator will bes

dλ v¯ s = Rs¯iss + s dt

(3.27)

The rotor voltage is zero, hence r

dλ v¯ r = 0 = Rr¯irr + r dt

(3.28)

The torque in an induction motor is cross product of the flux and current expressed as space vectors. And can be written as T¯ e = Kλrr × ¯irr = KLm¯irs × ¯isr

(3.29)

Here × sign is used for vector product and K = 23 P. Similarly the torque equation in terms of stator variables can be written as T¯ e = Kλss ×¯iss (3.30) and it can also be expressed in terms of currents as – T¯ e = K1¯isr × ¯iss

3.5

(3.31)

Principle of Vector Control

Vector control in induction motor is implemented by independently controlling the torque component and the field component of the stator current through a coordinated change in the supply voltage amplitude, phase, and frequency. The induction motor with vector control is able to give performance comparable to a dc machine. The field orientation control concept implies that the current components supplied to the machine should be split into flux component and torque component. The flux component of the current is oriented in phase with the rotor flux vector, and the torque component is oriented in quadrature with it. For vector control of induction motor its model is considered in a synchronously rotating d-q reference frame, where the sinusoidal quantities

56

3 Vector Control of Induction Motor Drives

behave like dc quantities in steady state. The torque of the induction motor Te in terms of space phasor quantities can be obtained as T¯ e = K¯ids¯iqs Or Te = KT λr iqs .

(3.32)

In induction motor the stator current phasor i¯s produces the rotor flux λr and the torque. The component of stator current producing the flux is in phase with rotor flux λr , or ids of the stator current is analogous to field current of dc machine. The current iqs is responsible for production of torque. The dc machine like performance is only possible if ids is responsible for the production of the total flux and aligned to the direction of the flux vector. There are diﬀerent ways of implementing the vector control strategy according to the choice of the reference frames for the space vectors. In an induction motor, there are three distinct flux space phasors: air gap flux, stator flux, and rotor flux. The air gap flux rotates at synchronous speed, and the rotor flux rotates at slip speed with respect to rotor. Vector control can be performed with respect to any of these flux space phasors by attaching d-axis of the reference system to the respective flux space phasor direction. The most convenient and common choice of reference frame for the vector control of an induction motor is the reference frame attached to the space vector representing the total flux linkage λr of the rotor. This is a rotating reference frame, rotating with the total flux linkage space vector of the rotor. Since the reference frame is the field frame, this method of speed control is also known as “field oriented control”. The simple scheme is shown in Fig. 3.5 Where inverter is shown to have vector controller with two command current inputs i∗ds and i∗qs These currents are the d axis and q axis components ia

ib ic 3

Switch ing

Inverter

2

e jT

i *ds

i *qs

Flux control

Torque control

Fig. 3.5 Field oriented control scheme

Motor

3.5 Principle of Vector Control

57

vs is

I Ts

Tf

Tr

Or

T sl Rotor ref

stator

Fig. 3.6

of stator current in a synchronously rotating reference frame. The three stator currents ias , ibs , and ics are transformed into the d-q components in stator reference frame. These are further transformed into field reference frame by the vector rotator block. It is assumed that field reference frame is at an angle θf with respect to stationary (stator) reference frame Figure 3.6. This angle will be varying with time as the rotor flux linkage space vector λr rotates and is referred as field angle. If θr is the angle between (figure 3.6) the stator current space vector and the reference field frame, and θsl is the slip angle between the rotor axis and the rotor field frame, then θf = θr + θsl

(3.33)

In terms of speeds θf =

(ωr + ωsl )dt =

ωs dt

(3.34)

These transformed currents are compared with the command current inputs i∗ds and i∗qs . The error signals obtained are amplified and used to control the flux and torque. These currents are transformed from field frame to stator frame using inverse transformation. Once the d-q components of current in stator frame are known, these can be converted into a-b-c components by 2/3 transformation. These currents are compared with the actual motor currents and used to control the switching of inverter. The inverter is a current regulated three-phase inverter. The switching control regulates the values of these currents so that they conform to the reference values. The decoupled control of these currents results in high level of dynamic response as attained by dc motor.

58

3 Vector Control of Induction Motor Drives

Although it is possible to orient the current Issd with rotor flux axis, or stator flux axis, or with air gap flux axis for vector control. However, rotor flux orientation gives natural decoupling control, whereas the stator flux or air gap flux orientation results in coupling and requires compensation for decoupling. There are essentially two methods of vector control; the direct or feed back method, and the indirect or feed forward method. In direct method, the flux linkage signal is acquired directly, either by installing special sensors to sense the field on the machine or by extracting the flux linkage space vector directly by flux model. In indirect method, the relative speed of the flux linkage space vector with respect to the rotor is determined and integrated to obtain the angle of movement of the field with respect to the rotor. This angle is added to the measured angle moved by rotor to obtain θr . Since the position of flux linkage space vector is obtained indirectly, this method is known as indirect vector control or feed forward control method.

3.6

Direct Vector Control

The direct vector control of induction motor where the flux linkage signal is acquired directly by sensors requires a specially designed motor equipped with Hall eﬀect sensors or coils. Here it will be assumed that the position of rotor flux linkage vector λr is known, and it is at an angle of θf from the stationary reference frame as shown in figure 3.6. The stator current is makes an angle θs with the stator reference frame, and an angle of θT with rotor flux linkage axis. This current is responsible for the production of rotor flux λr and the torque Te . The component of current producing the rotor flux has to be in phase with λr . Thus, if is is resolved into d- axis and q- axis components in rotor flux linkage frame; is cosθT = if is the field producing component and the component along q axis iT = is sinθT is the torque producing component. Since is space phasor rotates at synchronous speed, and the rotor flux linkage space phasor has a speed equal to the sum of rotor speed and slip speed, (which is equal to synchronous speed), the relative speed between it and rotor field is zero. Thus the currents if and iT are dc quantities and can be ideally used as control variables. A block diagram of the direct vector control scheme is shown in figure 3.7. The desired speed is given as the speed reference ω∗r . This speed is compared with the actual rotor speed ωr . This error is amplified in speed control amplifier, which may be a PI controller. The output of the speed controller serves as the reference input to the inner torque loop. The reference torque is compared with the prevailing value of torque to get torque error. This error is processed to generate the reference torque producing component of stator current i∗T . In the same way the rotor flux linkage reference λ∗r is also obtained from the rotor speed by an absolute –value function generator. λ∗r is kept constant at 1 p.u. for speed in the range of 0 to 1 pu. For speeds beyond 1 p.u. it is varied ( flux linkages decreased) as a function of rotor speed. This block is

3.6 Direct Vector Control

Magnitude And angle

Zr

59

Zr

Fig. 3.7 Direct vector control sceme

shown as field weakening block in the Fig. 3.7, and it ensures constant power operation of the motor. As the rotor flux linkages are reduced for the same value of torque component of the current, the electromagnetic torque is reduced, and since the speed will increase (because of flux weakening) constant power is obtained. The rotor flux linkage reference is compared with the rotor flux linkages obtained from the model of the machine. The errors generated from the flux comparator is used to generate reference flux producing components of stator current i∗f . The phasor addition of i∗f and i∗T is the stator current reference phasor i∗s . The angle between i∗f and i∗T is the torque angle reference θ∗T . If angle θ∗f can be obtained the angle between rotor flux axis and stator axis θs is known. Knowing of i∗s and θ∗f , the stator phase current references i∗a , i∗b , and i∗c can be obtained. It becomes now possible through PWM inverter control to force the motor currents to follow these reference currents. Thus in direct vector control the feed back variables θ∗f , Te , and λr are required. These variables can be obtained in the flux and torque processor block. Depending on the measured variables used, the processor will have the following quantities as input. 1. Line to line voltages, and stator currents, or 2. induced emf from flux measuring coils, and stator currents. From these inputs the values of θ∗f , Te , and λr are computed as described below.

3.6.1

Direct Vector Control Sensing Line Voltages and Currents (Rotor Flux)

The computation of the values of θ∗f , Te , and λr are possible using terminal voltages and currents by computing either rotor, or stator flux linkages. If

60

3 Vector Control of Induction Motor Drives

rotor flux linkage method is used the torque and flux calculator will have the following algorithm. Suppose the line currents measured are ia , ib , and ic , and q-axis is aligned with phase ‘a’- axis, then the d-q components of stator current in stator reference frame are obtained as given below isqs = −

1 1 ibs + ics 3 3

(3.35)

and

2 1 1 isds = ias − ibs − ics = ias (3.36) 3 3 3 Similarly if the voltages measured are vab , vbc , and vca and phase voltages are va , vb and vc , then 1 1 vbs + vcs , vsqs = − (3.37) 3 3 and

1 1 2 (3.38) vsds = vas − vbs + vcs 3 3 3 1 (3.39) = (vab + vac ) 3 From these equations rotor currents can be obtained using following equations as vds = (Rs + Ls p)ids + Lm pidr (3.40) and vqs = (Rs + Ls p)iqs + Lm piqr

(3.41)

from which vds =

1 { Lm

(vds −Rs ids )dt − Lsids }

(3.42)

vqs =

1 { Lm

(vqs −Rs iqs )dt − Lsiqs }

(3.43)

Once the stator and rotor currents are known, the torque, flux and field angle can be calculated easily using following relations. From equation 2.55 the torque is given by Te =

3P Lm (iqs idr −ids iqr ) 22

(3.44)

and λdr = Lr iqr + Lm iqs

(3.45)

λqr = Lr idr + Lm ids

(3.46)

3.6 Direct Vector Control

61

λdr λqr

(3.47)

λ2dr + λ2qr

(3.48)

tanθf = and, λr =

As can be seen from these equations, the motor control using this method depends heavily on motor parameters Rs , Ls , Lr and Lm , which ultimately depends on the magnetic saturation level.

3.6.2

Direct Vector Control Stator Flux Model

However, instead of using rotor flux based model, stator flux linkages and stator currents can also be used to calculate the torque. Going back to the equations 3.40, and 3.41 vds = (Rs + Ls p)ids + Lm pidr = Rs ids + pλds

(3.49)

vqs = (Rs + Ls p)iqs + Lm piqr = Rs iqs + pλqs

(3.50)

Or λds =

λqs =

λs =

(vds −Rs ids )dt

(3.51)

(vqs −Rs iqs )dt

(3.52)

λ2ds + λ2qs

(3.53)

The angle between stator flux and the d axis of the stator reference frame θfs is given by

λqs θfs = tan−1 and torque (3.54) λds 3P (i λds −ids λqs ) (3.55) 2 2 qs In this method the algorithm depends only on the stator resistance, but the accuracy with which the stator flux linkages can be determined is low when stator voltages are small. Thus at low speeds this method is not accurate. Te =

3.6.3

Direct Vector Control Sensing Induced EMF and Currents

Instead of using voltages, if induced emfs can be obtained, then the errors due to variation in stator resistance can be avoided. In order to obtain the values of emf induced, two sets of flux sensing coils or Hall sensors are placed in

62

3 Vector Control of Induction Motor Drives

stator slots with 90 electrical degree displacement between them. The output of these sensing coils can be directly processed in the logic circuits used for vector control If the d and q axis emfs are represented by eds and eqs , then the stator flux linkages in d and q axis can be represented as λds =

eds .dt

(3.56)

λqs =

eqs .dt

(3.57)

where λds = Ls ids + Lm idr

(3.58)

λqs = Ls iqs + Lm iqr

(3.59)

Or

λ2ds + λ2qs

λqs θfs = tan−1 λds λs =

(3.60) (3.61)

and torque 3P (i λds −ids λqs ) 2 2 qs If rotor flux linkages are used the rotor currents can be obtained as Te =

(3.62)

idr =

λ− ds Ls ids Lm

=

1 ( Lm

(eds dt − Ls ids )

(3.63)

iqr =

λ− qs Ls iqs Lm

=

1 Lm

(eqs dt − Lsiqs )

(3.64)

From the rotor currents thus obtained, and the stator currents measured from sensors, the rotor flux linkages can be obtained as λdr = Lr idr + Lm ids

(3.65)

λqr = Lr iqr + Lm iqs

(3.66)

and Or λr =

λ2dr + λ2qr

and θf = Te =

tan

−1

λqr λdr

(3.67)

3P Lm (iqs idr −ids iqr ) 22

(3.68) (3.69)

3.7 Direct Vector Control with VSI Using Space Vector Modulation (SVM)

63

The disadvantage of using induced emf in vector control is due to the emf induced at low speed is very small. Thus the calculations of rotor currents and rotor flux linkages using these values of emf results in erroneous values of these variables. Also the installation of flux sensors adds to the complexity of the system and untidy look with number of wires coming out of the motor.

3.7

Direct Vector Control with VSI Using Space Vector Modulation (SVM)

The implementation of vector control is possible either with a current source inverter CSI or voltage source inverter VSI. The control of inverter to obtain the phase currents as desired following switching techniques can be used. 1. PWM 2. Hysteresis 3. Space vector modulation PWM and hysteresis techniques are very widely used in CSI and VSI inverters. Recently space vector modulation method of inverter control for vector control drive has been implemented and is described here. A three-phase induction motor supplied from a voltage source inverter has been shown in Figure 3.1. In this method torque and stator flux is computed from measured stator voltages and currents and is compared with the command values of these variables. The error signals are applied to PID controller and limiter to generate the command values of torque and flux components of the current. These command currents are compared with the transformed d-q components of measured stator currents. The d-q current errors set the required voltage commands veqs ∗ and veds ∗. These voltages can be converted to voltage commands in a-b-c frame by 2/3 transformation. These values of voltages are generated by the inverter using space vector modulation as described here. The voltage supplied to the motor is vab , vbc and vca , where vab = va − vb vbc = vb − vc

}

(3.70)

vac = va − vc Here va , vb and vc are the voltages between points a, b, and c with respect to negative point of dc supply respectively. The VSI has six switches T1 , T2 , T3 , − − T6, The switches in one leg say T1 and T4 , do not conduct simultaneously and therefore they can be combined as one switch S1 . S1

= 1 for T1 on and T4 oﬀ

S1

= 0 for T1 oﬀ and T4 on

64

3 Vector Control of Induction Motor Drives Table 3.1 Summary of inverter switching states

State

Switch state

Van

Vbn

Vcn

Space voltage vector

0

S1 = 0, S2 = 0, S3 = 0

0

0

0

V0 (000)

1

S1 = 1, S2 = 0, S3 = 0

2Vd 3

−Vd 3

−Vd 3

V1 (100)

2

S1 = 1, S2 = 1, S3 = 0

Vd 3

Vd 3

−2Vd 3

V2 (110)

3

S1 = 0, S2 = 1, S3 = 0

−Vd 3

2Vd 3

−Vd 3

V3 (010)

4

S1 = 0, S2 = 1, S3 = 1

−2Vd 3

Vd 3

Vd 3

V4 (011)

5

S1 = 0, S2 = 0, S3 = 1

−Vd 3

−Vd 3

2Vd 3

V5 (001)

6

S1 = 1, S2 = 0, S3 = 1

Vd 3

−2Vd 3

Vd 3

V6 (101)

7

S1 = 1, S2 = 1, S1 = 1

0

0

0

V7 (111)

Similarly for other legs the switches S2 and S3 will have either 1 or 0 value depending on switch positions as S2 = 1 for T3 ‘ON’ and T6 ‘OFF’ S2 = 0 for T3 ‘OFF’ and T6 ‘ON’

and

S3 = 1 for T5 ‘ON’ and T2 ‘OFF’ S3 = 0 for T5 ‘OFF’ and T2 ‘ON’ Thus for three switches S1 , S2 , andS3 , there are eight possible combinations or switching states 000, 111, where 0 indicates open and 1 as closed condition of the switch. For these eight switching states, the impressed voltages to the motor are tabulated in Table 3.1. If the motor is supposed to have an isolated neutral’ n’ then rotating space vector corresponding to the voltagesvan, vbn andvcn is defined as Vs then it can also be tabulated as in table 3.1.

2 −j2π 2π j 3 (3.71) Vs (t) = 3 3 Van (t)+ Vbn (t)e + Vcn (t)e Also van = (vab − vca )/3 (3.72) vbn = (vbc − vab )/3

(3.73)

vcn = (vca − vbc )/3

(3.74)

and

3.7 Direct Vector Control with VSI Using Space Vector Modulation (SVM)

65

V2 (110)

3

V*

1

D

V1 (100)

4

6 5

V5 (001)

V6 (101)

Fig. 3.8 Space vector of three-phase inverter

The inverter has six active states when voltage is applied to the load. There are two zero states when no voltage is applied as the motor terminals are short circuited by the upper or lower switches. The sets of phase voltages for each switching state can be combined using equation. The vector Vs can be represented graphically as shown in figure 3.8. The voltage vector Vs which is represented by Vs = vqs + j vds has a six active vectors located at π/3 angle apart and describe a hexagon boundary. For three phase square wave inverter the vector sequence is V1 to V6 and each switch is active for π/3 degrees and there is no V0 and V7 vector.

66

3 Vector Control of Induction Motor Drives

In order to control the magnitude and phase angle of inverter voltage it is important to time the voltage space vectors V0 to V7 . The timing of eight voltage space vectors V0 to V7 is obtained by PWM inverters in either open loop or closed loop PWM strategy. In open loop space-vector PWM control the reference voltage space –vector of the motor is considered as single entity and not phase by phase. The reference voltage space-vector is sampled say once for every switching period. The phase of the reference voltage space-vector identifies the nearest two non-zero voltage vectors. Thus these two voltage can be applied one at a time for a fraction of the switching period depending on the magnitude required. For remaining switching period V0 or V7 is applied. For example if the reference voltage vector V∗ shown in figure 3.8 is to be generated, a convenient way to generate the PWM output is to use the adjacent vectors V1 and V2 . If Va and Vb are the components of V∗ in the directions of V1 and V2 then (3.75) V∗ = V2a + V2b + 2Va Vb cos π3 = Va + Vb If V∗ makes an angle α with V1 then V∗ sin( π3 −α) = Va sin π3 V∗ sinα = Vb sin Or

π 3

(3.76) (3.77)

π 2 Va = √ V∗ sin( − α) 3 3

(3.78)

2 Vb = √ V∗ sinα. 3

(3.79)

and

If Tc is the switching period during which the average output of the inverter must match the reference voltage and V1 and V2 are the voltages are the voltage vectors (100) and (110) then Ta Tb and Vb = V2 Tc Tc

(3.80)

Vb Va Tc and Tb = Tc V1 V2

(3.81)

Va = V1 and Ta =

V∗ = Va + Vb + (V0 or V7 )[Tc − (Ta + Tb )]

(3.82)

[Tc − (Ta + Tb )] = T0

(3.83)

V∗ Tc = V1 Ta + V2 Tb + (V0 or V7 )T0

(3.84)

If Then

3.7 Direct Vector Control with VSI Using Space Vector Modulation (SVM)

67

V*

Fig. 3.9 Under modulated space vector

The construction of symmetrical pulse pattern for two consecutive Tc can be obtained using switching sequence of V0 (T0 /2)...V1 (Ta )..V2 (Tb )...V7 (T0 /2)...V7 (T0 /2)..V2 (Tb )...V1 (Ta )...V0 (T0 /2).

It has been found that symmetrical pulse pattern produces minimal harmonics in the output. The time Tc can also be divided in the following sequence V0 (T0 /3)...V1 (Ta /3)..V2 (Tb /3)...V2 (Tb /3)..V1 (Ta /3)..V7 (T0 /3)...V0 (T0 /3)...

This type of modulation is known as modified space vector PWM and gives better results. The space –vector PWM produces high performance but requires computation of the reference voltage space vector V∗ on line. So far the reference voltage vector has been assumed to be within the circle shown in Fig. 3.9. or giving under modulation. As the reference voltage increases the timings for zero voltage phasors V0 and V7 goes on decreasing till it becomes zero and the inverter works as normal six step square wave inverter. When reference voltage vector is outside the circle, (in over modulation region), special techniques are required (ref).

3.7.1

Torque Control

Torque control is performed by comparing the command torque to the torque calculated from the measured values of stator flux and stator currents asTe =

3P (λds iqs − λqs ids ) 22

(3.85)

The torque error is processed and current space phasor error is used to obtain the switching states of the inverter. The information of current space phasor error ∆is and its derivative dtd ∆is along with previous voltage space vector, and the emf vector E position is used to determine appropriate non-zero or zero voltage vector of the inverter. A table of optimal switchings may be obtained based on the machine equation in stator coordinates.

68

3 Vector Control of Induction Motor Drives

qe

Oqr

0

Osqr

i sqs

qs

is

Tf

T sl

Or de

Ze

Zr

ds

Fig. 3.10 Phasor diagram for indirect vector control

3.8

Indirect Vector Control or Feed Forward Control

The indirect vector control method is similar to direct vector control except that In indirect method, the relative speed of the rotor flux linkage space vector with respect to the rotor is determined and integrated to obtain the angle of movement of the field with respect to the rotor. This angle is added to the measured angle moved by rotor to obtain θr . Indirect vector control is very popular in industrial applications. The phasor diagram shown in Figure 3.10 can be used to explain the principles involved in indirect vector control. Assume that rotor field reference frame is at an angle θf with respect to stationary (stator) reference frame Figure (3.10). This angle will be varying with time as the rotor flux linkage space vector λr rotates and is referred as field angle. The rotor rotates at an angular speed of ωr and the rotor field rotates at an angular speed of ωsl with respect to rotor. The synchronous speed of the field is given by ωs = ωr + ωsl andθf =

ωs dt =

(ωr + ωsl )dt

(3.86)

3.8 Indirect Vector Control or Feed Forward Control

69

Now for indirect vector control, the induction machine will be represented in the synchronously rotating reference frame. The motor is also assumed to be supplied through a current source inverter so that the stator currents can be directly controlled. For indirect vector control the control equations can be derived with the help of d-q model of the motor in synchronous reference frame as given below. The rotor equations are given below with superscript e to indicate synchronous reference frame dλedr + Rr iedr − ωsl λeqr = 0 dt

(3.87)

In the following equations superscript e is ignored. dλqr + Rr iqr − ωsl λdr = 0 dt

(3.88)

λdr = Lr idr + Lm ids

(3.89)

λqr = Lr iqr + Lm iqs

(3.90)

Where

From these equation rotor currents can be obtained in terms of stator currents as λdr − Lmids (3.91) idr = Lr iqr =

λqr − Lmiqs Lr

(3.92)

Substituting these values of rotor currents in equations for flux linkages we get dλdr Rr Lm Rr ids −ωsl λqr = 0 (3.93) + λdr − dt Lr Lr dλqr Rr Lm + λqr − Rr iqs −ωsl λdr = 0 dt Lr Lr

(3.94)

The rotor flux linkage can be assumed to be aligned with de axis, such that dλ λqr = 0 and dtqr = 0 and total rotor flux λr = λdr . Substituting these values in equations ——the new equations are obtained as Lm Lr dλr dλr Rr + λr = + λr = Lm ids Rr ids or dt Lr Lr Rr dt and

Lm Rr iqs = Lr λ r

ωsl

(3.95)

(3.96)

70

3 Vector Control of Induction Motor Drives

The field producing component of the stator current if = ids =

Lr dλr 1 ) (λr + Lm Rr dt

(3.97)

and the torque producing component of stator current iT = iqs = ωsl

Lr λ r Lm Rr

(3.98)

and the torque is given by Te =

3 P Lm λr iqs 2 2 Lr

where KT =

= KT λr iT

3 P Lm 2 2 Lr

(3.99)

(3.100)

the torque is proportional to the rotor flux linkages and the q axis component of stator current. If rotor flux linkages are kept constant, then the torque can be controlled by the q axis component of stator current. The stator current in abc frame can be obtained once the d and q component are known.

where

[iabc ] = [T]−1 [idq ]

(3.101)

2 cos θ f cos(θf − 2π ) cos(θf + 2π ) 3 3 [T] = 2π 3 sin θ f sin(θf − 2π 3 ) sin(θf + 3 )

(3.102)

As can be seen from these equations, the torque can be controlled if the information about θf and is is available. An indirect vector control scheme block diagram is shown in Figur 3.11. The speed error is generated from the speed reference ω∗r and actual speed ωr obtained through a position or speed sensor transducer. The reference torque T∗e is obtained as a function of speed error with the help of a speed error processor. In the same way the rotor flux linkage reference λ∗r is also obtained from the rotor speed by an absolute –value function generator. λ∗r is kept constant at 1 p.u. for speed in the range of 0 to 1 pu. For speeds beyond 1 p.u. it is varied as λ∗r =

ωr(rated) λr (rated) |ωr |

(3.103)

T∗e KT λ∗r

(3.104)

That means the flux is weakened above the rated speed so that the output power is kept constant. From the reference or desired values of the torque and rotor flux, the command values of if , iT and ωsl is produced by the current source indirect vector controller as i∗T =

3.8 Indirect Vector Control or Feed Forward Control

71

Or

Z

* r

i *qs

* e

T

Zr

i

PI + limiter

O*r

Tf

i

Slip calculator

³

Space vector PWM

Motor

Tf Lm 1 Tr s

Or

Rotor flux to station ary

* ds

i

* qs

* ds

Station ary to rotor flux

a-b-c/ d-q

Z sl

encoder

Ze

Zr

Fig. 3.11 Block diagram of Indirect Vector control

i∗f = and ω∗sl =

Lr dλ∗ 1 (1 + ) r Lm Rr dt

Lm Rr i∗T , and θ∗sl = Lr λ∗r

(3.105) ω∗sl dt

(3.106)

Also θf = θr + θsl From the torque and flux producing components of stator current and rotor field angle θf , the reference d –q axes currents are obtained as ∗ ∗ iqs iT cos θ f sinθf (3.107) = i∗f −sinθf cos θ f i∗ds From d-q component of stator current the line currents can be obtained as ⎡ ∗ ⎤ ⎡ 1 0 1⎤ ⎡ ∗ ⎤ iqs √ ias 3 ⎥ −1 − 1 ⎦ ⎣ i∗ ⎦ ⎣ i∗ ⎦ = ⎢ (3.108) ⎣ 2 2 bs ds 3 i∗cs 0 − 21 1 2 Selecting the q axis along the ‘a’ axis of the stator current, the stator currents can be written as ⎫ i∗as = i∗qs = |i∗s | sin θs ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ 2π ∗ ∗ ibs == |is | sin(θs − 3 ) (3.109) ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ i∗cs == |i∗s | sin(θs + 2π ) 3

72

3 Vector Control of Induction Motor Drives

These three phase stator current commands are generated as derived in equations (3.109) and can be implemented as shown in the flow chart in figure 3.12. These current commands are applied to the inverter with suitable

START

Select torque and flux commands Read motor parameters

calculate

i *f , i *T , Z sl*

computei *s , T *T , T*sl

computei*qs , i *ds

compute i a , i b and i c

STOP

Fig. 3.12 Flow chart for indirect vector control

3.8 Indirect Vector Control or Feed Forward Control

73

i '' qs i qs

i ' qs

i '' ds

i ' ds i ds

Fig. 3.13 Detuning eﬀect due to rotor mismatch

control so that the currents are forced by the inverter to follow the command values. The output of the inverter is applied to the motor. In indirect vector control ω∗sl and θ∗sl are function of the machine parameters. It is therefore, desirable that these parameters match the actual parameters of the machine at all operating conditions to reduce errors in the vector control. Since the rotor resistance changes with temperature and the leakage inductance changes with the magnitude of the stator current, the detuning of slip speed is a serious disadvantage in indirect vector control. The eﬀect of detuning of rotor resistance and corresponding coupling eﬀect can be explained from phasor diagram shown in Figur 3.13. Here Rr is actual rotor resistance and Rˆ r is the estimated resistance used to determine ω∗sl and θ∗sl . If Rˆ r is lower than Rr , the slip frequency ω∗sl will be lower than the actual value, and the q-d axis will be moved backward in position shown in figure as ′ ′ ′ iqs −ids . Now if the torque of the motor increases, the current iqs will increase. Since this current will have a component along d axis, it will result in increase in value of ids , thus increasing the flux. In this case the commanded value of torque and the flux both will be more than the actual value. Similarly, if the ˆ r is more than actual resistance Rr , the torque and the flux will be resistance R less than the actual value. During torque transients therefore, an oscillation is caused both in rotor flux linkages and torque response. The torque has a

74

3 Vector Control of Induction Motor Drives

settling time equal to the rotor time constant which is of the order of 0.5 seconds.

3.9

Case Study 1

Simulation of Vector Control Drive in MATLAB/SIMULINK Before simulating the drive in MATLAB/SIMULINK the induction motor is represented by system equations with following assumptions. 1. Air gap flux is assumed to be distributed Sinusoidally. 2. The motor is operating in linear magnetic region. 3. The stator windings are star connected with isolated neutral. 4. A two pole motor is assumed. The stator currents and rotor currents are represented in their own axis by the space vectors as2π

4π

2π

4π

is = ias ej0 + ibs ej( 3 ) + ics ej( 3 ) ir = iar ej0 + ibr ej( 3 ) + icr ej( 3 ) Similarly the space vectors for stator and rotor voltages and flux linkages can also be expressed. Transformation to a Common Reference Frame If rotor field axis is taken as reference, and it is at an angle of θ f from stationary axis, then irs = iss e−jθf In order to express the torque in terms of rotor flux space vector and stator current space vector, the d- axis of reference frame is attached to rotor flux axis, and q –axis is 90 degree ahead. This reference frame moves at the same speed as rotor field. The electromagnetic torque Te is given byL i

Te = KT λr iqs and slip speed ωsl = Tmr λqsr where Tr is rotor time constant. The rotor flux and the d component of current space vector are related by the following equation: r Tr dλ dt + λr = Lm ids and the slip speed is related to quadrature component of stator current space vector as –

L i

ωsl = Tmr λqsr . From these two equations if the reference currents i∗ds and i∗qs are known the rotor flux and slip speed can be obtained.

References

75

References [1] Boldea, I., Nasar, S.A.: Electric Drive. CRC Press, Boca Raton (1999) [2] Vithayathil, J.: Power Electronics, Principles and Applications. McGraw-Hill Inc., New York (1995) [3] Krishnan, R.: Electric Motor Drives, Modeling Analysis and Control. Prentice Hall, Englewood Cliﬀs (2001) [4] Bose, B.K.: Modern Power Electronics and AC Drives. Pearson Education Inc., London (2002) [5] Sathikumar, S., Vithayathil, J.: Digital simulation of field-oriented control of induction motor. IEEE Trans. Ind. Electron. IE-31, 141–148 (1984) [6] Vas, P.: Sensorless Vector Control and Direct Torque Control. Oxford University Press, Oxford (1998) [7] Bodson, M., Chiasson, J.N., Novotnak, R.T.: A Systematic Approach to Selecting Flux References for Torque Maximization in Induction Motors. IEEE Trans. on Control System Tech. 3(4), 388–397 (1995) [8] Erdman, W.L., Hoft, R.G.: Induction machine field orientation along air gap and stator flux. IEEE Trans. Energy Conversion 5, 115–121 (1990) [9] Gastli, A., Takeshita, T., Matsui, N.: A new stator-flux-oriented speedsensorless control algorithm for general purpose induction motor drive. Trans. Inst. Elect. Eng. Jpn. D 114-D(1), 9–16 (1994) [10] Blaschke, F.: The principles of field orientation as applied to the new transvector closed loop control system for rotating field machines. Siemens Review 34, 217–220 (1972) [11] Hotz, J., Bube, E.: Field oriented asynchronous PWM for high performance AC machine drives operating at low switching frequency. IEEE Trans. on Industry Applications IA-27(3), 574–581 (1991) [12] Kjaer, P.C., Kjellqvst, T., Delaloye, C.: Estimation of field current in Vector controlled Synchronous Machine Variable Speed Drives Employing Brushless Asynchronous Exciters. IEEE Trans. on Industry Applications 41(3), 834–840 (2005)

Chapter 4

Direct Torque Control and Sensor-Less Control of Induction Machine

4.1

Introduction

The application of vector control method for variable speed induction motor drives has been described in chapter 3. Generally, a closed loop vector control scheme results in a complex control structure as it consists of the following components; 1. 2. 3. 4. 5. 6. 7.

PID controller for motor flux and toque Current and/or voltage decoupling network Complex coordinate transformation Two axis to three axis transformation voltage or current modulator Flux and torque estimator PID speed controller

In a direct torque control system introduced by Takahashi and Depenbrock independently in 1986, the first five components are replaced by two hysteresis comparators and a selection table1−3 . This method therefore, results in highly simplified control structure compared to vector control. In the vector control scheme, it is assumed that the controllable power source can force any desired wave shape and value of current into the stator winding. But in practical circuits an inverter can produce only seven discrete space vector values of the actuating variable. In most cases none of these values is exactly equal to the desired instantaneous value of the space vector. Although by using high switching frequency in a PWM inverter the desired curves of the actuating variable can be suﬃciently approximated. However, for high power drives, the switching frequency can not be more than 200-300 Hz due to economic reasons. Thus, in high power drives, it is very diﬃcult to apply a current wave

M. Ahmad: High Performance AC Drives, Power Systems, pp. 77–96. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

78

4 Direct Torque Control and Sensor-Less Control of Induction Machine

shape of desired magnitude and shape. The vector control therefore, can not provide very fast control required in many drives. In direct torque control instantaneous values of torque and flux are calculated from primary variables and controlled independently by using an optimum switching table. The controllers for a direct torque control drive do not require complex coordinate transformation essential in all vector controlled drives. Instead the decoupling of non-linear ac motor structure is obtained by the use of ‘on –oﬀ’ control of inverter switches. The voltage vector is selected from the inverter feeding the motor with the help of hysteresis controllers.

4.1.1

Sensorless Control

The vector controlled ac motor drives require speed or position sensors. The presence of sensors results in many disadvantages in terms of cost, reliability, motor size, and noise immunity. The sensorless drives have therefore been developed for precise control of torque and speed. In these drives the speed of the motor is estimated from the applied voltage, line current and frequency. The direct torque control and sensorless drives are very popular now and is a subject of discussion in this chapter. Sensorless drives are now well established in those industrial applications where persistent operation at lower speed is not required.

4.2

Direct Torque Control Basics

The direct torque control is diﬀerent from vector control in such a way, that it does not control the flux through the current control, but it directly controls the flux itself. The direct torque control is also diﬀerent from vector control in the sense that the reference frame here is stator flux instead of rotor flux used in vector control. Direct torque control is in fact an extension of direct vector control and direct self control. Direct vector control theory is quite well known and has been used in industrial drives. As in case of Vector control, in direct torque control method also, the flux and the torque are either measured or estimated and used as feedback signals for the controller. The input to the direct toque controller are the torque error, error in magnitude of the stator flux space vector, and the angle of the stator flux space vector, from which the states of the power switches are determined. Based on this information, a certain voltage vector or combination of voltage vectors is directly applied to the inverter with a certain average timing. This gives the induction motor drives a very fast response.

4.2 Direct Torque Control Basics

4.2.1

79

Torque and Flux Control

The torque in an induction motor in stator reference frame can be expressed as3P s s λi (4.1) Te = 22 ss In order to apply flux control the stator current iss is to be replaced by rotor flux vector from following expressions.

Also λs =

λss = Ls is + Lm ir

(4.2)

λr = Lr ir + Lm is

(4.3)

Lm λr + (Ls Lr − L2m )is Lr

(4.4)

From these equations the torque can be written asLm 3P λr λs 2 2 Lr (Ls Lr − L2m )

(4.5)

3P Lm |λr | |λs | sinγ 2 2 Lr (Ls Lr − L2m )

(4.6)

Te = The magnitude of torque is Te =

Where γ is the angle between the stator and rotor flux, known as torque angle. In direct torque control the stator flux is a state variable that is controlled by stator voltage using the following equation dλs = vs − Rs is dt

(4.7)

Here vs is the inverter output voltage applied to the motor. If the motor is supplied through a voltage source inverter, the inverter voltage vectors with six active vectors, and two zero vectors are shown in Figure 4.1. From equation 4.7, neglecting stator resistance the stator flux isλs =

vs dt

(4.8)

For six step operation, the inverter output voltage consists of a cyclic and symmetric sequence of active vectors, so that in accordance with Eq. 4.8 stator flux moves with constant speed along a hexagonal path. The application of zero vectors stops the flux, but does not change its path.

80

4 Direct Torque Control and Sensor-Less Control of Induction Machine

V2 (110)

3

V*

1

D

V1 (100)

4

6 5

V5 (001)

V6 (101)

Fig. 4.1 Six active states and two zero states of three-phase inverter

For production of torque the Eq. 4.6 is important, which clearly indicates that the relative angle between the rotor flux and stator flux aﬀects the torque. Suppose the rotor flux λr is moving slowly in the anticlockwise direction, if stator flux is moved in clockwise direction by the active voltage vector the angle γ increases rapidly and torque is increased. On the other hand if zero voltage vector is used to stop the stator flux the torque angle and the torque, both will decrease. Thus by cyclic switching of active and zero vectors the torque of the motor can be easily controlled.

4.3 DTC Control Strategy

81

O sref

Flux and torque estimator

Os W

Inverter

Voltage vector selector

W ref

Fig. 4.2 Block diagram of DTC scheme

4.3

DTC Control Strategy

The basic concept of direct torque control as enumerated by Depenbrock and Takahashi is summarized below. The speed of the induction machine mainly depends on the angular speed of the rotating magnetic field. In steady state this speed depends on the number of poles and the frequency of the supply connected to stator. The magnitude of the magnetic flux depends on the voltage to frequency ratio. As shown in Eq. 4.6, if the rotor flux remains constant and stator flux is changed incrementally by the stator voltage, there is corresponding change in angle γ and an incremental change in the torque. Figure 4.2 shows the basic blocks of a direct torque control scheme. There are three blocks that process the information that is applied to the inverter supplying power to the motor. These blocks are:(i) Torque and flux processor (ii) Optimal switching logic block, and (iii) Adaptive motor model block. The torque and flux processing is performed in a hysteresis block. In the torque and flux processor block the reference torque is compared with the actual torque, and the reference flux with the actual flux. The actual values of these quantities are obtained from the adaptive motor model. When the actual torque value drops below its diﬀerential hysteresis limit, the torque status output goes high. Similarly, if the actual torque value rises above the diﬀerential hysteresis limit, the torque status output goes low. The same function is performed by the flux comparator. The upper and lower diﬀerential limit switching points for both torque and flux are determined by the hysteresis window input. This input is used to vary the diﬀerential hysteresis limit windows, such that the switching frequencies of the power output devices are maintained within the range of 1.5 to 3.5 KHz. The voltage vector selector block is an ASIC block that processes the torque status output and flux status output. The function of the optimum

82

4 Direct Torque Control and Sensor-Less Control of Induction Machine

switching logic is to select the appropriate stator voltage vector that will satisfy both the torque status output and flux status output. As described in chapter 3, there are eight switching states of a three-phase inverter, out of which two are zero states. The optimal switching table of the inverter is obtained using the torque and flux comparator outputs, and the position of the stator flux linkage space vector. For controlling the torque of the motor, the inverter switching states are selected based on the following criterion. The torque developed is proportional to the cross product of stator flux vector λs and the rotor flux vector λr . The stator flux vector is kept constant, and the torque is controlled by varying the angle between the stator flux vector and rotor flux vector. Since the time constant of rotor is much higher than the stator time constant, the rotor flux does not change much during the time of interest. If an increase in motor torque is required due to increase in load torque, the optimal switching logic selects a stator (inverter) voltage vector such that the angle between stator flux vector and rotor flux vector increases. When a decrease in torque is required, the optimum switching logic selects a zero voltage vector, which allows both stator flux and produced torque to decay naturally. If stator flux decays below its normal lower limit, the flux status output requests an increase in stator flux. If the torque status output is low, anew stator voltage vector is selected that tends to increase the flux but reduces the angle γ between the stator and rotor flux vectors. The combination of torque and flux comparators, combined with optimal switching logic eliminates the need for traditional PWM modulator. This presents major advantage, as the small signal delay associated with modulator are eliminated. The Adaptive motor model is required to calculate the actual values of stator flux, torque produced, reference speed, and frequency. The actual flux and torque values are critical to direct torque control. These values are therefore calculated every 25 microsecond. The reference speed and frequency are required by the outer speed loop and are calculated every millisecond. The motor adaptive model is developed from the motor parameters and from the measured stator currents, link voltage, and power switches positions The DSP is responsible for calculating all motor variables required in DTC. Determination of actual stator flux is the most important step in the adaptive motor model. the following two equations are used to determine the stator flux. (4.9) λs = (vs − Rs is )dt and λs = Ls is + Lm ir

(4.10)

The initial estimate of the flux is obtained from first equation which is then fine tuned using second equation. Once the flux is determined the torque is calculated using the following equation

4.3 DTC Control Strategy

83

Te = Kλs × is

(4.11)

In order to obtain the values of actual frequency and speed, the rotor flux vector and rotor flux angle is calculated using the following equationsλr =

Lr Lm (λs − σLs is )

= λrd + jλrq

(4.12)

where σ is the leakage factor. Rotor flux angle θr = arctan

λrq λrd

(4.13)

The complete block diagram of direct torque control drive is shown in Figure 4.2. The speed control block compares the reference speed with the actual speed feedback from the motor adaptive model. This block contains the traditional PID controller. The output of this block is the reference torque (speed) for the torque controller. The torque controller has one more input as absolute torque reference. If the drive is used for speed control the speed (torque) reference is used. If torque control drive is used then only the absolute torque reference is used. The reference torque is compared with the actual torque in the hysteresis torque controller as discussed earlier. The flux level reference and actual frequency feedback are the inputs to flux reference control block. The actual frequency feedback is used to provide frequency sensitive flux manipulation. The flux reference is compared with the actual flux in flux hysteresis comparator as shown in Figure 4.2. The block showing switching frequency control is used to limit the switching frequency within a minimum and maximum level. The minimum frequency is decided on the basis of voltage waveform, and the maximum frequency is decided based on the switching frequency of the power devices. The function of the switching frequency control is to vary the size of hysteresis windows of the flux and torque comparators to limit the switching frequencies within 1.5 to 3.5 K Hz. Compared to Vector control ( FOC) scheme, the DTC scheme has the following features. • • • •

There are no current loops; hence the current is not regulated directly. Coordinate transformation is not required. There is no separate voltage pulse width modulation Stator flux vector and torque estimation is required.

Depending on how the switching sectors are selected, two diﬀerent DTC switching schemes are used. The scheme proposed by Takahashi operates with circular stator flux vector path, whereas the scheme of Depenbrock operates with hexagonal stator flux vector path. The two switching sector selections are shown in Figure 4.3.

84

4 Direct Torque Control and Sensor-Less Control of Induction Machine

sectorIII

sectorII

V3 V2

V4

sectorI

sectorIV V1

V5

sectorV

V6

sectorVI

(a)

V3

V2

V4 V1

V5 V6 (b)

Fig. 4.3 (a). Circular stator flux vector path (b). Hexagonal stator flux vector path

4.4 Switching Table Based DTC Scheme

85

O sref

Flux and torque estimator

Os W

PWM Inverter

Voltage vector selector

W ref

Fig. 4.4 Switching Table based DTC scheme

4.4

Switching Table Based DTC Scheme

The Switching table based direct torque control scheme with circular stator flux vector path is shown in Figure 4.4. The command stator flux and torque values are compared with actual flux and torque values in hysteresis flux and torque controllers respectively. The flux controller is a two- level comparator while the torque controller is a three-level comparator. If the output of torque hysteresis comparator is denoted by τ, then τ= -1 means that the actual value of the torque is above the reference value and outside the hysteresis limit. τ=1 means that the actual value of torque is less than the reference value and outside the hysteresis limit. τ=0 means the torque is within the hysteresis limit. The flux hysteresis comparator output is denoted by φ. If φ=0, it means that the actual value of the flux linkages is above the reference value and outside the hysteresis limit. φ=1 means the actual value of the flux linkage is below the reference value and outside the hysteresis limit. These digitized variables representing the output of torque and the stator flux region R obλ tained from the angular position γ = Arctan( λsq ) create a digital word which sd is used as address for accessing EPROM. The EPROM contains the selection table as shown in Table 4.1. Table 4.1 φ=1

φ=0

τ=1 τ=0 τ = −1 τ=1 τ=0 τ = −1

R(1) V2 (110) V(000) V6 (101) V3 (110) V(000) V5

R(2) V3 (010) V(000) V1 V3 (010) V(000) V6

R(3) V4 (011) V(000) V2 V4 (011) V(000) V1

R(4) V5 (001) V(000) V3 V5 (001) V(000) V2

R(5) V6 (101) V(000) V4 V6 (101) V(000) V3

R(6) V1 (100) V(000) V5 V1 (100) V(000) V4

86

4 Direct Torque Control and Sensor-Less Control of Induction Machine

The characteristic features of this scheme are summarized below. •

The stator flux and current waveforms are nearly sinusoidal; the harmonic content depends on the flux and torque controller hysteresis bands. Excellent dynamic torque behavior. The inverter switching frequency is determined by the flux and torque hysteresis bands.

• •

Number of modifications has been made on the basic ST-DTC scheme to improve its performance by using modified switching table. Torque ripples can be reduced by dividing the sampling periods into two or three equal intervals thereby increasing the switching voltage vectors to 12 or 56. Also rotor flux amplitude may be controlled to increase the overload torque capability.

4.4.1

Direct Self Control Scheme

The block diagram of DSC scheme proposed by Depenbrock is shown in Figure 4.5. The flux and torque estimator block has voltage and currents as input. The output of this block is the flux and torque. From the flux values thus obtained, the flux comparators generate digitized variables dA , dB and dC which correspond to active voltage vectors for six-step operation of the inverter. The hysteresis torque controller generates the digitized value d0 that

W

O* O

W*

Fig. 4.5 Depenbrock scheme for DTC

4.5 Sensorless Control of Induction Motor

87

determines the zero state duration. The control algorithm for constant flux region is therefore of the formIf d0 =1, then inverter switching states are SA = dB , SB = dC and SC = dA that means active vector is selected. For d0 = 0 zero vector is selected, ie. SA = 0, SB = 0, SC = 0 or SA = 1, SB = 1, SC = 1 The main features of DSC scheme are summarized below. • The DSC scheme has a PWM operation in constant flux region and six step operation in flux weakening region. • Non-sinusoidal current and flux waveforms in both the regions. • Stator flux vector moves in hexagonal path in PWM operation also. • The inverter frequency is lower than in Switching Table based DTC. • However the behavior of DSC based DTC can be obtained from Switching Table based DTC scheme by properly selecting the hysteresis band of stator flux comparator.

4.4.2

Main Features of DTC

The main features of DTC are summarized below• DTC operates with closed torque and flux loops, but without current controllers. • DTC needs stator flux and torque estimation. It is therefore insensitive to rotor parameters. • DTC is basically a motion sensorless scheme. • DTC has simple and robust control structure.

4.5

Sensorless Control of Induction Motor

Vector control schemes of induction motor require speed or position sensor. The speed sensor has several disadvantages in terms of cost, reliability, drive size, and noise immunity. As real time computation costs are continuously decreasing, speed and position estimation can be performed by using software based state estimation techniques. Various approaches have been proposed in the literature for estimation of speed using stator voltage, phase currents and frequency measurements. Techniques for obtaining the speed information of an induction motor without using the speed encoder can be broadly classified as1. Open loop speed control with slip compensation. 2. Closed loop control with speed estimation

88

4 Direct Torque Control and Sensor-Less Control of Induction Machine

In the first case motor synchronous speed is regulated and the estimated slip frequency is used to compensate for load changes. In second method, the motor speed is estimated and used as a feedback signal for closed loop speed regulation. Based on the methods of implementation following techniques can be used for sensorless control of induction motor drives employing vector/ direct torque control. 1. 2. 3. 4. 5. 6.

Slip frequency calculation method Speed estimation using state equations Flux estimation method Model reference adaptive systems (MRAS) Observer ( Kalman, Luenberger) based methods Artificial intelligence methods for speed estimation

Slip Frequency Calculation Methods The slip frequency of an induction motor is the diﬀerence between the stator frequency and the electrical frequency corresponding to rotor speed. By calculating the slip frequency, the speed of the motor can be obtained. The slip frequency ωsl is related to stator frequency as ωsl = ωe −ωr

(4.14)

Where, ωe is the stator frequency, and ωr is the frequency of rotor. A number of methods have been suggested for calculation of slip frequency. In one method5 the rotor frequency is calculated directly from the phase lag between the stator voltage and the stator current. The slip frequency is then calculated using motor parameters and stator current. From the steady state equations obtained from the equivalent circuit of the induction motor the following relationship between the rotor frequency ωr and the phase angle φ is obtained. Dω2r −E (4.15) φ = arctan (Aω2r + Bωr + C) Where A=R1 L22 ; B=ω1 L2m R2 ; C=R1 R22 ; D= ω1 (L2 L2m − L1 L22 ); and E=ω1 L1 R22 . From above equation a curve between slip frequency and phase angle can be obtained. The Eq. (4.15) is valid for limited range of slip frequency and can be used when fast dynamic response is not required. Speed Es timation Using State Equations The slip frequency is obtained as a function of rotor EMF obtained in stationary reference frame. In this method only measurement of phase voltages and currents is required6,7 The state equations are modified to express speed in terms of motor parameters and measured quantities.

4.5 Sensorless Control of Induction Motor

89

The stator and rotor voltage equations of the induction machine in the stationary reference frame can be written asvds = Rs ids +

d λds dt

(4.16)

vqs = Rs iqs +

d λqs dt

(4.17)

0 = Rr idr +

dλdr + ωr λqr dt

(4.18)

0 = Rr iqr +

dλqr − ωr λdr dt

(4.19)

3P (iqs λds −ids λqs ) 22

(4.20)

Te = and

dωr 1 = (Te − TL ) (4.21) dt M From the above dynamic equations, the expression for slip frequency can be obtained as Lm eqr .iqs + edr .ids ωsl = ωe .Rr (4.22) Lr e2dr + e2qr Where the rotor EMFs can be expressed in terms of inductances as Lm Rr dλdr = edr = Rr ids − λdr − ωr λqr (4.23) dt Lr Lr dλqr eqr = = dt

Rr Lm Rr iqs − λqr + ωr λdr Lr Lr

(4.24)

Model Reference Adaptive System Model reference Adaptive Systems approach makes a comparison of two machine models of diﬀerent structure, that estimate the same state variable on the basis of diﬀerent set of input variables. One model does not include speed and is called the reference model. The other, which includes the speed also, is called the adjustable model. The error between the two models is used to derive an adoption model that produces the estimated speed for the adjustable model. The adjustments in the adjustable models are made so that the error between the two models vanishes to zero. A block diagram for estimation of speed using MRAS technique is shown in Fig. 4.6. The reference model is obtained as stator voltage equations. The inputs to this model are motor stator voltage and current signals. The output is calculated in the form of rotor

90

4 Direct Torque Control and Sensor-Less Control of Induction Machine

IM

i sqs i sds

Reference model

ª *s º «O dr » «O sqr » ¬ ¼

s Lr °ªv ds º ªR s VL s S ®« s » « Lm ° 0 ¯¬«v qs ¼» ¬

Stator Eq

s º ªi ds º ½ ° R s VL s S »¼ «¬«i sqs »¼» °¾¿

0

Osdr

Osqr

X

Adaptive model

X

X

I M

Y

Oˆ sdr

Rotor Equations

Oˆ sqr Estimated speed

ˆr Z

Fig. 4.6 Speed estimation for MRAS system

flux vector λsdr and λsqr . The adaptive model of the motor is obtained using stator current inputs and an estimated speed signal (assuming it is available). ˜ s and λ ˜ s . If the The output from this model is also the rotor flux linkages λ qr dr estimated speed is correct the fluxes calculated from the reference model and the adaptive model must have the same value. An adaptation algorithm with ˜ r so that the error ξ = 0. The PI PI control can be used to tune the speed ω control is implemented as˜ r = ξ(Kp + ω and

KI ) s

˜ s − λs λ ˜s ξ = (λsqr λ dr dr qr )

(4.25)

(4.26)

Luenberger Speed Observer A device that estimates or observes state variables of a system is called a state observer. A state observer utilizes measurements of the system inputs and outputs and a model of the system based on diﬀerential or diﬀerence equations. Three main quantitative state observers are: Luenberger observer, adaptive observer and Kalman filter. In the deterministic case, when no random noise is present, the Luenberger observer and its extensions are used for time-invariant systems with known parameters. The equation for the Luenberger observer contains a term that corrects the current state estimates by an amount proportional to the prediction error: the estimation of the current output minus the actual measurement. An observer is basically an estimator

4.5 Sensorless Control of Induction Motor

91

that uses full or partial plant model, and a feedback loop with measured plant variables. In the case of speed observer first the stator currents isds and isqs and the rotor fluxes λsdr and λsqr are calculated through a full order Luenberger observer based on stator and rotor equations in stator coordinates. The rotor voltage equations can be written from voltage model asvsdr = 0 = isdr Rr +

d s (λ ) + ωr λsqr dt dr

(4.27)

vsqr = 0 = isqr Rr +

d s (λ ) + ωr λsdr dt qr

(4.28)

Also λsdr = Lm isds + Lr isdr

(4.29)

λsqr = Lm isqs + Lr isqr

(4.30)

By eliminating isdr the following equation is obtained. d s Rr Lm Rr s λ = − λsdr − ωr λsqr + i dt dr Lr Lr ds

(4.31)

d s Lm Rr s Rr λqr = − λsqr + ωr λsdr + i dt Lr Lr qs

(4.32)

Similarly for q axis

The stator currents isds and isqs , in terms of machine parameters can be expressed as 2 Lm Rr + L2r Rs s Lm Rr s L m ωr s 1 s d s (i ) = − λ + λ + v (4.33) ids + dt ids σLs L2r σLs L2r dr σLs Lr qr σLs ds 2 Lm Rr + L2r Rs s d s Lm Rr s L m ωr s 1 s (iiqs ) = − λdr + λqr + v iqs − 2 2 dt σLs Lr σLs Lr σLs Lr σLs qs

(4.34)

2

Where σ = 1 − LLsmLr . The state variable equations can now be written in the form of ˙ = [A] [X] + [B] [U] X

where

(4.35)

T [X] = isds isqs λsds λsqs

(4.36)

T [U] = vsds vsqs 0 0

(4.37)

92

4 Direct Torque Control and Sensor-Less Control of Induction Machine

Fig. 4.7 Block Diagram of Luenberger method of speed control

⎡

⎢ ⎢ ⎢ [A] = ⎢ ⎢ ⎣

−(L2m Rr +L2r Rs ) σLs L2r

0 L m Rr Lr

0 −(L2m Rr +L2r Rs ) σLs L2r

0 L m Rr Lr

0 ⎡

1 σLs

⎢ 0 [B] = ⎢ ⎣ 0 0

0 1 σLs

(Lm Rr ) σLs L2r −Lm ωr σLs Lr − RLrr

ωr

⎤

⎥ ⎥ 0 ⎦ 0

Lm ωr σLs Lr

⎤

⎥ ⎥ ⎥ ⎥ ⎥ −ωr ⎦

L m Rr σLs L2r

(4.38)

Rr Lr

(4.39)

Fig. 4.7 shows the block diagram of Luenberger observer using the above machine model. The estimated values are shown with ∧ . The output current ∧

∧

ds

qs

signals i and i are obtained from the following equation (Here superscript ‘s’ is not written) ⎤ ⎡ ⎡∧⎤ ids

i ⎥ 1000 ⎢ ⎢ ds ⎥ ⎢ iqs ⎥ ⎣∧⎦ = ⎣ λdr ⎦ 0100 i qs λqr

(4.40)

The input voltage signals vds andvqs are measured from stator terminal of the machine. As shown in Fig. 4.8, the speed adaptation algorithm utilizes the speed adaptive flux observer obtained from the machine model. The observer equation is given by d ˆ ˆX ˆ + BVs + G(ˆis − is ) (X) = A dt

(4.41)

4.5 Sensorless Control of Induction Motor

93

W

v dc i qs

Z r*

Z

v qs

v ds

r

i

va vb vc

ia ib

ds

Fig. 4.8 Block Diagram of Extended Kalman Filter method of speed control

where ˆis = [ids iqs ] and G observer gain matrix. The gain matrix G multiplied with the error signal e = ˆis − is applies the corrective signal such that e becomes zero. If speed signal ωr in matrix A is known the fluxes and currents can be solved from the state equations. The speed adaptive flux observer permits estimation of the unknown speed ωr . The estimation error in the stator currents and rotor fluxes is expressed by the equation: d ˆ (e) = (A + GC)e − ∆AX dt

(4.42)

Where ˆ e = X − X,

ˆ − A, and ∆ωr = ω ˆ r − ωr ∆A = A

(4.43)

To derive the speed adaptive algorithm the following Lyapunov function is considered: ˆ r − ωr )2 (ω V = eT e + (4.44) λ Using Lyapunov’s theorem the following adaptation scheme for speed estimation can be obtained, ˆr dω ˆ qr − eiqs λ ˆ dr )Lm /(σLs Lr ) = λ(eids λ dt

(4.45)

here eids = ids − ˆids and eiqs = iqs − ˆiqs . The gain matrix is so chosen as to make the speed adaptive flux observer stable. In this method the estimation error becomes large at low speeds due to the eﬀects of stator and rotor resistance variation.

94

4 Direct Torque Control and Sensor-Less Control of Induction Machine

Extended Kalman Filter Observer The Luenberger observer is a deterministic observer, and is applicable to linear time-invariant systems In the extended Kalman filter, (EKF) the state transition and observation models need not be linear functions of the state but may instead be (diﬀerentiable) functions. In EKF the state variables are only the stator currents and the magnetizing current. The block diagram of EKF algorithm is shown in Fig. 4.8. The induction motor model in stationary reference frame with stator currents ids , iqs and rotor fluxes λdr , λqr as state variables is as foillowsdX = AX + BU (4.46) dt and Y = CX (4.47) Where

T X = ids iqs λdr λqr ωr

(4.48)

T Y = ids iqs

(4.49)

⎡

1 σLs

⎢ 0 [B] = ⎢ ⎣ 0 0 ⎡

⎢ ⎢ ⎢ ⎢ [A] = ⎢ ⎢ ⎢ ⎣

−(L2m Rr +L2r Rs ) σLs L2r

0 L m Rr Lr

0 1 σLs

⎤

⎥ ⎥ 0 ⎦ 0

0 −(L2m Rr +L2r Rs ) σLs L2r

0 L m Rr Lr

0 0

0 C=

10000 01000

(4.50)

(Lm Rr ) Lm ωr σLs Lr σLs L2r Lm ωr (Lm Rr ) − σLs Lr σL L2 s r − RLrr −ωr ωr − RLrr

0

0

⎤ 0⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎦ 0

(4.51)

(4.52)

In the dynamic model of an induction motor, if the dimension of state vector is increased, by adding the angular speed of motor, then state model is of fifth order, as well it becomes non-linear. In this case the speed of the rotor is considered as a state and a parameter. The extended Kalman filter algorithm is calculated using a microprocessor, and the system is expressed in a discrete form. The discrete model of the induction motor with noise sources is X(k + 1) = Ad X(k) + Bd U(k) + V(k)

(4.53)

Yk) = Cd X(k) + W(k)

(4.54)

References

95

Where W (k) and V (k) are zero mean white Gaussian noise vectors of Y(k) and X(k) respectively. Both W (k) and V (k) are independent of Y(k) and X(k) respectively. The statistics of noise and measurements are given by three covariance matrices, Q, R and P. The system noise vector covariance matrix, and system state vector covariance matrix P are 5 × 5 matrices, whereas the measurement noise vector covariance matrix is 2 × 2 only. The EKF has two main stages: the prediction stage and filtering stage. In prediction stage, the next predicted values of states X∗ (k + 1) are calculated by the machine model and the previous values of estimated states. The prediction of state is given by the following equation ∗

∗

X (k + 1) = X (k) +

tk=1

f[x(t), U(t)dt

(4.55)

tk

Here U(t) is assumed to remain constant during tk to tk+1 . The predicted state covariance matrix P∗ (k + 1) is obtained using the system noise covariance vector Q.

References [1] Depenbrock, M.: Direct Self-control (DSC) of Inverter Fed Induction Machine. IEEE Transactions on Power Electronics 3(4), 420–429 (1988) [2] Takahashi, I., Ohmori, Y.: High Performance Direct Torque Control of an Induction Motor. IEEE Transactions on Industry Applications 25(2), 257–265 (1989) [3] Takahashi, I., Noguchi, T.: A new quick response and high eﬃcient control strategy of an induction motor. IEEE Transaction Industry Application 22(5), 457–464 (1986) [4] Ludke, I., Jane, M.G.: A comparative study of high Performance speed Control strategies for voltage Source PWM Inverter fed Induction Motor Drives. In: Seventh International Conference on Electric Machines and Drives, UK (September 1995) [5] Baader, U., Depenbrock, M.: Direct Self Control (DSC) of inverter fed induction machine: A basis for Speed control without speed measurement. IEEE Transaction Industry Application 28, 581–588 (1992) [6] Nash, J.N.: Direct torque control, Induction motor vector control, without an encoder. IEEE Transaction Industry Application 33(2), 333–341 (1997) [7] Buja, G.S., Kazmierkowski, M.P.: Direct torque control of PWM Inverter -Fed AC motors- A survey. IEEE Transaction Industrial Electronics 51(4), 744–757 (2004) [8] Bose, B.K.: Modern Power Electronics and AC Drives. Pearson Education Inc., London (2002) [9] Buja, G.: A new control strategy of Induction Motor drives: The direct flux and torque control. IEEE Industrial Electronics Newsletter 45, 14–16 (1998)

96

4 Direct Torque Control and Sensor-Less Control of Induction Machine

[10] Holtz, J.: Sensorless Speed and position control of Induction Motors: Tutorial. In: IEEE Industrial Electronics Annual Conference, IECON (NovemberDecember 2001) [11] Ohtami, T., Takada, N., Tanaka, K.: Vector Control of Induction Motor without Shaft encoder. IEEE Transactions on Industry Applications 28(1), 157–165 (1992) [12] Holz, J.: Sensorless Position Control of Induction Motors-an emerging technology. IEEE Transactions on Industrial Electronics 45(6), 840–852 (1998) [13] Jansen, P.J., Lorenz, R.D., Novotony, D.W.: Observer based direct field orientation and comparison of alternative methods. IEEE Transactions on Industry Applications 30(4), 945–953 (1994) [14] Kim, Y.R., Sul, S.K., Park, M.H.: Speed sensorless vector control of induction motor using extended Kalman filter. IEEE Transaction Industry Applications 30(5), 1225–1233 (1994)

Chapter 5

Control of Permanent Magnet Machine (PM)

5.1

Introduction

While induction motors are the most commonly used motors for most of the simple applications, much attention is currently being given to permanent magnet motors, specially as servo drives. Permanent magnet AC (PMAC) machines use magnets instead of windings to produce the air gap magnetic field. The replacement of rotor winding by magnets results in simplification in construction, reduction in losses, and improvement in eﬃciency. With the availability of Samarium –Cobalt and neodymium-iron –boron materials for permanent magnets the replacement of dc motors and induction motors with PM motors is on the increase. The PM motors are basically synchronous machines and can be operated at unity power factor, thus making them more eﬃcient than induction motors. The cost of PM motor is more than the induction motor due to cost of the magnet, but due to its high eﬃciency the running cost is smaller. The PM motors are broadly classified on the basis of direction of field flux as (1) Radial field (2) Axial field In radial field machines the direction of flux is along the radius of the machine. These machines are more common than axial field machines in which the direction of flux is parallel to the rotor shaft. Axial field machines are finding applications in high performance drives due to their high power density and acceleration. The permanent magnet motors are also classified as(a) Permanent Magnet Synchronous Motor (PMSM) or sinusoidally excited PM motors (b) Trapezoidally excited PM motors, or Brushless DC motors (BLDM) or (BLDC), or simply switched PM motors. A sinusoidal machine can be a surface permanent magnet (SPM) type or an interior or buried permanent magnet (IPM) type. These motors with their M. Ahmad: High Performance AC Drives, Power Systems, pp. 97–128. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

98

5 Control of Permanent Magnet Machine (PM)

Fig. 5.1 Interior permanent, magnet motor

rotor geometries are shown in Figure 5.1. In PM motors the rotor has an iron core that may be solid or may be made of punched laminations. Thin permanent magnets are mounted on the surface of this core using adhesives. In the SPM machines the magnets are mounted on surface of the outer periphery of rotor laminations. This arrangement provides highest air gap flux density but lacks mechanical robustness. These machines can not be used for high speed applications. Also surface type configuration is only possible with radial flux magnetization. In interior PM machines the magnets are placed in the middle of the rotor laminations, resulting in robust construction. These machines are more suitable for high speed drives. The stator of the PMSM has conventional distributed three phase winding located in the stator slots. Magnets of alternately opposite magnetization produce radially directed flux density across the air gap. This flux density then interacts with the stator currents to produce torque. The stator of a trapezoidal PMAC motor has concentrated windings in the stator and a rotor with wide pole arc. For two pole motor, the rotor magnets extend around approximately 180 degree peripherally. The diﬀerences in construction between these two classes of machines result in diﬀerent control requirements. The electrical supply system of switched PM motors is designed to provide a rectangular pulse of current that can be switched sequentially to pairs of the three stator terminals as shown in Figure 5.2. The motor is normally supplied through a 3-phase inverter and the six switches are sequentially switched in pairs. The basic waveforms of applied current are shown in

5.1 Introduction

99 i

i

a

b

c

Fig. 5.2 Power supply of switched PM motor

Ip

S T1 T1

T2

S T2

Fig. 5.3 Phase current waveform

Figure 5.3. Although the waveform is trapezoidal, but it is 3-phase in nature with 120 degree phase displacement between each phase. As shown in the Figure 5.2 the current i is switched to enter phase a, and leave phase c. The torque is produced by the action of this current with the magnetic field of the rotor. The supply to phase b and c is connected in sequence, using switches from signals obtained from position sensors. Six steps of switching cycles are required per revolution for a two-pole motor. At any instant therefore, only two-thirds of the stator produces torque. Since all PMAC motors are used for variable speed applications, some form of power electronic converter is required to transform the fixed frequency into variable frequency. Although, cyclo-converters and matrix converters can directly convert fixed frequency into variable frequency, the common method is to use two stage ac-dc-ac conversions. For three phase machines the basic converter topology is same for switched PM or sinusoidal PM motors. The diﬀerence in the voltage waveform is due to the method of switching used.

100

5 Control of Permanent Magnet Machine (PM)

Both voltage source and current source inverters are used to supply power to PMAC motors. Voltage source inverters are more common due to the following reasons. The cost, size and weight of electrolytic capacitor in the dc link are significantly lower than dc link inductor for the same rating of voltage and current source inverters respectively. Also, the new gate controlled switching devices such as IGBT are more suited for voltage source inverter configurations. The PMSM or sinusoidal PMAC machines have distributed windings on the stator to provide sinusoidal distribution of field. Also, these machines have salient pole structure in the rotor, whereas the trapezoidal PMAC machines have non salient structure. The IPM machine has a smaller eﬀective air gap compared to SPM machine, and has stronger armature reaction eﬀect. The PMAC machines are particularly attractive in drive applications that require simple control and minimum number of sensors. However, the drives with trapezoidal excitation have the disadvantages of higher torque ripples, and limited range of speed in constant power type of applications. The PMAC machines have found applications in several drives as described here. The drives requiring high dynamic response such as machine tool, and robotic actuator drives use PMAC machines because of their high power density. The absence of rotor losses also accounts for higher eﬃciency required by servo drives specially used in low speed range. PMAC machines are also finding applications in wide range of commercial and residential applications. The ability to control speeds in blowers; compressors have made these motors more attractive than induction machines. Another application area for sinusoidal PMAC machines is in electric vehicles because they require wide range of constant power operation. The BLDC PM motor is extremely common in wide variety of low power applications such as computers and oﬃce machinery.

5.2

Design Considerations

The advances in permanent magnet materials, power electronic devices and control strategies have given rise to a great variety of PMSM designs. There is a choice for the designer on selection of magnetic materials and configurations, the number of poles and the placement of conduction cage winding. The choice of magnetic material is important from economic and performance considerations. Alnico is used in small power machines because of low coercivity, and nonlinear demagnetization curve. Ferrite is low in cost, has linear demagnetization characteristics, but has low remnance. These materials are therefore suitable for cost factor is more important and high performance is not required. Samarium –Cobalt has high residual flux density and coercivity, but has high cost and is more suitable for high performance servo drives. Neodymium-Boron-Iron material is found to have properties equivalent to Samarium-Cobalt with low material cost. The problems due to corrosion of

5.3 Modeling of PMSM

101

iron must be given due consideration in selecting this material for Permanent magnet. The geometry of rotor poles also plays an important role in the performance of the motor. The surface magnet motor results in a small rotor diameter with low inertia and are suitable for servo drives with high dynamic performance. In interior magnet motors the q-axis inductance is more than daxis inductance. The saliency in rotor produces significant reluctance torque apart from the normal torque due to permanent magnets. These motors have the advantages of mechanical robustness and smaller air gap and can have applications in constant power mode of operations. Higher number of pole pairs result in the lower ratio between the pole pitch and the rotor radius and some constructions can become better than others. In PM motors the space in rotor is very limited, proper utilization of magnetic material is therefore very vital. The stator punchings are made of low loss coated modern electrical steels with laser welded stackings. Good designs require the optimum use of material by operating the motor at highest energy density. The motors for variable speed operation are supplied through inverters; there is no need to have starting conduction cage torque. The presence of cage windings is helpful in reducing the distortion and spikes of line to line voltage. The damper windings can also reduce the commutation overlap. However, damper windings are not provided if the motor is supplied from a voltage source inverter, because the damper windings can provide a path for the flow of harmonic currents from the inverter.

5.3

Modeling of PMSM

The stator of the PMSM and wound rotor synchronous motor is similar. Also, there is no diﬀerence between the back EMF produced by a permanent magnet and that produced by an excited coil. The mathematical model of PMSM is therefore, similar to that of wound rotor synchronous motor. The following assumptions are made in deriving the model here. 1. Saturation is neglected ( although, it can be accounted by parameter changes); 2. The back EMF is sinusoidal; 3. Eddy current and hysteresis losses are negligible. The rotor reference frame is chosen here because the position of rotor magnets determines the induced EMFs and the current and torque of the machine independent of stator voltage and currents. In induction machine the rotor fluxes are not independent of stator voltages or currents; therefore any frame of reference is suitable for dynamic modeling of induction machines. The stator flux linkage equations in rotor reference frame arevrds = Rs irds + pλrds−ωr λrqs

(5.1)

102

5 Control of Permanent Magnet Machine (PM)

vrqs = Rs irqs + pλrqs + ωr λrds

(5.2)

where, Rs = Rd = Rq . And the d and q axes stator flux linkages are – λrqs = Lq irqs

(5.3)

λrds = Ld irds + Lmirdr

(5.4)

Here Ld and Lq are the d axis and q axis winding inductances respectively, and Lm is the mutual inductance between the stator winding and rotor magnets. Since, the rotor has no winding but only magnets and the rotor flux is along d axis. This flux is modeled as Lm irdr . Substituting the values of flux linkages in equations 5.1 and 5.2, the voltage equations obtained are r r Rs + pLd −ωr Lq ids vds 0 = + (5.5) vrqs ωr L d Rs + pLq irqs ωr Lm irdr and Torque is given by Te =

3P r r (λ i −λr ir ) 2 2 ds qs qs ds

(5.6)

Which, in terms of inductances is given by Te =

3P (Lm irdr irqs + (Ld −Lq )irds irqs 22

(5.7)

Here, the rotor flux linkage is considered constant except for temperature effects, as it is produced by permanent magnets. With increase in temperature the residual flux density and therefore flux linkages are reduced. Corrections may be provided in the value of rotor flux linkages for temperature in evaluating the performance of these machines.

5.4

Modeling of Brushless DC Motor

The brushless dc motor has three-phase windings on the stator and a permanent magnet on the rotor. The current in the stator winding is electronically commutated and sequentially switched on. The flux distribution in a PM brushless dc motor is trapezoidal. The d-q rotor reference frame model developed for PM synchronous motor is therefore not valid. For PM BLDC motor the model is therefore derived in phase variables. Since both the magnet and the steel retaining sleeves have high resistance, the induced currents in the rotor due to stator harmonic fields are neglected. Usually, damper windings are not a part of PM BLDC motor; damping is provided by inverter control.

5.4 Modeling of Brushless DC Motor

103

The circuit equations for the three phases in phase variables are⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ vas Rs 0 0 ias Laa Lab Lac ia eas ⎣ vbs ⎦ = ⎣ 0 Rs 0 ⎦ ⎣ ibs ⎦ + p ⎣ Lba Lbb Lbc ⎦ ⎣ ib ⎦ + ⎣ ebs ⎦ vcs ics Lca Lcb Lcc ic ecs 0 0 Rs

(5.8)

Where Rs is the resistance of stator winding per phase, assumed to be equal for all three phases. The induced EMFs eas , ebs , ecs have trapezoidal waveform. If there is no change in the rotor reluctance with angle because of a non-salient rotor then Laa = Lbb = Lcc = Ls ; And Lab = Lba = Lac = Lca = Lbc = Lcb = M

(5.9)

Substituting these values in equation 5.8, the voltage equation is ⎡

⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ vas Rs 0 0 ias L MM ia eas ⎣ vbs ⎦ = ⎣ 0 Rs 0 ⎦ ⎣ ibs ⎦ + p ⎣ M L M ⎦ ⎣ ib ⎦ + ⎣ ebs ⎦ vcs 0 0 Rs ics ic MML ecs

(5.10)

For balanced system ia + ib + ic = 0 or Mib + Mic = −Mia

(5.11)

The equation 5.10 can be written as – ⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ vas Rs 0 0 ias (L − M) 0 0 ia eas ⎣ vbs ⎦ = ⎣ 0 Rs 0 ⎦⎣ ibs ⎦ + p ⎣ 0 ⎦ ⎣ ib ⎦ + ⎣ ebs ⎦ (L − M) 0 vcs 0 0 Rs ics ic 0 0 (L − M) ecs (5.12) The equation 5.12 is similar to the armature equation of a dc machine. Thus, this machine has been named as brushless dc machine. The electromagnetic torque developed by the motor is given by Te = [eas ias + ebsibs + ecs ics ]

1 ωr

(5.13)

Since the back EMFs, eas , ebs and ecs are non-sinusoidal as shown in Figure 5.14, the nature of variation of mutual inductance between rotor and stator is also non-sinusoidal. It is therefore not easily possible to have a transformation from a-b-c reference frame to d-q reference frame. In PM BLDC motor therefore, no transformation is used and equations in phase variables are used for finding the dynamic performance. The equivalent circuit from the dynamic equations can be obtained as shown in Figure 5.15.

104

5 Control of Permanent Magnet Machine (PM)

vs

is

Zr

G

Orr

Ds Tr

Fig. 5.4 Phasor diagram

The induced emf can be expressed in terms of the peak value of the voltage induced, which is given by Ep = (Blv)N = (Blrωr )N = λp ωr

(5.14)

Where N is the number of conductors in series per phase, l is the length of conductor, r is the radius of rotor, v is the velocity, and B is the flux density. The equation of motion for the drive is given asJ

dωr + Bωr = Te −Tl dt

(5.15)

Where J is the moment of inertia of the drive, B is coeﬃcient of friction, and Tl is load torque. 3P Te = Lm irdr is (5.16) 22 It is clear from Equation 5.16 that, if torque angle is kept at 90 degrees, and flux is kept constant the torque can be controlled by controlling the magnitude of stator current. From the dynamic equations of PMSM, the equivalent circuit can be obtained as shown in Figure 5.5. In steady state

5.4 Modeling of Brushless DC Motor

Rd

105

Ld

i rds

-

v rds

Z r Orqs +

i rqs

Rq r v qs

Lq +

Z r O rds

Fig. 5.5 Equivalent circuit of PMSM from dynamic equations

conditions the currents id andiq are constant. The dynamic equivalent circuit of Figure 5.5 can be reduced to steady state circuit shown in Figure 5.6. The vector control scheme of PMSM is shown in Figure 5.7. A position transducer is connected to the shaft of the motor. The rotor position θr is continuously sensed by this transducer. The torque reference is a function of speed error and the speed controller is a PI controller. If fast response is desired PID controller may be used. The torque reference can be used to calculate the reference currents i∗ds and i∗qs . Then the Transformation from dq to abc is used to produce reference currents i∗as , i∗bs and i∗cs . The motor currents ia , ib and ic are also measured by current sensors. Three PWM current controllers produce the triggering signals of the inverter. By independently adjusting the amplitude of the currents and their phase angle with respect to the dc voltage of the inverter, the torque and flux can be controlled independently.

106

5 Control of Permanent Magnet Machine (PM)

Rd

Ld

i rds

-

v rds

Z r Orqs +

i rqs

Rq

Lq +

r v qs

Z r O rds

Fig. 5.6 Equivalent circuit of PMSM from steady state equations

Te*

Z r*

Zr

i *ds e jT r

i *qs

i *a i *b i *c

Zr

Tr

Fig. 5.7 Block diagram of Vector control of PMSM

5.5 Drive Operation with Inverter i*q

Zr*

Zr

107

e jT

i*d

Fig. 5.8 Block diagram of Vector control of PMSM

In another scheme shown in Figure 5.8, the motor currents are regulated in rotor control the torque.

5.5

Drive Operation with Inverter

A trapezoidal PM machine is basically a surface magnet non-salient pole machine that induces three-phase trapezoidal voltage waves at the machine terminal due to concentrated full pitch windings in the stator, as shown in Figure 5.16. Ideally, the crest of each waveform of the back EMF should be ≥ 1200 electrical to maximize the smoothness of the resulting output torque, and back EMF is proportional to the speed. If a current with fixed amplitude I is applied to the machine phase when the back EMF of that phase is cresting, the instantaneous power converted by this phase into mechanical power is Ef .I = Te .ωr , where Te is the instantaneous torque developed by that phase. These motors are specifically designed to develop nearly constant torque when excited with six -step switched current waveforms. These motors are operated in the self control mode and require a three-phase PWM inverter in the front end as shown in Figure 5.17. The inverter that excites the PM BLDC motor has two primary responsibilities. Electronic commutation: that is to direct excitation to proper phases at each time instant to maintain synchronization, and to maximize output torque. Second function of the inverter is the current regulation to control the torque. The inverter can be operated in the following two modes: • 120 degree angle switch-on mode • Voltage and current control PWM mode • current control with half wave converter operation

108

5.5.1

5 Control of Permanent Magnet Machine (PM)

120 Degree Angle Switch-On Mode

Rectangular current control is applied to BLDC motor due to concentrated full-pitch windings in the stator. To reduce torque pulsations rectangular current is needed in these motors. Basically it is an electronically commutated dc motor and has performance similar to a dc motor. The inverter circuit has six switches S1 − S6 connected as shown in Figure 5.17. These switches are turned on in a sequence in such a way that each switch conducts for 120 degrees. Two switches, one from the upper group, and the other from the lower group conduct at any instant of time. The waveforms of voltages and currents are shown in Figure 5.19. The total time period corresponding 360degrees is divided into six groups each of 60 degrees. If switches S1 and S6 are on say at t=0, supply voltage Vd is connected to phase a and b connected in series. A current Id will be applied in phase a in positive direction and in phase b in negative direction. After π/3 interval S6 is turned oﬀ and S2 is turned on. Now S1 and S2 are conducting, and the voltage Vd is applied across phases a and c connected in series. The current Id will continue through phase a but -Id will now be applied to phase c. After π/3 the switches S1 is turned oﬀ and S3 is turned on. The switches are turned on and oﬀ after every π/3 angle and each switch remains on for 2π 3 period in the sequence given below: S1 S6

S1 S2

S2 S3

S3 S4

S4 S5

S5 S6

S1 S6

A position sensor is required to identify the switching instants and commutation of switches from S1 −S6 in the sequence shown above. Assuming instantaneous current commutation, at any time two phase windings connected in series are carrying current. Consequently, only two third of the motor capacity is utilized. The power flow to the machine remains constant through the complete cycle. The inverter here works as position sensitive electronic commutator. Since the machine has similarity with a dc motor it is called brushless dc motor. To reverse the speed the sequence of the switches is changed such that 180 degree phase shifting is applied. It is also possible to have 180-degree conduction for each switch using the same hardware at high speeds, when all the three phases conduct at the same time.

5.5.2

Voltage and Current Control PWM Mode

In order to control the current and voltages at the machine terminal, it is possible to control the switches of the inverter shown in Figure 5.17, in PWM chopping mode. There are two possible chopping modes; freewheeling mode (FW), and feedback (FB) mode. In FB mode both the switches are turned on

5.5 Drive Operation with Inverter

109

and oﬀ to control the machine average current, and the corresponding average voltage. In FW mode, only one device is switched on and oﬀ, the other device is kept on throughout. For example in the circuit shown in Figure 5.17 all the upper switches S1 , S2 and S3 may be switched on sequentially while switches S4 , S5 , and S6 are switched on and oﬀ during 120 degrees in sequence.

5.5.3

Current Control with Half Wave Converter Operation

The inverter circuit described so far requires six switches and diodes for a three-phase motor. Since cost minimization of PM BLDC motor is a crucial factor in large number of applications, it is important to reduce the cost of inverter. Half-wave converter topologies provide a possibility of designs with less than six number of switches and thereby reducing the cost. Although it is possible to design an inverter with single switch per phase but has the disadvantage of not being able to operate in all four quadrants. Hence, configurations with more than one but less than two switches per phase are designed and used in these motors. One converter configuration with four switches and diodes is shown in Figure 5.18. Switch T, diode D, inductor L, and capacitor C forma step down chopper power stage to the inverter, through which input voltage is varied. Since there is only one switch in each phase winding, the current through it is uni-directional. The operation of motor in first quadrant can be explained as follows: Suppose the motor moves in clockwise direction when the supply is in the phase sequence of a-b-c of motor phase windings. When switch T1 is turned on supply voltage Vd is connected to phase a. For this condition the equivalent circuit is shown in Figure 5.19 (a). When switch T1 is turned oﬀ to regulate the current in phase a, the current passes through diode D1 , source voltage Vd , and capacitor C. Voltage (Vi − Vdc ) is applied across the machine phases as shown in Figure 5.19 (b). waveforms of voltage, current, and power are shown in Figure 5.20. The average air- gap power and input power are positive indicating motoring operation. In order to reverse the direction of motion of the motor (third quadrant), the phase energization sequence is changed to a-c-b. If the motor is to be operated in regenerating mode negative torque can be generated by turning on T1 during the negative constant emf period. This results in negative torque and air–gap power in the machine and the direction of motion is positive. Thus the motor works in fourth quadrant with regenerative braking, and transferring kinetic energy stored in motor to the supply. Based on the rotor position information and the polarity of the commanded current, appropriate machine phase is turned on. If the current in the phase increases beyond a current window over the reference current, the phase switch is modulated. As the machine input voltage is controlled through the

110

5 Control of Permanent Magnet Machine (PM)

chopper switch T, the phase switch is rarely modulated to regulate the phase current. The phase switch is turned oﬀ only during commutation of phase. Advantages (i) Only four switches and diodes are required for three-phase PMBLDC motor and four quadrant operation can be obtained. (ii) Since switches are in series with the phase windings, shoot through faults are avoided. (iii) If one switch fails the motor can still be operated with other two switches. (iv) The power switches rating is equal to the source voltage rating only. (v) Only two isolated supplies are required. The disadvantage is low eﬃciency due to two stage power conversion.

5.6

Speed Control Using PWM Inverter

A closed loop speed control system for PM BLDC motor fed from PWM inverter is shown in Figure 5.21. A set of three low-cost Hall sensors are placed on the stator side at the edge of rotor poles as absolute position sensors to generate three 120 degree angle phase shifted square waves. These pulses are in phase with the respective phase voltage waves. A decoder circuit is used to convert these waves into six step waves. As shown in Figure 5.21, the speed control loop compares the speed of the motorωr with the reference speed ω∗r and generates command reference current I∗d . A decoder circuit produces the reference phase currents. The actual phase currents then follow the commanded currents by hysteresis band current control. At any instant, two phase currents are enabled, one with positive polarity and other with negative polarity. For example the decoder circuit has enabled phase a with positive command current and phase b with negative current. Switch S1 in phase a, and switch S6 in phase b are turned on simultaneously to increase the currents ia and − ib . When these currents exceed the hysteresis band, both the switches are turned oﬀ simultaneously.

5.7

Vector Control of PMSM

In order to achieve high performance equivalent to that of dc machine vector control can be applied to PMSM The vector control of PMSM is simpler as compared to the vector control of induction motor described in chapter 3. The main reason for this simplicity is from the fact that the rotor magnet flux is fixed to the direct axis of the rotor reference frame and the angle between magnet flux and stator reference axis is equal to the rotor angle (θr = ωr t)

5.7 Vector Control of PMSM

111

that can be easily measured using a position sensor. The vector control of PMSM can be derived from its dynamic model presented in 5.3. The vector controller is designed starting from the d-q model of the machine as follows. If rotor reference frame is used, the stator currents and voltages in d-q frame are given as ⎡ ⎤ ias r 2π 2π 2 sinωr t sin ωr t− 3 sin ωr t + 3 ids ⎣ ibs ⎦ = (5.17) 2π irqs 3 cosωr t cos ωr t− 2π 3 cos ωr t + 3 ics Where ias , ibs and ics are balanced three phase stator currents and ωr is the rotor speed. Similarly, the voltages in d-q axis can also be obtained. As shown in Figure 5.4, δ is the angle between the rotor field and stator current phasor is ., the stator currents therefore can be represented as ias = is sin (ωr t + δ) ibs = is sin ωr t + δ − 2π 3 ics = is sin ωr t + δ + 2π 3

(5.18) (5.19) (5.20)

The d and q components of stator current are related to stator current phasor by r ids cosδ (5.21) = i s irqs sinδ These currents are constant in rotor reference frame, since δ is constant for given torque and rotor rotates at synchronous speed. The electromagnetic torque can be obtained by substituting the values of irds an irqs in Eq. 5.7. 3P 1 2 r Te = (Ld −Lq )is sin 2δ + Lmidr is sin δ 22 2

(5.22)

As can be seen from equation 5.22 the torque mainly depends on is sinδ. Thus, the q axis current is equivalent to armature current of a dc machine. For δ = π2 , the torque becomes equal to Te =

3P Lm irdr is 22

(5.23)

It is clear from Equation 5.23 that, if torque angle is kept at 90 degrees, and flux is kept constant the torque can be controlled by controlling the magnitude of stator current. From the dynamic equations of PMSM, the equivalent circuit can be obtained as shown in Figure 5.5. In steady state conditions the currents id and iq are constant. The dynamic equivalent circuit of Figure 5.5 can be reduced to steady state circuit shown in Figure 5.6. The vector control scheme of PMSM is shown in Figure 5.7. A position transducer is connected to the shaft of the motor. The rotor position θr

112

5 Control of Permanent Magnet Machine (PM)

is continuously sensed by this transducer. The torque reference is a function of speed error and the speed controller is a PI controller. If fast response is desired PID controller may be used. The torque reference can be used to calculate the reference currents i∗ds and i∗qs . Then the Transformation from dq to abc is used to produce reference currents i∗as , i∗bs and i∗cs . The motor currents ia , ib and ic are also measured by current sensors. Three PWM current controllers produce the triggering signals of the inverter. By independently adjusting the amplitude of the currents and their phase angle with respect to the dc voltage of the inverter, the torque and flux can be controlled independently. In another scheme shown in Figure 5.8, the motor currents are regulated in rotor reference frame. The position sensor is used to get the rotor position as in earlier scheme. The three phase motor currents are measured and transformed into rotor reference frame using rotor position feedback. These currents are compared with the reference currents and controlled in rotor reference frame using integral controller. In this scheme continuous transformation from rotor d − q axis to stator frame is required; hence the rotor position sensor is essential to the proper operation of such drives. This constraint is basically a disadvantage of this scheme. From torque equation 5.23, it is clear that both irdr and irqr must be controlled for precise control of developed torque by the motor. For surface magnet synchronous motor, because of large air gap Ld ≈ Lq , so that irdr must be maintained at zero level. For interior magnet motor Lq Ld , that means irdr and irqr must be controlled simultaneously to control the torque. The following operating modes are possible.

5.8

Operating Modes

Since in PMSM the drive command variable is the reference torque as required by a speed controller, the reference currents i∗ds and i∗qs are chosen from the torque equation. Thus we require one more equation to obtain i∗ds and i∗qs . This additional information is obtained using an optimization criterion such as: i) ii) iii) iv)

5.8.1

Constant Torque angle control Unity power factor control Maximum torque per ampere control Flux weakening control

Constant Torque Angle Control

In figure 5.4, it is shown that the torque angle δ is the angle between the stator current phasor and the rotor field. In constant torque angle control this angle is maintained at 90 degrees. Under this condition the direct component of the stator current phasor irds (flux producing component) is made equal to zero,

5.8 Operating Modes

113

and only torque producing component irqs is present. The torque equation as given in equation 5.22 becomes Te =

3P r r (λ i ) 2 2 ds qs

(5.24)

3P r (λ i ) 2 2 ds s

(5.25)

Since irds = 0 the current irqs = is Te =

The torque per unit current is given asTe 3P r λ = is 2 2 ds

(5.26)

In order to derive constant torque angle control criteria, it is better to normalize the torque expression and express it as function of normalized stator current components. Defining the base torque as 3P r (λ ib ) 2 2 ds

(5.27)

is Te = = isn Teb ib

(5.28)

Teb = The normalized torque Ten =

The Eq.(5.28) shows that the normalized torque is equal to the stator current vector in per unit. The phasor diagram where the stator resistance is neglected is shown in Figure 5.9. The vector control block diagram is shown in Figure 5.10. The stator command current irqs ∗ = is ∗ is obtained from the speed control loop. Its polarity is positive for motoring mode and negative for generating mode. This current is in the rotor reference frame which is converted to stator frame using rotor position signals cosθe and sinθe as shown. The inverter is controlled to provide the stator currents equal to the commanded currents.

5.8.2

Unity Power Factor Control

The performance equations for unity power factor control are obtained from the phasor diagram shown in Figure 5.4. In unity power factor control the currents ids and iqs are controlled in such away that the torque angle δ as a function of motor variables result in cosφ =1. The stator current in rotor reference frame can be written as irqs = is sinδ irds = is cosδ

114

5 Control of Permanent Magnet Machine (PM)

v rqs

vs is

I

G

Zr Or

Fig. 5.9 Phasor diagram for constant torque angle control

Z

i *ds

Te*

* r

e

Zr

jT r

i *qs

0

i *a i *b i *c

Zr

Tr

Fig. 5.10 Block diagram of Vector control of PMSM

5.8 Operating Modes

115

The power factor cosφ can be obtained from the power input the motor P = 3vs is cosφ = Te ωr Or cosφ =

T e ωs and vs = ωr λs 3vs is

(5.29)

i2ds + i2qs . also is = If the value of cos φ =1 is substituted in Eq. 5.29 and the values of torque and stator voltages are substituted in terms of motor parameters Lds Lqs − 1 ids iqs

(5.30) 1 = 2 2 ds ids + i2qs + i2qs + i2ds LLqs Or

−1 1= 1 2 1 2 2 Lds 1 2δ + tan + 1 + Lqs tan2δ Lds Lqs

(5.31)

From Eq. (5.31) the torque angle δ can be computed.

5.8.3

Maximum Torque Per Ampere Control

One of the common method for obtaining i∗d and i∗q from the torque command T∗e is to optimize the torque per unit stator current. This method of control results in optimum utilization of inverter and maximizing the eﬃciency of the motor. But this does not mean best transient response. The normalized torque per unit of stator current is given by 1 (5.32) Ten = isn sinδ + Ldn − Lqn isn sin2δ (p.u.) 2 The torque per ampere is given by Ten 1 = sinδ + Ldn − Lqn isn sin2δ isn 2

(5.33)

The maximum value Tisnen is obtained by diﬀerentiating (Eq. 5.33) with respect to δ and equating to zero. The control of the drive is implemented in the same way as shown in Figure 5.10. The speed control loop generates the torque command T∗e . From T∗e signal command currents i∗d and i∗q are generated using function generators. The function generators use the relationship between stator current

116

5 Control of Permanent Magnet Machine (PM)

and torque for maximum torque per ampere condition. The absolute position signal θr is used to convert i∗d and i∗q into stationary stator currents ∗ ∗ i∗, a , ib and ic .

5.8.4

Flux Weakening Control

In the PWM mode current control when the motor is working in constant torque region, if the speed is increased the supply voltage has to be increased. The peak value of voltage available from the inverter is 2Vπ d where Vd is the dc link voltage of inverter. If the speed is to be increased beyond the maximum value obtainable by the inverter rating, field weakening control is applied. The basic idea of flux weakening operation can be understood from the relationship between maximum speed and stator current magnitude. In order to reduce the flux the demagnetizing component of stator current is obtained based on the maximum current and voltage limit of the inverter. The normalized value of stator voltage under steady state condition is given by – 2 p.u. (5.34) v2sn = ω2rn (1 + Ldnirdsn )2 + Lqn irqsn Here the voltage and current phasors are the maximum values that can be supplied by the inverter, and can be assumed to be constant. Since the flux is controlled by the direct axis component of stator current isdn , it is varied keeping isn constant. From Eq. 5.34 it is clear that it has only two variables ωrn and isdn . Assuming that the rotor speed is available for feedback, the isdn can be obtained. Once isdn is known and isn is fixed based on inverter rating, isqn can be obtained. Then the stator command currents can easily be obtained by d-q to a-b-c transformation. The torque angle δis obtained as –

r iqsn −1 (5.35) δ = tan irdsn

5.9

Direct Torque Control of PM Motor

In direct torque control of induction machines it is assumed that torque is directly proportional to applied current. For PM machines this assumption may not be true since it accounts only one source of torque generation. A PMSM can be thought as a synchronous machine with constant excitation current. The following diﬀerences must be considered. • The stator inductance of a PMSM is quite low. • The quadrature axis inductance is greater or equal to the direct axis inductance. • Generally no damping windings are used.

5.9 Direct Torque Control of PM Motor

117

In PM synchronous machines there are three sources of electromagnetic torque generation. These are: • Cogging torque; • Reluctance torque; and • Mutual torque. Cogging torque is produced by the interaction of rotor magnetic field and stator slots and is independent of stator current. Reluctance torque is present due to the diﬀerence in the air gap and does not depend on current. The mutual torque is the only component that depends on the stator current and rotor magnetic field. In surface mounted PM machines the air gap is large, thus the eﬀect of stator current on the rotor field is minimum. These machines also have low saliency (uniform air gap), hence the reluctance torque and cogging torques are also minimum. The block diagram of switching table based DTC scheme is shown in Figure 5.11. The direct torque control is obtained basically by triggering one or more voltage vectors in the PWM voltage source inverter that supplies PMSM. The table of optimum switching is obtained based on the principle that the stator flux space vector is varied in the same direction as the applied voltage space vector. To sense the stator flux space phasor and torque errors; estimation of these variables is essential. The key element of a direct torque controlled drive is the estimation of the stator flux linkage. In the earlier

e jT

Te*

Zr*

Te

Zr

i*d

O*s

Fig. 5.11 Block diagram of Vector control of PMSM

118

5 Control of Permanent Magnet Machine (PM)

method for induction machine the stator flux linkage was estimated using the voltage model given bydλs = vs − Rs is dt

(5.36)

and torque is given by3P r (λ i ) (5.37) 2 2 ds s Unfortunately this model fails at low frequency and compensation has been used to overcome this problem. In synchronous machines it is possible to calculate the stator flux linkage using the current model of the machine. However, the rotor flux linkages in a synchronous machine are created by the field current or permanent magnets which are independent of stator quantities. It can only be determined if the rotor position is known, the measurement of rotor angle is thus unavoidable. The basic principle of DTC is to select proper voltage vectors using a predefined switching table. The selection is based on the hysteresis control of the stator flux linkage and the torque. The command stator flux and torque values are compared with in hysteresis flux and torque controllers respectively. The flux controller is a two- level comparator while the torque controller is a threelevel comparator. If the output of torque hysteresis comparator is denoted by τ, then τ= -1 means that the actual value of the torque is above the reference value and outside the hysteresis limit. τ=1 means that the actual value of torque is less than the reference value and outside the hysteresis limit. τ=0 means the torque is within the hysteresis limit. The flux hysteresis comparator output is denoted by λs . If λs =0, it means that the actual value of the flux linkages is above the reference value and outside the hysteresis limit. λs =1 means the actual value of the flux linkage is below the reference value and outside the hysteresis limit. The space vector PWM inverter is used to select proper voltage vectors from the output of flux and torque comparator. The inverter has eight permissible switching states, out of which six are active and two zero or inactive states. The voltage vector plane is divided into six sectors so that each voltage vector divides each region in two equal parts as shown in Figure 5.12. In each sector four of the six non-zero voltage vectors along with zero vectors may be used. All the possibilities can be tabulated in the form of a switching table presented in Table 5.1. The voltages for three legs of the inverter are shown in two groups. If the switch of positive leg is conducting it is shown by 1, and if negative leg switch is on it is shown as 0. For example V1 (100) means switch of positive group of leg 1 is conducting, and negative group of switches of leg 2 and 3 are on. The stator flux controller imposes the time duration of the active voltage vectors, which move the stator flux along the reference trajectory. The torque controller determines the time duration of zero voltage vectors, which keep the motor torque within the defined hysteresis band. At every sampling time Te =

5.10 Sensorless Control of PM Motor

119

O

O 1

1

Fig. 5.12 Voltage vector selection for sector I Table 5.1 R(1) λs = 1

λs = 0

R(2)

R(3)

R(4)

R(5)

R(6)

τ=1

V2 (110) V3 (010) V4 (011) V5 (001) V6 (101) V1 (100)

τ=0

V(000) V(000) V(000) V(000) V(000) V(000)

τ = −1

V6 (101)

V1

V2

V3

V4

V5

τ=1

V3 (010) V4 (011) V5 (001) V6 (101) V1 (100) V2 (110)

τ=0

V(000) V(000) V(000) V(000) V(000) V(000)

τ = −1

V5

V6

V1

V2

V3

V4

the voltage vector selection block selects the proper switching states. The scheme presented in table 5.1 is as suggested by Takahashi and Noguchi.

5.10

Sensorless Control of PM Motor

In PMSM the ideal back EMF is sinusoidal; the sinusoidal currents in the windings are therefore required to produce constant torque with very low ripple. The motor requires continuous rotor position feedback to apply the sinusoidal currents from the inverter. The drawbacks of using a mechanical shaft position sensor are well known, and drives without a shaft position sensor are preferred in many applications.

120

5 Control of Permanent Magnet Machine (PM)

Te*

Z r*

is

iT

Zr

i *a i *b i *c

Zr

Os

Fig. 5.13 Block diagram of Vector control of PMSM

ep e as 0

0

180

0

- ep ep

e bs 0

0

180

0

- ep

ep e cs

0

0

180

0

- ep

Fig. 5.14 Waveform of Back emf in BLDC

There are generally two methods of rotor position estimation using the mathematical model of the system. In surface mounted PM motor drives the output voltage of PWM inverter is integrated to obtain the stator flux and calculate the rotor position from the current angle. This scheme works in open loop and is vulnerable to parameter uncertainty and voltage drift. A low pass filter is therefore used in place of pure integrator, which results in delay in estimated rotor position at low speeds. The emf space vector can be obtained from the measurements of stator line to line voltages and stator phase currents using the following equation eL = vL − Rs iL

5.10 Sensorless Control of PM Motor

121

Ra Rb ea

eb

ec Rc

Fig. 5.15 Equivalent circuit of PMBLDC motor

ep e as 0

0

180

0

- ep ep e bs 0

0

180

0

- ep

ep e cs

0

0

180

0

- ep

Fig. 5.16 Waveform of nduced emf in BLDC

122

5 Control of Permanent Magnet Machine (PM)

S1

Ra

S3

S2

Rb ea

eb

ec Rc

S4

S5

Fig. 5.17 BLDC with front end PWM inverter

L B

A

C

T

Vdc

Cd

C

D

T1

T2

T3

Fig. 5.18 BLDC with 4 switch PWM inverter

The flux linkage space vector is obtained from the equation λL =

eL dt

The space angle of flux can be obtained from its real and imaginary components as

λLI −1 θλL = tan λLR Once the flux space angle is obtained, synchronized control of phase current can be achieved depending on which method of control is desired. If unity power factor control is desired, the current space phasor is must lead stator phase flux vector by 90 degrees, or it should lead λL by 60 degrees.

5.10 Sensorless Control of PM Motor

123

i as

i as Rs

Vi

Rs

Vi Ls

Ls ep

ep

D1

Vdc (a)

(b)

Fig. 5.19 (a). Switch T1 ON (b). Switch T1 OFF

Or

π 3 In interior PM SM the phase inductance varies appreciably as a function of rotor position. The inductance of the motor in q-axis is much larger than the inductance along d –axis. The phase inductance of an IPM motor can be calculated from the measured voltages and currents. The calculated phase inductance is then used to estimate the rotor position with the help of look up table. The algorithm looks for a value of the phase inductance in the table that is closest to the calculated value. The inductances for all three phases are stored in a lookup table. In another method the variation of self-inductance with rotor position is obtained by injecting a variable frequency sinusoidal signal into one winding of the machine and measuring the terminal voltage and current. The phase winding self inductance is measured to give approximate model of the coil in terms of equivalent resistance and inductance. The variation of inter-winding mutual inductance is the second method which is more robust is a closed loop observer using measured currents as feedback. θS = θλL +

5.10.1

Position Information from Measurement of Voltages and Currents

This is one of the simplest methods where stator voltage and current signals are used to construct a flux linkage position signal, through which the phase angle of the stator current is controlled. The block diagram of the drive is shown in Figure 5.13. The stator flux linkage space vector is obtained from the measurement of current and voltages using the following equations

124

5 Control of Permanent Magnet Machine (PM)

λds =

(vds − Rs ids )dt

(5.38)

λqs =

(vqs − Rs iqs )dt

(5.39)

(λ2ds + λ2qs)

(5.40)

|λs | =

cos(θr + δ) = sin(θr + δ) = ω=

λds |λs |

λqds |λs |

(vqs − iqs Rs )λds − (vds − ids Rs )λqs dθr = dt |λ2s |

(5.41) (5.42) (5.43)

Eq.(5.43) is used to get the estimated value of the speed of the motor. The d-q component of current is obtained from the phase currents using usual transformation. Although, unity power factor control method is used in this example, any other power factor control can be applied with slight modification. The estimation of speed using voltages and currents is not valid for very low speeds. Hence this method gives speed control beyond 10% of base speed.

5.10.2

Position Information from Measurement of Inductance Variation

In interior PM SM the phase inductance varies appreciably as a function of rotor position. The inductance of the motor in q-axis is much larger than the inductance along d axis. The phase inductance of an IPM motor can be calculated from the measured voltages and currents. The self inductance of a, b, and c phase of the stator winding of a two pole motor can be represented in the same way as for salient pole synchronous machine described in chapter 2 and is given by Laa = L1 + L2 cos2θr

(5.44)

2π ) (5.45) 3 2π (5.46) Lcc = L1 + L2cos(2θr − ) 3 Where L1 = Self inductance plus magnetizing inductance of phase a due to fundamental air gap flux. L2 = self inductance component of stator due to the rotors position dependent flux, and θr is rotor electrical angle . The mutual inductance also varies with rotor position and is given as Lbb = L1 + L2 cos(2θr +

5.11 Sensorless Control of BLDC Motor

π Lab = −L3 − L2 cos2(θr + ) 6

125

(5.47)

π (5.48) Lbc = −L3 − L2 cos2(θr − ) 2 π Lac = −L3 − L2cos2(θr − ) (5.49) 6 The calculated phase inductance is then used to estimate the rotor position with the help of look up table. The algorithm looks for a value of the phase inductance in the table that is closest to the calculated value. The inductances for all three phases are stored in a lookup table. In another method the variation of self-inductance with rotor position is obtained by injecting a variable frequency sinusoidal signal into one winding of the machine and measuring the terminal voltage and current. The phase winding self inductance is measured to give approximate model of the coil in terms of equivalent resistance and inductance. The second method which is more robust is a closed loop observer using measured currents as feedback. Extended Kalman Filter method provides a good method for determining the speed estimation. The observer senses the machine terminal voltages and currents and estimates the position θr and speed signal ωr . The machine model of the motor can be in stationary frame or synchronously rotating frame.

5.11

Sensorless Control of BLDC Motor

The sensorless control of BLDC is simple because only two devices conduct at a time. One of the common method for sensorless control is by sensing the induced terminal voltage. The terminal voltages Vag , Vbg and Vcg of the motor with respect to the ground can easily be measured as shown in Fig. 5.17. The phase voltages Van , Vbn and Vcn can be obtained from these voltages by subtracting the DC bias voltage (0.5 Vd ) from them. A dc blocking capacitor is used to subtract the bias voltage. The phase to neutral voltages thus obtained are trapezoidal in nature. The induced emf of the phase provides the information on zero crossing and on when the emf reaches the constant region, indicating when that phase is to be energized. These voltages are then integrated to obtain near triangular waves. The integrator output corresponding to thirty degrees from the zero crossing point can be termed as threshold value used in energizing a phase. This threshold is independent of the rotor speed. The only disadvantage of this method is that stand still or near zero speed , there is no or very small induced emf which means no information about position is available. A separate method to generate control signal at near zero speed is therefore required.

126

5 Control of Permanent Magnet Machine (PM)

ep

e as 0

ip 0

v as

Pa

Fig. 5.20 Wave forms for I quadrant operation

Z r*

1 k

Zr ¦

Fig. 5.21 PMBLDC motor drve scheme

Third Harmonic Induced EMF Detection Another method to determine the rotor position and to generate the control signals is to detect the third harmonic induced emf in stator windings If a three phase four wire star connected system is available, third harmonic induced emf can easily be obtained with five resistors. A balanced

5.11 Sensorless Control of BLDC Motor

Rd

S1

127

S3

S2

ea

eb

Vd ec

Rd S4 S5

Fig. 5.22 Connections to obtain third harmonic voltage

Van

VON

Fig. 5.23 Utilization of third harmonic voltage for switching of inverter

wye-connected resistor bank of resistance R is connected at the output of Inverter to create a neutral point as shown in Fig. 5.22. Two resistors Rd are connected at dc source to create a mid point of the dc supply. The third harmonic voltage is available between the machine neutral point n and resistor bank neutral N. The third harmonic voltage wave vnN is nearly triangular. If machine neutral is not available the third harmonic voltage wave can be obtained between mid point of dc supply O and neutral point N. The third harmonic voltage wave is integrated to obtain a near sinusoidal waveform lagging the triangular wave by π/2 angle (Fig. 5.23). The zero

128

References

crossing point from negative to positive of this sine wave gives commutation instants of the upper devices of the inverter. (S1 , S3 and S5 ) Fig. 5.17. The positive to negative zero crossing gives the commutation instants of the lower devices. The zero crossing points of trapezoidal phase voltage wave is necessary for synchronization purposes.

References [1] Krishnan, R.: Electric Motor Drives, Modeling Analysis and Control. Prentice Hall, Englewood Cliﬀs (2001) [2] Jahns, T.M.: Motion control with permanent magnet AC machines. Proceedings of IEE 82, 1241–1252 (1994) [3] Pillay, P., Krishnan, R.: Modeling, simulation and Analysis of Permanent magnet motor drives part I, The permanent magnet synchronous motor drive. IEEE Transaction Industry Application 25, 265–273 (1989) [4] Pillay, P., Krishnan, R.: Modeling, simulation and Analysis of Permanent magnet motor drives part II, The brushless DC motor drive. IEEE Transaction Industry Application 25, 274–279 (1989) [5] Rahman, M.A., Zhou, P.: Analysis of Brushless Permanent magnet Synchronous motors. IEEE Transactions on Industrial Electronics 43(2), 256–267 (1996) [6] Bose, B.K.: High Performance Inverter fed Drive System of an Interior Permanent magnet Synchronous machine. IEEE Transactions on Industry Application 24, 987–997 (1988) [7] Jahns, T.M.: Torque production in permanent magnet synchronous motor drives with rectangular current excitation. IEEE Trans. on Ind. Application 20, 803– 813 (1984)

Chapter 6

Switched Reluctance Motor Drives (SRM)

6.1

Introduction

The principle of operation of reluctance machines are known for a long time. In 1838 a locomotive was driven by such a motor in Scotland. The stepper motor which also works on the principle of variable reluctance was patented by CL Walker in 1920. The basic concepts of switched reluctance motors in present form were first introduced by SA Nasar in his paper in Proceedings of IEE in 1969. During late 70’s and 80’s, as fast switching devices became available, the applications of SRM have increased manifold. Due to their simple and rugged topology these motors are finding applications in consumer appliances, auto industries, and defence. Presently Ford motor company is using SRM in power assisted steering system. The slow development in commercialization of these motors was mainly due to the need to have specialized design, and requirement of a sensor to control the speed. The other disadvantages are the torque ripple and acoustic noise. Now these disadvantages have been minimized by improving the design, sensorless drives have been developed, and many manufacturers are producing the SRM, making it easily available motor. The switched reluctance motors have many advantages, e.g. high eﬃciency, can be designed for ratings from few watts to M watts and can be employed in harsh working environments.

6.2

Construction

The switched reluctance machines have salient poles in stator and rotor (doubly salient). These machines do not have any field winding or permanent magnets on the rotor. Salient poles on the stator carry concentrated coils which are excited sequentially through dc voltage pulses. The rotor is passive and is made up of laminated magnetic material without windings or permanent magnet. The number of poles in the stator Ps = 2mq; where m is the number of phases and each phase is made of concentrated coils placed on M . A h mad: High Performance AC Drives, Power Systems, pp. 129–160. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

130

6 Switched Reluctance Motor Drives (SRM)

Fig. 6.1 8/6 Switched Reluctance Motor

2q stator poles. The number of poles on rotor is the salient poles = Pr . SR machines can have many topologies. These can be single phase, two- phase, three-phase, four-phase or even more than four phases. However, the most common configurations are 6/4 three phase, and 8/6 four phase machines as shown in Figure 6.1 (a) and (b). If torque ripples are to be reduced further, five phase and six phase motor are also used. Similarly, if q=2 or 3, then for three phase machines 12/8, or 18/12 topologies are obtained. The selection of stator and rotor pole arc angles βs and βr is an important part of the design of reluctance machines. The standard design normally has the stator pole arc angle βs slightly less than the rotor pole angle βr . The magnetic circuits of SRM are symmetrical and have almost zero mutual flux linkages among its phase windings even under saturated conditions. The self inductance only of the phase windings is responsible for production of torque. Thus, these motors have high fault tolerant properties compared to any other machine. The Switched reluctance machines are simple and rugged by construction. The stator and rotor of the SR machine are both made of stacked steel laminations. The electrical windings on the stator are wound concentrically. Each stator pole winding is connected with that of diametrically opposite pole to form a stator phase. The basic principle of operation of all reluctance type machines is same i.e. the stator and rotor poles seek the minimum reluctance position. In this position the stator excited flux becomes maximum. Now, when one phase of the SR motor is excited, the nearest rotor pole will tend to align itself with the direction of the developed magnetic field. This position is known as aligned position. To achieve continuous rotation, the stator phase currents are switched ‘on’ and ‘oﬀ’ in each phase in a sequence.

6.2 Construction

131

Since the movement of rotor is made possible by switching of currents in the stator winding and variation of reluctance; hence these machines are named Switched Reluctance Motors. The SRM can be constructed as axial or radial field machine. The axial field SRM is ideal for applications where the total length may be a constraint, such as in a ceiling fan or in a propulsion application. Radial field SRMs are designed to have either short or long flux path. Short flux path SRM has the advantage of lower core losses due to shorter flux path. However, they have a disadvantage of having higher mutual inductance. The switching sequence in all SR machines depends on the position of rotor poles; therefore, some form of rotor position sensing is essential for the proper operation of SR motors. The requirement of position sensors added to the complexity and cost to the system, and was the main impediment in the application of these motors. However, several sensorless control methods have been reported in the literature during last 15 years, making these motors suitable for many applications requiring variable speed drives. Single phase SR motors are also of great interest in the field of high speed and automotive applications because of their simple construction and low manufacturing cost. Single phase SR motor is capable of operating at very high speeds because of its robust construction. The speed of this motor is not limited by the motor geometry but due to the inverter switching speed The single–phase SRMs have 2/2, 4/4, 6/6 and 8/8 stator and rotor poles. The stator winding is energized when the rotor poles are unaligned, and switched oﬀ when the stator and rotor poles are aligned. The rotor keeps moving in the aligned condition due to stored kinetic energy whereas torque is produced in unaligned condition. The single phase SR motor has a problem at the starting if the poles are aligned at the standstill. This problem is overcome by placing a permanent magnet on the stator to pull the rotor away from alignment. The disadvantage of this method is additional cost of the magnet and the requirement of space. Other methods of starting include shifting of one pole 0 from its normal position by 20P , where P is number of stator poles, or by shaping of stator poles to provide graded air gap.

6.2.1

Linear SR Motors

Switched reluctance motors are now available as linear or rotary machines. Rotary machines are further classified as axial field or radial field machines. A radial field machine has its magnetic field path that is perpendicular to the shaft, or along the radius of cylindrical shaft. In axial machine the flux path is along the axis of the rotor. Linear SR motors are considered more suitable for machine tool drives as these do not require gears or rotary to linear motion converters. A linear SRM may have windings on stationary part or on the moving part of the machine. The fixed part is called stator or Track and the moving part is calleda translator. There are two distinct configurations of linear SR motors, similar to axial and radial flux rotary SR machines. The linear SR machines are classified as longitudinal and transverse flux machines.

132

6 Switched Reluctance Motor Drives (SRM)

T1

S1

T2

S2

T3

T1 4

S3

Fig. 6.2 Three-phase longitudinal linear SR motor

Figure 6.2 shows the longitudinal flux and transverse flux machine having windings on the stator. The active stator machine has the advantage of having static windings that can be easily connected to the supply. The longitudinal flux machine is a linear counter part of radial flux rotary machine. This machine is mechanically robust and simple in construction and has lower eddy current losses as the flux is in the same direction as the motion of translator. In transverse flux machine the flux is perpendicular to the direction of motion of translator; an emf is therefore induced in the core. There is significant eddy current loss in these motors due to this induced emf. A linear SR motor may be constructed either with two stators and one translator or two translators and one stator. These motor are known as double-sided linear SR motor. This motor can not produce a net levitation force whereas a single sided linear SR motor can provide net levitation force that can be used in Maglev systems.. The single-sided linear SR motor has

6.3 Basic Principle of Operation

133

higher inductance and lower force density compared to double-sided linear SR motor. The principle of operation of linear SR machines is the same as that of rotary SR motor. When voltage is applied to a pair of stator windings connected in series, the translator tends to move to align itself with the magnetic flux axis of the excited stator phase winding. In this position, known as fully aligned position, the phase inductance is maximum. By switching stator windings in sequence, the translator can be moved forward or backward.

6.3

Basic Principle of Operation

The torque production in a SR motor can be explained from its magnetization characteristics shown in Figure 6.3. Assuming linear magnetic circuit, the flux linkage is proportional to the phase current for any rotor position. As shown in the Figure 6.3, the inductance L at any position θ is λ/i = constant, and is independent of current. As the motor rotates, each stator phase undergoes a cyclic variation of inductance. The variation of self inductance of a stator phase with rotor position is shown in Figure 6.4. During certain portion of electrical cycle, the inductance has positive slope with θ, while during another part of the cycle the inductance variation has negative slope; and in the remaining part of the cycle the inductance has a flat profile. As can be seen in the Fig. 6.4, the inductance L has a maximum value when rotor pole axis is in direct alignment with the stator pole axis. Inductance L has minimum value in the fully unaligned position. Thus the inductance is a function of position only, and is independent of the current.

O Flux linkage

Algned positon Rotor postion

T drgrees

Unaligned positon

Current i

Fig. 6.3 Magnetization characteristic of linear SR motor

134

6 Switched Reluctance Motor Drives (SRM)

La

L as Lu

T1

T 2 T3

T4

Fig. 6.4 Inductance Vs Rotor position profile

Four distinct regions can be identified in the inductance vs position plot shown in Figure 6.4. These distinct regions correspond to ranges of rotor pole positions relative to stator pole positions. In region A, starting with rotor angle θ1 , the rotor pole tip starts to come under the stator pole and moves to overlap till at an angle θ2 , when overlap of poles is complete. In this region the inductance L increases with rotor position and is maximum at θ2 which is fully aligned position. A current impressed in the winding during this region produces a positive torque. In region B from θ2 to θ3 , movement of rotor poles does not alter the complete overlap of the stator pole. The flux path is therefore mainly through stator and rotor laminations and inductance L is constant at its maximum value. Since inductance remains constant no torque is generated even if supply is connected to the stator. In region C from θ3 to θ4 the forward rotor tip coincides with the end of stator pole. The rotor is moving away from the overlapping region. The inductance therefore decreases with the rotor movement till it is minimum. If current is applied in the winding during this time, negative torque is produced. The rotor therefore will move in the opposite direction. The direction of current in the winding is immaterial, as the rotor always moves in the direction, so as to seek the minimum reluctance position. In region D there is no overlapping of stator and rotor poles, the flux path is through the air gap only. The inductance therefore, remains constant at the minimum value and no torque can be produced. It is not possible to achieve the ideal inductance profiles as shown in Figure 6.4 due to saturation. The eﬀect of saturation is to make the inductance profile curved near the top. F the machine works in saturated region it will result in less power and torque output.

6.4

Design of SR Machine

Generally, the conventional machines are designed starting from the output equation. In order to design a SR motor the specifications given are the required power output in h.p., speed in rpm, allowable peak phase current in

6.4 Design of SR Machine

135

Ampere, and the supply voltage (volts). Once the speed and power output is fixed the torque is automatically fixed. The starting point for the design of SR motor is the selection of frame size. The frame size may be selected on the basis of equivalent induction motor. If, during the design the frame size is found to be small next higher size may be selected. Similarly, if the size is found to be large a lower dimension frame may be selected. The preliminary selection of frame size automatically fixes the outer diameter D0 of the stator asD0 = (Framesize − 3) × 2

6.4.1

(6.1)

Selection of Poles

In selection of poles it is preferred to have ratio between stator and rotor poles to be a non-integer. Common combinations of stator and rotor poles in industrial design are; 6/4, 8/6, 12/8 and 12/10. The selection of poles is influenced by the converter power switches, their cost and control requirements, and the torque ripples. Increasing the stator poles increases the cost of motor due to increased cost of inverter. Similarly, increasing the rotor poles result in increased core losses due to increased stator frequency.

6.4.2

Stator and Rotor Pole Angle Selection

The stator and rotor pole angle selection are important variables in the design of SR motor. The primary selection criteria for the stator and rotor pole arcs are based on self starting requirement and shaping of static torque vs. rotor position characteristics. The guidelines for the selection of pole arc angles are available in in3,5,6,9 and briefly given below. 1. The stator pole arc angle βs is less than rotor pole arc angle βr . 2. The eﬀective torque zone is less than the stator pole arc angle βs , but greater than the stroke angle ε. The stroke angle is defined as – ε=

4π Ns Nr

(6.2)

If βs is less than ε then the motor may not start at certain rotor positions. The inductance profile as shown in Figure 6.4 and the reason why the motor will not start under certain conditions can be explained from the stator and rotor pole arc geometry shown in Figure 6.5. As shown in the fig. 6.4, the inductance profile of a phase repeats every 2π Nr radian. In the region between angles 0 to θ1 , the stator and rotor pole arcs do not overlap. The inductance therefore has the minimum value Lu . The rotor has to traverse an angle θ1 = Nπr − β2s − β2r for the tip of the rotor angle to come under the tip of stator pole. From θ1 to θ2 , the rotor and stator poles overlap to certain extent, and

136

6 Switched Reluctance Motor Drives (SRM)

Es

S Nr Er (a) Rotor position from unaligned position 0 to

T1

Es

Er (b )Rotor position from to

T1 to T 2

Es

Er T 2 to T 3

(c )Rotor position from to

Es

Er (d )Rotor position from to

T 3 to T 4 Es

Er (e )Rotor position from to

T 4 to T 5

Fig. 6.5 Movement of rotor and its position at diﬀerent instants

6.4 Design of SR Machine

137

completely overlap at θ2 . The angle moved by the rotor is θ2 −θ1 = βs . The inductance is steadily increased to the maximum value at θ2 . From θ2 to θ3 the angle traveled is βr −βs , and there is a complete overlap of the rotor and stator poles. The inductance therefore, remains at the maximum value La . From angle θ3 to θ4 the rotor starts to move away from the stator and completely separates at θ4 . The inductance therefore, decreases steadily and becomes minimum at θ4 . The distance traveled from θ3 to θ4 is equal to βs . The same cycle is repeated. The equation for torque developed by the motor can be written asTe =

i2 dL 2 dθ

(6.3)

Where i is the current in the stator winding. As can be seen from Eq. (6.3), the direction of current has no eﬀect on the direction of torque. Positive torque is produced when rate of change of inductance is positive. In order to get better understanding the process of starting, two successive phase inductance profiles are shown in Figure 6.6. As can be seen in the figure, the inductance of phase ‘b’ reaches its maximum value exactly εdegrees after phase ‘a’ inductance reaches the maximum value. The inductance of phase ‘b’ will remain at minimum value till θb1 = θ1 + ε. Now for θ2 = θ1 + βs = θb1 + βs − ε there will not be any problem in starting as at all times one or the other phase of the motor has rising inductance. However, if βs < ε, then phase b will have rising inductance after phase has reached its maximum value. This may cause problem in starting of the motor.

La L as Lu

T1

T2 T3

T 4 T5

T 2 T3

T4

Lb

L bss Lu

T1

Fig. 6.6 Inductance Vs Rotor Position profile of phase a and b

138

6 Switched Reluctance Motor Drives (SRM)

3. The third condition in the design of stator and rotor pole is based on higher torque generation. 2π > βs + βr then rotor and stator poles will overlap even in unaligned conIf N r dition thereby increasing the value of inductance under unaligned condition causing lower torque generation.

6.4.3

Selection of Dimensions

The SR motor is normally used for variable speed drives. It is therefore appropriate to design the motor for a specific base speed. At base speed, the motor is expected to deliver the rated torque at the rated output power. The selection of dimension for the bore diameter is started by assuming it to be equal to the frame size. The stack length can be initially chosen to be equal to the distance between the mounting holes in a foot mounted machine. With the selection of preliminary values of outer diameter D0 , stack length L, bore diameter D and pole pitches βs and βr the design is continued as follows. From B-H characteristics of the material used in stator and rotor stampings the knee point is noted. The maximum flux density in any part of the machine is generally fixed to this value. Normally the maximum flux density occurs at the stator poles. Thus the flux density at stator poles is taken as Bmax and flux density in other parts is designed based on this value. The air gap is determined from constraints imposed by the manufacturing techniques employed. The design of stator winding requires the number of turns per phase and conductor size. The conductor size is selected such that the available winding space can be filled. The number of turns I calculated for a given current of the motor based on its power rating.

6.5

Converters for SR Machine

Unlike induction motors or d.c. motors the reluctance motor cannot run directly from an a.c. or d.c. supply. A certain amount of control and power electronics must be present. The power converter is the electronic commutator, controlling the phase currents to produce continuous motion. The control circuit monitors the current and position feedback to produce the correct switching signals for the power converter to match the demands placed on the drive by the user. The purpose of the power converter circuit is, to provide some means of increasing and decreasing the supply of current to the phase winding. Since the torque developed by SR motors is independent of current polarity, the SR motors operate on the principle of unipolar current. The current therefore can be supplied to the phase winding with the help of only one switch per phase connected in series with the winding. By turning on and oﬀ this switch the flow of current can be regulated. The design of converters

6.5 Converters for SR Machine

139

for SR motors has to be based on the special requirements of such motors. Since the mutual coupling between phases is almost negligible, stored magnetic field energy can create problem during commutation of a phase. The stored magnetic field energy has to be provided a path during commutation: otherwise excessive voltage can develop across the winding and may result in the failure of the semiconductor switch connected in series with the winding. Diﬀerent converter configurations have been suggested for proper utilization of this stored energy. Few configurations of converters commonly used in SR motor drives are presented here.

6.5.1

Asymmetric Bridge Converter

In this section the operation of popular two switches per phase asymmetric bridge converter is described. This converter is simple but requires (m+1) gate drive power supplies for m phase motor. Figure 6.7 shows an eight switch asymmetrical converter. To supply a current to a particular phase say phase a switches T1 and T2 are turned on. If the current in the winding exceeds the commanded value, either T1 or T2 is turned oﬀ. The energy stored in the winding of phase a is dissipated by the freewheeling current either through D1 or D2 . The voltage across the winding becomes zero (neglecting the diode and switch voltage drops). The waveforms of the current and voltages are shown in Figure 6.8. Suppose the current at phase a during positive inductance slope is to be provided is Ip . A current i∗a is commanded at the positive slope of inductance, which is compared with the phase current ia in a current feedback loop. The current error is processed in a hysteresis controller with a current window of ∆i. When the current error is greater than ∆i, switch T1 or T2 is turned oﬀ (usually T2 ). The current starts to fall down and when the current error is + ∆i, switch T2 is turned on. When the current command becomes zero, both T1 and T2 are turned oﬀ simultaneously. During this period, the voltage across the winding is - Vdc when D1 and D2 are conducting, and equal to zero after the diodes stop conducting. The voltage across T2 when T1 is on

T3

T1

T5

D1

T7

D3

D5

Vdc D7

D6

D2 T2

D4

T4

D8 T6

Fig. 6.7 Eight switch asymmetrical converter

T8

140

6 Switched Reluctance Motor Drives (SRM)

L as

i *a

ia ON

T1 OFF ON

T2 OFF

Vdc Va

- Vdc

Fig. 6.8 Waveforms of eight switch converter

and T2 is oﬀ, is equal to Vdc . The power switches and diodes therefore must be rated for more than Vdc . Advantages of this converter are: 1. Vdc , 0 or -Vdc can be applied to the phase winding. 2. Two or even three phases can be turned on regardless of the current in other phases, thereby negative torque for reversing the motor can be generated instantly. 3. If one switch is damaged the converter can still supply power (reduced) to the motor through other three phases.

6.5 Converters for SR Machine

141

The disadvantages are:1. 8 switches and 8 diodes are required for four phase drive. 2. 8 gate drive circuits and 5 gate drive power supplies are required. This results in increased cost and size of the drive.

6.5.2

Six Switch Converter

Another converter which is commonly used for SR motors with even number of phases works with 1.5 switches per phase. The six switch converter has four phase switches and one common upper switch for phases a and c, along with one common upper switch for phases b and d. a six switch converter for four phase SR motor is shown in Figure 6.9. The switches T5 and T6 carry twophase currents in one cycle of operation. These switches therefore must have higher current ratings than the other four (T1 , T2 , T3 and T4 ) switches. The windings of the motor are also grouped as ac, and bd. The diﬀerent grouping of non-successive phases guaranties the independent control of their currents. For supplying current to phase a, the switches T5 and T1 are turned on. The diodes D1 and D3 form a circuit similar to 8 switch converter for releasing the stored energy. The operation of the converter is similar to 8 switch converter. The waveforms for this converter are simlar to the waveforms of eight switch converter shown in Figure 6.8. Advantages:1. Only six switches and six diodes are required. 2. Only three power supplies for gate drive circuits are required. 3. Vdc , 0 or -Vdc can be applied to the phase winding. Disadvantages:1. Phases a and c or phases b and d can not be connected simultaneously. 2. The direction of rotation of the motor can not be reversed until currents in all the windings become zero.

T5

T6 D3

D1

Vdc

D4 a

D5

D6

T2 T1

d

b

c

T3

Fig. 6.9 Six switch converter for SRM drive

T4

142

6 Switched Reluctance Motor Drives (SRM)

6.5.3

Variable dc Link Buck Converter

A variable dc link converter circuit3,11 for four-phase SR motor is shown in Figure 6.10. The switch T, diode D, inductor L, and capacitor C form the buck converter power stage which is basically a step down chopper. This stage varies the input voltage Vdc to a desired input level Vi to be applied to the windings. This stage also provides isolation required for faster commutation of current, while limiting the voltage rating to the source voltage level. When phase switch T1 is turned on, the voltage Vi is applied to phase a of the machine. To regulate the current in the winding, switch T1 is turned oﬀ, and the current freewheels through the diode D, source voltage Vdc and capacitor C, assuming that switch T is oﬀ. During this time voltage equal to Vi − Vdc is applied to the phase winding. Energy stored in capacitor C is suﬃcient to turn on the next phase. In this way independence of various machine phases is maintained. 6.5.3.1

Operational Modes

Modes of operation for the circuit can be derived by isolating each machine phase mode, and then combining all the modes. Mode 1: Switch T is on, Diode D is oﬀ, Switch T1 and diode D1 are oﬀ, phase current ia = 0 and iL = ic > 0. Mode 2: Switch T is oﬀ, Diode D is on, Switch T1 and diode D1 are oﬀ, ia = 0 and iL = ic > 0. Mode 3: Switch T is oﬀ, Diode D is oﬀ, Switch T1 and diode D1 are oﬀ, ia = 0, iL = 0. During this mode no phase is energized, and motor is not connected to supply. Mode 4: Switch T is on, Diode D is oﬀ, Switch T1 is on, and diode D1 is oﬀ, ia > 0, ic > 0.

T1

D1

D2 T2

C

Vdc

Fig. 6.10 Variable dc link converter

T3

D3

6.5 Converters for SR Machine

143

During this mode phase a is energized. The converter equation can be written as :Vdc − Vi diL = (6.4) dt L dVi ic iL − (ia + ib + ic + id ) = = (6.5) dt C C Here all the four currents are considered. However, if the currents in other phases have already been commutated, only phase a current is to be considered. For phase a the voltage equation can be written as – Vi = ia Rs +

dλa dt

(6.6)

The current ia can be obtained from the flux linkage vs rotor position relation. Mode 5: Switch T is on, Diode D is oﬀ, Switch T1 is oﬀ, and diode D1 is on, ia > 0, ic > 0. During Mode 5, the current in phase a is being regulated or commutated. The equations for the converter in this mode are:diL dt

by-

=

Vdc −Vi L

and

dVi dt

= iCc =

iL −(ia +ib +ic +id ) C

and the voltage equation is given

dλa (6.7) dt Negative voltage equal to (Vi − Vdc ) is applied to phase a during this mode. (Vi − Vdc ) = ia Rs +

Mode 6: Switch T is oﬀ, Diode D is on, Switch T1 is oﬀ, and diode D1 is on ia > 0, ic > 0. In this mode also the current in phase a is being regulated or commutated and current through main switch T is regulated. The equations can be written as −Vi diL = (6.8) dt L dVi ic iL − (ia + ib + ic + id ) = = dt C C a and voltage equation is (Vi − Vdc ) = ia Rs + dλ dt .

Mode 7: Switch T is oﬀ, Diode D is oﬀ, Switch T1 is oﬀ, and diode D1 is on, ia > 0, ic = 0. During this mode phase a is energized and current through the main switch a T is regulated. Positive voltage is Vi equal to Vi = ia Rs + dλ dt applied to the machine. Mode 8: Switch T is oﬀ, Diode D is oﬀ, Switch T1 is on, and diode D1 is oﬀ, ia > 0, iL = 0, ic > 0.

144

6 Switched Reluctance Motor Drives (SRM)

This mode occurs when the machine is switched oﬀ. The equations are: −(ia +ib +ic +id ) dVi diL Positive voltage Vi is applied to phase a Vi = dt = 0, dt = C dλa ia Rs + dt . Mode 9: Switch T is oﬀ, Diode D is oﬀ, Switch T1 is on, and diode D1 is oﬀ, ia > 0, iL = 0, ic = 0. This mode also occurs when the motor is switched oﬀ. Negative voltage is a applied to phase a and the voltage is equal to(Vi − Vdc ) = ia Rs + dλ dt . If phase b of the machine is energized, while phase a is still conducting, or is being commutated, six additional modes similar to modes 4 to 9 will occur. During this time the currents in phase a and b will be positive. Since the mutual inductances between the phases in SR can be neglected, then modes 1 to 9 can be considered as independent modes. 6.5.3.2

Design of Converter Circuit

The chopper circuit in the variable dc link converter plays an important role in the operation of SR motor drive. The inductor L is designed on the basis of the ripple current rating that allows minimum energy storage to enable faster charging of the capacitor C. The maximum ripple current in the capacitor can be expressed asVdc = 2Ip (6.9) ∆ic = Lfc Where Ip is the maximum allowed ripple current per phase. The value of inductor L can be obtained as– L=

Vdc H 2fc Ip

(6.10)

The capacitor C is designed so that the complete energy stored in it can be transferred to the machine in n cycles of the converter. Or 2Vi Ip 1 (6.11) CV2i = n 2 fc Or C=n

4Ip Vi f c

The minimum value of capacitance C is obtained when Vi = Vdc . Or 4Ip C=n Vdc fc

(6.12)

(6.13)

The value of n is decided on the basis of commutation time of the current and the number of times the switch is operated.

6.6 Buck-Boost Converter

145

Example 6.1 For an 8/6 SR motor running at 1500 rpm determine the time for which each phase of the Buck converter is turned on. Also if main switch is turned on and oﬀ twice what is the carrier frequency fc of the inverter. Solution 360 0 For 8/6 machine each phase is on for ( 360 6 − 8 ) = 15 15 1 =1.666 m.sec. The time to traverse 150 t= 360 × 1500 60

fc = nt , Since main switch is turned on and oﬀ 2 times, n=2 2 fc = 1.666×10 = 1200 Hz. −3 The value of n is decided on the basis of desired performance. Higher value of n is required for high performance drive. 6.5.3.3

Merits and Demerits of Buck Converter

The merits of the buck converter are: 1. Only (m+1) switches and diodes are required for four quadrant operation of m phase motor. 2. Phases can be operated independently. Thus the operation equivalent to 2m switches is obtained with m+1 switches. 3. Only 2 logic supplies are required because all the machine phase switch emitters are tied together. 4. A very small capacitor and inductor are required in the chopper power stage. 5. The peak voltage rating of the power devices is equal to the dc link voltage Vdc only. The demerits of this converter are: 1. The rating of chopper circuit has to be equal the motor power rating, thus it is not suitable for high power drives. 2. The losses in the system are higher due to additional power processing stage. 3. Full negative dc voltage can not be applied to the phases, hence commutation time is larger.

6.6

Buck-Boost Converter

Another form of variable dc link voltage converter circuit requiring m+1 switches and diodes12 is shown in Figure 6.11. In this converter also the switches are connected in series with the phase windings which prevents a shoot through fault. The switch T, diode D, inductor L, and capacitor C form the buck-boost front end power stage converter circuit. The machine voltage Vi can be varied from zero to about twice the dc source voltage Vdc The application of increased input voltage to the motor accelerates the build

146

6 Switched Reluctance Motor Drives (SRM)

D1

Vdc

Cd D2

D3

D T1

T2

T3

D4 T4

Fig. 6.11 Variable dc link converter with m, +1 switches

up of the current in the machine phases. This is a clear advantage over the buck converter topology. The modes of operation are similar to the modes of buck converter. When machine phase (say phase a) is to be energized, the phase switch T1 is turned on, thus applying the voltage Vi to machine phase a. During this mode T is on, D is oﬀ and D1 is also oﬀ. In order to regulate the current in phase a, switch T1 is turned oﬀ. The current is routed through the freewheeling diode D1 , the dc source voltage Vdc and phase a winding. The current continues to flow through this path even when switch T is turned oﬀ. Negative voltage equal to -Vdc is applied to the phase winding during this time. The energy stored in the capacitor C will be able to cater the turning on of phase b when switch T is oﬀ. Thus independence between phases is maintained in this topology also. The main disadvantage of this converter is due to the voltage rating of phase switches which must be equal to Vdc + Vi .

6.7

Analytical Model of SR Machine

The mathematical model of SR machine is highly nonlinear due to magnetic saturation. The key characteristics of a SR motor can be described by flux linkage as a function of phase current and rotor position. However, it is difficult to express this relationship mathematically due to eﬀect of saturation, hysteresis, and the double saliency of construction. The analytical model of SR machine is obtained using per unit system as Taking the rated values of machine voltage Vb volts θb and speed nb in rpm as base values. The base value of position angle θb taken as θb = π. The rotor aligned position is assigned as θ = 0 and unaligned position at θ = π. The current that produces the base flux linkage at aligned position is taken as base current. The equation of flux linkage in per unit system can be written mathematically as-

6.7 Analytical Model of SR Machine

147

λ(i,θ) = λu (i) + {λa(i)−λu (i)}.f(θ)

(6.14)

Where λa and λu are aligned and unaligned flux linkages depending only on phase currents. f(θ} is a function that depends on the rotor position only. Neglecting the eﬀect of saturation and hysteresis, the flux linkage at the unaligned position can be assumed to be linear, and can be expressed asλ u = Lu i

(6.15)

For flux linkage in aligned position the following four conditions must be met. dλa dλa = La , = Lu (6.16) λa |i=0 = 0, λa |i=1 = 1, di i=0 di i=∞

From these conditions, the aligned flux linkage can be written as – λa (i) = Lu .i + (La − Lu)

i (1 + Ks) .i

(6.17)

Where Ks is the saturation factor given byKs =

La − 1 1 − Lu

(6.18)

Equation (6.14) has to satisfy the following conditions λ(i,θ)θ=0 = λa (i)

(6.19)

λ(i,θ)θ=1 = λu (i)

(6.20)

f(θ)θ=0 = 1andf(θ)θ=1 = 0

(6.21)

and Also angular function

If overlap eﬀect of rotor and stator poles is considered with θk representing the eﬀective overlap position, the angular function can be written as – θ 1 1 + cos (6.22) .θb for |θ| ≤ θk f(θ) = 2 2 θk f(θ) = 0 for |θ| ≥ θk

(6.23)

Where θb = π is the base value of the position angle. The inductances La and Lu and the overlap angle θk can be obtained by Finite Element method calculation or by measurement. The general expression for torque produced by one phase is ∂W (6.24) T= ∂θ i=constant

148

6 Switched Reluctance Motor Drives (SRM)

Where W is the magnetic co-energy defined as W′ =

i

λdi

(6.25)

0

The co-energy in terms of the analytical model derived above is 1 W′ = Lu i2 + Wi .f(θ) 2

(6.26)

Where Wi depends only on current. Then instantaneous torque is obtained as T = .Wi .f′ (θ) (6.27)

6.8

Control of SR Motor Drive

The control of SR motor is complicated due to the fact that the machine winding inductance is a function of excitation current as well as the rotor position. In order to have simple control not requiring high performance applications, the excitation current may be assumed constant, and inductance maybe considered as a function of rotor position only. The basic principle of SR motor control is explained here. The inductance profile of phase a of the motor is shown in Figure 6.12. The current to phase a is applied at the instant the inductance starts increasing. The curve for the torque for this phase is also shown. An average torque will result due to the combined value of instantaneous torques produced by all the machine phases. The SR motor produces discrete pulses of torque, and continuous torque can be produced by suitable design of overlapping inductance profile. However, overlapping of inductance results in reduced power density, and increased complexity in control. The average torque of the motor can be controlled by controlling the magnitude of phase current, or by varying the angle θd . In order to reduce the torque ripples, the angle θd is kept constant, and the torque is controlled by varying the phase current. A current controller for low performance, closed loop speed control drive is presented first. To ensure instantaneous torque production it is essential that the current is applied immediately at the instant of increasing inductance. Since in an R-L circuit the current can not rise or fall instantaneously, it is essential that the voltage should be applied in advance for starting the current. Also the commutation should also be applied in advance to bring the current to zero before the inductance slope becomes negative. For low cost SRM drives Hall position sensors are used to trigger the turn oﬀ in a conducting phase, and at the same time turn on the incoming phase. The actual waveforms will therefore, be as shown in Figure 6.13. The angle by which the applied

6.8 Control of SR Motor Drive

L as La Lu

T2 T3

T1

T4

i as

Fig. 6.12 Inductance profile and torque production n SR motor

L as

La Lu

T1

T 2 T3

T4

i *as

i as

T

Fig. 6.13 Actual current waveform

149

150

6 Switched Reluctance Motor Drives (SRM)

voltage or the commutation is advanced depends on the magnitude of the peak value of phase current, and the rotor speed. The current in the winding is modulated by turning on or oﬀ the phase switch within the hysteresis band of ±∆i. Starting of SRM drives under load poses problems due to number of proximity sensor signals can provide information about which phase has to be turned on but not exactly the initial position. A separate speed signal is necessary for safe starting under load.

6.9

General Purpose SRM Drive with Speed/Position Sensor

For applications in moderately low power and dynamic performance a simple low cost Switched Reluctance motor drive is presented here. The block diagram of an SRM drive along with the controller is shown in Figure 6.14. The reference speed is compared with the actual value of the speed. The speed error is processed through a PI controller and limiter to determine the torque command. From torque command the current command signals are obtained using a torque constant. This torque constant is actually valid for linearalized inductance vs. rotor position for a particular value of current. The current command is compared with motor currents, and the error signals are processed with triangular carrier frequency to generate pulse width modulation control signals. These control signals are used to switch the power transistors supplying power to the motor. The switch to be turned ‘on’ or ‘oﬀ’ is selected by a switch select circuit using actual rotor position information.

ω

* r

PI controller

ω

Torque command

Current controller

PWM converter signals

ia Power converter

i

b

IM

r

Rptor position

Fig. 6.14 Block Diagram of general purpose SRM drive with speed/position sensor

6.9 General Purpose SRM Drive with Speed/Position Sensor

151

The absolute position information is obtained from the speed encoder connected to the shaft of the motor. This information is processed in a Programmable ROM which contains several data tables based on direction of rotation, and the advance angles. The zero reference point of the rotor position is the instant where the winding of phase ‘a’ is in alignment with the rotor. Based on the rotor position and the quadrant of operation the switching signals for phase ‘a’ are described below. For example for a 6/4 SRM for an advance angle and fall angle of zero degree, the switching signals should be changed at 00 , 300 , and 600 . For conditions where advance angle and fall angle are same, the conduction angle for each phase winding will still be 300 . For example if the fall angle and advance angle are 150 , the range of ‘on time’ for switching phase ‘a’ will be between 450 and 750 in the first quadrant in clockwise direction.

6.9.1

Design of Current Controllers

Since the SR motor is a non-linear device the design of a controller is more involved as compared to other ac machines. In one of the methods the motor model is linearized and the well-known linear control theory is applied. Although these linear schemes are simple, and real time tractable, the results are not accurate due to simplified models which are not so accurate. In linearized schemes, the voltage and torque equations of SR motor are linearized about the rated current and speed or the most likely operating point. The inductance of the windings is assumed to be constant and is taken as the mean value between the aligned inductance and unaligned inductance at the rated current. A non-linear controller enables linearization and decoupling of current loop, resulting in high performance. There are number of schemes for designing non-linear current controller as described briefly here. In one of the schemes feedback linearization is used that provides compensation for the magnetic non- linearities of the motor. In these schemes either the eﬀect of mutual inductance is not considered, or there are few schemes where the eﬀect is considered. In all these schemes a PWM method is used to force current command but sometimes a bang–bang current controller is used.

6.9.2

Torque Control

Torque control in all electric machines is required and is implemented by controlling the currents. However, in SR motor the relationship between the torque and current is non-linear and requires the information of rotor position also. The methods of torque control in SR motors depend on the number of phases conducting at a given instant and the current control capability. For a

152

6 Switched Reluctance Motor Drives (SRM)

single-phase excitation control where only one phase is excited at a time, the current flows in two windings, at the time of commutation of one phase, and initiation of another phase. If the current is controlled only in the incoming phase, the resultant torque is not constant and will have a trough. This results in ripples in torque which is not desirable in high performance drives. A number of techniques have been suggested to reduce the torque ripples. During commutation interval, linearly varying currents can be applied in both the phases. It does not overcome the problem of constant torque during commutation completely due to diﬀerent torque constants in two phases. However, the it reduces the demand for steep currents in the phases. If instead of using a single phase excitation two phases are excited at the same time, the torque is distributed to two conducting phases and the sum of the two torques is the motor torque. If mutual coupling between the two phases is also considered, the motor torque has three components; the torque due to the self inductances of the two phases, and the third component due to mutual inductance between the phases. In order to implement torque control, it is assumed that rotor position and stator instantaneous currents are available through measurement or estimation. The feedback currents are processed, conditioned and converted to digital form in an ADC. When a particular value of torque is requested, the torque distribution function controller provides the phase torque commands. From phase torque commands, the phase current commands are generated using rotor position into consideration. From these current commands the current controller provides the phase voltage commands. From these voltage commands the PWM circuit generates the gate drive signals of the power converter.

6.10

Direct Torque Control of SRM Drive

Because of double saliency in Switched reluctance motors these motors are not excited from normal ac voltages, thus the rotating electromagnetic field theory applicable to ac machines can not be applied here. Also, due to the motor’s non uniform torque output characteristics, high torque ripples are present unless some form of torque ripple reduction technique is employed. In order to get better performance with less sophistication, direct torque control method has been proposed. The normal method of direct torque control as used in ac machines assumes linear characteristics of motor, and balanced three phase ac supply. The SR motor however has a non-linear model and non-sinusoidal supply. The direct torque control for SR motor is therefore, not as simple as in other ac machines. The direct torque control of SR motor can be obtained from the following voltage equationv = Ri +

dλ(θ, i) dt

(6.28)

6.10 Direct Torque Control of SRM Drive

153

If the concept of co-energy for production of torque is applied to SR motor, the torque equation can be written as – ′

T=

∂W ] ∂θ i=const

(6.29)

This equation provides an insight for the control of torque in SR motor. The expression for instantaneous torque T can be written asT ≈i

∂λ(θ, i) ∂θ

(6.30)

The approximate value of torque as given by Eq. 6.30 is fairly accurate because the magnitude of torque is controlled using a hysteresis band. The Eq. 6.30 is used only to increase or decrease the torque and is not used to control the actual magnitude of the torque. It is also important to remember that in SR motors unipolar drives are normally used, and thus the current in motor phases is always positive. From Eq. (6.30) it is clear that the sign of torque depends on the sign of ∂λ ∂θ . In other words to produce positive torque, the stator flux amplitude must be increasing with respect to rotor position14 . In order to obtain negative torque, the change in stator flux should be negative with respect to rotor position. A positive value of ∂λ ∂θ may be considered as ‘flux acceleration’ whereas the negative value of ∂λ ∂θ may be defined as ‘flux deceleration’. If the magnitude of the flux must increase, with reference to rotor position, the stator flux vector must be ahead of the rotor with respect to the direction of rotor movement. Similarly, if the magnitude of the flux is to decrease relative to rotor position, then the stator flux vector must lag behind the rotor position. It is therefore possible to control the motor torque of an SR motor by accelerating or decelerating the stator flux vector with respect to the rotor position. In order to directly control the torque of an SR motor, the stator flux linkage of the motor is kept constant (within amplitude hysteresis band). The torque is controlled simply by accelerating or decelerating the stator flux vector with respect to rotor position. The stator flux linkage is kept constant by selecting one of the six-voltage vectors of the voltage source inverter as shown below. It is assumed here that a 3-phase inverter similar to the voltage source inverter used for direct torque control of induction motor is used. In Eq. 6.28, if stator voltage drop is neglected, then v∼ =

dλ(θ, i) dt

(6.31)

If time interval is suﬃciently small, it can be written as∆λ(θ, i) = v∆t

(6.32)

154

6 Switched Reluctance Motor Drives (SRM)

s1

a

s2

Fig. 6.15 Supply to phase a

In order to vary the torque of the motor, the acceleration or deceleration of stator flux vector with respect to rotor is obtained by applying the appropriate voltage vector from the inverter. In SR motor, the phases are excited by switched currents, which are completely independent of each other, thus the normal DTC technique applied in induction machine can not be applied here. An equivalent to voltage space vector used in induction machine can be defined for SR motor. Due to the salient pole structure of SR motor, the voltage space vector for each phase is defined as lying on the center axis of the stator pole. Hence the voltage space vectors for 3-phase SR motor are as shown in Fig. 6.14. Since only uni-directional current is applied, each motor phase can have only three possible voltage states. The three possible voltage states from Fig. 6.15 can be considered as:1. (state 1) when both the devices in one phase are conducting or positive voltage is applied 2. When only one of the devoice is conducting or zero voltage is applied (state 0), and 3. When both device are oﬀ and current freewheels through diode. The voltage across the motor phase is negative (state-1) In SR motor therefore, there are three possible states for each phase or there are total 27 possible configurations15,16. However, in order to apply six equal

6.10 Direct Torque Control of SRM Drive

Phase2

V2 (0,1. 1)

V3

155

Phase3

V1

( 1,10)

(1,0. 1) Phase 1

V4 (1,0.1)

V6 (1,1.,0) V5 (0, 1.1)

Fig. 6.16 Voltage vectors for DTC of SR motor

amplitude voltage vectors that are separated by π6 radians, as is normally applied in induction motors, six possible voltage vectors are obtained as shown in Fig. 6.16. These vectors are located at the center of a zone of π6 radian length. In this scheme the controller allows no other states. For the torque control one of these six possible states is chosen at a time to control the stator flux linkage within a hysteresis band. If an increase in torque is required, the voltage vector that advances the stator flux linkage in the direction of rotor movement is selected. It is therefore, possible to define a switching table for stator flux linkage and motor torque, as in normal direct torque control of induction motor. Also to control the torque and stator flux linkage within hysteresis band, the magnitude of individual stator phase flux linkage is resolved into a single stator flux linkage vector by transforming these vectors into a stationary orthogonal two axis α − β reference frame. In order to control the torque of the motor in DTC scheme using hysteresis band, torque feedback is also required. In this scheme actual measured value of torque of the motor for diﬀerent values of rotor position and stator current is stored as a lookup table. This avoids the complex computation of instantaneous torque in real time. However, the disadvantage of this scheme is requirement of position encoder and measurement of torque at diﬀerent rotor positions before using the motor.

156

6.11

6 Switched Reluctance Motor Drives (SRM)

Sensorless Control of SRM Drives

In SR motor drives the application of current to the stator winding requires the information of rotor position. The rotor position information is estimated from measurement of current and voltages. A number of methods of rotor position estimation are being developed for sensorless operation as reported in the literature8,10,17,18,19. The desirable feature of a sensorless scheme is that it should require only terminal measurements and does not require additional hardware or memory. In addition it also desired to have reliable operation over the entire speed and torque range, while maintaining high resolution and accuracy. The main idea behind all the sensorless schemes is based on the fact that the mechanical time constant of the SR motor drive is much larger than its electrical time constant. It is therefore, possible to recover the encoded position information that is stored in the form of flux linkage, inductance, back emf etc. Knowledge of the magnetic characteristics of an SRM plays an important role in determining the rotor position indirectly for sensorless control. The earlier methods almost all of them therefore, have used the inductance variation information to detect the rotor position.

6.11.1

Position Information from Inductance Variation

Since the phase inductance of the SR motor varies significantly between the aligned and unaligned rotor positions, the inductance variation information can be used to determine the rotor position. The relationship between rotor position and inductance is unique for a particular value of excitation current over half rotor pole pitch. Therefore, information of rotor position is stored in the form of table between measured or estimated values of inductance vs. rotor position for each value of excitation current. In one of the schemes the rise time or the fall time of the current which are proportional to incremental inductance are measured. In current rise time method, the currents at the beginning of on time of the switch, and oﬀ time are measured. the diﬀerence between these currents is incremental rise in current ∆i, and the time for which the device is on is ton . From these values along with the applied voltage the flux linkages are estimated. The incremental inductance δL is the incremental change in flux linkage due to incremental change in current which is obtained as follows. The applied voltage to the conducting phase ‘a’ is given by – va = Ra ia + = Ra ia +

dλa (θ, i) dt

δλa dθ δλa dia . + . δθ dt δia dt

(6.33) (6.34)

6.11 Sensorless Control of SRM Drives

157

Now the rate of rise of current for one carrier or PWM switching can be ∆i obtained by taking di = ∆i and dt=ton or di dt = ton . Also the emf induced is given byδλ dθ . (6.35) δθ dt From these equations the ‘on’ time ton can be obtained as δλ ∆i ton = . (6.36) δi va − Rs ia − e

Here the term δλ δi . represents incremental inductance, and can be written as δL. The Equation(6.36) can be used to estimate the value of incremental inductance δL as – δλ va − Rs ia − e (6.37) δL = .= ton δi ∆i e=

Once the value of inductance has been estimated, the rotor position information can be obtained from the stored value. The main problem with this method is the error that is introduced because of variation in stator resistance due to skin eﬀect and heating. Also in this method mutual coupling between the phases has been neglected. The mutual coupling can introduce error in the estimation of inductance when currents are overlapping in the machine phases. Another method of estimating from the equation (6.37) is rotor position va −Rs ia −e is constant. Thus the incremental based on the assumption that ∆i inductance is directly proportional to the current rise time ton . The incremental inductance can also be estimated from the time taken by the current to fall from i1 toi2 in time tf . Or −Rs ia − e (6.38) δL = tf ∆i Also if the same assumptions are made as in current rise method the incremental inductance can be assumed to be directly proportional to current fall time. The drawback of both the schemes is that it is not accurate at high speeds but gives good results at low speeds. In spite of drawbacks of the current rise or fall method it has the advantage of not using any external circuitry to estimate rotor position. The rotor poison can be estimated by processing the measured current and rise or fall time of the current only.

6.11.2

Estimation Based on Inductance Measurement with External Signal Injection

There are two methods for speed estimation based on measurement of inductance with external signal injection; the constant current or flux linkage

158

6 Switched Reluctance Motor Drives (SRM)

Os

Is Fig. 6.17 Flux linkage vs. phase current curve

method and demodulation technique. The constant current/flux linkage method is based on the property of linear characteristics of the motor for small values of excitation current and flux linkages. Fig. 6.17 shows the relationship between flux linkages and phase current for diﬀerent rotor positions. For small value of phase current Is , which is assumed as constant the flux linkages vs. rotor position can be obtained from Fig. 6.17. As shown in Fig. 6.18, for constant value of phase current equal to Is , the flux linkage vs. rotor position is linear. Using this information a sensorless scheme for the control of SRM is shown in Fig. 6.19. As shown in the block diagram a sample and hold circuit samples the flux linkage λ = (v − Rs i)dt for phase current equal to Is .This value of flux linkage is used to extract the rotor position stored in memory. The memory has rotor position value stored corresponding to the flux linkages as obtained from Fig. 6.18. This estimated rotor position is used to estimate the motor speed. The estimated rotor position and speed along with measured currents and desired speed is applied to a control block. The error in speed signal is used to generate torque reference, from which the current reference is generated. The current reference command for individual phases is generated with the help of rotor position information. The PWM converter placed after the control block is switched based on the phase current references. As shown in Fig. 6.18, there are two values of λ for each value of θ. It is therefore necessary at the time of starting to use two phases to clearly identify the

References

159

O i is

T1

T2

Fig. 6.18 Flux linkage vs. rotor position

Fig. 6.19 Sensorless speed control of SRM drive

rotor position Once the motor starts running only one phase is suﬃcient to identify the rotor position. This method of rotor position estimation is not suitable for high currents, due to mutual inductance between phases and saturatuion.

References [1] Miller, T.J.E.: Brushless Permanent magnet and Reluctance Motor Drives. Clarendon Press, Oxford (1989) [2] Miller, T.J.E.: Switched Reluctance Motors and their Control. Magna Physics, Oxford (1992)

160

6 Switched Reluctance Motor Drives (SRM)

[3] Krishnan, R.: Switched Reluctance Motor Drives. CRC Press, USA (2001) [4] Nasar, S.A.: D.C. Switched Reluctance Motor. Proceedings of IEE 116(6), 1048 (1969) [5] Radun, A.V.: Design Considerations for the Switched Reluctance Motor. IEEE Transactions on Industry Applications 31(5), 1079–1087 (1994) [6] Krishnan, R., Arumugam, R., Lindsay, J.F.: Design Procedure for Switched Reluctance Motors. IEEE Transactions on Industry Applications 24(3), 456– 461 (1988) [7] Miller, T.J.E.: Converter volt-ampere Requirements of the Switched Reluctance Motor Drives. IEEE Transactions on Industry Applications 21(5), 1136– 1144 (1985) [8] Corda, J., Stephenson, J.M.: An Analytical Estimation of the Minimum and Maximum Inductances of a Double-salient Motor. In: Proceedings of International Conference on Stepping Motors and Systems, Leeds, U.K., pp. 50–59 (1979) [9] Vijayraghvan, P.: Design of Switched Reluctance motors and Development of Universal controller for Switched Reluctance and Permanent Magnet Brushless DC Motor drives. Ph.D. Dissertation Virginia Polytechnic Institute and State University (2001) [10] Ehsani, M., Fahimi, B.: Position Sensorless Control of Switched Reluctance Motor drives. IEEE Transactions on Industrial Electronics 49(1), 40–48 (2002) [11] Krishnan, R.: A Novel Converter Topology for Switched Reluctance Motor Drives. In: 27th Annual IEEE PESC Conf., Baveno, Italy, pp. 1811–1816 (1996) [12] Vukosavic, S., Stefanovic, V.R.: SRM inverter topologies: A comparative evaluation. IEEE Transactions on Industry Applications 27, 1034–1047 (1991) [13] Lawrenson, P.J., Stephenson, J.M., Blenkinson, P.T., Corda, J., Fulton, N.N.: Variable-speed switched reluctance motors. Proceedings of IEE 127(4), pt. B, 253–265 (1980) [14] Cheok, A.D., Fukuda, Y.: A New Torque and Flux Control Method for Switched Reluctance Motor Drives. IEEE Transactions on Power Electronics 17(4), 543–558 (2002) [15] Jeong, B.H., Lee, K.Y., Na, J.D., Cho, G.B., Baek, H.L.: Direct Torque Control or the 4-phase Switched Reluctance motor drives. In: ICEMs 2005, Proceedings of 8th International Conference on Electrical Machines and Systems, September 2005, vol. 1, pp. 524–528 (2005) [16] Guo, H.-J.: Considerations of direct torque control for switched Reluctance Motors. In: IEEE International Symposium on Industrial Electronics 2006, July 2006, vol. 3, pp. 2321–2325 (2006) [17] Fahimi, B., Suresh, G., Ehsani, M.: Review of sensorless control Methods in Switched Reluctance Motor Drives. In: IEEE Industry Applications Conference 2000, vol. 3, pp. 1850–1857 (2000) [18] Ehsani, M., Fahimi, B.: Elimination of Position sensors in Switched Reluctance motor Drives: State of the art and Future Trends. IEEE Transactions on Industrial Electronics 49(1), 40–47 (2002) [19] Lee, D.-H., Kim, T.-H., Ahm, J.-W.: A simplified Novel Sensorless Control of SRM. In: IEEE Industry Applications Conference 41st IAS Annual Meeting 2006, vol. 4, pp. 2001–2005 (2006)

Chapter 7

Control of Multiphase AC Motor Drives

7.1

Introduction

Three-phase AC motor has been used as the main driving machine for a very long period. The main reason for the use of three-phase machines has the availability of three-phase AC supply from the mains. However, with the availability of power electronic devices of high power rating, the control of drives is always from the inverter supplying power to the motor. If a motor is supplied from a dc link inverter, the number of phases can take any value without any problem. This has resulted in interest in multi-phase motor drives recently. When compared to three-phase machines, drives with higher than three-phases oﬀer a number of advantages. The most important advantage that resulted in the development of first five-phase motor in 1969is the reduction in amplitude and an increase in the frequency of torque pulsations1 . The other advantages of multi-phase drives are:1. Reduction of per phase inverter rating for the same output power of the motor. This is great advantage for very high power drives because it eliminates the need for parallel connection of switching devices. 2. Reduction in the rotor harmonic losses due to absence of certain time harmonic currents. 3. Possibility of higher torque density per ampere current, as space harmonic fields can also contribute to produce torque.

7.2

Modeling of a Five Phase Induction Motor

For determining the steady state characteristics of an induction motor having ‘n’ phases, per phase equivalent circuit can always be used1,2 . However, for determining the dynamic response, which is essential in high performance variable speed drives, dynamic modeling of the motor is required. In a fivephase machine the phases have magnetic axes that are 72-electrical degrees apart. The rotor may be squirrel cage type or will have similar winding as in M. Ahmad: High Performance AC Drives, Power Systems, pp. 161–173. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

162

7 Control of Multiphase AC Motor Drives

the stator. The model of a five phase machine under the usual assumptions made for three-phase machine in terms of voltage and flux linkages can be written in matrix form as[vsabcde ] = [Rs ][isabcde ] +

d s [λ ] dt abcde

d r [λ ] =0 dt abcde [λsabcde ] = [Ls ][isabcde ] + [Lsr ][irabcde ]

[vrabcde ] = [Rr ][irabcde ] +

[λrabcde ] = [Lr ][irabcde ] + [Lrs ][isabcde ]

(7.1) (7.2) (7.3) (7.4)

Eq. 7.1 and 7.2 can be written as – [vsabcde ] = [Rs ][isabcde ] + Lss

d d s [iabcde ] + [Lsr irabcde ] dt dt

d r d [i ] + [Lsr isabcde ] = 0 dt abcde dt T [vsabcde ] = vsa vsb vsc vsd vse

[vrabcde ] = [Rr ][irabcde ] + Lrr

[isabcde ] = isa [irabcde ] = ira

isb

isc

isd

ise

irb

irc

ird

ire

T

T

(7.5) (7.6) (7.7) (7.8) (7.9)

The stator and rotor inductances can be represented in the matrix form similar to a three-phase machine as – ⎡ ⎤ Laas Labs Lacs Lads Laes ⎢ Lbas Lbbs Lbcs Lbds Lbes ⎥ ⎢ ⎥ ⎥ (7.10) [Ls ] = ⎢ ⎢ Lcas Lcbs Lccs Lcds Lces ⎥ ⎣ Ldas Ldbs Ldcs Ldds Ldes ⎦ Leas Lebs Lecs Leds Lees Where Laas = Lbbs = Lccs = Ldds = Lees are the self inductances of phases a,b,c,d,e., Labs = Lbas , Lacs = Lcas . . . . . . . . . . . . . . . . . . . . . . . . etc. are the mutual inductances between phase ‘a’ and phases b, c, d, and e. For five-phase machine taking α = 2π 5 , the inductance matrix of Eq. (7.10) can be written as – ⎡ ⎤ Lls + M M cos α M cos 2α M cos 2α M cos α ⎢ M cos α Lls + M M cos α M cos 2α M cos 2α ⎥ ⎢ ⎥ [Ls ] = ⎢ M cos 2α M cos α L + M M cos α M cos 2α ⎥ (7.11) ls ⎢ ⎥ ⎣ M cos 2α M cos 2α M cos α ⎦ M cos α Lls + M M cos α M cos 2α M cos 2α M cos α Lls + M

7.2 Modeling of a Five Phase Induction Motor

similarly the rotor inductance matrix can be ⎡ Laar Labr Lacr ⎢ Lbar Lbbr Lbcr ⎢ [Lr ] = ⎢ ⎢ Lcar Lcbr Lccr ⎣ Ldar Ldbr Ldcr Lear Lebr Lecr ⎡

Llr + M ⎢ M cos α ⎢ [Lr ] = ⎢ ⎢ M cos 2α ⎣ M cos 2α M cos α

M cos α Llr + M M cos α M cos 2α M cos 2α

M cos2α M cos α Llr + M M cos α M cos2α

163

written asLadr Lbrdr Lcdr Lddr Ledr

⎤ Laer Lber ⎥ ⎥ Lcer ⎥ ⎥ Lder ⎦ Leer

M cos 2α M cos 2α M cos α Llr + M M cos α

(7.12)

⎤ M cos α M cos 2α ⎥ ⎥ M cos 2α ⎥ ⎥ M cos α ⎦ Llr + M

(7.13)

Suppose at any instant, the magnetic axis of rotor phase ‘a’ makes an angle θ with reference to the magnetic axis of phase ‘a’ of the stator winding The mutual inductances between stator and rotor windings can be obtained as follows⎡ ⎤ cosθ cos(θ + α) cos(θ + 2α) cos(θ−2α) cos(θ−α) ⎢ cos(θ−α) cosθ cos(θ + α) cos(θ + 2α) cos(θ−2α) ⎥ ⎢ ⎥ ⎢ cosθ cos(θ + α) cos(θ + 2α) ⎥ [Lsr ] = M ⎢ cos(θ−2α) cos(θ−α) ⎥ ⎣ cos(θ + 2α) cos(θ−2α) cos(θ−α) cosθ cos(θ + α) ⎦ cos(θ + α) cos(θ + 2α) cos(θ−2α) cos(θ−α) cosθ (7.14) Also

[Lrs ] = [Lsr ]T

(7.15)

The motor torque can be expressed as –

s dL P s iabcde abcde T r T [iabcde ] [iabcde ] Te = irabcde 2 dθ Or

(7.16)

dL P s sr r T [i [i Te = ] ] 2 abcde dθ abcde

(7.17)

Here the stator voltage and current equations are described using axes of reference fixed to the stator, whereas rotor equations use axes fixed to the rotor. In order to simplify the model, the time varying inductances are changed to constant inductances using following transformation. The matrix [A] as given below is used for power invariant transformation. ⎤ cosθs cos (θs − α) cos (θs − 2α) cos (θs + 2α) cos (θs + α) ⎢ −sinθs −sin (θs − α) −sin (θs − 2α) −sin (θs + 2α) −sin (θs + α) ⎥ ⎥ 2⎢ ⎢ 1 ⎥ (7.18) cos2α cos4α cos4α cos2α [As ] = ⎢ ⎥ 5⎣ 0 sin2α sin4α −sin4α −sin2α ⎦ ⎡

√1 2

√1 2

√1 2

√1 2

√1 2

164

7 Control of Multiphase AC Motor Drives

Transformation of rotor variables is obtained using the same transformation matrix except that θs is replaced with β, where β = θs − θ. Here β is the instantaneous angular position of d- axis of common reference frame with respect to magnetic axis of phase ‘a’ of the rotor. The transformation matrix for rotor is – ⎤ ⎡ cosβ cos (β − α) cos (β − 2α) cos (β + 2α) cos (β + α) ⎢ −sinβ −sin (β − α) −sin (β − 2α) −sin (β + 2α) −sin (β + α) ⎥ ⎥ 2⎢ ⎢ 1 cos2α cos4α cos4α cos2α ⎥ [Ar ] = ⎥ ⎢ 5⎣ 0 sin2α sin4α −sin4α −sin2α ⎦ √1 2

√1 2

√1 2

√1 2

√1 2

(7.19) Here θs = ωs dt and β = θs − θ = (ωs − ωr )dt and ωr is the instantaneous angular speed of the rotor. Using these transformations the equations in d-q-x-y-0 domain can be written as

vsdqxy0 = [As ] [vsabcde ]

isdqxy0 = [As ] [isabcde ]

(7.20) λsdqxy0 = [As ] [λsabcde ] r vdqxy0 = [Ar ] [vrabcde ] r idqxy0 = [Ar ] [vrabcde ] r λdqxy0 = [Ar ] [vrabcde ]

7.3

Machine Model in Arbitrary Reference Frame

The stator voltage equations for the machine in arbitrary reference frame can be written as – vsd = Rs isd −ωs λsq + pλsd vsq = Rs isq + ωs λsd + pλsq vsx = Rs isx + pλsx vsy

(7.21)

= Rs isy + pλsy

vs0 = Rs is0 + pλs0 Similarly for rotor voltages the equations are – vrd = Rr isd −ωs λsq + pλsd vrq = Rr irq + (ωs − ωr )λrd + pλrq vrx = Rr irx + pλrx

(7.22)

7.3 Machine Model in Arbitrary Reference Frame

165

vry = Rr iry + pλry vr0 = Rr ir0 + pλr0 The flux linkages λsd , λsq , λrd , and λrq can be written after transformation asλsd = (Lsl + 2.5M)isd + 2.5Mird λsq = (Lsl + 2.5M)isq + 2.5Mirq r r s λrs d = (Ll + 2.5M)id + 2.5Mid

λrq = (Lrl + 2.5M)irq + 2.5Misq λsx = Lls isx λsy = Lls isy

λs0 = Lls is0

(7.23)

λrx = Llr irx λry = Llr iry λr0 = Llr ir0 Substituting the value Lm = 2.5M λsd = (Lsl + Lm )isd + Lmird λsq = (Lsl + Lm )isq + Lm irq

λsx = Lls isx

λsy = Lls isy

λs0 = Lls is0

λrd = (Lrl + Lm )ird + Lmisd

(7.24)

λrq = (Lrl + Lm )irq + Lm isq

λrx = Llr irx

λry = Llr iry

λr0 = Llr ir0

The torque equation can be written after transformation as – Te =

5P r s s r M id iq − id iq 2

(7.25)

Or Te = PLm ird isq − isd irq

(7.26)

And in terms of flux linkages it can be written asTe =

5 P Lm r s λd iq − λrq isd 2 2 Llr + Lm

(7.27)

Thus the diﬀerence between the five-phase machine model and the three phase machine model is the presence of x-y components in Equation (7.21) and (7.22). The d-q components and x-y components in stator circuit are fully decoupled from one another. Also, since the rotor winding is short circuited the x-y components can not exist. The zero sequence components in both

166

7 Control of Multiphase AC Motor Drives

stator and rotor can be ignored if the stator winding is star connected and rotor winding is short circuited. In order to apply vector control in five-phase induction motor, the same principle of rotor field oriented control (RFOC) can be used, as for a three- phase induction motor.

7.4

Vector Control of Five-Phase Induction Motor

As in case of three-phase induction motors, direct and indirect methods of vector control of five-phase induction motor can be applied. In direct vector control, direct sensing of air gap flux vector is required. In indirect method, the rotor flux is estimated using the stator vector. The indirect vector control is more commonly used and is described here3,4 . The equation (7.27) gives the electromagnetic torque in terms of flux linkages which is similar to the equation for dc machines. Suppose the rotor flux linkages in the synchronously rotating reference frame is entirely in d-axis so that λsqr = 0 The torque Equation (7.27) is then reduced toTe =

5 P Lm r s λ i 2 2 Llr + Lm d q

(7.28)

This equation is similar to the equation of a dc shunt machine. Thus the torque and rotor flux can be controlled independently by regulating the d and q components of stator current together with slip frequency with the constraint of λsqr = 0. The indirect rotor flux oriented controller is shown in Fig. 7.1.

Fig. 7.1 Indirect Rotor Flux oriented controller

7.5 Five-Phase Inverters

167

The commanded values of torque and rotor flux are obtained from the following equations4 Lr Te∗ (7.29) i∗qs = 5P Llr + Lm λ∗dr i∗ds =

1 ∗ Lr d ∗ λ λ + Lm dr Rr Lm dt dr

(7.30)

T∗ 4 Rr e 2 5P (λdr )

(7.31)

And sωs =

7.5

Five-Phase Inverters

A five-phase motor is generally supplied from a five-phase voltage source inverter with PWM control. Fig. 7.2 shows a power circuit topology of a five-phase voltage source inverter. Each switch in the circuit consists of two power semiconductor devices, connected in anti- parallel. One of these devices is fully controllable semiconductor such as IGBT, or bipolar transistor, while the second one is a diode. The input to the inverter is a dc voltage which is assumed to be constant. Also the load which in this case is a motor is assumed to be star connected. The inverter output voltages are shown in Fig. 7.3 with lower case symbols (a,b,c,d,e). The leg voltages are represented by symbols in capital letters (A,B,C,D,E). The relationship between the machine star connected phase voltages and inverter leg voltages is given by-

Vdc A

B

a

b

C

D

c

d

Fig. 7.2 Five-phase voltage source inverter

E

e

168

7 Control of Multiphase AC Motor Drives q axis

v 4 phase

v 5 phase

v3 phase

v13 phase

v14 phase

v12 phase

v 23 phase

v 24 phase

v15 phase

v 2 phase

v 25 phase v 22 phase S /5

v26 phase

v 6 phase v16 phase

v1phase

v11 phase

d axis

v21 phase v 27 phase v30 phase v 28 phase v17 phase

v 7 phase v18 phase

v8 phase

v 20 phase

v 29 phase

v19 phase

v10 phase

v9 phase

Fig. 7.3 Five phase inverter

va =

1 [4vA − vB − vC − vD − vE ] 5

1 [4vB − vA − vC − vD − vE] 5 1 vc = [4vC − vA − vB − vD − vE] (7.32) 5 1 vd = [4vD − vA − vB − vC − vE] 5 1 ve = [4vE − vA − vB − vC − vD ] 5 Although a number of PWM techniques are available to control a three-phase VSI, the same is not true for five-phase inverters. A specific problem faced by multiphase machines, is that generation of certain low order harmonics in the inverter output can result in very large stator current harmonics5.In a five-phase machine if the voltage contains 3rd and 7th harmonics, the stator current will have large 3rd and 7th harmonics as these are limited by stator leakage impedance only. It is therefore important; that the multiphase inverter must have as close sinusoidal voltage waveforms as possible. The Space vb =

7.5 Five-Phase Inverters

169

Vector Pulse Width Modulated Scheme (SVPWM)7 is found to provide good sinusoidal output voltages as described here.

7.5.1

SVPWM Five-Phase Voltage Source Inverter

Space Vector PWM (SVPWM) inverters are used in direct torque control of induction machines. A five-phase voltage source inverter (Fig. 7.2) has a total 25 = 32, space vectors, of which 30 are active vectors and 2 zero state vectors (0 and 31). If the five voltage outputs when the inverter is at each of these states is transformed to d-q domain, it will produce a five dimensional space. The five dimensional space are divided into three groups – d-q (main vector space), x-y (auxiliary vector space), and single dimensional (zero sequence). For star connected system the zero vector is zero. The space vectors after transformation can be represented as2 vdq = vd + jvq = (va + avb + a2 vc + a∗2vd + a∗ ve ) 5

(7.33)

2 vxy = vx + jvy = (va + a2 vb + a∗ vc + avd + a∗2 ve ) 5

(7.34)

2π 4π 4π ∗ 2 ∗2 wherea = exp( j 2π 5 ),a = − exp( j 5 ), a = exp( j 5 ),a = (− j 5 ) The 30 active vectors of Eq.(7.32) and Eq.(7.33) form three decagons for all possible inverter states. The five-phase VSI space vectors in d-q and xy plane are shown in Fig. 7.3. The phase voltage space vectors for all 32 switching states in d-q plane are summarized in Table 7.1 Table 7.2 presents the phase voltage space vectors in x-y plane.

Table 7.1 Phase voltage space vectors in d-q plane Space vector

Magnitute of space vector

v1phase to v10phase (large) v21phase to v30phase (small)

2 5 vdc 2v 5 dc 2 5 vdc

v31phase to v32phase

0,0

v11phase to v20phase (medium)

2 cos ( π5 ) exp( jkπ 5 ), k = 0,1,2 - - - - - - - 9 exp ( jk 5 ), k = 0,1,2 - - - - - - - 9 jkπ 2 cos ( 2π 5 ) exp( 5 ), k = 0,1,2 - - - - - - - 9

From Fig. 7.3 it is clear that the outer decagon space vectors of d-q plane map into inner decagon of x-y plane. Also the inner decagon of d-q plane maps into outer decagon of x-y plane. The middle decagon of x-y plane maps into the middle decagon of x-y plane. Further the phase sequence of a,b,c,d,e of dq plane corresponds to a,c,e,b,d sequence of x-y plane. An ideal space vector modulator for five- phase inverter must satisfy a number of requirements. If the switching frequency is to be kept constant, each switch can change

170

7 Control of Multiphase AC Motor Drives Table 7.2 Phase voltage space vectors in d-q plane Space vector

Magnitute of space vector

v1phase to v10phase (small) v21phase to v30phase (large)

2 5 vdc 2v 5 dc 2 5 vdc

v31phase to v32phase

0,0

v11phase to v20phase (medium)

jkπ 2 cos ( 2π 5 ) exp( 5 ), k = 0,1,2 - - - - - - - 9

exp ( jk 5 ), k = 0,1,2 - - - - - - - 9 2 cos ( π5 ) exp( jkπ 5 ), k = 0,1,2 - - - - - - - 9

state only two times (on to oﬀ and oﬀ to on) during the entire switching period. The second requirement that must be fulfilled is that RMS value of fundamental output phase voltage must be equal to RMS value of d-q space vector. Also in order to have the sinusoidal waveform the x-y components must be minimized. In order to determine the nature of output voltage a number of Space Vector PWM schemes are possible. SVPM Using Large Vectors Only Direct torque control of five-phase induction motor has been proposed by Xu, et al. using SVPWM with large vectors only8 . The input voltage vector to the motor is synthesized from two active neighboring vectors and zero space vectors. It is found that with large vectors only apart from desired d-q components, x-y components are also produced, which predominantly contains the 3rd harmonics. It is also found that the largest possible fundamental peak output voltage that can be achieved using this scheme corresponds to the radius of largest circle that can be inscribed within the decagon. The maximum fundamental peak output voltage Vmax is π π 2 Vmax = ( )2 cos( ) cos( )VDC 5 5 10

= 0.61554VDC

(7.35)

It is observed that the output phase voltages contain a considerable amount of 3rd and 7th harmonics. These harmonic components are due to the x-y components of the space vectors and these will exist regardless of the reference voltage value. SVPWM Using Medium and Large Vectors If the number of applied active space vectors for VSI with odd phase number is one less than the number of inverter phases, the performance of the inverter is greatly improved. This means that one needs to apply four active vectors rather than two.

7.6 Five-Phase Permanent-Magnet Motor Drives

7.6

171

Five-Phase Permanent-Magnet Motor Drives

Permanent magnet synchronous machines are becoming attractive in many industrial applications due to their high torque to inertia ratio, higher efficiency, and power density. Also, interest in Multi-phase motor drives has considerably increased during last few years. The main applications of multiphase machines are in electric traction, electric vehicles, and hybrid vehicles. Multi-phase motor drives have many advantages including higher reliability and power density, and lower torque pulsations at higher frequencies Mathematical Model of the Five-Phase PM Motor The stator voltage equations are given by Vs = Rs Is +

dλs dt

(7.36)

Where Rs, Is and λs are stator resistance, current and flux linkages matrices respectively. The air gap flux linkages are represented by – λs = λss + λsr

(7.37)

λs = Lss Is + Lsr Ir

(7.38)

Or Lss is the stator inductance matrix which contains the self and mutual inductances of the stator phases and varies with rotor position. Lsr is the mutual inductance matrix between the stator winding and virtual rotor winding. The rotor voltage equation is given by Vr = Rr Ir +

dλr dt

(7.39)

Where rotor flux linkages matrix is given byλr = λrr + λrs

(7.40)

λr = Lrr Ir + Lrs Is

(7.41)

Or Here permanent magnet is modeled as coil with constant current Ir = Ifd which depends on the flux density of magnets. Rr and Lrr is the resistance and self inductance of virtual coil. The transformation matrix to transform variables in five-phase system to d-q frame rotating at an arbitrary angular velocity is

172

7 Control of Multiphase AC Motor Drives

⎤ cosθs cos (θs − α) cos (θs − 2α) cos (θs + 2α) cos (θs + α) ⎢ −sinθs −sin (θs − α) −sin (θs − 2α) −sin (θs + 2α) −sin (θs + α) ⎥ ⎥ 2⎢ ⎢ 1 ⎥ cos2α cos4α cos4α cos2α Ar = ⎢ ⎥ 5⎣ 0 sin2α sin4α −sin4α −sin2α ⎦ ⎡

√1 2

√1 2

√1 2

√1 2

√1 2

(7.42) 2π α= 5 The d-q reference frame is attached to rotor The stator flux linkages along d-q axis are – (7.43) λsd = Ld ids + λm λsq = Lq iqs

(7.44)

The stator voltage equations are – vsd = Rs isd −ωs λsq + pλsd vsq = Rs isq −ωs λsd + pλsq vsx

(7.45)

= Rs isx + pλsx

vsy = Rs isy + pλsy vs0 = Rs is0 + pλs0 The torque 5P s s (7.46) M λd iq − λsq isd 2 It can be seen from Eq.(7.46) that only the interaction between stator flux space vector in d-q subspace and permanent magnet flux produces the torque. In order to control the torque therefore, the voltage space vectors are selected to control the stator flux. The stator flux can be obtained from the equation Te =

vs = Rs is +

dλs dt

(7.47)

References [1] Klingshirin, E.A.: High Phase order Induction Motors-Part I and Part II Description and Theoretical considerations. IEEE Transaction on Power Apparatus and Systems PAS-102(1), 47–59 (1983) [2] Singh, G.K.: Multi-phase Induction Machine Drive Research- A Survey. Electric Power System Research 61, 139–147 (2002) [3] Toliyat, H.A.: Analysis and Simulation of Five-phase Induction Motor Drives. IEEE Transactions on Power Electronics 13(4), 751–756 (1998) [4] Toliyat, H.A., Rahimian, M.M., Lipo, T.A.: Analysis and Modeling of ﬁvephase converters for Adjustable speed drive applications. In: Fifth European Conference on Power Electronics and Applications, vol. 5, pp. 194–199 (1993)

References

173

[5] Zhao, Y., Lipo, T.A.: Space Vector PWM control of dual three-phase Induction Machine using vector space decomposition. IEEE Transaction on Industry Applications 31(5), 1100–1109 (1995) [6] Kelly, J.W., Stangas, E.G., Miller, J.M.: Multiphase space vector pulse width modulation. IEEE Transactions on Energy Conversion 18(2), 259–264 (2003) [7] Desilva, P.S.N., Fletcher, J.E., Williams, B.W.: Development of space vector modulation strategies for ﬁve-phase voltage source inverters. In: Proceedings IEE Power Electronics, Machines and Drives Conference, PEMD, Edinburgh, UK, pp. 650–655 (2004) [8] Xu, H., Toliyat, H.A., Peterson, L.J.: Five-phase Induction Motor drives with DSP based Control system. IEEE Transaction on Power Electronics 17(4), 524–533 (2002) [9] Persa, L., Toliyat, H.A.: Sensorless Direct Torque Control of Five-phase Interior Permanent Magnet Drives. IEEE Transaction on Industry Applications 40, 992–999 (2004) [10] Persa, L., Toliyat, H.A.: Five-phase Permanent Magnet Motor Drives. IEEE Transaction on Industry Applications 41, 30–37 (2005)

Chapter 8

Fuzzy Logic and Neural Network Applications in AC Drives

8.1

Introduction

The concept of Fuzzy Logic (FL) was conceived by Lotfi Zadeh, a professor at the University of California at Berkley in 1965. He presented it not as a control methodology, but as a way of processing data by allowing partial set membership rather than crisp set membership or non-membership. This approach to set theory was not applied to control systems until the 70’s due to insuﬃcient small-computer capability prior to that time. Professor Zadeh in his paper presented a theory that if people without precise, numerical information input, are capable of highly adaptive control, the machines can also be made to work in this manner. He proposed that if feedback controllers could be programmed to accept noisy, imprecise input, they would be much more eﬀective and perhaps easier to implement. Explained in most simple terms, FL is a problem-solving control system methodology that lends itself to implementation in systems ranging from simple, small, embedded microcontrollers to large, networked, multi-channel PC or workstation-based data acquisition and control systems. It can be implemented in hardware, software, or a combination of both. FL provides a simple way to arrive at a definite conclusion based upon vague, ambiguous, imprecise, noisy, or missing input information. FL’s approach to control problems is similar to how a person would make decisions from imperfect information, and much faster. A fuzzy control system is diﬀerent from the conventional control system in the sense that its control is not based on the mathematical model of the plant. Instead the FL model is empirically based, relying on the experience and intuition of a human plant engineer, and /or on the designer, and researcher of the plant. The advantage of FL is for complex plants such as nuclear reactors that can not be expressed in terms of reasonably good mathematical model. Instead, a plant engineer may have very good knowledge of the working of the plant and control process. The fuzzy control does not require any model of the plant. It is based on plant operator experience and heuristics. Fuzzy control is basically adaptive, is easy to apply, and gives robust performance M. Ahmad: High Performance AC Drives, Power Systems, pp. 175–188. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com

176

8 Fuzzy Logic and Neural Network Applications in AC Drives

for plant parameter variation. The application of fuzzy logic in many complex processes is found to give superior performance as compared to conventional PID controllers. The basic features of fuzzy logic and expert system are the same. It is therefore, often called as “fuzzy expert system”. In expert system the knowledge is organized in the form of set of IF-THEN production rules. The IF statement is the condition of the system, and THEN statement defines the action. The FL system also works in the same way utilizing IF and THEN statements. The diﬀerence here is that the knowledge or information for the IF rule in FL is not precise. As discussed earlier, the processing stage is based on a collection of logic rules in the form of IF-THEN statements, where the IF part is called the “antecedent” and the THEN part is called the “consequent”. Typical fuzzy control systems have dozens of rules. In general the Fuzzy systems will have rules of the form IF x= A, and y = B, Then z = C A, B, and C have linguistic values eg. LOW, MEDIUM, HIGH, COLD, WARM, HOT etc. Consider a rule for a thermostat: IF (temperature is "cold") THEN (heater is "high") This rule uses the truth value of the “temperature” input, which is some truth value of “cold”, to generate a result in the fuzzy set for the “heater” output, which is some value of “high”. This result is used with the results of other rules to finally generate the crisp composite output. Obviously, the greater the truth value of “cold”, the higher the truth value of “high”, though this does not necessarily mean that the output itself will be set to “high”, since this is only one rule among many. In practice, the fuzzy rule sets usually have several antecedents that are combined using fuzzy operators, such as AND, OR, and NOT, though again the definitions tend to vary: AND, in one popular definition, simply uses the minimum weight of all the antecedents, while OR uses the maximum value. There is also a NOT operator that subtracts a membership function from 1 to give the “complementary” function. Fl can supplement an Expert System, and many times ES and FL are combined to solve complex problems. FL has been successfully applied in process control, control of washing machines, automatic cameras, and industrial air conditioners etc.

8.2

Basic Principle

The input variables in a fuzzy control system are in general mapped into by sets of membership functions, known as “fuzzy sets”. The process of converting a crisp input value to a fuzzy value is called “fuzzification In fuzzy sets, a particular object has a degree of membership in a given set that may be

8.2 Basic Principle

177

hot

cold 1

mild

0 20

40

60

80

Temprature in c Fig. 8.1 Motor temprature as fuzzy variable

anywhere in the range of 0 (not member of set) to 1(completely in the set). Thus FL works with multi-valued logic between 0 to 1. A crisp relation represents the presence or absence of association, interaction, or inter-connection between the elements of two or more sets. In fuzzy systems the degree of association can be represented by membership grade. The membership grade indicates the strength of the relation present in the set. The significance of fuzzy variables is that they facilitate gradual transitions between states. Consequently it is possible to deal with observation and measurement with uncertainties. Since fuzzy variables capture measurement uncertainties as part of experimental data, they are more tuned to reality than crisp variables. An interesting statement of Albert Einstein ; So far as laws of mathematics refer to reality, they are not certain, and so far as they are certain, they do not refer to reality is very much applicable to fuzzy sets and fuzzy logic. For example a Fuzzy set for stator temperature of motor can have values as COLD, MILD,and HOT where each is represented as triangular or straight line segment membership function as shown in Fig 8.1. The fuzzy set can have further subdivision as VERY COLD, COLD, MILD HOT, VERY HOT etc. The membership function is generally taken as triangular, but it can also be trapezoidal or Gaussian. Considering the Fig 8.1 that shows a relationship between membership function and Temperature, the temperature of 55◦ C, it is in set COLD MF =0.3 and in set MILD it is 0.5. The membership functions can either be defined by mathematical equation or look up table. The numerical interval for which the fuzzy variable is defined is called universe of discourse (for fig. 8.1, temperature 20 to 90 degree). The basic properties of Boolean logic of Union, Intersection and complement are also valid for fuzzy logic.

178

8.3

8 Fuzzy Logic and Neural Network Applications in AC Drives

Fuzzy System

A Fuzzy system basically involves following four steps: • • • •

Fuzzification or conversion of crisp input variables into fuzzy variables A fuzzy rule base A fuzzy inference engine Defuzzification module

8.3.1

Fuzzification

In order to convert the crisp data into fuzzy variables, first measurements are made for all the variables that describe the relevant conditions of the process to be controlled. Next these measurements are converted into appropriate fuzzy sets to express measurement uncertainties In most cases, these fuzzy sets are fuzzy numbers, which represent linguist labels such as negative large (NL), negative medium(NM), negative small (NS), approximately zero (Z), positive small (PS), positive medium(PM), and positive large(PL). The linguistic states are usually represented by fuzzy sets with triangular membership functions, but it can also be trapezoidal or Gaussian These membership functions are defined for appropriate range of the measured variable.

8.4

Fuzzy Rule Base

The fuzzified measurements are used by the inference engine to evaluate the control rules stored in the fuzzy rule base. In this step, the knowledge pertaining to the given control problem is formulated in terms of a set of fuzzy inference rules. There are two principal ways in which relevant inference rules can be determined. One way is to use the knowledge of experienced human operators. Other method is to obtain them from empirical data by suitable learning methods using neural networks. This will be more clear when an actual problem of speed control is discussed.

8.5

Fuzzy Inference Engine

Inferences regarding output variables are obtained in inferenceengine where measurements of input variables are combined with the relevant fuzzy information rules. There are number of methods for obtaining inferences from input variables and fuzzy rules. Some of these methods are described here.

8.5.1

Mandani Type

Mandani, one of the pioneers in the application of FL in problems of control systems proposed this method. If X and Y are the input variables, Z is the

8.6 Defuzzification Methods

179

output variable and NS, ZE, and PS are the fuzzy sets then three rules of the fuzzy system which has three input fuzzy sets and two output sets can be described asRule 1: If X is NS and Y is ZE THEN Z is PS. Rule 2: If X is ZE and Y is ZE THEN Z is ZE. Rule 3: If X is ZE and Y is PS THEN Z is NS. It nay be noted here that all these rules have an AND operator.

8.6

Defuzzification Methods

While designing a fuzzy controller it is important to select a suitable method for defuzzification. The purpose of defuzzification is to convert the conclusions obtained from the inference engine, which are in the form of fuzzy output to crisp output. A number of defuzzification methods are available in the literature. Three commonly used methods are described here.

8.6.1

Centre of Area (COA) Method

The centre of area method is also known as centre of gravity method or centroid method. In this method the crisp output Z0 of the variable Z is taken to be the geometric centre of the output fuzzy value C(Z) area. Where C(Z) is obtained by taking the union of all the contributions of rules whose values are >0. The general formula for COA method of defuzzification is given by C(Z).ZdZ Z0 = (8.1) C(Z)dZ In case of discrete values of Z { Z1 , Z2 .........Zn} the value of Z0 is obtained from the expression n

∑ C(Zk )Zk Z0 =

k=1 n

(8.2)

∑ C(Zk )

k=1

If Z0 is not equal to any value in the universal set, the value closest to it is taken as crisp value. The COA method divides the graph of membership function C into two equal sub-areas and is very commonly used. For example Figure 8.2 shows simple fuzzy output for two rule system. using COA method as1. 1 + 2. 32 + 3. 32 + 4. 23 + 5. 31 + 6. 13 + 7. 31 = 3.7 (8.3) Z0 = 3 1 2 2 2 1 1 1 3+3+3+3+3+3+3

180

8 Fuzzy Logic and Neural Network Applications in AC Drives 2 3 1 3

1 3

0

1

2

3

4

5

6

7

Fig. 8.2 Mean of Maxima method

8.6.2

Mean of Maxima Method

Mean of Maxima (MOM) method is defined for discrete value functions only. The defuzzified value of the function is the average of all values in crisp set. For example if the crisp value of Z is Z0 , and the height membership component of the variable at this value is Zm , then for M such values the crisp value is given byM Zm (8.4) Z0 = ∑ m=1 M where Zm is the mth element in the universe of discourse, where the output membership function (MF) is at the maximum value, and M = the number of such elements. As shown in Figure 8.2, the output Z0 = 3. If more than one (M) such maxima are present then the added values will be divided by the number M.

8.6.3

Centre of Maxima Method

This method is simplification of COA method as it considers only the height of each contributing membership function at the mid-point of the base. Considering the same system shown in Figure 8.2 Z0 =

8.7

3. 32 + 5. 31 2 3

+ 13

= 3.67

(8.5)

Speed Control of Induction Motor Using Fuzzy Controller

Fuzzy logic control may be used to identify and control non-linear dynamic systems through an approximation in a wide range of non-linear functions to any desired degree of accuracy. Fuzzy logic controllers may be used to design speed and flux controllers, and to improve the observer system and voltage model for exact stator flux and speed computation.

8.7 Speed Control of Induction Motor Using Fuzzy Controller

181

In order to apply a fuzzy controller for the control of an electric drive the following general procedure may be applied. • The first consideration of using a fuzzy controller is to analyze whether the drive will require a fuzzy controller or not. A fuzzy controller may not have any advantage in many simple drives. • Second step is to obtain all the information from system engineer. • Using the model of the drive develop a simulation program with conventional control and the performance. • Identify the input and output variables of the fuzzy system. • Formulate the fuzzy sets and select the corresponding MF shape of each • Formulate the rule table. • Simulate the system with fuzzy controller., iterate the fuzzy sets and rule table until the performance is optimized. In order to demonstrate the application of fuzzy control in an induction motor drive, a simple vector control drive is shown in Fig. 8.3. The controller observes the pattern of the speed loop error and correspondingly updates the output DU so that the actual speed ωr corresponds to ω∗r . There are two input signals to fuzzy controller, the error E=ω∗r − ωr , and the change in dE CE error CE= dE dt . In a discrete system dt = Ts . Here Ts is sampling time. If Ts is dE constant CE is proportional to dt . In a vector controlled drive the controller output DU is ∆i∗qs . This diﬀerential current is summed or integrated to obtain the control current i∗q . The action of a fuzzy PI controller can be written in the form of equation K1 E + K2CE = DU

(8.6)

Where K1 and K2 are non-linear coeﬃcients or gain factors. The above equation can be integrated in the form

d dt

Z r* +

-

Zr

K1 Edt +

K2 CEdt =

(8.7)

DU

CE Fuzzy controlle r

DU

³

U

E

Fig. 8.3 Fuzzy speed controller

VC and inverter

IM

182

8 Fuzzy Logic and Neural Network Applications in AC Drives

Or putting CE= dE dt U = K1

Edt + K2 E

(8.8)

Equation (8.8) represents a fuzzy P-I controller with non-linear gain factors.

8.7.1

Formation of Fuzzy Set and Fuzzy Rules for the System

From the physical operation principle of the system, a simple control rule in fuzzy logic can be written as – If E is near zero (ZE)and CE is positive small (PS), Then the controller output DU is small negative (NS). After formulating the fuzzy set next step is to select the MF, which for this example is considered to be triangular MF. The fuzzy sets are defined as NL, NM, NS, Z, PS, PM, and PL. The universe of discourse of all variables, covering the whole region is expressed in per unit values. All the MFs are asymmetrical near origin. but symmetrical for positive and negative values of the variables. Seven MFs are chosen for E in p.u.and CE (p.u.) signals, and nine for output. Thus there are 7 × 7 = 49 rules that can be formed. these rules can be tabulated in the form of a table shown. e(pu) ce(pu) NL NL NL NM NL NS NL Z NL PS NM PM NS PL Z

NM

NS

Z

PS

PM

PL

NL NL NL NM NS Z PS

NL NL NM NS Z PS PM

NM NM NS Z PS PM PL

NS NS Z PS PM PL PL

NS Z PS PM PL PL PL

Z PS PM PL PL PL PL

The fuzzy controller is designed from the rule matrix as follows: 1. If both e(pu) and ce(pu) are zero, then present control setting is maintained 2. 2. If e(pu) is not zero but is approaching this value then also control setting is maintained. 3. However if e(pu) is not zero and is increasing, then control signal is changed depending on the magnitude and sign of e(pu) and ce(pu) so the e(pu) approaches zero.

8.8 Neural Network Based Control

8.8

183

Neural Network Based Control

8.8.1

Artificial Neuron

Artificial Neural Network (ANN) is a branch of artificial Intelligence (AI) that emulates the human thinking process. An ANN tries to emulate the biological neurons of the human brain in a limited way by an electronic circuit or computer program. It is composed of large number of highly interconnected processing elements (neurons) working in unison to solve problems. ANNs like people learn through experience Neural networks have the ability to derive meaningful information from complicated or imprecise data. ANN is particularly suitable for solving pattern recognition and image processing problems which are diﬃcult to solve using digital computers. The basics of ANN were known even before the advent of modern digital computer. The first artificial neuron was produced in 1943 by McCulloch and Walter Pitts. But the technology available at that time was not suitable for using these neurons to any useful work. Renewed interest in neural network was generated after the presentation of Hopfield before National Academy of Science in 1982. Since 1990 the neural network has found applications in various problems solving, requiring artificial intelligence. An artificial neuron tries to emulate the biological neuron present in human brain. Fig 8.4 shows the structure of artificial neuron. It is a device with

Fig. 8.4 Model of artificial Neuron

184

8 Fuzzy Logic and Neural Network Applications in AC Drives

Fig. 8.5 Diﬀerent mathematical functions

many inputs and one output. The input signals X1 , X2 , X3 . . . . are normally continuous variables but can be discrete pulses also. The input signals are multiplied by weights. These weights can be positive or negative. The summing node accumulates all the input weighted signals which is then passed to output through a mathematical function which determines the activation of neuron. The mathematical function can be linear, step function, signum type or non-linear continuously varying type, such as sigmoid, hyperbolic, or Gaussian type. These functions also known as transfer function are shown in Fig. 8.5. The most simple of these functions is linear function, where the output varies linearly with the input but saturates at ± 1. The step activation function has logical value1 S 0 and 0 for S less than 0. Other functions are also shown in Fig. 8.5. The commonly used activation functions are nonlinear, continuously varying types between two asymptotic values 0 and 1 or -1 and +1. These functions are sigmoid and the hyperbolic tan function.

8.8 Neural Network Based Control

185

These functions are diﬀerentiable, and the derivative dF/dS is maximum at S=0.

8.8.2

Artificial Neural Network

In order to emulate the function of human brain a large number of artificial neurons are interconnected. These networks are classified as feed forward and feedback type. In feed forward network, signals from neuron to neuron flow only in the forward direction. In feedback ANN the signals can flow in forward as well as backward direction. These networks may have diﬀerent philosophy and diﬀerent principles but these networks are capable of learning. Learning is the process by which a neural network acquires ability to carry out certain tasks by adjusting its internal parameters according to some learning scheme. Depending on the particular neural architecture considered, the learning can be supervised or unsupervised.

8.8.3

Feed Forward Neural Networks

Feed-forward neural networks consist of one or more layers of nonlinear processing elements or units. A simple one hidden layer feed-forward network with inputs X1, X2, X3−−−−−−−Xn and output Y is shown in Fig. 8.6. The network as shown is divided into layers. The input layer consists of inputs to the network. The input layer is followed by hidden layer, which consists of number of neurons. Placed in parallel. Each neuron performs the weighted summation of inputs. and sends an output in the form of nonlinear function σ, called neuron function. The output of the neurons in hidden layer is summed up with diﬀerent weights to get the network output. This summation on the output is called output layer. Fig. 8.6 shows a single output layer problem. The size of the input and output layers is defined by the number of inputs and outputs of the network. However, the number of hidden layer neurons are to be specified when the network is defined. The operation of the network can be divided into two phases. 8 .8 .3 .1

T raining of Neural Network

A feed-forward neural network maps a set of input vectors to set of output vectors. For a given set of inputs X, the output Y is given by Y=T(X) Where, T is a nonlinear operator which depends on the architecture of neural network considered. A neural network can be trained to perform certain task by appropriately creating a training set. Basically a training set is a set of output - input vector pairs (Yk, Xk), k = 1,2−−−m. This input/output pattern matching is possible if appropriate weights are selected.

186

8 Fuzzy Logic and Neural Network Applications in AC Drives Hidden layer 4 W94

X1

1

5

9

Y1

Input X2

X3

2

output

6

3

7

10 Y2

8

W8 10 weights

error

desired output Z1

Z2

Fig. 8.6 Feed -forward neural network

In general an ANN can be trained using any one of the following three methods of learning. (1) supervised learning, that is taking the help of training algorithm; (2) unsupervised or self learning; and (3) reinforced learning. In supervised learning the network initially assigns the weights in hidden layer arbitrarily. The output pattern thus obtained is compared with the desired pattern, and the weights are adjusted by an algorithm till the error becomes negligible. Such training is continued for large number of input – output patterns. At the end of the learning process. The network should be capable of doing not only the problems it has learnt but should be able to tackle new problems also. In unsupervised learning, the system learns by itself when it is exposed to number of inputs and it makes its own classification for inputs. This is very much similar to learning by a child who sees new objects and learns its names. In reinforced learning the process of learning is verified by an expert before using. 8.8.3.2

Back Propagation Learning Algorithm

The back propagation method is the most commonly used method for multilayer feedforward neural network. In Fig. 8.6, the hidden layer is assigned random positive and negative weights at the beginning. For a given input

8.8 Neural Network Based Control

187

function the output is calculated (forward pass) and compared with the desired output pattern. The weights are then adjusted using cost function given by the squared diﬀerence between the calculated output and desired output. The cost function for a set of inputs is generated, and is minimized by a gradient descent method changing the weights one at a time starting from the output layer (back propagation). Let us consider ‘m’ processing elements in the input layer, ‘l’ in hidden layer, and ‘n’ in output layer, and an input –output system (x,d). The processing element q in the hidden layer receives a net input of q = ∑ vqi xj

j = 1....m.

(8.9)

It produces an output zq = a ∑ vqi xj

(8.10)

This output is applied as input to output layer. The net input for processing element ‘i’ in the output layer is given by Neti = ∑ wiq zq = ∑ wiq a(∑ vqi xj )

q = 1....l

(8.11)

The output at the ith processing element is yi = a ∑ wiq a(∑ vqi xj )

q = 1....l

(8.12)

This Eq indicates the forward propagation of the input signals through the layer of neurons. The squared output error signal for all the output layer neurons is 1 i = 1...n (8.13) Ep = ∑ (di − yi )2 2 The weights in the hidden to output connections are updated based on gradient descent method as (8.14) ∆Wiq = −ηδEp δWiq = −ηδoi Zq

(8.15)

Where δoi is the error signal of ‘i’th node in output layer., and η is learning rate. The weight update equation is then given as (8.16) Wiq (k + 1) = Wiq (k) + ηδEp δWiq (k) The weights are iteratively updated for all the P training patterns. The network is assumed to have suﬃcient learning when the total error E summed over the pattern P becomes very small (nearly zero). Since the iterative process propagates the error backward in the network, it is therefore known as back propagation algorithm.

188

In order to ensure that the error converges to global minimum, a momentum term is added in the equation 8.16. It is also possible to improve the performance of back propagation algorithm by making the learning rate small i.e. η(k + 1) = uη(k) u ≺ 1.0 (8.17) The process of training neural networks is very laborious and time consuming. The time involved becomes more and more if number of neurons in hidden layer are increased, or the number of hidden layer is more than one. The training of neural networks is therefore done oﬀ line by a computer program. This means that the weights are fixed when solving an actual problem In many applications such as electric drives, the network has to emulate nonlinear time varying functions and will require continuous training. This type of ANN is called adaptive network fast and improved version of back propagation algorithms are now available using DSP to tune the ANN weights. Although majority of applications use back propagation types of networks for solving the problems, Recurrent Neural Networks are now being suggested for some special types of problems. The RNN uses feedback from the output layer to an earlier layer, that is why it is also known as feedback network.

References [1] Bose, B.K.: Modern Power Electronics and AC drives. Pearson Prentice Hall International, London (2002) [2] Zadeh, L.A.: Fuzzy logic. Computer 21(4), 83–93 (1988) [3] Bose, B.K.: Fuzzy logic and neural network applications in Power electronics and motor control. Proceedings IEEE 82, 1303–1323 (1994) [4] Buja, G.S.: Neural network implementation of a fuzzy logic controller. In: Proceedings of International conference on Industrial Electronics control Instrumentation, IECON, vol. 1, pp. 414–417 (1993) [5] Kosc, P., Fedak, V., Profumo, F.: AC drives for high performance applications using Fuzzy logic controllers. In: Power Conversion Conference 1993, Yokohama, Japan, pp. 695–701 (1993) [6] Ross, T.: Fuzzy logic with Engineering Applications, 2nd edn. Wiley, Chichester (2004) [7] Bose, B.K.: Neural Network Applications in Power Electronics and Motor Drives— An Introduction and Perspective. IEEE Transaction on Power Electronics 54, 14–33 (2007) [8] Cirrincione, M.: Control and diagnosis of electrical drives: Some applications by using neural networks. In: Proceedings IEEE joint symposia on Intelligence and Systems, pp. 210–217 (1998) [9] Karanayil, B., Rahman, M.F., Grantham, C.: Induction motor parameter determination technique using artificial neural networks. In: International Conference on Electrical Machines and Systems, pp. 793–798 (2008)