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Control of Synchronous Motors

Control of Synchronous Motors

Edited by Jean-Paul Louis

First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Adapted and updated from Commandes classiques et avancées des actionneurs synchrones published 2010 in France by Hermes Science/Lavoisier © LAVOISIER 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 and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA

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© ISTE Ltd 2011 The rights of Jean-Paul Louis to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. ____________________________________________________________________________________

Library of Congress Cataloging-in-Publication Data Commandes classiques et avanceés des actionneurs syncrones. English Control of synchronous motors / edited by Jean-Paul Louis. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-273-2 1. Actuators--Automatic control. 2. Synchronization. I. Louis, Jean-Paul, 1945- II. Title. TJ223.A25C66 2011 621--dc22

2011013014

British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-273-2 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Jean-Paul LOUIS

Chapter 1. Synchronous motor controls, Problems and Modeling . . . . . . 1 Jean-Paul LOUIS, Damien FLIELLER, Ngac Ky NGUYEN and Guy STURTZER

1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2. Problems on the synchronous motor control . . . . . . . . . . . . . . . . 2

1.2.1. The synchronous motor control, a vector control . . . . . . . . . . . 2 1.2.2. Direct/inverse model and modeling hypotheses . . . . . . . . . . . . 3 1.2.3. Control properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3. Descriptions and physical modeling of the synchronous motor . . . . . 6 1.3.1. Description of the motor in preparation for its modeling . . . . . . . 6 1.3.2. Hypotheses on the motor . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.3. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.4. Main transformation matrices . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.5. Physical model of the synchronous motor . . . . . . . . . . . . . . . 10 1.3.6. The two levels voltage inverter . . . . . . . . . . . . . . . . . . . . . . 12 1.3.7. Model of the mechanical load . . . . . . . . . . . . . . . . . . . . . . 13

1.4. Modeling in dynamic regime of the synchronous motor in the natural three-phase a-b-c reference frame . . . . . . . . . . . . . . . . . . . . . 14

1.4.1. Model of the machines with non-salient poles and constant excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.2. Exploitation of the model in the a-b-c reference frame in sinusoidal steady state, electromagnetic torque . . . . . . . . . . . . . . . . 18 1.4.3. Extensions to the case of non-sinusoidal field distribution machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

vi Control of Synchronous Motors

1.5. Vector transformations and dynamic models in the α-β and d-q reference frames (sinusoidal field distribution machines with non-salient and salient poles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5.1. Factorized matrix modeling . . . . . . . . . . . . . . . . . . . . . . . . 24 1.5.2. Concordia transformation: α-β reference frame . . . . . . . . . . . . 25 1.5.3. Park transformation, application to the synchronous salient pole motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.5.4. Note on the torque coefficients . . . . . . . . . . . . . . . . . . . . . . 30

1.6. Can we extend the Park transformation to synchronous motors with non-sinusoidal field distributions? . . . . . . . . . . . . . . . . . . . . . . 31 1.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.8. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1.8.1. Numerical values of the parameters . . . . . . . . . . . . . . . . . . . 39 1.8.2. Nomenclature and notations . . . . . . . . . . . . . . . . . . . . . . . 40 1.8.3. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Chapter 2. Optimal Supply and Synchronous Motors Torque Control: Designs in the a-b-c Reference Frame . . . . . . . . . . . . . . . . . . 49 Damien FLIELLER, Jean-Paul LOUIS, Guy STURTZER and Ngac Ky NGUYEN

2.1. Introduction: problems of the controls in a-b-c . . . . . . . . . . . . . . . 49 2.2. Model in the a-b-c reference frame: extension of the steady state approach in transient regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.1. Case of sinusoidal field distribution machines . . . . . . . . . . . . . 50 2.2.2. Case of trapezoidal field distribution machines (brushless DC motor) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.3. Note on the electromagnetic torque for non-sinusoidal machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.3. Structures of torque controls designed in the a-b-c reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.3.1. Case of the sinusoidal distribution machine . . . . . . . . . . . . . . 54 2.3.2. Extension to brushless DC motors (case of trapezoidal field distribution machines) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.4. Performances and criticisms of the control approach in the a-b-c reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.4.1. Case of a proportional control . . . . . . . . . . . . . . . . . . . . . . 57 2.4.2. Case of an integral and proportional (IP) current regulation . . . . . 62 2.4.3. Interpretation in Park components of the IP controller designed in a-b-c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.4.4. Advanced controllers: example of the resonant controller . . . . . . 72 2.4.5. Interpretation by Park transformation of the regulation by resonant controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Table of Contents vii

2.5. Generalization: extension of the supplies to the case of non-sinusoidal distribution machines . . . . . . . . . . . . . . . . . . . . . . . 78

2.5.1. Generalization of the modeling . . . . . . . . . . . . . . . . . . . . . . 79 2.5.2. A first (heuristic) approach of the solution . . . . . . . . . . . . . . . 80 2.5.3. First generalization: optimization of the Joule losses (without constraint on the zero-sequence component current) . . . . . . . 81 2.5.4. Application of this approach: optimization in the case where electromotive forces are sinusoidal . . . . . . . . . . . . . . . . . . . . . . . 83 2.5.5. Second generalization: optimization of the Joule losses with constraint (the zero-sequence component current must be equal to zero) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2.5.6. Geometrical interpretation of the two optimal currents . . . . . . . . 86

2.6. Use of Fourier expansion to obtain optimal currents. . . . . . . . . . . . 90 2.6.1. Interest of the Fourier expansion (FS) . . . . . . . . . . . . . . . . . . 90 2.6.2. Modeling by Fourier series (with complex coefficients) . . . . . . . 91 2.6.3. Properties of the results by the Fourier expansion . . . . . . . . . . . 92 2.6.4. First important case: the back-EMF only contains uneven order harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.6.5. Second important case: the back-EMF only contain even order harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.6.6. General case, even and uneven order harmonics . . . . . . . . . . . 94 2.6.7. Rules: to impose the torque, it is necessary to impose its different harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.6.8. General approach for the optimization (heuristic demonstration in one example) . . . . . . . . . . . . . . . . . . . . . . . . . . 95 2.6.9. General formulation of the optimization method . . . . . . . . . . . 99 2.6.10. An important example: the sinusoidal field distribution machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.6.11. Application: obtaining a constant torque . . . . . . . . . . . . . . . 107 2.6.12. Some results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

2.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.8. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

2.8.1. Digital parameters values . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.8.2. Nomenclature and notations . . . . . . . . . . . . . . . . . . . . . . . 113

2.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

Chapter 3. Optimal Supplies and Synchronous Motors Torque Controls. Design in the d-q Reference Frame . . . . . . . . . . . . . . . . . . . 119 Damien FLIELLER, Jean-Paul LOUIS, Guy STURTZER and Ngac Ky NGUYEN

3.1. Introduction: on the controls designed in the Park d-q reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

viii Control of Synchronous Motors

3.2. Dynamic model (case of the salient pole machine and constant excitation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.3. First approach to determine of optimal current references (d-q reference frame) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.4. Determination of the current controls designed in the d-q reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3.4.1. Principle of control by model inversion: example of the proportional controller with compensations . . . . . . . . . . . . . . . . . . 124 3.4.2. Self-control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.4.3. Some properties of efficient current regulation . . . . . . . . . . . . 128 3.4.4. Robustness problems of a proportional controller of the currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

3.5. New control by model inversion: example of an IP controller with compensations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.5.1. Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.5.2. Performances of the IP regulations for current loops . . . . . . . . . 137 3.5.3. Robustness of the IP controllers for the current loops . . . . . . . . 140 3.5.4. Conclusion on the controls performances in the d-q reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

3.6. Optimal supply of the salient poles synchronous motors; geometrical approach of the isotorque curves . . . . . . . . . . . . . . . . . . 143

3.6.1. General information: a general approach with the torque surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.6.2. Preliminaries 1: case of synchronous machines, with magnets, with non-salient poles and with spatial distribution of the sinusoidal field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.6.3. Preliminaries 2: case of synchronous machines with magnets, with non-salient poles and with spatial distribution of a non-sinusoidal field – first extension of the Park transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 3.6.4. Remark: Analogy with the p-q theory . . . . . . . . . . . . . . . . . . 153 3.6.5. 3D visualization, case of non-salient pole machines . . . . . . . . . 154 3.6.6. Generalization to the salient pole machines: case of synchronous magnet machines with sinusoidal field distribution . . . . . 155 3.6.7. Visualization: case of an excited synchronous machine with salient poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 3.6.8. Case of a reluctance synchronous machine . . . . . . . . . . . . . . . 159 3.6.9. Case of synchronous machines with variable reluctance and non-sinusoidal spatial field distribution: second extension of the Park transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 3.6.10. Visualization: torque surface of a reluctance synchronous machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Table of Contents ix

3.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 3.8. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

3.8.1. Numerical parameters values . . . . . . . . . . . . . . . . . . . . . . . 167 3.8.2. Nomenclature and notations . . . . . . . . . . . . . . . . . . . . . . . 167

3.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Chapter 4. Drive Controls with Synchronous Motors . . . . . . . . . . . . . . 173 Jean-Paul LOUIS, Damien FLIELLER, Ngac Ky NGUYEN and Guy STURTZER

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.2. Principles adopted for speed controls: case of IP controllers . . . . . . . 176 4.3. Speed controls designed in the a-b-c reference frame (application to a non-salient pole machine) . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4.3.1. General information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.3.2. IP speed controller with an IP current controller in the a-b-c reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 4.3.3. IP speed controller with a resonant current controller . . . . . . . . 183

4.4. Determination of the speed controls designed in the d-q reference frame (application to a salient pole machine) . . . . . . . . . . . . 184

4.4.1. General information . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 4.4.2. Introductory example: speed control with compensation or decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4.4.3. Discussion on the speed controls . . . . . . . . . . . . . . . . . . . . . 188 4.4.4. Examples of regulation choices. The interest of an IP controller: its limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.4.5. Examples of the regulation choices: IP controller with an anti-windup device . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4.4.6. Examples of regulation choices: IP controller with limited dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 4.4.7. Example of an advanced regulation: P controller associated with an integral observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.5. Note on position regulations . . . . . . . . . . . . . . . . . . . . . . . . . . 211 4.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 4.7. Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

4.7.1. Numerical values of the parameters . . . . . . . . . . . . . . . . . . . 216 4.7.2. Nomenclature and notations . . . . . . . . . . . . . . . . . . . . . . . 216

4.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

Chapter 5. Digital Implementation of Vector Control of Synchronous Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Flavia KHATOUNIAN and Eric MONMASSON

5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

x Control of Synchronous Motors

5.2. Classical, analog and ideal torque control of a synchronous motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

5.2.1. Calculation of the current regulators . . . . . . . . . . . . . . . . . . . 223 5.2.2. Determination of the current references . . . . . . . . . . . . . . . . . 224 5.2.3. Parameters of the studied synchronous motor . . . . . . . . . . . . . 225 5.2.4. Simulation results of the ideal analog vector control of synchronous motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

5.3. Digital implementation problem of the synchronous motor vector control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

5.3.1. The interfaces, sources of restrictions . . . . . . . . . . . . . . . . . . 227 5.3.2. Time diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5.3.3. Digital implementation constraints of the vector control of a synchronous motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

5.4. Discretization of the control system . . . . . . . . . . . . . . . . . . . . . 230 5.4.1. Choice of the sampling period . . . . . . . . . . . . . . . . . . . . . . 231 5.4.2. Choice of the sampling instant . . . . . . . . . . . . . . . . . . . . . . 232 5.4.3. Implementation of the digital control . . . . . . . . . . . . . . . . . . 233 5.4.4. Simulation of the control with discrete regulators . . . . . . . . . . . 236

5.5. Study of the delays introduced by the digital implementation of the vector control of the synchronous motor . . . . . . . . . . . . . . . . . . . 237

5.5.1. Simulation results after introduction of the delays in the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 5.5.2. Calculation of the new regulators after taking into account the delays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 5.5.3. Simulation after delays correction and system discretization . . . . 240

5.6. Quantization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 5.6.1. Quantization affecting the current measures . . . . . . . . . . . . . . 241 5.6.2. Quantization at the level of the position measure . . . . . . . . . . . 244 5.6.3. Calculation of the speed by digital differentiation . . . . . . . . . . . 245 5.6.4. Quantization in the vector PWM of the voltage inverter . . . . . . . 246

5.7. Delays in the reverse Park transformation . . . . . . . . . . . . . . . . . . 248 5.8. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 5.9. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Chapter 6. Direct Control of a Permanent Magnet Synchronous Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Jean-Marie RÉTIF

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 6.2. Model of the permanent magnet synchronous machine in the d-q reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

6.2.1. State modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Table of Contents xi

6.3. Conventional DTC with free switching frequency . . . . . . . . . . . . . 253 6.3.1. General principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 6.3.2. Experimental application of DTC . . . . . . . . . . . . . . . . . . . . 256

6.4. DTC at a fixed switching frequency . . . . . . . . . . . . . . . . . . . . . 258 6.4.1. Principle of the control . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 6.4.2. Development of the reference vector #Ψ . . . . . . . . . . . . . . . . 261 6.4.3. Experimental results of DTC on a period of fixed calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

6.5. Predictive direct control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 6.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 6.5.2. General principle of predictive direct control . . . . . . . . . . . . . 264 6.5.3. Application to the permanent magnet synchronous motor . . . . . . 265 6.5.4. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 6.5.5. Predictive direct control by model inversion . . . . . . . . . . . . . . 272

6.6. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 6.7. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

Chapter 7. Synchronous Machine and Inverter Fault Tolerant Predictive Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Caroline DOC, Vincent LANFRANCHI and Nicolas PATIN

7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 7.2. Topologies of three-phase fault tolerant machines . . . . . . . . . . . . . 284

7.2.1. Restriction of the short-circuit current of permanent magnet machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 7.2.2. Restriction of the fault to the phase at fault alone . . . . . . . . . . . 284

7.3. Topologies of fault tolerant converters . . . . . . . . . . . . . . . . . . . . 285 7.4. Fault tolerant controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

7.4.1. Modeling synchronous machines in preparation for fault tolerant control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 7.4.2. Simulation of synchronous machines with fault tolerant control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7.4.3. Predictive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 7.4.4. Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

7.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 7.6. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

Chapter 8. Characterization of Control without a Mechanical Sensor in Permanent Magnet Synchronous Machines . . . . . . . . . . . . . . . . . . 305 Maurice FADEL

8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 8.1.1. State observation and disturbance observer . . . . . . . . . . . . . . 306

xii Control of Synchronous Motors

8.1.2. Interaction of the dynamics of control and observation . . . . . . . 307 8.1.3. Poles placement for control and observation . . . . . . . . . . . . . . 310

8.2. Sensorless control of PMSM, thanks to an extended Kalman filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313

8.2.1. A brief reminder on the Kalman filter (KF) . . . . . . . . . . . . . . 313 8.2.2. Application to the PMSM case . . . . . . . . . . . . . . . . . . . . . . 315 8.2.3. Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318

8.3. Comparison with the MRAS (model reference adaptive system) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 8.4. Experimental results comparison . . . . . . . . . . . . . . . . . . . . . . . 323 8.5. Control without sensor of the PMSM with load torque observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

8.5.1. Control by state feedback on the currents . . . . . . . . . . . . . . . . 331 8.6. Starting the PMSM without a mechanical sensor . . . . . . . . . . . . . 334

8.6.1. Equilibriums of the system without a mechanical sensor . . . . . . 335 8.6.2. Analysis by simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 8.6.3. Modification of the control law for a global convergence . . . . . . 341

8.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 8.8. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

Chapter 9. Sensorless Control of Permanent Magnet Synchronous Machines: Deterministic Methods, Convergence and Robustness . . . . . . 347 Farid MEIBODY-TABAR and Babak NAHID-MOBARAKEH

9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 9.2. Modeling PMSMs for mechanical sensorless control . . . . . . . . . . . 350

9.2.1. State model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 9.2.2. Reduced-order model . . . . . . . . . . . . . . . . . . . . . . . . . . . 354

9.3. Convergence analysis of mechanical sensorless control laws . . . . . . 356 9.3.1. Proportional-type control law . . . . . . . . . . . . . . . . . . . . . . . 356 9.3.2. Variable structure control law . . . . . . . . . . . . . . . . . . . . . . 364

9.4. Estimation of the back-EMF vector . . . . . . . . . . . . . . . . . . . . . 371 9.5. Robustness of sensorless control of PMSM with respect to parameter uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

9.5.1. Uncertainty on the stator inductances . . . . . . . . . . . . . . . . . . 375 9.5.2. Uncertainty on the torque coefficient . . . . . . . . . . . . . . . . . . 377 9.5.3. Uncertainty on the stator resistance . . . . . . . . . . . . . . . . . . . 378

9.6. Sensorless control of PMSMs in the presence of uncertainties on the resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

9.6.1. Online estimation of the resistance . . . . . . . . . . . . . . . . . . . 387 9.6.2. Minimization of the sensitivity of the sensorless control with respect to the resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 392

Table of Contents xiii

9.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 9.8. Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.9. Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.10. Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Introduction

For the countless operations of its machine tools, its robots and its “special machines”, modern industrial production has a tremendous need for elements called “motors”: these machines must impose on tools in motion (generally in rotation) a torque, a speed or a position, all determined by a high level decisional element. Execution rapidity and precision are necessary for a high productivity of quality. Electric motors have thus taken a predominant position in the “drive control”. Consequently, they are found in modern industrial production – but also in many “general public” applications, even if in this book we will mainly discuss the professional applications. Indeed, they have a predominant position because of their maneuverability and their (relative but effective) ease of use. Hydraulic motors (for example) have much better performances in terms of “torque-mass ratio”, but they are much trickier to control.

Historically, direct current motors were the first to be used because – in some aspects – they were ideal: excellent performances in terms of speed and of implementation ease, on the converter level (a thyristor rectifier or a transistorized chopper are sufficient), as well as on the controls level. Indeed, the “electromagnetic torque” is proportional to the “armature current”. Therefore, a simple “current loop” imposes the torque, and then a “speed loop” is sufficient to produce a electronic “speed variator” (see on these topics Chapter 1 of [LOU 04b] written by J.-P. Hautier). The serious disadvantage of DC motors is the “mechanical collector” – the exact element which allows this ease of implementation. However, this mechanical “converter” could have fragilities (wear) and risks of accident in wet or dusty atmospheres. Moreover, the armature current circulates at the rotor. Therefore, cooling is not easy, which limits the motor performance, because the current and thus the mass torque cannot be very high.

Introduction written by Jean-Paul LOUIS.

xvi Control of Synchronous Motors

The development of power electronics – involving the extension of the use of the inverters – has made it possible to supply the alternating current machines as easily as the direct current machines. There is however an additional constraint: we must know the rotor position, and therefore it is necessary to install a mechanical sensor (or to fulfill this function by other means), in order to carry out the “electronic collector” operation using a “self-control system”.

This book belongs to a monograph series devoted to AC motors, and specifically discusses synchronous motor control, which has a prominent position among these motors. For a long time, the most widespread synchronous machine was the alternator, i.e. a generator. However, the operation is mainly in motor mode – even if during transients, it transfers to a generator mode. As long as we only had fixed frequency alternative sources, the synchronous motor could only turn at fixed speed. The development of power electronics completely changed this situation. Thanks to thyristor bridges (operating in “line commutated inverters”) self-controlled synchronous motors first appeared: they were especially used in high powers, for rolling trains, for example, or for traction (the first high-speed French trains). The component development (transistors, GTO) for the “forced switching” inverters facilitated the variable frequency supply of alternating current motors and led to their development in a wide range of applications. Lastly, the massive arrival of microprocessors led to powerful controls, thanks to sophisticated and complex algorithms, which were executed very quickly in real time.

The first alternating current machine used as a motor was the synchronous motor, mainly with a permanent magnet excitation. The use of a position sensor (or of an equivalent function) made the self-control machine possible and, thus, made it operate (almost) like a DC motor: the torque is indeed proportional to a current (known as the “q axis”, as we will see in Chapter 3). With respect to direct current motors, the synchronous motor has technical advantages. Initially, the “armature current” circulates at the stator. Therefore, cooling is easy. We can then have currents – thus torque-mass ratios – much higher than for DC motors. Also, there are no more fragility or safety problems because of the mechanical collectors, replaced by “electronic collectors” (without wear or sparks). The robustness becomes excellent. We understand that the components’ manufacturers (the motors themselves, inverters, and controllers) developed very efficiently with competitive product ranges.

These synchronous machines, thus used, have received various names: “synchronous self-control motor” or “electronic switching synchronous motor”. The industrial name is often “brushless DC motor”, or “DC motor without collector”1.

1. This term is often allocated to synchronous motors with trapezoidal field distribution, supplied with square wave currents, which will be discussed several times in this book.

Introduction xvii

The predominance of the synchronous motors with permanent magnet excitation is obvious, as most of the chapters of this book will testify. The reduction in manufacturing costs of high-efficiency permanent magnets is certainly at the origin of the scope expansion of these machines.

Competition has come from the other conventional AC motor, the induction motor. Thanks to its simple and robust rotor, this motor has the great advantage of being naturally more economical than the synchronous motor. This argument is industrially very strong. But the induction motor is more difficult to use for drive control. Much work has been undertaken in order to supply and control the induction motor with performances close to those of the synchronous motor (thanks to “vector control”). Books will be devoted to this motor in the framework of this monograph series. Let us only remark that the price decrease of the permanent magnets, obtained thanks to the efforts of the iron and steel industry, puts the economic advantage of the induction motor in perspective with respect to the increasingly used synchronous motor.

This book is part of a series published by ISTE-Wiley and Hermes-Lavoisier. Two books have already been published. They are devoted to modeling motors for their control [LOU 04a and LOU 04b]. Another volume has been devoted to the identification and the observation of electric machines [FOR 10]. One of the volumes presented the general methods relative to the control of electric machines [HUS 09], and another has presented the technological problems [LOR 03]. Electric motors control is associated with the static converter’s control (here, inverters), and in the past, works concerning electric machines control have especially taken an interest in the converter’s control (often very complex). Nowadays, with the development and progress of the technical realizations, there is a relative decoupling between these two activities, in particular when the inverter is at forced switching and controlled in pulse-width modulation (PWM.): another monograph specifically discusses these questions [MON 11], being centered on modulations and on current controls.

The program of this book is thus targeted at the presentation of the control laws of the “conventional” synchronous motors. Another monograph (forthcoming) will present control laws suitable for more “non-conventional” motors, which are often specific alternatives of synchronous motors. Conventional synchronous motors are defined by their respect of specific hypotheses making, in particular, the immediate use of the “Park transformation” (or d-q frame) possible. Those hypotheses are recalled in Chapter 1 of this book, by Jean-Paul Louis, Damien Flieller, Ngac Ky Nguyen and Guy Sturtzer. We can summarize them in a few words: linearity (without saturation), first harmonic (sinusoidal field distribution) and symmetry (or “circularity”). But very often, the motors installed in industry do not completely check all these hypotheses (for example non-sinusoidal or trapezoidal field

xviii Control of Synchronous Motors

distribution). Moreover, we present expansions, showing that there is an “extended” meaning to this “conventional” adjective. This first chapter summarily exposes the basic models, necessary for the design of the synchronous motor controls:

– models in the “natural” three-phase reference frame (or “a-b-c”);

– models in the two-phase Concordia reference frame (or “α β− ”);

– models in the rotor reference frame, or Park reference frame (or “d-q”);

– with extensions to some non-sinusoidal field distribution machines.

The crucial stage of the motor control (whatever its type), is torque control. Therefore, two chapters (Chapter 2 and 3), by Damien Flieller, Jean-Paul Louis, Guy Sturtzer and Ngac Ky Nguyen, are devoted to this fundamental question. The first problem to be solved is the establishment of “direct models” defining “inverse models” that are in fact the control laws. We then obtain algorithms that are “vector controls”, whose core is “self-control” (necessity to synchronize the currents with the back electromotive force (back-EMF), therefore in fine, with the position). This vocabulary was popularized by the induction machines control, but perfectly applies to the synchronous motor. There, the controls suitable for electrical drives is in agreement with the general methods of control science, such as the “input-output linearization with state feedback” that power electrical engineers naturally practice when they realize controls with “decoupling between the d and q axes”.

The vector controls show that to impose the torque, we must impose – and thus regulate – the currents. There are two large families of current regulations:

– there is the controls family regulating the three-phase currents in the natural a-b-c reference frame; these currents are those effectively measured;

– and there are controls regulating the currents in the Park “d-q” reference frame. These currents must be reconstructed by real time calculations.

The first solution is a priori technically simpler and was the first to be implanted. It has the advantage of working with real currents and thus of leading to immediate current (security) monitoring, but it is more difficult to obtain good results, because of the presence of static errors during the tracking of sinusoidal references – except specific strategies (one of them will be presented).

The second solution is naturally more efficient, because the current references in the d-q frame are “continuous”. But, as it required more real time calculations, it was popularized only when dedicated components appeared on the market.

These two approaches each have their advantages and disadvantages. Chapter 2 (for controls in the a-b-c reference frame) and Chapter 3 (for controls in the d-q

Introduction xix

reference frame) present and discuss them. For reasons of simplicity, the controls are presented in the case where inverters are piloted in PWM. [MON 11] gives other alternatives of current control (for example, “hysteretic controls”).

The second problem encountered by specialists is relative to the determination of the optimal form of the motors’ feed currents. Indeed, very often the motors manufacturers seek to obtain the best torque/mass ratio. This often leads them to machines without a sinusoidal field distribution. Then, the optimal currents (exactly supplying the required torque by minimizing the Joule losses) are no longer sinusoidal. Chapter 2 gives us very powerful analytical tools in the case of the natural a-b-c reference frame for the particular case of the non-salient pole machines (with cogging torque). Chapter 3 shows the possibility of a geometrical approach combined with the Park transformation, in order to define efficient solutions, in particular for salient pole machines (also with cogging torque).

The electric motor’s control has several borders: we have just skimmed over one of them, the converter’s control. There are other borders: the “position” and the “drive control”. In this last domain, Chapter 4, by Jean-Paul Louis, Damien Flieller, Ngac Ky Nguyen and Guy Sturtzer, mainly exposes examples of “electronic speed variators” with a synchronous motor. The “speed variator” is a control unit that is very common in industry. It must generally be “transparent” for the user. The aforementioned control unit imposes a speed reference coming from a higher hierarchical level. The motor must then have a speed response with extremely fast dynamics. The problem of torque control is the first stage, presented in Chapters 2 and 3 and illustrated by several solutions. The problem of the speed control is the second stage, largely depending on the mechanical load.

The mechanical load can be simple, purely inertial for example, with a constant load torque. This is the case generally considered in many studies, and it will be the case discussed here. But the reader must know that users often encounter much more complex cases. Let us quote two quite representative cases: variable inertia mechanical load (as in robotics or with unwinder-rewinder); load with elastic links, dry and viscous frictions difficult to identify or oscillating modes (as with rolling trains). We thus leave the domains specific to the electric motor’s control seen by power electrical specialists, because they estimate to have completed their task when they have carried out a good torque control. The “drive control” in complex cases comes to the general automatics applied to complex mechanical systems. A monograph has tackled these questions [HUS 09].

However, some drive control problems are coupled with specific properties of the electric motor. Chapter 4 is centered on these questions and written by Jean-Paul Louis, Damien Flieller, Ngac Ky Nguyen and Guy Sturtzer. The authors discuss in this chapter examples of axis control applied to the most conventional mechanical

xx Control of Synchronous Motors

system, because it is regarded as a generic example: constant inertia, viscous friction and load torque piecewise constant. They show that the strategy of torque control (such as it was presented in Chapters 2 and 3) has an influence on the performances of the speed control and that, consequently, the same control cannot have the same performances according to whether we associate it with a torque control in the a-b-c reference frame or with a torque control in the d-q domain.

We will also see that the conventional controls of the synchronous motors applied to this generic example have a great advantage. Indeed, with the traditional mechanical sensors, all the state variables are measurable, and therefore there are very efficient controls, in speed as well as in position. Thus, we will examine (by assuming that the machine is well controlled in torque, from the methods seen previously) several regulations and feedbacks approaches (P, IP controllers and introduction to the load observers). Some robustness aspects are examined.

Torque controls (and thus current controls) presented in Chapters 2 and 3 are modeled by conventional continuous equations: algebraic equations, differential equations, transfer functions. The controls described by these models are immediately transposable when similar components are used. But for a long time, implementations have been carried out with numerical technologies: microprocessors, specialized signal processors, FPGA. Another book has developed these aspects [LOR 03]. Digital technology introduces new problems. Chapter 5, by Flavia Khatounian and Eric Monmasson, discusses questions associated with the implementation and the digitization of current and speed controls of synchronous motors. This chapter considers points not discussed in the previous chapters: numerical regulations of current and PI-type speed, fast sampling frequency for current loops, and slower frequencies for speed and position regulations, with the recognition of the various constraints due to the technical realization.

Indeed, the concrete implementation imposes specific studies. It is necessary to model the interfaces and the sensors, in particular the position encoder. Then, element by element, we must study the phenomena to be taken into account in the framework of a numerical implementation: selection of the sampling frequency, delays due to the time taken by the various calculations and due to the PWM, quantization effects on the measures, problems due to the resolution of the incremental position encoder and to speed determination by numerical differentiation, control discretization, PWM cut, implementation of the reverse transformation of the reference voltage of the d and q axes with the question introduced by the difference between the angle used for the Park transformation and its real value. The chapter gives an original summary of these various problems, a summary not often presented explicitly in specialized books and papers. It gives in particular a very complete “time diagram” and precisely lists the various “critical periods” to be examined.

Introduction xxi

Torque controls (presented in Chapters 2, 3 and 5) and speed controls (Chapter 4) have been limited in practice to “vector control” controls associated with piloting the inverter in pulse width modulation. This approach can now be regarded as conventional and is very often used industrially. These approaches have the advantage of decoupling the static converter’s (inverter’s) control of the machines themselves. This decoupling is simple, which is a great advantage in the industrial domain. We cannot however guarantee that the overall outcome is optimal2. However, other approaches have appeared in the last few years, regarding the “predictive control” and the “direct torque control” (or “DTC”) concepts. They are “smart” controls seeking to optimize the motor-inverter association to obtain new properties.

Chapter 6, by Jean-Marie Rétif, first of all presents the torque “direct control” method. This method has especially been developed for the asynchronous machine. It is quite useful in high power when the machine is supplied by a relatively low frequency inverter. It is presented here in its version for the synchronous motor. By principle, the DTC uses heuristics based on known tendencies on flux and electromagnetic torque variation. The heuristics determine the inverter configuration, supplying the best voltage fluctuation to be supplied: the control thus associates the inverter modeling with the motor modeling. The control itself is in fine carried out by hysteresis controllers, therefore very fast ones. As a result, this method is likely to give the shortest possible torque response times. One of the constraints is that the arithmetic unit (generally a microprocessor) is constantly calculating to introduce a switching control, only when a threshold is crossed. This method is thus very constraining for the processor. In addition, it introduces variable switching frequencies, which can be undesirable.

This is why alternatives are introduced, for example fixed frequency DTC. It is preferable to generate a control based on an analytical model, rather than from heuristics built on evolution tendencies. Moreover, the current development of the control theories applied to the electric motors strives for a “hybrid approach”, for which the control is no longer the required voltage, but the inverter configuration. We then reach a control family increasingly used nowadays: the “predictive controls”. Chapter 6 thus shows an application to the synchronous motor of a “direct predictive control”. The conventional two levels voltage inverter has only eight distinct configurations and we can easily determine, by a model which is linearized

2. We once again have the classic problem opposing “local optimizations” and “global optimization”: the machine control can be optimized in itself and the converter control can also be optimized in itself (two local optimizations). This does not guarantee that the set machine-converter is optimally controlled (global optimization). This problem is nowadays well known by the electrical vehicle specialists, seeking to drastically minimize all the losses of the whole electric system destined for this embedded device.

xxii Control of Synchronous Motors

on a short horizon, the best configuration at a given moment. The author gives examples of very efficient strategies.

The predictive control does not lead to a single control, but to a control family with various properties and application fields. As it is very promising, another application chapter of this approach is presented. Chapter 7 gives an example of predictive controls tolerant to the inverter faults, by Caroline Doc, Vincent Lanfranchi and Nicolas Patin. This example shows that modern controls bring solutions to significant problems (control under fault conditions), which could hardly be dealt with by conventional approaches, such as the vector control conceived in the d-q axes, naturally presupposing that the machine and converter are normal. Therefore, the new controls strategies bring new services.

This is also the case for the two final chapters of this book. The conventional synchronous motor controls require a position sensor to carry out the self-control, even for a torque or a speed control. However, there are some cases where we wish to make controls “without sensor”, i.e. without mechanical position sensor. Various reasons explain this fact: economic, size or reliability reasons, or to be able to continue working in “degraded operation”, when the sensor signal disappears (failure, accident). These themes have been discussed for a long time, but are still open to discussion. Many solutions have been proposed and this is why this very important question is studied in two chapters of this book.

Maurice Fadel, in the Chapter 8, takes a look at the characterization of the control without mechanical sensor of the synchronous motors. Indeed, the position is no longer a measured variable, but a magnitude rebuilt by a real time calculation – in particular with an extended Kalman filter with respect to model reference control. This magnitude then has a certain dynamics influencing various motor controls, as well as the various observations, particularly of the load torque usually integrated in the control. This last question was summarily presented in Chapter 4 of this book. It was discussed thoroughly in Chapters 7 (by Maurice Fadel and Bernard de Fornel) and 8 (by Stephan Caux and Maurice Fadel) of another monograph [FOR 10].

Chapter 8 examines the different dynamics: of the position observation, of the load torque observation, of the speed control, compared with the inverter decoupling frequency. The point of view is non-linear and is concerned with the global stability of the set observer-control, which makes up the electronic speed and/or position variator.

Chapter 9, by Farid Meibody-Tabar and Babak Nahid-Mobarakeh, takes a more specific look at deterministic observation methods of the synchronous motors position. These methods often use an estimate of the electromotive force (emf), because it has important advantages (it requires the knowledge of only a small

Introduction xxiii

number of electric variables). The disadvantage of these methods lies in the convergence domain limit. There are thus important stability problems. This is why the point of view adopted in this chapter is basically non-linear, to bring guarantees to the global stability of a control with a position estimate. The main method is due to Matsui. This method is very interesting, but unfortunately has a limited convergence domain. In this chapter it is, however, shown that there are solutions to extend the control convergence domain. The authors thus study a methods family to estimate and observe the position and speed without a mechanical sensor. They also examine the control properties using this observation: stability, dynamics and robustness with respect to the parametric errors.

This book thus proposes a broad overview of the conventional (or almost conventional) synchronous motor control, from traditional methods (regulations with inverter controlled in PWM) to very promising advanced methods, such as direct controls and predictive methods. We give extensions of modeling and of methods, by stressing the very important question of the controls without mechanical sensors.

Other questions remain, but they do not directly concern the problems of torque, speed or position control of a synchronous motor supplied by a voltage inverter controlled in PWM. These questions relate to other supply modes, or to other nonconventional types of motors: we then talk about “special machines”. They are often – more or less – synchronous. It is logical, after the conventional synchronous motor control, to consider them. These questions must be the subject of a future monograph.

This work is dedicated to the memory of René Husson (Nancy) and Manual da Silva Garrido (Lisbon), who contributed to the quality of EGEM treatises ([HUS 09] and [LOU 04a], Chapter 1).

Bibliography: monograph series on control of electrical motors published by ISTE-Wiley and Hermes-Lavoisier

[FOR 10] DE FORNEL B., LOUIS J.-P, Electrical Actuators: Identification and Observation, ISTE, London and John Wiley & Sons, New York, 2010.

[HUS 09] HUSSON R. (ed.), Control Methods for Electrical Machines, ISTE, London and John Wiley & Sons, New York, 2009.

[LOR 03] LORON L. (ed.), Commande des systèmes électriques : perspectives technologiques, Hermès, Paris, 2003.

[LOU 04a] LOUIS J.-P (ed.), Modélisation des machines électriques en vue de leur commande, concepts généraux, Hermès, Paris, 2004.

xxiv Control of Synchronous Motors

[LOU 04b] LOUIS J.-P (ed.), Modèles pour la commande des actionneurs électriques, Hermès, Paris, 2004.

[MON 11] MONMASSON E. (ed.), Power Electronic Converters: PWM Strategies and Current Control Techniques, ISTE, London and John Wiley & Sons, New York, 2011.

Chapter 1

Synchronous motor controls, Problems and Modeling

1.1. Introduction

The tremendous importance of rotary synchronous motors in the industrial systems control has been recalled in the general introduction of this book. There is in the professional community a very important emulation to define control structures, simple to materially implant and to design and very efficient ([BOS 86, LEO 90, VAS 90, MIL 89, LEP 90, LAC 94, LAJ 95, GRE 97, LOU 99, STU 00b, LOU 04c, LOU 09]). Nowadays, we can consider that the control structures are based on some very solid basic principles that we will present. Of course, from one designer to another, many alternatives can appear (each manufacturer wants to have their own patents), but we can consider that the basic principles exploited in practice are those that will be covered in Chapters 2 and 3 of this book, which are devoted to the torque controls. The most important key concepts will be: “self-control”, torque control in the “natural” reference frame (often known as the a-b-c reference frame); torque control in the rotor reference frame (often known as “Park reference frame” or d-q reference frame). Indeed, when the torque control is carried out, it is easy to successively implant a speed control, to obtain a device usually called an “electronic speed variator” and then if necessary, a position control. These last questions will be tackled in Chapter 4. This chapter exposes a general modeling of the synchronous motors and is particularly used as introduction to Chapters 2 to 4.

Chapter written by Jean-Paul LOUIS, Damien FLIELLER, Ngac Ky NGUYEN and Guy STURTZER.

2 Control of Synchronous Motors

1.2. Problems on the synchronous motor control

1.2.1. The synchronous motor control, a vector control

The synchronous motor has much better performances than the direct current motor ([VAS 90, BEN 07, MUL 06]), but the counterpart has more sophisticated power electronics (an inverter instead of a rectifier or a chopper) and more complex control laws. Indeed, it is necessary to fulfill the “brush-collector” function via the converter control. This requires knowledge of the rotor flux direction. As it is interdependent with the rotor, a mechanical position sensor gives the necessary information. There are applications where we seek not to use a mechanical sensor, but this is the objective of Chapter 8 and Chapter 9. We will see that we must synchronize the currents on the position, which is the “self-control” function (see the note in section 1.4.2.2). Indeed, it is ideally necessary (here we simplify a little) to create a stator field in quadrature with the rotor field: this type of control thus completely deserves the term of “vector control”, a concept that was popularized by the induction motor’s control [LEO 90, CAR 95, CAN 00, ROB 07]. By this strategy we seek to precisely control those as synchronous motors (and consequently in fine as direct current motors, since – as we will see – there is a very strong analogy between the axis q of the synchronous motor and the armature of the direct current motor).

These vector controls can be conceived by various approaches. Theoretically, the most satisfactory approach uses the Park model (in the rotor reference frame, known as d-q). This is discussed in Chapter 3, but historically it was not the first to be industrially used. An approach in the “natural” model (in the three-phase stator reference frame, known as a-b-c) was initially largely used, especially for the non-salient pole machines, where a “three-single-phase” model (apparently simpler) is usable: this approach leads to a “vector control”, since we always seek to impose a given direction to the stator field. This approach is covered in Chapter 2.

This gives us the chance to recall that control modeling and control design are two different activities. We can regard as “logical” the use of a three-phase model written in the natural reference frame to design an a-b-c control, since there is a similarity between the variables of the model and the variables used to implant the control. In the same way, it is logical to use a Park model written in the d-q reference frame, to design a d-q control, with the same argument. But we can install a three-phase current regulation (thus in a-b-c) and use a d-q model to estimate the performances. This is what we will do in Chapter 2, where signals are alternating, because the variables in the d-q reference frame are “continuous” (constant in steady state). It is easier to estimate the performances of this type of variable, than the performances of the sinusoidal variables in steady state.

Problems and Modeling 3

Several approaches are thus possible. The approach in the three-phase reference frame has the advantage of respecting the effective magnitudes and therefore preserving the specific functioning of the inverter and supervising the effective current amplitudes (protection and security). This is also a practical approach to consider the non-sinusoidal field distribution machines, in order to preserve the effective form of the induced back electromotive forces (back-EMF)1. The implementation is also simpler, and can be done with a few programmable components [LOU 99]. It is thus normal that it is used and presented here (the subject of Chapter 2). This “natural” approach has disadvantages: it is more difficult to evaluate its dynamic performances and (correlatively) it is more difficult to have very good performances.

This is why we will devote Chapter 3 to controls design in the d-q reference frame: the implementation is certainly more complex (but really facilitated nowadays by the programmable modern components); on the other hand, the vector control and the controls’ design imposing the desired dynamics is direct and natural. We can easily obtain very good dynamic performances. Therefore, this approach is frequently used nowadays.

1.2.2. Direct/inverse model and modeling hypotheses

As shown in the previous section, the synchronous motor control is largely based on the mathematical machine model. We also have just seen that the a-b-c model and the d-q model “naturally” lead to controls of different structures (even if it is possible but not necessarily very easy – to go from one to the other). The modeling hypotheses will thus necessarily influence the control algorithms. Indeed, controls, under various names: “input-output linearization with state feedback” (automatics vocabulary), control “with d and q axes decoupling” (electrical engineering vocabulary), “inverse models” control, amount in fact to deducing the model control structure (known as “direct” [HAU 97]) of the motor. The structure, but not necessarily the controller: thus, these approaches show that we must (for example) control the currents (three-phase currents or Park components of the currents), but they do not tell how to control them: the designer can freely choose the controller type. However, these approaches with models give the references of the currents to be controlled, and the model used brings a quality to these references, according to the precision of the model used. Moreover, these are the references that must be synchronized with the help of the position sensor (“self-control”).

1. By convention, because we consider that the typical functioning is the motor functioning, we call the induced voltages: “counter electromotive forces” [WIL 05].

4 Control of Synchronous Motors

According to the machines’ characteristics and the precision of the models used to represent them, various alternatives will intervene:

– The first alternative relates to the difference between non-salient pole machines and salient pole machines. The difference comes from the geometries used to build these machines. Thus, in the case of the synchronous motors with “magnets installed on the surface”, the air gap is constant and the machine is with non-salient poles. It is then easy to use a three-phase model in the three-phase a-b-c reference frame. On the other hand, for “buried magnet” motors (in the rotor), the machine is with salient poles and the three-phase model is appreciably more complex, whereas the Park model (d-q) leads to a remarkably simple model. It is necessary to be aware that the simpler a model is, the easier it is to design efficient controls. Of course, we can give a model in d-q of a non-salient pole motor (the model is then even simpler), but we can see here that a constructive structure can have effects on the choice of modeling. The modeling itself has effects on the control design.

– The second important variant is also constructive: the alternating current machines are known as “well built” if they obey the Park hypotheses (i.e. to the hypotheses implicitly used for the conventional modeling in the d-q reference frame). It is necessary to distinguish the machines with “sinusoidal field distribution” and the ones with “non-sinusoidal distribution”. The most well known of these last machines is the “trapezoidal” back-EMF machine. These machines have excellent physical performances in terms of “mass torque”. They are thus very popular among designers and users. But a precise model is difficult to write and exploit for the control. It is then common to accept approximations:

- on the model level (accepting a model limited to the first harmonic, in order to find the model of the sinusoidal field distribution machines);

- or on the control level (supplying the stator by square wave currents, for example): we will show (Chapters 2 and 3) supply and control examples of sinusoidal field distribution machines with extensions to the non-sinusoidal cases. There too, modeling has effects on the machine control strategy.

– Another variant is the existence of saturation: the Park model assumes that we operate in linear (unsaturated) regime, which is legitimate for the machines with magnets installed on the surface, because they have a large air gap and they only saturate a little. But this hypothesis can be faulty with small air gaps machines. Obviously, a saturated model of the synchronous machine is extremely complex – and not easy to use for the controls’ design (let us recall that the conventional models of synchronous saturated machines are limited to the steady state, whereas the control laws are conceived with dynamic models). The conventional control laws are generally unaware of the saturation effect. However, there are extensions to the methods, in order to discuss the case of saturated machines, which will refer to them ([STU 01]).