Dynamics of Saturated Electric Machines

Vlado Ostovic

Dynamics of Saturated Electric Machines

With 250 Illustrations

Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong

Vlado Ostovic Studij Elektrotehnike University of Osijek 54000 Osijek, Yugoslavia

Library of Congress Cataloging-in-Publication Data

Ostovic, Vlado. Dynamics of saturated electric machines / Vlado Ostovic

p. cm. Bibliography: p. Includes index.

ISBN-13: 978-1-4613-8935-4 e-ISBN-13: 978-1-4613-8933-0 DOl: 10.1007/978-1-4613-8933-0

I. Electric machinery-Mathematical models. I. Title. TK2744.087 1989 621.31 '042-dc20 89-11414

Printed on acid-free paper.

© 1989 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1989

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag. 175 Fifth Avenue. New York. NY 10010. USA). except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names. trademarks, etc. in this pUblication, even if the former are not especially identified. is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act. may accordingly be used freely by anyone.

Camera-ready text provided by the author using the TPS system by Interleaf.

9 8 7 6 5 4 3 2 I

To Tea and Marko for their love, patience, and understanding

Preface

This book is a result of the author's work which was initiated about a decade ago and which, in the meantime, has resulted in his Ph.D. Thesis and several technical papers. The book deals with accurate modeling of electric machines during transient and steady states, a topic which has been usually avoided in the literature. The modeling techniques herein take into account all machine peculiarities, such as the type and connection of its windings, slotting, and saturation in the iron core. A special emphasis in the book is given to the exact physical interpretation of all phenomena which influence the machine's transient behavior.

Besides the Introduction, the book has five chapters. The second chapter describes basic concepts of the magnetic equivalent circuit theory and has examples of magnetic equivalent circuits of several types of machines with their node potential equations.

In the third chapter the transform matrices w' and w" of A.C. wind­ings are derived. These matrices playa very important role in the magnetic equivalent circuit theory because they connect the quantities from the ma­chine's magnetic equivalent circuit, branch fluxes, and mmfs with the ma­chine's phase currents and fluxes.

The additional equations which are needed to solve magnetic circuits are introduced in the fourth chapter. These equations had to be introduced because the node potential method, which is used in the solution of the magnetic equivalent circuits, does not carry information about those equiva­lent circuit branches in which the values of fluxes are known.

In the fifth chapter the relationships between algebraic equations of the magnetic equivalent circuits and the machine's differential equations are established. The connection and condition matrices are generated in this chapter. These matrices are functions of the machine's type of connection and the character of the source to which the machine is connected including possible faults in the machine (short-circuited or open phase).

viii

The reader should pay special attention to the sixth chapter because a new technique for the solution of the nonlinear system dynamics is intro­duced there. The usual approaches to the solution of nonlinear system dy­namics employ iterations of the nonlinear system of algebraic equations at each step of integration of the system of differential equations. These itera­tions are inevitable because they reflect the fact that the only way to solve a system of nonlinear algebraic equations is numerical. In the approach pre­sented in this chapter the iterations in the system of algebraic equations are avoided by converting the algebraic equations into differential equations. In other words, the unknowns of the system of algebraic equations became state variables in the system of differential equations. The price for this is an enlarged system of differential equations, along with a longer time for its solution. On the other hand, a significant amount of time is saved and the solution accuracy is increased by eliminating the iterations of the system of algebraic equations.

In Appendix ill some applications of the principles developed in the book are given. Transients in a squirrel cage induction machine and a per­manent magnet synchronous machine are calculated. The results obtained, especially those represented in three-dimensional plots, not only help illus­trate the potential of the magnetic equivalent circuit method, but also clarify some basic principles of operation of electric machines.

A part of the work on the book was done during the author's frequent visits to the Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC) group at the University of Wisconsin in Madison. Dr. Rich Schiferl, at that time a Ph.D. student at the UW in Madison, had written a program for graphical postprocessing which was used in generation of the two- and three-dimensional plots in the Appendix ill. He also proofread the manuscript of the book and had several useful suggestions.

Contents

1 Introduction 1

2 Magnetic Circuits 5

2.1 Analogies Between Electric and Magnetic Circuits . . . . . . 6 2.2 Permeances in Magnetic Circuits . . . . . . . . . . . . . . . . . . 9

2.2.1 Constant Permeances . . . . . . . . . . . . . . . . . . . . 9 2.2.2 Parametric Nonlinear Permeances . . . . . . . . . . . 29

2.2.2.1 Unskewed Machine with Different Number of Stator and Rotor Slots . . . . . . . . . . . . . . . . 34

2.2.2.2 Influence of Skewing . . . . . . . . . . . . . . . 36 2.2.2.3 Rotor Eccentricity . . . . . . . . . . . . . . . . . 43 2.2.2.4 Complete Air Gap Permeance . . . . . . . . . 47 2.2.2.5 Influence of Fringing Flux . . . . . . . . . . . . . 49

2.2.3 Inherently Nonlinear Permeances . . . . . . . . . . . . . . 50 2.2.3.1 B-H Curve Approximation . . . . . . . . . . . . . 54 2.2.3.2 Influence of Variable Geometry . . . . . . . . . . 61

2.2.4 Convention About Symbols . . . . . . . . . . . . . . . . . . 63 2.3 Sources in Magnetic Circuits . . . . . . . . . . . . . . . . . . . . . 64

2.3.1 Magnetomotive Force of a Coil in a High-Permeability Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.3.2 Magnetomotive Force of a Distributed Winding 71 2.3.3 Permanent Magnets in Magnetic Circuits . . . . . . . . 72 2.3.4 Convention About Symbols . . . . . . . . . . . . . . . . . 74

2.4 Force and Torque . . . . . .. ................ 75 2.5 Magnetic Equivalent Circuits of Electromagnetic Devices 86

2.5.1 Singly and Multiply Excited Iron Core . . . . . . . . . 86 2.5.2 Windings in Slots . . . . . . . . . . . . . . . . . . . 90 2.5.3 Elementary Pole Pair . . . . . . . . . . . . . . . . . . . 94

2.5.3.1 The Number of Stator Parallel Circuits Restriction . . . . . . . . . . . . . . . . . . . . . . 94

2.5.3.2 Stator Fractional Slot Winding Restriction 98 2.5.3.3 The Number of Rotor Slots per Pole Restriction 99 2.5.3.4 Summary of Conditions for the Number of

Elementary Pole Pairs . . . . . . . . . . . . . . . 99 2.5.3.5 The Number of Stator and Rotor Slots per

Elementary Pole Pair . . . . . . . . . . . . . . . . 100 2.5.4 Induction Machine . . . . . . . . . . . . . . . . . . . . . 100

x

2.5.5 Salient Pole, Wound Rotor Synchronous Machine .... . 2.5.6 Permanent Magnet Synchronous Machine ...... . 2.5.7 Stepper and Switched Reluctance Motor ........ . 2.5.8 Shaded Pole Induction Machine ............ . 2.5.9 Conductive Plate in External Magnetic Field

2.6 Magnetic Equivalent Circuit Solution Procedure 2.6.1 Node Potential Equations and Permeance Matrices 2.6.2 k-Winding, n-Limb Transformer ............ . 2.6.3 Induction Machine .................. .

2.6.3.1 Simple Model .................. . 2.6.3.2 Complex Model ................. .

2.6.4 Wound Rotor Synchronous Machine .......... . 2.6.5 Permanent Magnet Synchronous Machine ....... . 2.6.6 Stepper and Switched Reluctance Motor ....... . 2.6.7 Shaded Pole Induction Machine ............ .

3 Winding Transfonn Matrices

3.1 Winding Topology Matrix .................. . 3.1.1 Single-Layer Windings .................. .

3.1.1.1 Concentric Winding ............... . 3.1.1.2 Distributed Winding ............... .

3.1.2 Double-Layer Winding ................ . 3.2 Number of Turns per Coil Matrix .............. . 3.3 Coil Currents Matrix ...................... . 3.4 Slot Ampere-Turns Matrix

3.4.1 Single-Layer Winding 3.4.1.1 Concentric Winding 3.4.1.2 Distributed Winding

3.4.2 Double-Layer Winding 3.4.2.1 One-Layer Model 3.4.2.2 Two-Layer Model

3.5 Tooth Magnetomotive Force Matrix 3.6 Magnetomotive Force Transform Matrix w" ........ . 3.7 Coil Flux Matrix . . . . . . . . . . . . . . . ....... .

3.7.1 Single-Layer Winding .................. . 3.7.1.1 Concentric Winding . . . . . . . . . . . . . . . 3.7.1.2 Distributed Winding .............. .

3.7.2 Double-Layer Winding .............. . 3.7.2.1 One-Layer Model ................ . 3.7.2.2 Two Layer Model ................ .

3.8 Number of Turns per Elementary Phase Matrix ..... . 3.9 Elementary Phase .Flux Matrix ............... .

102 104 105 108 109 113 114 114 118 118 122 125 130 131 136

139

140 141 141 143 146 149 150 151 151 151 155 160 160 163 165 167 169 171 171 174 177 177 178 179 179

3.9.1 Single-Layer Windings 3.9.1.1 Concentric Winding 3.9.1.2 Distributed Winding

3.9.2 Double-Layer Windings 3.10 Flux Transform Matrix Wi . . . . . . .. . ..•....•

3.11 Summary of Winding Transform Matrices 3.12 Squirrel Cage Connection Matrices ............ .

4 Extended System of Machine Algebraic Equations

4.1 Squirrel Cage Induction Machine - A Simple Model 4.2 Squirrel Cage Induction Machine - A Complex Model 4.3 Wound Rotor Synchronous Machine ....... . 4.4 Permanent Magnet Synchronous Machine ........ . 4.5 Stepper and Switched Reluctance Motor .......... . 4.6 Shaded Pole Induction Motor ........... .

5 Machine System of Differential Equations and Complete System of Algebraic Equations . . . . . . . . . . . . . ...

5.1 Stator Differential Equations of an A.C. Machine 5.1.1 Voltage-Fed m-Phase Machine ...... .

5.1.1.1 Polygon Connection . . . . . . . . . . 5.1.1.2 Star without Common Neutral Connection 5.1.1.3 Star with Common Neutral Connection

5.1.2 Current-Fed m-Phase Machine ....... . 5.1.2.1 Polygon Connection .......... . 5.1.2.2 Star without Common Neutral Connection 5.1.2.3 Star with Common Neutral Connection

5.1.3 Summary of Stator Winding Connection Types 5.1.4 Voltage Equations of Other Types of A.C. Machines 5.1.5 Failures in a Voltage-Fed A.C. Machine Stator Winding

5.1.5.1 Voltage-Fed m-Phase Machine; Polygon Connection ................... .

5.1.5.2 Voltage-Fed m-Phase Machine; Star without Common Neutral Connection ....... .

5.1.5.3 Voltage-Fed m-Phase Machine; Star with Common Neutral Connection ........ .

5.1.6 Failures in a Current-Fed A.C. Machine Stator Winding 5.1.6.1 Current-Fed m-Phase Machine; Polygon

Connection .............. ...... . 5.1.6.2 Current-Fed m-Phase Machine; Star without

Common Neutral Connection

xi

180 180 181 183 184 186 187

190

192 205 210 212 212 215

218

221 222 222 226 231 234 235 238 242 245 248 248

249

251

253 253

254

258

xii

5.1.6.3 Current-Fed m-Phase Machine; Star with Common Neutral Connection ........ 260

5.2 Rotor Differential Equations of an A.C. Machine ....... 261 5.2.1 Squirrel Cage Winding Voltage Differential Equations 261 5.2.2 Wound Rotor Voltage Differential Equations ...... 264

6 Unique System of Machine Equations

6.1 Influence of Series-Added Impedance 6.1.1 Inductors in Series with a Machine 6.1.2 Capacitors in Series with a Machine ......... .

6.2 k-Winding, n-Limb Transformer .............. . 6.3 Induction Machine - a Simple Model ........... .

6.3.1 Voltage-Fed Machine ............. . . 6.3.2 Current-Fed Machine .............. .

6.4 Squirrel Cage Induction Machine - a Complex Model 6.4.1 Voltage-Fed Machine .................. . 6.4.2 Current-Fed Machine .................. .

6.5 Wound Rotor Synchronous Machine 6.6 Permanent Magnet Synchronous Machine 6.7 Step and Switched Reluctance Motor 6.8 Shaded Pole Induction Machine 6.9 Initial Conditions Definition

Appendix I - Matrix Operators

Appendix H - Vector and Matrix Differentiation

Appendix HI - Applications

265

270 271 273 276 287 294 296 299 305 306 306 313 315 322 325

329

334

345

ill.1. Squirrel Cage Induction Machine . . . . . . . . . . . . . ., 346 ill.1.1 Starting Characteristics of the Basic Machine ...... 348 ill.l.2 Basic Machine with Additional Inertia on the Shaft 360 ill.l.3 Basic Machine with a Decreased Number of Rotor

Slots and an Additional Inertia on the Shaft ...... 361 ill.l.4 Basic Machine with Linear B-H Curve and Additional

Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 ill.l.5 Wye-Connected Basic Machine with Additional Inertia 371 ill.l.6 Reversing of the Basic Machine with Additional Inertia 373 ill. I. 7 Basic Machine with Blocked Rotor . . . . . . . . . 375 TIL 1.8 Basic Machine with Full Pitch Stator Winding and

Additional Inertia . . . . . . . . . . . . . . . . . . . . . . 379 ill.1.9 Wye-Conn~cted Basic Machine with 4/6 Winding Pitch 382

xiii

III.1.10 Basic Machine with Unskewed Rotor and Additional Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

III.1.11 Unsymmetric Connection of the Basic Machine 390 III. 1. 12 Basic Machine in Steinmetz Connection . . . . . . 394 III.1.l3 Basic Machine with Additional Inertia Fed from a

Six-Step Voltage Source Inverter . . . . . . . . . 401 III.1.14 Wye-Connected Basic Machine with Additional

Inertia Fed from a Six-Step Voltage Source Inverter 407 III. 1. 15 Basic Machine with Additional Inertia Fed from a

Current Source . . . . . . . . . . . . . . . . . . . . . 408 III.1.16 Capacitor Braking of the Basic Machine . . . . . . 413 III.1.17 D.C. Current Braking of the Basic Machine 422

III. 2. Permanent Magnet Synchronous Machine. . . . . . . . . 425 III.2.1 Starting of the Basic Machine without Permanent

Magnets . . . . . . . . . .. .......... 426 III.2.2 Starting of the Basic Machine . . . . . . . . . . . . . 429 III.2.3 Loading of the Basic Machine . . . . . . . . . . . . . . 434 III.2.4 Heavy Loading - Loss of Synchronism . . . . . . .. 438

Bibliography 441

Index 444