ANALYSIS OF ELECTRIC MACHINERY AND DRIVE SYSTEMS

ANALYSIS OF ELECTRIC MACHINERY AND DRIVE SYSTEMS

IEEE Press 445 Hoes Lane Piscataway, NJ 08854 IEEE Press Editorial Board 2013 John Anderson, Editor in Chief Linda Shafer Saeid Nahavandi George Zobrist George W. Arnold David Jacobson Tariq Samad Ekram Hossain Mary Lanzerotti Dmitry Goldgof Om P. Malik Kenneth Moore, Director of IEEE Book and Information Services (BIS) A complete list of titles in the IEEE Press Series on Power Engineering appears at the end of this book.

ANALYSIS OF ELECTRIC MACHINERY AND DRIVE SYSTEMS THIRD EDITION Paul Krause Oleg Wasynczuk Scott Sudhoff Steven Pekarek IEEE PRESS

Copyright 2013 by Institute of Electrical and Electronics Engineers, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Krause, Paul C. Analysis of electric machinery and drive systems / Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek. Third edition. pages cm Institute of Electrical and Electronics Engineers. Includes bibliographical references and index. ISBN 978-1-118-02429-4 (cloth) 1. Electric machinery. 2. Electric driving. I. Wasynczuk, Oleg. II. Sudhoff, Scott D. III. Pekarek, Steven. IV. Institute of Electrical and Electronics Engineers. V. Title. TK2181.K72 2013 621.31'042 dc23 2012050394 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1

CONTENTS Preface xiii 1 THEORY OF ELECTROMECHANICAL ENERGY CONVERSION 1 1.1. Introduction 1 1.2. Magnetically Coupled Circuits 1 1.3. Electromechanical Energy Conversion 12 1.4. Elementary ac Machines 35 Reference 44 Problems 44 2 DISTRIBUTED WINDINGS IN AC MACHINERY 53 2.1. Introduction 53 2.2. Describing Distributed Windings 54 2.3. Winding Functions 64 2.4. Air-Gap Magnetomotive Force 67 2.5. Rotating MMF 71 2.6. Flux Linkage and Inductance 73 2.7. Resistance 76 2.8. Voltage and Flux Linkage Equations for Distributed Winding Machines 77 Reference 83 Problems 84 3 REFERENCE-FRAME THEORY 86 3.1. Introduction 86 3.2. Background 87 3.3. Equations of Transformation: Change of Variables 88 3.4. Stationary Circuit Variables Transformed to the Arbitrary Reference Frame 90 3.5. Commonly Used Reference Frames 97 v

vi CONTENTS 3.6. Transformation of a Balanced Set 98 3.7. Balanced Steady-State Phasor Relationships 99 3.8. Balanced Steady-State Voltage Equations 102 3.9. Variables Observed from Several Frames of Reference 105 3.10. Transformation Between Reference Frames 110 3.11. Specialty Transformations 111 3.12. Space-Phasor Notation 113 References 115 Problems 115 4 PERMANENT-MAGNET AC MACHINES 121 4.1. Introduction 121 4.2. Voltage and Torque Equations in Machine Variables 122 4.3. Voltage and Torque Equations in Rotor Reference-Frame Variables 125 4.4. Analysis of Steady-State Operation 127 4.5. Brushless dc Motor 129 4.6. Phase Shifting of Applied Voltages of a Permanent-Magnet ac Machine 134 4.7. Control of Stator Currents 138 References 140 Problems 140 5 SYNCHRONOUS MACHINES 142 5.1. Introduction 142 5.2. Voltage Equations in Machine Variables 143 5.3. Torque Equation in Machine Variables 149 5.4. Stator Voltage Equations in Arbitrary Reference-Frame Variables 149 5.5. Voltage Equations in Rotor Reference-Frame Variables 151 5.6. Torque Equations in Substitute Variables 157 5.7. Rotor Angle and Angle Between Rotors 158 5.8. Per Unit System 159 5.9. Analysis of Steady-State Operation 160 5.10. Stator Currents Positive Out of Machine: Synchronous Generator Operation 171 5.11. Computer Simulation 201 References 210 Problems 210

CONTENTS vii 6 SYMMETRICAL INDUCTION MACHINES 215 6.1. Introduction 215 6.2. Voltage Equations in Machine Variables 216 6.3. Torque Equation in Machine Variables 220 6.4. Equations of Transformation for Rotor Circuits 222 6.5. Voltage Equations in Arbitrary Reference-Frame Variables 224 6.6. Torque Equation in Arbitrary Reference-Frame Variables 229 6.7. Commonly Used Reference Frames 232 6.8. Per Unit System 233 6.9. Analysis of Steady-State Operation 235 6.10. Free Acceleration Characteristics 244 6.11. Free Acceleration Characteristics Viewed from Various Reference Frames 251 6.12. Dynamic Performance During Sudden Changes in Load Torque 257 6.13. Dynamic Performance During a Three-Phase Fault at the Machine Terminals 260 6.14. Computer Simulation in the Arbitrary Reference Frame 261 References 266 Problems 267 7 MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES AND TIME CONSTANTS 271 7.1. Introduction 271 7.2. Park s Equations in Operational Form 272 7.3. Operational Impedances and G( p) for a Synchronous Machine with Four Rotor Windings 273 7.4. Standard Synchronous Machine Reactances 276 7.5. Standard Synchronous Machine Time Constants 278 7.6. Derived Synchronous Machine Time Constants 278 7.7. Parameters from Short-Circuit Characteristics 283 7.8. Parameters from Frequency-Response Characteristics 290 References 295 Problems 297 8 ALTERNATIVE FORMS OF MACHINE EQUATIONS 299 8.1. Introduction 299 8.2. Machine Equations to Be Linearized 300 8.3. Linearization of Machine Equations 302

viii CONTENTS 8.4. Small-Displacement Stability: Eigenvalues 308 8.5. Eigenvalues of Typical Induction Machines 309 8.6. Eigenvalues of Typical Synchronous Machines 312 8.7. Neglecting Electric Transients of Stator Voltage Equations 313 8.8. Induction Machine Performance Predicted with Stator Electric Transients Neglected 318 8.9. Synchronous Machine Performance Predicted with Stator Electric Transients Neglected 322 8.10. Detailed Voltage Behind Reactance Model 325 8.11. Reduced Order Voltage Behind Reactance Model 332 References 333 Problems 335 9 UNBALANCED OPERATION AND SINGLE-PHASE INDUCTION MACHINES 336 9.1. Introduction 336 9.2. Symmetrical Component Theory 337 9.3. Symmetrical Component Analysis of Induction Machines 338 9.4. Unbalanced Stator Conditions of Induction Machines: Reference-Frame Analysis 339 9.5. Typical Unbalanced Stator Conditions of Induction Machines 346 9.6. Unbalanced Rotor Conditions of Induction Machines 351 9.7. Unbalanced Rotor Resistors 354 9.8. Single-Phase Induction Machines 358 9.9. Asynchronous and Unbalanced Operation of Synchronous Machines 368 References 375 Problems 375 10 DC MACHINES AND DRIVES 377 10.1. Introduction 377 10.2. Elementary dc Machine 377 10.3. Voltage and Torque Equations 384 10.4. Basic Types of dc Machines 386 10.5. Time-Domain Block Diagrams and State Equations 394 10.6. Solid-State Converters for dc Drive Systems 398 10.7. One-Quadrant dc/dc Converter Drive 400 10.8. Two-Quadrant dc/dc Converter Drive 418 10.9. Four-Quadrant dc/dc Converter Drive 421

CONTENTS ix 10.10. Machine Control with Voltage-Controlled dc/dc Converter 423 10.11. Machine Control with Current-Controlled dc/dc Converter 426 References 431 Problems 431 11 SEMI-CONTROLLED BRIDGE CONVERTERS 434 11.1. Introduction 434 11.2. Single-Phase Load Commutated Converter 434 11.3. Three-Phase Load Commutated Converter 445 11.4. Conclusions and Extensions 456 References 458 Problems 458 12 FULLY CONTROLLED THREE-PHASE BRIDGE CONVERTERS 460 12.1. Introduction 460 12.2. The Three-Phase Bridge Converter 460 12.3. Six-Step Operation 466 12.4. Six-Step Modulation 474 12.5. Sine-Triangle Modulation 477 12.6. Extended Sine-Triangle Modulation 483 12.7. Space-Vector Modulation 485 12.8. Hysteresis Modulation 489 12.9. Delta Modulation 492 12.10. Open-Loop Voltage and Current Regulation 493 12.11. Closed-Loop Voltage and Current Regulation 495 References 499 Problems 500 13 INDUCTION MOTOR DRIVES 503 13.1. Introduction 503 13.2. Volts-per-Hertz Control 504 13.3. Constant Slip Current Control 510 13.4. Field-Oriented Control 517 13.5. Direct Field-Oriented Control 521 13.6. Robust Direct Field-Oriented Control 523 13.7. Indirect Rotor Field-Oriented Control 528 13.8. Direct Torque Control 532 13.9. Slip Energy Recovery Drives 535

x CONTENTS 13.10. Conclusions 538 References 538 Problems 539 14 PERMANENT-MAGNET AC MOTOR DRIVES 541 14.1. Introduction 541 14.2. Voltage-Source Inverter Drives 542 14.3. Equivalence of Voltage-Source Inverters to an Idealized Source 543 14.4. Average-Value Analysis of Voltage-Source Inverter Drives 552 14.5. Steady-State Performance of Voltage-Source Inverter Drives 555 14.6. Transient and Dynamic Performance of Voltage-Source Inverter Drives 557 14.7. Case Study: Voltage-Source Inverter-Based Speed Control 562 14.8. Current-Regulated Inverter Drives 567 14.9. Voltage Limitations of Current-Regulated Inverter Drives 571 14.10. Current Command Synthesis 572 14.11. Average-Value Modeling of Current-Regulated Inverter Drives 576 14.12. Case Study: Current-Regulated Inverter-Based Speed Controller 578 References 581 Problems 581 15 INTRODUCTION TO THE DESIGN OF ELECTRIC MACHINERY 583 15.1. Introduction 583 15.2. Machine Geometry 585 15.3. Stator Windings 590 15.4. Material Parameters 593 15.5. Stator Currents and Control Philosophy 596 15.6. Radial Field Analysis 597 15.7. Lumped Parameters 602 15.8. Ferromagnetic Field Analysis 603 15.9. Formulation of Design Problem 609 15.10. Case Study 614 15.11. Extensions 618 Acknowledgments 619 References 620 Problems 621

CONTENTS xi Appendix A Trigonometric Relations, Constants and Conversion Factors, and Abbreviations 623 A.1. Basic Trigonometric Relations 623 A.2. Three-Phase Trigonometric Relations 624 A.3. Constants and Conversion Factors 624 A.4. Abbreviations 625 Appendix B Carter s Coefficient 626 Appendix C Leakage Inductance 629 References 635 Index 636

PREFACE Those familiar with previous editions of this book will find that this edition has been expanded and modified to help meet the needs of the electric machinery, electric drives, and electric power industries. Like previous editions, reference-frame theory is at the core of this book. However, new material has been introduced that sets the stage for machine design. In particular, in Chapter 2, the winding function approach is used to establish the rotating air-gap magnetomotive force and machine inductances, including end-turn winding effects. In addition, an introduction to machine design is set forth in Chapter 15. These two new chapters, combined with reference-frame theory-based machine analysis, add a significant dimension not found in other texts. Another major change is set forth in Chapter 8, wherein the standard linear and reduced-order machine equations are derived and a section has been added on the method of analysis referred to as voltage behind reactance. This new formulation of the machine equations is especially useful in the analysis and modeling of electric machines that are coupled to power electronic circuits. Consequently, this technique has become a useful tool in the electric power and electric drives industries. There are other, less major, changes and additions in this edition that warrant mentioning. In Chapter 1, the electromagnetic force (torque) equations are derived without the need of numerous, involved summations that have plagued the previous approach. This straightforward approach is made possible by the identification of a second energy balance relationship. Also, the chapter on reference-frame theory has been augmented with transformations that apply when the three-phase currents, currents, and flux linkages sum to zero. Although this is not the case if a third harmonic is present, it is quite common, and the transformations are helpful in cases where the neutral is not accessible, and only the line-to-line voltages are available. Calculation of operational impedances is given in Chapter 7. Added to this material is a generalized approach of determining machine parameters from machine measurements. An interesting combination of Park s approach to the derivation of the torque relationship and reference-frame theory is set forth in Chapter 6. In the previous editions the synchronous machine was analyzed assuming positive current out of the machine, convenient for the power system engineer. Unfortunately, this approach is somewhat frustrating to the electric drives engineer. The chapter on synchronous machines has been modified in an attempt to accommodate both drive and power system engineers. In particular, the analysis is first carried out with positive currents into the machine and then with the current direction reversed. xiii

xiv PREFACE However, whenever power system operation or system fault studies are considered, positive current is assumed out of the machine consistent with power system convention. The remaining chapters, including the chapters on electric drives, as well as the chapters on converters, have been updated to include recent advances in analysis and converter control. Also, the analysis of unbalanced operation covered in the first edition but not in the second, has been simplified and is presented in Chapter 9. We have spent a major part of our professional careers dealing with electric machines and drives. We are not only coauthors but colleagues and good friends. With the close working relationship that existed during the preparation of this manuscript, an ordering of the coauthors based on contribution would be difficult if not impossible; instead, the ordering is by age only. P aul Krause Oleg Wasynczuk Scott Sudhoff Steven Pekarek West Lafayette, Indiana May 2013