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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 George W. Arnold Ekram Hossain Om P. Malik Saeid Nahavandi David Jacobson Mary Lanzerotti George Zobrist Tariq Samad Dmitry Goldgof 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 speciﬁ cally disclaim any implied warranties of merchantability or ﬁ tness 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 proﬁ t 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: Analysis of electric machinery and drive systems / Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Krause, Paul C. Pekarek. – Third edition. 1. Electric machinery. Steven. TK2181.K72 2013 621.31'042–dc23 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1 pages cm “Institute of Electrical and Electronics Engineers.” Includes bibliographical references and index. ISBN 978-1-118-02429-4 (cloth) IV. Institute of Electrical and Electronics Engineers. 2. Electric driving. I. Wasynczuk, Oleg. II. Sudhoff, Scott D. III. Pekarek, V. Title. 2012050394

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

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

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

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

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