ANALYSIS OF ELECTRIC MACHINERY AND DRIVE SYSTEMS

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1 ANALYSIS OF ELECTRIC MACHINERY AND DRIVE SYSTEMS

2 IEEE Press 445 Hoes Lane Piscataway, NJ 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.

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

4 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) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at 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) , outside the United States at (317) or fax (317) 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 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 (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.K '042 dc Printed in the United States of America

5 CONTENTS Preface xiii 1 THEORY OF ELECTROMECHANICAL ENERGY CONVERSION Introduction Magnetically Coupled Circuits Electromechanical Energy Conversion Elementary ac Machines 35 Reference 44 Problems 44 2 DISTRIBUTED WINDINGS IN AC MACHINERY Introduction Describing Distributed Windings Winding Functions Air-Gap Magnetomotive Force Rotating MMF Flux Linkage and Inductance Resistance Voltage and Flux Linkage Equations for Distributed Winding Machines 77 Reference 83 Problems 84 3 REFERENCE-FRAME THEORY Introduction Background Equations of Transformation: Change of Variables Stationary Circuit Variables Transformed to the Arbitrary Reference Frame Commonly Used Reference Frames 97 v

6 vi CONTENTS 3.6. Transformation of a Balanced Set Balanced Steady-State Phasor Relationships Balanced Steady-State Voltage Equations Variables Observed from Several Frames of Reference Transformation Between Reference Frames Specialty Transformations Space-Phasor Notation 113 References 115 Problems PERMANENT-MAGNET AC MACHINES Introduction Voltage and Torque Equations in Machine Variables Voltage and Torque Equations in Rotor Reference-Frame Variables Analysis of Steady-State Operation Brushless dc Motor Phase Shifting of Applied Voltages of a Permanent-Magnet ac Machine Control of Stator Currents 138 References 140 Problems SYNCHRONOUS MACHINES Introduction Voltage Equations in Machine Variables Torque Equation in Machine Variables Stator Voltage Equations in Arbitrary Reference-Frame Variables Voltage Equations in Rotor Reference-Frame Variables Torque Equations in Substitute Variables Rotor Angle and Angle Between Rotors Per Unit System Analysis of Steady-State Operation Stator Currents Positive Out of Machine: Synchronous Generator Operation Computer Simulation 201 References 210 Problems 210

7 CONTENTS vii 6 SYMMETRICAL INDUCTION MACHINES Introduction Voltage Equations in Machine Variables Torque Equation in Machine Variables Equations of Transformation for Rotor Circuits Voltage Equations in Arbitrary Reference-Frame Variables Torque Equation in Arbitrary Reference-Frame Variables Commonly Used Reference Frames Per Unit System Analysis of Steady-State Operation Free Acceleration Characteristics Free Acceleration Characteristics Viewed from Various Reference Frames Dynamic Performance During Sudden Changes in Load Torque Dynamic Performance During a Three-Phase Fault at the Machine Terminals Computer Simulation in the Arbitrary Reference Frame 261 References 266 Problems MACHINE EQUATIONS IN OPERATIONAL IMPEDANCES AND TIME CONSTANTS Introduction Park s Equations in Operational Form Operational Impedances and G( p) for a Synchronous Machine with Four Rotor Windings Standard Synchronous Machine Reactances Standard Synchronous Machine Time Constants Derived Synchronous Machine Time Constants Parameters from Short-Circuit Characteristics Parameters from Frequency-Response Characteristics 290 References 295 Problems ALTERNATIVE FORMS OF MACHINE EQUATIONS Introduction Machine Equations to Be Linearized Linearization of Machine Equations 302

8 viii CONTENTS 8.4. Small-Displacement Stability: Eigenvalues Eigenvalues of Typical Induction Machines Eigenvalues of Typical Synchronous Machines Neglecting Electric Transients of Stator Voltage Equations Induction Machine Performance Predicted with Stator Electric Transients Neglected Synchronous Machine Performance Predicted with Stator Electric Transients Neglected Detailed Voltage Behind Reactance Model Reduced Order Voltage Behind Reactance Model 332 References 333 Problems UNBALANCED OPERATION AND SINGLE-PHASE INDUCTION MACHINES Introduction Symmetrical Component Theory Symmetrical Component Analysis of Induction Machines Unbalanced Stator Conditions of Induction Machines: Reference-Frame Analysis Typical Unbalanced Stator Conditions of Induction Machines Unbalanced Rotor Conditions of Induction Machines Unbalanced Rotor Resistors Single-Phase Induction Machines Asynchronous and Unbalanced Operation of Synchronous Machines 368 References 375 Problems DC MACHINES AND DRIVES Introduction Elementary dc Machine Voltage and Torque Equations Basic Types of dc Machines Time-Domain Block Diagrams and State Equations Solid-State Converters for dc Drive Systems One-Quadrant dc/dc Converter Drive Two-Quadrant dc/dc Converter Drive Four-Quadrant dc/dc Converter Drive 421

9 CONTENTS ix Machine Control with Voltage-Controlled dc/dc Converter Machine Control with Current-Controlled dc/dc Converter 426 References 431 Problems SEMI-CONTROLLED BRIDGE CONVERTERS Introduction Single-Phase Load Commutated Converter Three-Phase Load Commutated Converter Conclusions and Extensions 456 References 458 Problems FULLY CONTROLLED THREE-PHASE BRIDGE CONVERTERS Introduction The Three-Phase Bridge Converter Six-Step Operation Six-Step Modulation Sine-Triangle Modulation Extended Sine-Triangle Modulation Space-Vector Modulation Hysteresis Modulation Delta Modulation Open-Loop Voltage and Current Regulation Closed-Loop Voltage and Current Regulation 495 References 499 Problems INDUCTION MOTOR DRIVES Introduction Volts-per-Hertz Control Constant Slip Current Control Field-Oriented Control Direct Field-Oriented Control Robust Direct Field-Oriented Control Indirect Rotor Field-Oriented Control Direct Torque Control Slip Energy Recovery Drives 535

10 x CONTENTS Conclusions 538 References 538 Problems PERMANENT-MAGNET AC MOTOR DRIVES Introduction Voltage-Source Inverter Drives Equivalence of Voltage-Source Inverters to an Idealized Source Average-Value Analysis of Voltage-Source Inverter Drives Steady-State Performance of Voltage-Source Inverter Drives Transient and Dynamic Performance of Voltage-Source Inverter Drives Case Study: Voltage-Source Inverter-Based Speed Control Current-Regulated Inverter Drives Voltage Limitations of Current-Regulated Inverter Drives Current Command Synthesis Average-Value Modeling of Current-Regulated Inverter Drives Case Study: Current-Regulated Inverter-Based Speed Controller 578 References 581 Problems INTRODUCTION TO THE DESIGN OF ELECTRIC MACHINERY Introduction Machine Geometry Stator Windings Material Parameters Stator Currents and Control Philosophy Radial Field Analysis Lumped Parameters Ferromagnetic Field Analysis Formulation of Design Problem Case Study Extensions 618 Acknowledgments 619 References 620 Problems 621

11 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

13 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

14 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

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