Analysis of electric machinery and drive systems

ANALYSIS OF

ELECTRIC MACHINERY

AND DRIVE SYSTEMS

IEEE Press

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appears at the end of this book.

IEEE PRESS

ANALYSIS OF

ELECTRIC MACHINERY

AND DRIVE SYSTEMS

THIRD EDITION

Paul Krause

Oleg Wasynczuk

Scott Sudhoff

Steven Pekarek

Copyright © 2013 by Institute of Electrical and Electronics Engineers, Inc. All rights reserved.

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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

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

CONTENTS

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 Coeffi cient 626

Appendix C Leakage Inductance 629

References 635

Index 636

Those familiar with previous editions of this book will fi nd that this edition has been

expanded and modifi ed 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 signifi -

cant 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 identifi cation 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, cur￾rents, and fl ux 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 measure￾ments. 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. Unfor￾tunately, this approach is somewhat frustrating to the electric drives engineer. The

chapter on synchronous machines has been modifi ed in an attempt to accommodate

both drive and power system engineers. In particular, the analysis is fi rst carried out

with positive currents into the machine and then with the current direction reversed.

PREFACE

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 conven￾tion. 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 fi rst edition

but not in the second, has been simplifi ed 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 diffi cult if not impossible;

instead, the ordering is by age only.

P aul K rause

O leg W asynczuk

S cott S udhoff

S teven P ekarek

West Lafayette, Indiana

May 2013