0-Pulse Width Modulation for Power Converters

(1)

Pulse Width Modulation

For Power Converters

(2)

IEEE Press

445 Hoes Lane Piscataway, NJ 08854 IEEE Press Editorial Board Stamatios V. Kartalopoulos,Editor in Chief M.Akay J. B. Anderson R. J. Baker J. E. Brewer M.E. El-Hawary R.J. Herrick D.Kirk R. Leonardi M. S. Newman M.Padgett

w.

D.Reeve S. Tewksbury G. Zobrist

Kenneth Moore,Director ofIEEE Press

Catherine Faduska,Senior Acquisitions Editor

John Griffin,Acquisitions Editor

Anthony VenGraitis,Project Editor

Books of Related Interest from the IEEE Press

Electric Power Systems: Analysis and Control

Fabio Saccomanno

2003 Hardcover 728pp 0-471-23439-7

Power System Protection

P. M. Anderson

1999 Hardcover 1344pp 0-7803-3472-2

Understanding Power Quality Problems: Voltage Sags and Interruptions

Math H. J. Bollen

2000 Hardcover 576pp 0-7803-4713-7

Electric Power Applications ofFuzzy Systems

Edited by M. E. El-Hawary

1998 Hardcover 384pp 0-7803-1197-3

Principles ofElectric Machines with Power Electronic Applications, Second Edition

M. E. El-Hawary

2002 Hardcover 496pp 0-471-20812-4

Analysis ofElectric Machinery and Drive Systems, Second Edition

Paul C. Krause, Oleg Wasynczuk, and Scott D. Sudhoff

(3)

Pulse Width Modulation

For Power Converters

Principles and Practice

D. Grahame Holmes

MonashUniversity Melbourne, Australia

Thomas A. Lipo

University of Wisconsin Madison, Wisconsin

IEEE Series on Power Engineering, Mohamed E. El-Hawary, Series Editor

+IEEE

IEEE PRESS

ffiWlLEY-~INTERSCIENCE

(4)

No part of this publication maybereproduced, 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) 750-4744, 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, e-mail: [email protected].

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representation 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 please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. 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, however, may not be available in electronic format.

Library ofCongress Cataloging-in-Publication Data is available.

Printed in the United States of America. ISBN 0-471-20814-0

(5)

1.4.2 Phase ShiftModulation for Single-Phase Inverter

17

1.4.3 Voltage Control with a Double Bridge 19

1.5 Current Source/Stiff Inverters 21

1.6 Concept of a Space Vector 24

1.6.1 d-q-OComponents for Three-Phase Sine Wave Source/

Load 27

1.6.2 d-q-OComponents for Voltage Source Inverter Operated

in Square-Wave Mode 30

1.6.3 Synchronously Rotating Reference Frame 35

1.7 Three-Level Inverters 38

1.8 Multilevel Inverter Topologies 42

1.8.1 Diode-Clamped Multilevel Inverter 42

1.8.2 Capacitor-Clamped Multilevel Inverter 49 1.8.3 Cascaded Voltage Source Multilevel Inverter 51

(6)

vi 1.9 Chapter 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 Contents

1.8.4 Hybrid Voltage Source Inverter 54

Summary 55

Harmonic Distortion...•.57

Harmonic Voltage Distortion Factor 57

Harmonic Current Distortion Factor 61

Harmonic Distortion Factors for Three-Phase Inverters 64

Choice of Performance Indicator 67

WTHD of Three-Level Inverter 70

The Induction Motor Load 73

2.6. I Rectangular Squirrel Cage Bars 73

2.6.2 Nonrectangular Rotor Bars 78

2.6.3 Per-Phase Equivalent Circuit 79

Harmonic Distortion Weighting Factors for Induction Motor

Load 82

2.7.1 WTHD for Frequency-Dependent Rotor Resistance 82 2.7.2 WTHD Also Including Effect of Frequency-Dependent

Rotor Leakage Inductance 84

2.7.3 WTHD for Stator Copper Losses 88

Example Calculation of Harmonic Losses 90

WTHD Normalization for PWM Inverter Supply 91

Summary 93

Chapter 3 Modulation of One Inverter Phase Leg 95

3.1 Fundamental Concepts ofPWM 96

3.2 Evaluation ofPWM Schemes 97

3.3 Double Fourier Integral Analysis of a Two-Level Pulse

Width-Modulated Waveform 99

3.4 Naturally Sampled Pulse Width Modulation 105

3.4.1 Sine-Sawtooth Modulation l 05

3.4.2 Sine-Triangle Modulation 114

3.5 PWM Analysis by Duty Cycle Variation 120

3.5.1 Sine-Sawtooth Modulation 120

(7)

Contents 3.6 3.7 3.8 3.9 3.10 Chapter 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Vl1

Regular Sampled Pulse Width Modulation 125

3.6.1 Sawtooth Carrier Regular Sampled PWM 130

3.6.2 Symmetrical Regular Sampled PWM 134

3.6.3 Asymmetrical Regular Sampled PWM 139

"Direct" Modulation 146

Integer versus Non-Integer Frequency Ratios 148

Review of PWM Variations 150

Summary 152

Modulation of Single-Phase Voltage Source Inverters 155

Topology of a Single-Phase Inverter 156

Three-Level Modulation of a Single-Phase Inverter 157

Analytic Calculation of Harmonic Losses 169

Sideband Modulation 177

Switched Pulse Position 183

4.5.1 Continuous Modulation 184

4.5.2 Discontinuous Modulation 186

Switched Pulse Sequence ~ 200

4.6.1 Discontinuous PWM - Single-Phase Leg Switched 200

4.6.2 Two-Level Single-Phase PWM 207

Summary 211

Chapter 5 Modulation of Three-Phase Voltage Source Inverters 215

5.1 Topology of a Three-Phase Inverter (VSI) 215

5.2 Three-Phase Modulation with Sinusoidal References 216

5.3 Third-Harmonic Reference Injection 226

5.3.1 Optimum Injection Level. 226

5.3.2 Analytical Solution for Third-Harmonic Injection 230

5.4 Analytic Calculation of Harmonic Losses 241

5.5 Discontinuous Modulation Strategies 250

5.6 Triplen Carrier Ratios and Subharmonics 251

5.6.1 Triplen Carrier Ratios 251

(8)

viii 5.7 Chapter 6 6.1 6.2 6.3 6.4 6.5

6.6

6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 Contents Summary 257

Zero Space Vector Placement Modulation Strategies 259

Space Vector Modulation 259

6.1.1 Principles of Space Vector Modulation 259 6.1.2 SYM Compared to Regular Sampled PWM 265 Phase Leg References for Space Vector Modulation 267

Naturally Sampled SVM 270

Analytical Solution for SVM 272

Harmonic Losses for SVM 291

Placement of the Zero Space Vector 294

Discontinuous Modulation 299

6.7.1 1200_{Discontinuous Modulation} ₂₉₉

6.7.2 600_{and 30}0_{Discontinuous Modulation} ₃₀₂

Phase Leg References for Discontinuous PWM 307 Analytical Solutions for Discontinuous PWM 311

Comparison of Harmonic Performance 322

Harmonic Losses for Discontinuous PWM 326

Single-Edge SYM 330

Switched Pulse Sequence 331

Summary 333

Chapter 7 Modulation of Current Source Inverters 337

7.1 Three-Phase Modulators as State Machines 338

7.2 Naturally Sampled CSI Space Vector Modulator 343

7.3 Experimental Confirmation 343

7.4 Summary 345

Chapter 8 Overmodulation of an Inverter ...•...349

8.1 The Overmodulation Region 350

8.2 Naturally Sampled Overmodulation of One Phase Leg of an

(9)

Contents 8.3 8.4 8.5 8.6 8.7 Chapter 9 9.1 9.2 9.3 9.4 9.5 9.6 ix

Regular Sampled Overmodulation of One Phase Leg of an

Inverter 356

Naturally Sampled Overmodulation of Single- and Three-Phase

Inverters 360

PWM Controller Gain during Overmodulation 364

8.5.! Gain with Sinusoidal Reference 364

8.5.2 Gain with Space Vector Reference 367

8.5.3 Gain with 60° Discontinuous Reference 37!

8.5.4 Compensated Modulation 373

Space Vector Approach to Overmodulation 376

Summary 382

Programmed Modulation Strategies 383

Optimized Space Vector Modulation 384

Harmonic Elimination PWM 396

Performance Index for Optimality 411

Optimum PWM 416

Minimum-Loss PWM 421

Summary 430

Chapter 10 Programmed Modulation ofMultilevel Converters 433

10.1 Multilevel Converter Alternatives 433

10.2 Block Switching Approaches to Voltage Control 436 10.3 Harmonic Elimination Applied to Multilevel Inverters 440

10.3.1 Switching Angles for Harmonic Elimination Assuming

Equal Voltage Levels 440

10.3.2 Equalization of Voltage and Current Stresses 441 10.3.3 Switching Angles for Harmonic Elimination Assuming

Unequal Voltage Levels 443

10.4 Minimum Harmonic Distortion 447

10.5 Summary 449

Chapter 11 Carrier-Based PWM of Multilevel Inverters 453

(10)

x Contents

Overmodulation of Cascaded H-Bridges 465

PWM Alternatives for Diode-Clamped Multilevel Inverters 467

Three-Level Naturally Sampled PO PWM 469

11.4.1 Contour Plot for Three-Level PD PWM 469 11.4.2 Double Fourier Series Harmonic Coefficients 473 11.4.3 Evaluation of the Harmonic Coefficients 475 11.4.4 Spectral Performance of Three-Level PD PWM 479 Three-Level Naturally Sampled APOD or POD PWM 481

Overmodulation of Three-Level Inverters 484

11.5 11.6

11.7 Five-Level PWM for Diode-Clamped Inverters 489 11.7.1 Five-level Naturally Sampled PO PWM 489 11.7.2 Five-Level Naturally Sampled APOD PWM 492

11.7.3 Five-Level POD PWM 497

11.8 PWM of Higher Level Inverters 499

11.9 Equivalent PD PWM for Cascaded Inverters 504

11.10 Hybrid Multilevel Inverter 507

11.11 Equivalent PO PWM for a Hybrid Inverter 517

11.2 11.3 11.4

11.12 Third-Harmonic Injection for Multilevel Inverters 519 11.13 Operation of a Multilevel Inverter with a Variable Modulation

Index 526

11.14 Summary 528

Chapter 12 Space Vector PWM for Multilevel Converters 531

12.1 Optimized Space Vector Sequences 531

12.2 Modulator for Selecting Switching States 534

12.3 Decomposition Method 535

12.4 Hexagonal Coordinate System 538

12.5 Optimal Space Vector Position within a Switching Period 543-12.6 Comparison of Space Vector PWM to Carrier-Based PWM 545 12.7 Discontinuous Modulation in Multilevel Inverters 548

(11)

Contents xi

Chapter 13 Implementation of a Modulation Controller 555

13.1 Overview of a Power Electronic Conversion System 556

13.2 Elements of a PWM Converter System 557

13.2.1 VSI Power Conversion Stage 563

13.2.2 Gate Driver Interface 565

13.2.3 Controller Power Supply 567

13.2.4 I/O Conditioning Circuitry 568

13.2.5 PWM Controller 569

13.3 Hardware Implementation of the PWM Process 572 13.3.1 Analog versus Digital Implementation 572

13.3.2 Digital Timer Logic Structures 574

13.4 PWM Software Implementation 579

13.4.1 Background Software 580

13.4.2 Calculation of the PWM Timing Intervals 581

13.5 Summary 584

Chapter 14 Continuing Developments in Modulation 585

14.1 Random Pulse Width Modulation 586

14.2 PWM Rectifier with Voltage Unbalance 590

14.3 Common Mode Elimination 598

14.4 Four Phase Leg Inverter Modulation 603

14.5 Effect of Minimum Pulse Width 607

14.6 PWM Dead-Time Compensation 612

14.7 Summary 619

Appendix 1 Fourier Series Representation of a Double Variable Con-trolled Waveform 623 Appendix 2 Jacobi-Anger and Bessel Function Relationships 629

A2.1 Jacobi-Anger Expansions 629

A2.2 Bessel Function Integral Relationships 631

(12)

xii Contents

Appendix 4 Overmodulation of a Single-Phase Leg 637

A4.1 Naturally Sampled Double-Edge PWM 637

A4.1.1 Evaluation of Double Fourier Integral for Overmodulated

Naturally Sampled PWM 638

A4.1.2 Harmonic Solution for Overmodulated Single-Phase Leg

under Naturally Sampled PWM 646

A4.1.3 Linear Modulation Solution Obtained from

Overmodulation Solution 647

A4.1.4 Square-Wave Solution Obtained from Overmodulation

Solution 647

A4.2 Symmetric Regular Sampled Double-Edge PWM 649 A4.2.1 Evaluation of Double Fourier Integral for Overmodulated

Symmetric Regular Sampled PWM 650

A4.2.2 Harmonic Solution for Overmodulated Single-Phase Leg under Symmetric Regular Sampled PWM 652 A4.2.3 Linear Modulation Solution Obtained from

Overmodulation Solution · 653

A4.3 Asymmetric Regular Sampled Double-Edge PWM 654 A4.3.1 Evaluation of Double Fourier Integral for Overmodulated

Asymmetric Regular Sampled PWM 655

A4.3.2 Harmonic Solution for Overmodulated Single-Phase Leg under Asymmetric Regular Sampled PWM 660 A4.3.3 Linear Modulation Solution Obtained from

Overmodulation Solution 661

Appendix 5 Numeric Integration of a Double Fourier Series Representa-tion of a Switched Waveform 663

A5.1 Formulation of the Double Fourier Integral 663 A5.2 Analytical Solution of the Inner Integral 666 A5.3 Numeric Integration of the Outer Integral 668

Bibliography 671

(13)

Preface

The work presented in this book offers a general approach to the development of fixed switching frequency pulse width-modulated (PWM) strategies to suit hard-switched converters. It is shown that modulation of, and resulting spec-trum for, the half-bridge single-phase inverter forms the basic building block from which the spectral content of modulated single- phase, three-phase, or multiphase, two-level, three-level, or multilevel, voltage link and current link converters can readily be discerned. The concept of harmonic distortion is used as the performance index to compare all commonly encountered modulation algorithms. In particular, total harmonic distortion (THO), weighted total har-monic distortion (WTHD), and harhar-monic distortion criterion specifically designed to access motor copper losses are used as performance indices.

The concept of minimum harmonic distortion, which forms the underlying basis of comparison of the work presented in this book, leads to the identifica-tion of the fundamentals ofPWM as

Active switch pulse width determination.

Active switch pulse placement within a switching period. Active switch pulse sequence across switching periods.

The benefit of this generalized approach is that once the common threads of PWM are identified, the selection of a PWM strategy for any converter topology becomes immediately obvious, and the only choices remaining are to trade-off the "best possible" performance against cost and difficulty of imple-mentation, and secondary considerations. Furthermore, the performance to be expected from a particular converter topology and modulation strategy can be quickly and easily identified without complex analysis, so that informed trade-offs can be made regarding the implementation of a PWM algorithm for any particular application. All theoretical developments have been confirmed either by simulation or experiment. Inverter implementation details have been included at the end of the text to address practical considerations.

Readers will probably note the absence of any closed loop issues in this text. While initially such material was intended to be included, it soon became apparent that the inclusion of this material would require an additional volume. A further book treating this subject is in preparation.

(14)

Acknowledgments

The authors are indebted to their graduate students, who have contributed greatly to the production of this book via their Ph.D. theses. In particular the important work of Daniel Zmood (Chapter 7), Ahmet Hava (Chapter 8) and Brendan McGrath (Chapter 11) are specifically acknowledged. In addition, numerous other graduate students have also assisted with the production of this book both through their technical contributions as well as through detailed proof-reading of this text. The second author (Lipo) also wishes to thank the David Grainger Foundation and Saint John's College of Cambridge University for funding and facilities provided respectively. Finally, we wish to thank our wonderful and loving wives, Sophie Holmes and Chris Lipo, for nuturing and supporting us over the past five years as we have written this book.

(15)

Nome.nclature

Generic Variable Usage Conventions

Variable Format Meaning

F CAPITALS: peak AC or average DC value

I

LOWER CASE: instantaneous value

<f> BRACKETED: low-frequency average value

1

OVERBAR: space vector (complex variable)

It

DAGGER: conjugate of space vector

I

BOLD LOWER CASE: column vector

F BOLD CAPITAL: matrix

IT

TRANSPOSED VECTOR: row vector

Specific Variable Usage Definitions

Variable Meaning Page First Used

a, b,c Phase leg identifiers for three phase inverter 9

- '21t/3

a Complex vector

el

34

y Third-harmonic component magnitudeM3/M 227

A_mn,

«:

Coefficients of Fourier expansion 102

-

-C_mn _{Complex Fourier coefficient Cmn = Amn +jBmn} 102

Ok' k=I,2.. Diode section of inverter switch 7

e_az Motor EMF w.r.t. DC bus midpoint 169

la'!b,le Generic variables ina-b-creference frame 26

las,lbs,les Generic variables ina-b-creference frame referenced

29 to load neutral (star) point

r,

Frequency of carrier waveform 112

1

0 Frequency of fundamental component 112

(16)

xvi Nomenclature

Variable Meaning Page First

Used

is

S ._{tationary space vector qs -}fS _J~s_ds ₃₄

I

qdO Vector[fqs,fds,fOsY 36

s s _{Generic variables in}_d-q-Q_{stationary reference frame} ₂₆ fq,fd'fo

s s _{Stationaryreference frame}_{(d-q-{) )}_variables

refer-fqs,fds,fos ₂₉

enced to load neutral (star) point

f{x,y) Unit cell variable 100

HDF Harmonicdistortion factor 248

i_a,i_b,i_e Three phase Iine currents 13

Ide DC link current 13

I_h RMS value of the overall harmonic currents 172

- _{Instantaneous harmonic current over internal k}

ih, k 385

tli_a Ripple component of current in phase a 170

j _~ 34

In(x) Bessel function of ordernand argumentx 110

L Number of multilevel inverter voltage levels 434

L I _{Thevenin equivalent stator leakage inductanceof}

81

1

inductionmotor

La Effectivemotor inductance of one phase 170

mk, k=I,2..ora,b,c Inverterswitching functions 14

m.n Harmonic index variables 102

M Modulation index (modulation depth) 92

M₃ Modulationindex for third harmonic 227

n Negative inverter DC rail 9

n Harmoniccomponent number 18

p Positive inverter DC rail 9

(17)

Nomenclature xvii

p pthcarrier interval 131

p Pulse ratio 250

p Pulse number 384

Ph.cu Harmonic copper loss 173

q Charge 26

q m+_n(roo/roc) 137

R Rotating transformation matrix 36 r_t, Thevenin Equivalent stator resistance of induction

motor 81

Re Equivalent load resistance 172

RMS Root mean square 10

-Voltage space vector corresponding to three-phase

SVx' x =I, ... ,7

31 inverter states

-Current space vector corresponding to three-phase

SCx,x =1, ... ,7 ₃₃₈

inverter states

Sk,k=I,2.. Inverter switch 31

Tc Carrier interval 99

Tk 'k=I,2.. Transistor section of inverter switch 7 T Transformation matrix 37 THD Total harmonic distortion 58

T₀ Period of fundamental waveform 100

T._I Switching time of inverter switch"i" 218

~T Carrier period - life 158

u per unit EMF - ea!Vdc 170

U Unbalance factor 597

(18)

xviii Variable Meaning Nomenclature Page First Used Vab' Vbc' Vca Vaz' Vbz' Vcz s s s vqs'vds: vOs WTHD WTHD2 WTHOI WTHOO x(t) y(l) y' z Z(P)

Line-to-line(I-I) voltages for a three phase inverter

Phase voltages with respect to DC link midpoint Stationary reference frame (d-q-O) voltages Voltage between load neutral and negative DC bus Peak magnitude of fundamental voltage component DC link voltage

One-half the DC link voltage

Space vector magnitude or phase voltage amplitude Amplitude of positive and negative phase voltages Target output space vector

Peak inputI-I voltage

RMS voltage

Weighted total harmonic distortion Weighted THO for rotor bar losses Weighted TH 0 for stator losses

Weighted THO normalized to base frequency Pulse width

Time variable corresponding to modulation angular frequency 0)ct = 21tfct

Rising and falling switching instants for phase leg Time variable corresponding to fundamental angular frequency 0)ot = 21tfot

(0

Variable for regular sampling:y - .-£(x - 21tp)

(0

C

DC bus midpoint (virtual) Load impedance 11 14 28 23 13 7 5 35 595 260 226 57 63 85 89 92 146 99 128 99 131 9 16

(19)

Nomenclature XIX

a Phase shift delay 17

a Skin depth 76

a l Amplitude of modulating function 178

aI' aI' ... ,a_2N Switching angles for harmonic elimination 397

badvance Advance compensation for PWM sampling delay 581

e

_c Phase offset angle of carrier waveform 99

eo Phase offset angle of fundamental component 99 eo(k) Phase offset angle of fundamental component at ₅₈₁

sampling timek

A, Flux linkage 17

cPmp'~mn Phase angle of positive and negative sequence phase 595

voltages respectively

'l' Overmodulation angle 353

(oc Angular frequency of carrier waveform 99

(00 Angular frequency of fundamental component 7