Pulse Width Modulation For Power Converters
IEEE Press 445 Hoes Lane Piscataway, NJ 08854
IEEE Press Editorial Board Stamatios V. Kartalopoulos, Editor in Chief
M. Akay
M. E. El-Hawary
M. Padgett
J. B. Anderson R. J. Baker J. E. Brewer
R. J. Herrick
w. D. Reeve
D.Kirk R. Leonardi M. S. Newman
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 2002 Hardcover 624pp 0-471-14326-X
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
A JOHN WILEY & SONS, INC., PUBLICATION
Copyright © 2003 by the Institute of Electrical and Electronics Engineers, Inc. All rights reserved. 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) 750-4744, 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
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Library ofCongress Cataloging-in-Publication Data is available. Printed in the United States of America. ISBN 0-471-20814-0 10 9 8 7 6 5 4 3
Contents Preface
xiii
Acknowledgments
xiv
Nomenclature
xv
Chapter 1 Introduction to Power Electronic Converters 1.1
Basic 1.1.1 1.1.2 1.1.3
1.2
Voltage Source/Stiff Inverters 7 1.2.1 Two-Phase Inverter Structure 7 1.2.2 Three-Phase Inverter Structure 8 1.2.3 Voltage and Current Waveforms in Square-Wave Mode ..9
1.3
Switching Function Representation of Three-Phase Converters 14
1.4
Output Voltage Control 1.4.1 Volts/Hertz Criterion
17 17
1.4.2 Phase ShiftModulation for Single-Phase Inverter
17
1.4.3
Converter Topologies Switch Constraints Bidirectional Chopper Single-Phase Full-Bridge (H-Bridge) Inverter
1
Voltage Control with a Double Bridge
2 2 4 5
19
1.5
Current Source/Stiff Inverters
21
1.6
Concept of a Space Vector 24 1.6.1 d-q-O Components for Three-Phase Sine Wave Source/ Load 27 1.6.2 d-q-O Components 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 1.8.1 Diode-Clamped Multilevel Inverter 1.8.2 Capacitor-Clamped Multilevel Inverter 1.8.3 Cascaded Voltage Source Multilevel Inverter
42 42 49 51 v
vi
Contents
1.8.4 1.9
Hybrid Voltage Source Inverter
Summary
54 55
Chapter 2 Harmonic Distortion ...............................................................•.57 2.1
Harmonic Voltage Distortion Factor
57
2.2
Harmonic Current Distortion Factor
61
2.3
Harmonic Distortion Factors for Three-Phase Inverters
64
2.4
Choice of Performance Indicator
67
2.5
WTHD of Three-Level Inverter
70
2.6
The Induction Motor Load 2.6. I Rectangular Squirrel Cage Bars 2.6.2 Nonrectangular Rotor Bars 2.6.3 Per-Phase Equivalent Circuit
73 73 78 79
2.7
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
2.8
Example Calculation of Harmonic Losses
90
2.9
WTHD Normalization for PWM Inverter Supply
91
2.10
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 WidthModulated Waveform 99
3.4
Naturally Sampled Pulse Width Modulation 3.4.1 Sine-Sawtooth Modulation 3.4.2 Sine-Triangle Modulation
105 l 05 114
3.5
PWM Analysis by Duty Cycle Variation 3.5.1 Sine-Sawtooth Modulation 3.5.2 Sine-Triangle Modulation
120 120 123
Contents
Vl1
3.6
Regular Sampled Pulse Width Modulation 3.6.1 Sawtooth Carrier Regular Sampled PWM 3.6.2 Symmetrical Regular Sampled PWM 3.6.3 Asymmetrical Regular Sampled PWM
125 130 134 139
3.7
"Direct" Modulation
146
3.8
Integer versus Non-Integer Frequency Ratios
148
3.9
Review of PWM Variations
150
3.10
Summary
152
Chapter 4 Modulation of Single-Phase Voltage Source Inverters
155
4.1
Topology of a Single-Phase Inverter
156
4.2
Three-Level Modulation of a Single-Phase Inverter
157
4.3
Analytic Calculation of Harmonic Losses
169
4.4
Sideband Modulation
177
4.5
Switched Pulse Position 4.5.1 Continuous Modulation 4.5.2 Discontinuous Modulation
183 184 186
4.6
Switched Pulse Sequence ~ 200 4.6.1 Discontinuous PWM - Single-Phase Leg Switched 200 4.6.2 Two-Level Single-Phase PWM 207
4.7
Summary
Chapter 5 Modulation of Three-Phase Voltage Source Inverters
211
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 5.3.1 Optimum Injection Level. 5.3.2 Analytical Solution for Third-Harmonic Injection
226 226 230
5.4
Analytic Calculation of Harmonic Losses
241
5.5
Discontinuous Modulation Strategies
250
5.6
Triplen Carrier Ratios and Subharmonics 5.6.1 Triplen Carrier Ratios 5.6.2 Subharmonics
251 251 253
viii
Contents
5.7
Summary
Chapter 6 Zero Space Vector Placement Modulation Strategies
257
259
6.1
Space Vector Modulation 6.1.1 Principles of Space Vector Modulation 6.1.2 SYM Compared to Regular Sampled PWM
259 259 265
6.2
Phase Leg References for Space Vector Modulation
267
6.3
Naturally Sampled SVM
270
6.4
Analytical Solution for SVM
272
6.5
Harmonic Losses for SVM
291
6.6
Placement of the Zero Space Vector
294
6.7
Discontinuous Modulation 6.7.1 1200 Discontinuous Modulation 6.7.2 600 and 300 Discontinuous Modulation
299 299 302
6.8
Phase Leg References for Discontinuous PWM
307
6.9
Analytical Solutions for Discontinuous PWM
311
6.10
Comparison of Harmonic Performance
322
6.11
Harmonic Losses for Discontinuous PWM
326
6.12
Single-Edge SYM
330
6.13
Switched Pulse Sequence
331
6.14
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 Inverter 351
ix
Contents
8.3
Regular Sampled Overmodulation of One Phase Leg of an Inverter
356
8.4
Naturally Sampled Overmodulation of Single- and Three-Phase Inverters 360
8.5
PWM 8.5.! 8.5.2 8.5.3 8.5.4
8.6
Space Vector Approach to Overmodulation
376
8.7
Summary
382
Controller Gain during Overmodulation Gain with Sinusoidal Reference Gain with Space Vector Reference Gain with 60° Discontinuous Reference Compensated Modulation
Chapter 9 Programmed Modulation Strategies
364 364 367 37! 373
383
9.1
Optimized Space Vector Modulation
384
9.2
Harmonic Elimination PWM
396
9.3
Performance Index for Optimality
411
9.4
Optimum PWM
416
9.5
Minimum-Loss PWM
421
9.6
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 11.1
PWM of Cascaded Single-Phase H-Bridges
453 453
Contents
x
11.2
Overmodulation of Cascaded H-Bridges
465
11.3
PWM Alternatives for Diode-Clamped Multilevel Inverters
467
11.4
Three-Level Naturally Sampled PO PWM 11.4.1 Contour Plot for Three-Level PD PWM 11.4.2 Double Fourier Series Harmonic Coefficients 11.4.3 Evaluation of the Harmonic Coefficients 11.4.4 Spectral Performance of Three-Level PD PWM
469 469 473 475 479
11.5
Three-Level Naturally Sampled APOD or POD PWM
481
11.6
Overmodulation of Three-Level Inverters
484
11.7
Five-Level PWM for Diode-Clamped Inverters 11.7.1 Five-level Naturally Sampled PO PWM 11.7.2 Five-Level Naturally Sampled APOD PWM 11.7.3 Five-Level POD PWM
489 489 492 497
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
11.12 Third-Harmonic Injection for Multilevel Inverters
519
11.13 Operation of a Multilevel Inverter with a Variable Modulation Index 526 11.14 Summary
Chapter 12 Space Vector PWM for Multilevel Converters
528
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
12.8
Summary
550
xi
Contents
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 13.2.1 VSI Power Conversion Stage 13.2.2 Gate Driver Interface 13.2.3 Controller Power Supply 13.2.4 I/O Conditioning Circuitry 13.2.5 PWM Controller
557 563 565 567 568 569
13.3
Hardware Implementation of the PWM Process 13.3.1 Analog versus Digital Implementation 13.3.2 Digital Timer Logic Structures
572 572 574
13.4
PWM Software Implementation 13.4.1 Background Software 13.4.2 Calculation of the PWM Timing Intervals
579 580 581
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 Controlled Waveform 623 Appendix 2 Jacobi-Anger and Bessel Function Relationships
629
A2.1
Jacobi-Anger Expansions
629
A2.2
Bessel Function Integral Relationships
631
Appendix 3 Three-Phase and Half-Cycle Symmetry Relationships
635
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 Representation 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
Index
715
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 spectrum 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 harmonic distortion (WTHD), and harmonic 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 identification 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 implementation, 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 tradeoffs 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. xiii
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.
xiv
N ome.nclature Generic Variable Usage Conventions Variable Format
Meaning
F
CAPITALS: peak AC or average DC value
I
LOWER CASE: instantaneous value
BRACKETED: low-frequency average value
1 It
OVERBAR: space vector (complex variable)
I
BOLD LOWER CASE: column vector
F
BOLD CAPITAL: matrix
IT
DAGGER: conjugate of space vector
TRANSPOSED VECTOR: row vector
Specific Variable Usage Definitions Page First Used
Meaning
Variable
a, b, c
Phase leg identifiers for three phase inverter
-
'21t/3
9
el
a
Complex vector
y
Third-harmonic component magnitude M3/M
227
Coefficients of Fourier expansion
102
A mn , «: -
34
--
C mn
Complex Fourier coefficient C
Ok' k=I,2 ..
Diode section of inverter switch
eaz
la'!b,le las,lbs,les
mn
= A
mn
+jB
mn
Motor EMF w.r.t. DC bus midpoint
102 7 169
Generic variables in a-b-c reference frame
26
Generic variables in a-b-c reference frame referenced to load neutral (star) point
29
r,
Frequency of carrier waveform
112
10
Frequency of fundamental component
112 xv
xvi
Nomenclature
Variable
is I qdO s s fq,fd'fo s s fqs,fds,fos
Meaning
Page First Used
Stationary . qs - J~s ds space vector fS
34
Vector [fqs,fds,fOsY
36
Generic variables in d-q-Q stationary reference frame
26
Stationaryreference frame (d-q-{) ) variables referenced to load neutral (star) point
29
Unit cell variable
100
HDF
Harmonic distortion factor
248
i a, i b , i e
Three phase Iine currents
13
Ide
DC link current
13
Ih
RMS value of the overall harmonic currents
172
ih , k
Instantaneous harmonic current over internal k
385
tli a
Ripple component of current in phase a
170
f{x,y)
~
j In(x) L
Bessel function of order n and argument x
110
Number of multilevel inverter voltage levels
434
L1
Thevenin equivalent stator leakage inductanceof inductionmotor
La
Effectivemotor inductance of one phase
I
mk, k=I,2 .. ora,b,c m.n
34
Inverter switching functions Harmonic index variables
81 170 14 102
M
Modulation index (modulation depth)
92
M3
Modulationindex for third harmonic
227
n
Negative inverter DC rail
n
Harmoniccomponent number
p
Positive inverter DC rail
p
p = dldt, time derivative operator
9 18
9 16
Nomenclature
xvii
Meaning
Variable
Page First Used
p
pth carrier interval
131
p
Pulse ratio
250
p
Pulse number
384
Harmonic copper loss
173
Ph.cu q
Charge
q
m + n(roo/ro c)
R
Rotating transformation matrix
36
rt
Thevenin Equivalent stator resistance of induction motor
81
Re
Equivalent load resistance
,
RMS
-
SVx' x = I, ... ,7
SCx,x = 1, ... ,7
26 137
172
Root mean square
10
Voltage space vector corresponding to three-phase inverter states
31
Current space vector corresponding to three-phase inverter states
338
Sk,k=I,2 ..
Inverter switch
31
Tc
Carrier interval
99
T k ' k=I,2..
T THD
Transistor section of inverter switch
7
Transformation matrix
37
Total harmonic distortion
58
T
Period of fundamental waveform
100
T.I
Switching time of inverter switch "i"
218
0
~T
Carrier period -
u
per unit EMF -
U
Unbalance factor
vas' Vbs' Vcs
life
ea!Vdc
Phase voltages with respect to load neutral
158 170 597 11
xviii
Nomenclature
Variable
Page First Used
Meaning
Vab' Vbc' Vca
Line-to-line (I-I) voltages for a three phase inverter
11
Vaz' Vbz' Vcz s s s vqs' vds: vOs
Phase voltages with respect to DC link midpoint
14
Stationary reference frame (d-q-O) voltages
28
Voltage between load neutral and negative DC bus
23
Peak magnitude of fundamental voltage component
13
DC link voltage
7
One-half the DC link voltage
5
Space vector magnitude or phase voltage amplitude
35
Amplitude of positive and negative phase voltages
595
Target output space vector
260
Peak input I-I voltage
226
RMS voltage
57
WTHD
Weighted total harmonic distortion
63
WTHD2
Weighted THO for rotor bar losses
85
WTHOI
Weighted TH 0 for stator losses
89
WTHOO
Weighted THO normalized to base frequency
92
Pulse width x(t)
146
Time variable corresponding to modulation angular frequency 0) ct = 21tfct Rising and falling switching instants for phase leg
y(l)
Time variable corresponding to fundamental angular frequency 0) ot = 21tfot
99
128
99
(0
y' z Z(P)
Variable for regular sampling: y - .-£(x - 21tp) (0
DC bus midpoint (virtual) Load impedance
131
C
9 16
Nomenclature
XIX
Variable
Meaning
Page First Used
a
Phase shift delay
17
a
Skin depth
76
al
Amplitude of modulating function
178
Switching angles for harmonic elimination
397
Advance compensation for PWM sampling delay
581
aI' aI' ... , a 2N badvance
ec
Phase offset angle of carrier waveform
99
eo
Phase offset angle of fundamental component
99
eo(k)
A,
cP mp '
~mn
'l'
Phase offset angle of fundamental component at sampling time k Flux linkage
17
Phase angle of positive and negative sequence phase voltages respectively
595
Overmodulation angle
353
(oc
Angular frequency of carrier waveform
(00
Angular frequency of fundamental component
roo/roc
581
Fundamental to carrier frequency ratio
99 7 106
