POWER
`POWER
`ELECTRONICS
`ELECTRONICS
`
`I
`
`I
`
`0000000
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 001
`
`

`

`ABOUT THE AUTHORS
`
`Ned Mohan is a professor in the Department of Electrical Engineering at the University
`of Minnesota, where he holds the Oscar A. Schott Chair in Power Electronics. He has
`worked on several power electronics projects sponsored by the industry and the electric
`power utilities, including the Electric Power Research Institute. He has numerous pub(cid:173)
`lications and patents in this field.
`
`Tore M. Undeland is a Professor in Power Electronics in the Faculty of Electrical
`Engineering and Computer Science at the Norwegian Institute of Technology. He is also
`Scientific Advisor to the Norwegian Electric Power Research Institute of Electricity
`Supply. He has been a visiting scientific worker in the Power Electronics Converter
`Department of ASEA in Vaasteras, Sweden, and at Siemens in Trondheim, Norway, and
`a visiting professor in the Department of Electrical Engineering at the University of
`Minnesota. He has worked on many industrial research and development projects in the
`power electronics field and has numerous publications.
`
`William P. Robbins is a professor in the Department of Electrical Engineering at the
`University of Minnesota. Prior to joining the University of Minnesota, he was a research
`engineer at the Boeing Company. He has taught numerous courses in electronics and
`semiconductor device fabrication. His research interests are in ultrasonics, pest insect
`detection via ultrasonics, and micromechanical devices, and he has numerous publications
`in this field.
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 002
`
`

`

`POWER
`ELECTRONICS
`Converters, Applications,
`and Design
`
`SECOND EDITION
`
`NED MOHAN
`Department of Electrical Engineering
`University of Minnesota
`Minneapolis, Minnesota
`
`TORE M. UNDELAND
`Faculty of Electrical Engineering and Computer Science
`Norwegian Institute of Technology
`Trondheim, Norway
`
`WILLIAM P. ROBBINS
`Department of Electrical Engineering
`University of Minnesota
`Minneapolis, Minnesota
`
`JOHN WILEY & SONS, INC.
`New York Chichester Brisbane Toronto Singapore
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 003
`
`

`

`Acquisitions Editor
`Developmental Editor
`Marketing Manager
`Senior Production Editor
`Text Designer
`Cover Designer
`Manufacturing Manager
`Illustration Coordinator
`
`Steven M. Elliot
`Sean M. Culhane
`Susan Elbe
`Savoula Amanatidis
`Lynn Rogan
`David Levy
`Lori Bulwin
`Jaime Perea
`
`This book was typeset in Times Roman by The Clarinda Company, and printed and bound
`by Hamilton Printing Company. The cover was printed by NEBC.
`
`Recognizing the importance of preserving what has been written, it is a policy of
`John Wiley & Sons, Inc. to have books of enduring value published in the United States
`printed on acid-free paper, and we exert our best efforts to that end.
`
`PSpice is a registered trademark of MicroSim Corporation.
`MATLAB is a registered trademark of The MathWorks, Inc.
`
`Copyright © 1989, 1995 by John Wiley & Sons, Inc.
`
`All rights reserved. Published simultaneously in Canada.
`
`Reproduction or translation of any part of this work beyond that permitted by Sections 107
`and 108 of the 1976 United States Copyright Act without the permission of the copyright owner
`is unlawful. Requests for permission or further information should be addressed to the
`Permissions Department, John Wiley & Sons, Inc.
`
`Library of Congress Cataloging in Publiclllion Data:
`
`Mohan, Ned.
`Power electronics: converters, applications, and design / Ned
`Mohan, Tore M. Undeland, William P. Robbins.-2nd ed.
`p.
`cm.
`Includes bibliographical references and indexes.
`ISBN 0-471-58408-8 (cloth)
`I. Power electronics. 2. Electric current converters. 3. Power
`I. Undeland, TOre M.
`11. Robbins, William P.
`semiconductors.
`III. Title.
`TK7881.15.M64 1995
`621.317-dc20
`
`94-21158
`CIP
`
`Printed in the United States of America.
`
`10 9 8 7 6 5 4 3 2 I
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 004
`
`

`

`To OUf Families . . .
`Mary, Michael, and Tara
`Mona, Hilde, and Arne
`Joanne and Jon
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 005
`
`

`

`Momentum Dynamics Corporation
`Exhibit 1 01 8
`Page 006
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 006
`
`

`

`PREFACE
`
`SECOND EDITION
`
`The first edition of this book was published in 1989. The basic intent of this edition
`remains the same; that is, as a cohesive presentation of power electronics fundamentals for
`applications and design in the power range of 500 kW or less, where a huge market exists
`and where the demand for power electronics engineers is likely to be. Based on the
`comments collected over a five-year period, we have made a number of substantial
`changes to the text. The key features are as follows:
`
`• An introductory chapter has been added to provide a review of basic electrical and
`magnetic circuit concepts, making it easier to use this book in introductory power
`electronics courses.
`• A chapter on computer simulation has been added that describes the role of com(cid:173)
`puter simulations in power electronics. Examples and problems based on PSpice@
`and MATLAB@ are included. However, we have organized the material in such a
`way that any other simulation package can be used instead or the simulations can
`be skipped altogether.
`• Unlike the first edition, the diode rectifiers and the phase-controlled thyristor con(cid:173)
`verters are covered in a complete and easy-to-follow manner. These two chapters
`now contain 56 problems.
`• A new chapter on the design of inductors and transformers has been added that
`describes easy-to-understand concepts for step-by-step design procedures. This
`material will be extremely useful in introducing the design of magnetics into the
`curriculum.
`• A new chapter on heat sinks has been added.
`
`ORGANIZATION OF THE BOOK
`
`This book is divided into seven parts. Part 1 presents an introduction to the field of power
`electronics, an overview of power semiconductor switches, a review of pertinent electric
`and magnetic circuit concepts, and a generic discussion of the role of computer simula(cid:173)
`tions in power electronics.
`Part 2 discusses the generic converter topologies that are used in most applications.
`The actual semiconductor devices (transistors, diodes, and so on) are assumed to be ideal,
`thus allowing us to focus on the converter topologies and their applications.
`Part 3 discusses switch-mode dc and uninterruptible power supplies. Power supplies
`represent one of the major applications of power electronics.
`
`vii
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 007
`
`

`

`viii PREFACE
`
`Part 4 considers motor drives, which constitute another major applications area.
`Part 5 includes several industrial and commercial applications in one chapter. An(cid:173)
`other chapter describes various high-power electric utility applications. The last chapter in
`this part of the book examines the harmonics and electromagnetic interference concerns
`and remedies for interfacing power electronic systems with the electric utilities.
`Part 6 discusses the power semiconductor devices used in power electronic converters
`including diodes, bipolar junction thyristors, metaI-oxide-semiconductor (MaS) field
`effect transistors, thyristors, gate tum-off thyristors, insulated gate bipolar transistors, and
`MaS-controlled thyristors.
`Part 7 discusses the practical aspects of power electronic converter design including
`snubber circuits, drive circuits, circuit layout, and heat sinks. An extensive new chapter
`on the design of high-frequency inductors and transfonners has been added.
`
`PSPICE SIMULATIONS FOR TEACHING AND DESIGN
`
`As a companion to this book, a large number of computer simulations are available
`directly from Minnesota Power Electronics Research and Education, P.O. Box 14503,
`Minneapolis, MN 55414 (Phone/Fax: 612-646-1447) to aid in teaching and in the design
`of power electronic systems. The simulation package comes complete with a diskette with
`76 simulations of power electronic converters and systems using the classroom (evalua(cid:173)
`tion) version of PSpice for IBM-PC-compatible computers, a 261-page detailed manual
`that describes each simulation and a number of associated exercises for home assignments
`and self-learning, a 5-page instruction set to illustrate PSpice usage using these simula(cid:173)
`tions as examples, and two high-density diskettes containing a copy of the classroom
`(evaluation) version of PSpice. This package (for a cost of $395 plus a postage of $4
`within North America and $25 outside) comes with a site license, which allows it to be
`copied for use at a single site within a company or at an educational institution in regular
`courses given to students for academic credits.
`
`SOLUTIONS MANUAL
`
`As with the first edition of this book, a solutions manual with completely worked-out
`solutions to all the problems is available from the publisher.
`
`ACKNOWLEDGMENTS
`
`We wish to thank all the instructors who have allowed us this opportunity to write the
`second edition of our book by adopting its first edition. Their comments have been most
`useful. We are grateful to Professors Peter Lauritzen of the University of Washington,
`Thomas Habetler of the Georgia Institute of Technology, Daniel Chen of the Virginia
`Institute of Technology, Alexander Emanuel of the Worcester Polytechnic Institute, F. P.
`Dawson of the University of Toronto, and Marian Kazimierczuk of the Wright State
`University for their helpful suggestions in the second edition manuscript. We express our
`sincere appreciation to the Wiley editorial staff, including Steven Elliot, Sean Culhane,
`Lucille Buonocore, and Savoula Amanatidis, for keeping us on schedule and for many
`spirited discussions.
`
`Ned Mohan
`Tore M. Undeland
`William P. Robbins
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 008
`
`

`

`CONTENTS
`
`INTRODUCTION
`PART 1
`Chapter 1 Power Electronic Systems
`1-1 Introduction
`3
`1-2 Power Electronics versus Linear Electronics
`1-3 Scope and Applications
`7
`1-4 Classification of Power Processors and Converters
`1-5 About the Text
`12
`1-6 Interdisciplinary Nature of Power Electronics
`1-7 Convention of Symbols Used
`14
`Problems
`14
`References
`15
`
`4
`
`13
`
`9
`
`Chapter 2 Overview of Power Semiconductor Switches
`2-1 Introduction
`16
`2-2 Diodes
`16
`18
`2-3 Thyristors
`20
`2-4 Desired Characteristics in Controllable Switches
`2-5 Bipolar Junction Transistors and Monolithic Darlingtons
`2-6 Metal- Oxide- Semiconductor Field Effect Transistors
`2-7 Gate-Tum-Off Thyristors
`26
`2-8 Insulated Gate Bipolar Transistors
`2-9 MOS-Controlled Thyristors
`29
`2-10 Comparison of Controllable Switches
`2-11 Drive and Snubber Circuits
`30
`2-12 Justification for Using Idealized Device Characteristics
`32
`Summary
`Problems
`32
`References
`32
`
`27
`
`29
`
`24
`25
`
`31
`
`Chapter 3 Review of Basic Electrical and Magnetic Circuit Concepts
`3-1 Introduction
`33
`3-2 Electric Circuits
`3-3 Magnetic Circuits
`Summary
`57
`Problems
`58
`References
`60
`
`33
`46
`
`1
`3
`
`16
`
`33
`
`ix
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 009
`
`

`

`x CONTENTS
`
`Chapter 4 Computer Simulation of Power Electronic Converters
`and Systems
`4-1 Introduction
`61
`4-2 Challenges in Computer Simulation
`4-3 Simulation Process
`62
`64
`4-4 Mechanics of Simulation
`4-5 Solution Techniques for Time-Domain Analysis
`4-6 Widely Used, Circuit-Oriented Simulators
`69
`4-7 Equation Solvers
`72
`Summary
`74
`Problems
`74
`References
`75
`
`62
`
`65
`
`PART 2 GENERIC POWER ELECTRONIC CIRCUITS
`Chapter 5 Line-Frequency Diode Rectifiers: Line-Frequency ac -+
`Uncontrolled dc
`5-1 Introduction
`79
`80
`5-2 Basic Rectifier Concepts
`82
`5-3 Single-Phase Diode Bridge Rectifiers
`100
`5-4 Voltage-Doubler (Single-Phase) Rectifiers
`5-5 Effect of Single-Phase Rectifiers on Neutral Currents in Three-Phase,
`101
`Four-Wire Systems
`103
`5-6 Three-Phase, Full-Bridge Rectifiers
`5-7 Comparison of Single-Phase and Three-Phase Rectifiers
`5-8 Inrush Current and Overvoltages at Tum-On
`112
`5-9 Concerns and Remedies for Line-Current Harmonics and Low Power
`113
`Factor
`Summary
`113
`114
`Problems
`References
`116
`117
`Appendix
`
`112
`
`Chapter 6 Line-Frequency Phase-Controlled Rectifiers and
`Inverters: Line-Frequency ac - Controlled dc
`6-1 Introduction
`121
`6-2 Thyristor Circuits and Their Control
`6-3 Single-Phase Converters
`126
`6-4 Three-Phase Converters
`138
`6-5 Other Three-Phase Converters
`Summary
`153
`Problems
`154
`References
`157
`158
`Appendix
`
`122
`
`153
`
`Chapter 7 dc-dc Switch-Mode Converters
`7 -1 Introduction
`161
`7-2 Control of dc-de Converters
`
`162
`
`61
`
`77
`
`79
`
`121
`
`161
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 010
`
`

`

`CONTENTS xi
`
`164
`7-3 Step-Down (Buck) Converter
`172
`7-4 Step-Up (Boost) Converter
`178
`7-5 Buck-Boost Converter
`184
`7-6 CUk dc-dc Converter
`7-7 Full Bridge dc-dc Converter
`7-8 dc-dc Converter Comparison
`Summary
`196
`Problems
`197
`References
`199
`
`188
`195
`
`200
`
`249
`
`299
`
`301
`
`202
`
`Chapter 8 Switch-Mode dc-ac Inverters: dc ~ Sinusoidal ac
`8-1 Introduction
`200
`8-2 Basic Concepts of Switch-Mode Inverters
`8-3 Single-Phase Inverters
`211
`8-4 Three-Phase Inverters
`225
`8-5 Effect of Blanking Time on Output Voltage in PWM Inverters
`8-6 Other Inverter Switching Schemes
`239
`8-7 Rectifier Mode of Operation
`243
`Summary
`244
`Problems
`246
`References
`248
`
`236
`
`252
`
`Chapter 9 Resonant Converters: Zero-Voltage and/or Zero-Current
`Switchings
`9-1 Introduction
`249
`9-2 Classification of Resonant Converters
`9-3 Basic Resonant Circuit Concepts
`253
`9-4 Load-Resonant Converters
`258
`9-5 Resonant-Switch Converters
`273
`9-6 Zero-Voltage-Switching, Clamped-Voltage Topologies
`9-7 Resonant-dc-Link Inverters with Zero-Voltage Switchings
`9-8 High-Frequency-Link Integral-Half-Cycle Converters
`289
`Summary
`291
`Problems
`291
`References
`295
`
`280
`287
`
`PART 3 POWER SUPPLY APPLICATIONS
`
`Chapter 10 Switching dc Power Supplies
`10-1 Introduction
`30 I
`301
`10-2 Linear Power Supplies
`10-3 Overview of Switching Power Supplies
`10-4 dc-dc Converters with Electrical Isolation
`10-5 Control of Switch-Mode dc Power Supplies
`10-6 Power Supply Protection
`341
`344
`10-7 Electrical Isolation in the Feedback Loop
`10-8 Designing to Meet the Power Supply Specifications
`Summary
`349
`
`302
`304
`322
`
`346
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 011
`
`

`

`xii CONTENTS
`
`Problems
`References
`
`349
`351
`
`Chapter 11 Power Conditioners and Uninterruptible Power
`Supplies
`11-1 Introduction
`354
`11-2 Power Line Disturbances
`11-3 Power Conditioners
`357
`11-4 U ninterruptib1e Power Supplies (UPSs)
`Summary
`363
`Problems
`363
`References
`364
`
`354
`
`358
`
`PART 4 MOTOR DRIVE APPLICATIONS
`Chapter 12
`Introduction to Motor Drives
`12-1 Introduction
`367
`12-2 Criteria for Selecting Drive Components
`Summary
`375
`Problems
`376
`References
`376
`
`368
`
`Chapter 13 de Motor Drives
`13-1 Introduction
`377
`13-2 Equivalent Circuit of dc Motors
`377
`380
`13-3 Permanent-Magnet dc Motors
`13-4 dc Motors with a Separately Excited Field Winding
`13-5 Effect of Armature Current Waveform
`382
`13-6 dc Servo Drives
`383
`13-7 Adjustable-Speed dc Drives
`Summary
`396
`Problems
`396
`References
`398
`
`391
`
`381
`
`Induction Motor Drives
`Chapter 14
`14-1 Introduction
`399
`400
`14-2 Basic Principles of Induction Motor Operation
`14-3 Induction Motor Characteristics at Rated (Line) Frequency
`and Rated Voltage
`405
`14-4 Speed Control by Varying Stator Frequency and Voltage
`14-5 Impact of Nonsinusoidal Excitation on Induction Motors
`14-6 Variable-Frequency Converter Classifications
`418
`14-7 Variable-Frequency PWM-VSI Drives
`419
`14-8 Variable-Frequency Square-Wave VSI Drives
`14-9 Variable-Frequency CSI Drives
`426
`14-10 Comparison of Variable-Frequency Drives
`
`425
`
`427
`
`406
`415
`
`354
`
`365
`367
`
`377
`
`399
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 012
`
`

`

`CONTENTS xiii
`
`428
`14-11 Line-Frequency Variable-Voltage Drives
`14-12 Reduced Voltage Starting ("Soft Start") of Induction Motors
`14-13 Speed Control by Static Slip Power Recovery
`431
`Summary
`432
`433
`Problems
`434
`References
`
`430
`
`Chapter 15 Synchronous Motor Drives
`15-1 Introduction
`435
`435
`15-2 Basic Principles of Synchronous Motor Operation
`15-3 Synchronous Servomotor Drives with Sinusoidal Waveforms
`15-4 Synchronous Servomotor Drives with Trapezoidal Waveforms
`15-5 Load-Commutated Inverter Drives
`442
`15-6 Cycloconverters
`445
`Summary
`445
`446
`Problems
`References
`447
`
`439
`440
`
`PART 5 OTHER APPLICATIONS
`Chapter 16 Residential and Industrial Applications
`16-1 Introduction
`451
`16-2 Residential Applications
`16-3 Industrial Applications
`Summary
`459
`459
`Problems
`459
`References
`
`451
`455
`
`460
`
`Chapter 17 Electric Utility Applications
`17-1 Introduction
`460
`17-2 High-voltage dc Transmission
`471
`17-3 Static var Compensators
`17-4 Interconnection of Renewable Energy Sources and Energy Storage
`475
`Systems to the Utility Grid
`17-5 Active Filters
`480
`Summary
`480
`481
`Problems
`482
`References
`
`Chapter 18 Optimizing the Utility Interface with Power
`Electronic Systems
`18-1 Introduction
`483
`484
`18-2 Generation of Current Harmonics
`485
`18-3 Current Harmonics and Power Factor
`18-4 Harmonic Standards and Recommended Practices
`18-5 Need for Improved Utility Interface
`487
`
`485
`
`435
`
`449
`451
`
`460
`
`483
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 013
`
`

`

`xiv CONTENTS
`
`18-6 Improved Single-Phase Utility Interface
`18-7 Improved Three-Phase Utility Interface
`18-8 Electromagnetic Interference
`500
`Summary
`502
`Problems
`503
`References
`503
`
`488
`498
`
`PART 6 SEMICONDUCTOR DEVICES
`
`Chapter 19 Basic Semiconductor Physics
`19-1 Introduction
`507
`19-2 Conduction Processes in Semiconductors
`19-3 pn Junctions
`513
`19-4 Charge Control Description of pn-Junction Operation
`19-5 Avalanche Breakdown
`520
`Summary
`522
`Problems
`522
`References
`523
`
`507
`
`518
`
`Chapter 20 Power Diodes
`20-1 Introduction
`524
`20-2 Basic Structure and /- V Characteristics
`20-3 Breakdown Voltage Considerations
`526
`531
`20-4 On-State Losses
`20-5 Switching Characteristics
`20-6 Schottky Diodes
`539
`Summary
`543
`Problems
`543
`References
`545
`
`535
`
`524
`
`546
`
`Chapter 21 Bipolar Junction Transistors
`21-1 Introduction
`546
`21-2 Vertical Power Transistor Structures
`21-3 /- V Characteristics
`548
`21-4 Physics of BJT Operation
`21-5 Switching Characteristics
`21-6 Breakdown Voltages
`562
`21-7 Second Breakdown
`563
`21-8 On-State Losses
`565
`21-9 Safe Operating Areas
`Summary
`568
`Problems
`569
`References
`570
`
`550
`556
`
`567
`
`Chapter 22 Power MOSFETs
`22-1 Introduction
`571
`22-2 Basic Structure
`571
`
`505
`
`507
`
`524
`
`546
`
`571
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 014
`
`

`

`574
`22-3 1- V Characteristics
`576
`22-4 Physics of Device Operation
`581
`22-5 Switching Characteristics
`22-6 Operating Limitations and Safe Operating Areas
`Summary
`593
`Problems
`594
`References
`595
`
`587
`
`Chapter 23 Thyristors
`23-1 Introduction
`596
`23-2 Basic Structure
`596
`597
`23-3 1- V Characteristics
`599
`23-4 Physics of Device Operation
`603
`23-5 Switching Characteristics
`23-6 Methods of Improving dildt and dv/dt Ratings
`Summary
`6/0
`Problems
`611
`References
`612
`
`608
`
`Chapter 24 Gate Tum-Off Thyristors
`24-1 Introduction
`613
`24-2 Basic Structure and 1-V Characteristics
`24-3 Physics of Turn-Off Operation
`614
`24-4 GTO Switching Characteristics
`616
`24-5 Overcurrent Protection of GTOs
`623
`Summary
`624
`Problems
`624
`References
`625
`
`613
`
`Insulated Gate Bipolar Transistors
`Chapter 25
`25-1 Introduction
`626
`25-2 Basic Structure
`626
`25-3 1-V Characteristics
`628
`25-4 Physics of Device Operation
`25-5 Latchup in IGBTs
`631
`25-6 Switching Characteristics
`25-7 Device Limits and SOAs
`Summary
`639
`Problems
`639
`References
`640
`
`634
`637
`
`629
`
`Chapter 26 Emerging Devices and Circuits
`26-1 Introduction
`641
`26-2 Power Iunction Field Effect Transistors
`26-3 Field-Controlled Thyristor
`646
`26-4 IFET -Based Devices versus Other Power Devices
`26-5 MOS-Controlled Thyristors
`649
`
`641
`
`648
`
`CONTENTS
`
`xv
`
`596
`
`613
`
`626
`
`641
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 015
`
`

`

`xvi CONTENTS
`
`656
`26-6 Power Integrated Circuits
`26-7 New Semiconductor Materials for Power Devices
`Summary
`664
`Problems
`665
`References
`666
`
`661
`
`PART 7 PRACTICAL CONVERTER DESIGN
`CONSIDERATIONS
`Chapter 27 Snubber Circuits
`27-1 Function and Types of Snubber Circuits
`27-2 Diode Snubbers
`670
`678
`27-3 Snuber Circuits for Thyristors
`27-4 Need for Snubbers with Transistors
`27-5 Turn-Off Snubber
`682
`27-6 Overvoltage Snubber
`686
`27-7 Turn-On Snubber
`688
`27 -8 Snubbers for Bridge Circuit Configurations
`27-9 GTO Snubber Considerations
`692
`Summary
`693
`Problems
`694
`References
`695
`
`669
`
`680
`
`691
`
`Chapter 28 Gate and Base Drive Circuits
`28-1 Preliminary Design Considerations
`696
`28-2 dc-Coupled Drive Circuits
`697
`28-3 Electrically Isolated Drive Circuits
`28-4 Cascode-Connected Drive Circuits
`28-5 Thyristor Drive Circuits
`712
`28-6 Power Device Protection in Drive Circuits
`28-7 Circuit Layout Considerations
`722
`Summary
`728
`Problems
`729
`References
`729
`
`703
`710
`
`717
`
`Chapter 29 Component Temperature Control and Heat Sinks
`29-1 Control of Semiconductor Device Temperatures
`730
`29-2 Heat Transfer by Conduction
`731
`29-3 Heat Sinks
`737
`29-4 Heat Transfer by Radiation and Convection
`Summary
`742
`Problems
`743
`References
`743
`
`739
`
`Chapter 30 Design of Magnetic Components
`30-1 Magnetic Materials and Cores
`744
`30-2 Copper Windings
`752
`
`667
`669
`
`696
`
`730
`
`744
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 016
`
`

`

`756
`
`754
`30-3 Thennal Considerations
`30-4 Analysis of a Specific Inductor Design
`30-5 Inductor Design Procedures
`760
`30-6 Analysis of a Specific Transfonner Design
`30-7 Eddy Currents
`771
`779
`30-8 Transfonner Leakage Inductance
`780
`30-9 Transfonner Design Procedure
`30-10 Comparison of Transfonner and Inductor Sizes
`Summary
`789
`Problems
`790
`References
`792
`
`767
`
`CONTENTS xvii
`
`789
`
`Index
`
`793
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 017
`
`

`

`CHAPTER 3
`
`REVIEW OF BASIC
`ELECTRICAL AND
`MAGNETIC CIRCUIT
`CONCEPTS
`
`3-1
`
`INTRODUCTION
`
`The purpose of this chapter is twofold: (1) to briefly review some of the basic definitions
`and concepts that are essential to the study of power electronics and (2) to introduce
`simplifying assumptions that allow easy evaluation of power electronic circuits.
`
`3-2 ELECTRIC CIRCUITS
`
`An attempt is made to use Institute of Electrical and Electronics Engineers (IEEE) stan(cid:173)
`dard letter and graphic symbols as much as possible. Moreover, the units used belong to
`the International System of Units (SI). The lowercase letters are used to represent instan(cid:173)
`taneous value of quantities that may vary as a function of time. The uppercase letters are
`used to represent either the average or the rms values. As an example, a voltage Voi and
`its average value Voi are shown in Fig. 1-4b. A value that is average or rms may be stated
`explicitly or it may be obvious from the context.
`The positive direction of a current is shown explicitly by a current arrow in the circuit
`diagram. The voltage at any node is defined with respect to the circuit ground, for
`example, Va is the voltage of node a with respect to ground. The symbol Vab refers to the
`voltage of node a with respect to node b, where, Vab = va - Vb'
`
`3-2-1 DEFINITION OF STEADY STATE
`
`In power electronic circuits, diodes and semiconductor switches are constantly changing
`their on or off status. Therefore the question arises: When is such a circuit in steady state?
`A steady-state condition is reached when the circuit waveforms repeat with a time period
`T that depends on the specific nature of that circuit.
`
`33
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 018
`
`

`

`34
`
`CHAPTER 3 REVIEW OF BASIC ELECTRICAL AND MAGNETIC CIRCUIT CONCEPTS
`
`3-2-2 AVERAGE POWER AND nns CURRENT
`
`Consider the circuit of Fig. 3-1, where the instantaneous power flow from subcircuit 1 to
`subcircuit 2 is
`
`p(t) = vi
`Both v and i may vary as a function of time. If v and i waveforms repeat with a time period
`T in steady state, then the average power flow can be calculated as
`
`(3-1)
`
`Pay = ~ iT p(t) dt = ~ iT vi dt
`
`(3-2)
`
`Under the conditions stated earlier, if subcircuit 2 consists purely of a resistive load,
`then v = Ri and in Eq. 3-2
`
`I IT
`P = R -
`T
`av
`
`o
`In tenns of the nns value I of the current, the average power flow can be ex(cid:173)
`pressed as
`
`Pdt
`
`(3-3)
`
`Pay = RP
`A comparison of Eqs. 3-3 and 3-4 reveals that the nns value of the current is
`
`1 ~ ~~ J: j' • dt
`
`(3-4)
`
`(3-5)
`
`which shows the origin of the tenn root-mean-square.
`If i is a constant dc current, then Eqs. 3-4 and 3-5 are still valid with the average and
`the nns values being equal.
`
`3-2-3 STEADY-STATE ae WAVEFORMS WITH SINUSOIDAL
`VOLTAGES AND CURRENTS
`
`Consider the ac circuit of Fig. 3-2a, with an inductive load under a steady-state operation,
`where
`
`i = V2I cos(wt -
`v = V2V cos wt
`and V and I are the nns values. The v and i wavefonns are plotted as functions of wt in
`Fig. 3-2h.
`
`(3-6)
`
`<1»)
`
`3-2-3-1 Phasor Representation
`
`Since both v and i vary sinusoidally with time at the same frequency, they can be
`represented in a complex plane by means of the projection of the rotating phasors to the
`horizontal real axis, as shown in Fig. 3-2c. Conventionally, these phasors rotate in a
`
`--+-
`
`+
`v
`
`Figure 3-1
`
`Instantaneous power flow.
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 019
`
`

`

`i --O R
`
`u"V
`-
`
`L
`
`3-2 ELECTRIC CIRCUITS
`
`35
`
`[S;J ~ --Reference
`
`I
`
`V=V!.Jl°
`
`-jlq ---------! I=ICio
`
`Source
`
`Load
`
`(a)
`
`(c)
`
`o
`
`rot
`
`Figure 3-2 Sinusoidal steady state.
`
`(b)
`
`counterclockwise direction with an angular frequency w, and their nns values (rather than
`their peak values) are used to represent their magnitudes:
`I::: le-j~
`
`V::: VeJ'° and
`
`(3-7)
`
`Considering Eq. 3-6, the phasor diagram in Fig. 3-2c corresponds to the time instant
`when v attains its positive-maximum value.
`In Eq. 3-7 V and I are related by the complex load impedance Z ::: R + jwL ::: Z~
`at the operating frequency w in the following manner:
`
`I ::: ~ ::: V~o ::: ~ e-j~ = le-j~
`Z
`Ze'~ Z
`
`where I = VIZ.
`
`3-2-3-2 Power, Reactive Power, and Power Factor
`
`The complex power S is defined as
`S ::: V" = veJO . lej~ = Vlej~ = Sej~
`
`(3-8)
`
`(3-9)
`
`Therefore, the magnitude of the complex power, which is also called the apparent power
`and is expressed in the units volt-amperes, is
`S = VI
`
`(3-10)
`
`The real average power P is
`P = Re[S] = VI cos ~
`
`(3-11)
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 020
`
`

`

`CHAPTER 3 REVIEW OF BASIC ELECTRICAL AND MAGNETIC CIRCUIT CONCEPTS
`
`which is expressed as a product of V and the current component Ip = I cos <1>, which is
`in phase with the voltage in the phasor diagram of Fig. 3-2c. The out-of-phase component
`is Iq = I sin <1>. The in-phase current component iiI) and the out-of-phase current
`component iiI) can be expressed as
`ip(l) = Vupcos wI = (Vu cos <I»cos wI
`
`(3-12)
`
`and
`
`(3-13)
`where i(l) = iiI) + iit). These two current components are plotted in Fig. 3-2b.
`It should be noted that ip and iq result in instantaneous power flow components PI =
`v'ip and P2 = v'iq, where P = PI + P2. Both PI and P2 pulsate at 2w, twice the source
`frequency w. Here PI has an average value given by Eq. 3-11; the average value of P2 is
`~o.
`In the phasor diagram of Fig. 3-2c, only Ip (=1 cos <1»
`is responsible for the power
`transfer, notIq (=1 sin <1». It is common to define a quantity called reaclive power Q with
`the units of var (volt-ampere-reactive) using Iq • Defining the complex power S = P + jQ
`and using Eqs. 3-9 and 3-10,
`
`(3-14)
`
`An inductive load shown in Fig. 3-2a has a positive value of <1>, where the current lags
`the voltage. In accordance with Eq. 3-14, an inductive load draws positive vars, also
`called lagging vars. Conversely, a capactive load draws negative vars, also called leading
`vars (in other words, it supplies positive vars to the electrical system).
`The physical significance of S, P, and Q should be understood. The cost of most
`electrical equipment such as generators, transformers, and transmission lines increases
`with S = VI, since their electrical insulation level and magnetic core size depend on V and
`their conductor size depends on I. Power P has a physical significance since it represents
`the rate of useful work being performed plus the power losses. In most situations, it is
`desirable to have the reactive power Q be zero.
`Based on the above discussion, another quantity called the power faclor is defined,
`which is a measure of how effectively the load draws the real power:
`P P
`Power factor = S = VI = cos <I>
`
`(3-15)
`
`which is dimensionless. Ideally, the power factor should be 1.0 (that is, Q should be zero)
`to draw power with a minimum current magnitude and hence minimize losses in the
`electrical equipment and possibly in the load .
`
`• Example 3-1 An inductive load connected to a 120-V, 6O-Hz ac source draws
`1 kWat a power factor of 0.8. Calculate the capacitance required in parallel with the load
`in order to bring the combined power factor to 0.95 (lagging).
`
`Solution
`
`For the load:
`
`PL = 1000 W
`1000
`= 1250 VA
`SL = -
`0.8
`QL = y,"SZ"---p"""Z = 750 V A (lagging)
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 021
`
`

`

`3-2 ELECTRIC CIRCUITS
`
`37
`
`Therefore, the complex power of the load is
`SL = PL + jQL
`= 1000 + j750 VA
`
`The reactive power drawn by a capacitor is represented as - jQc because the
`capacitor current leads the voltage by 90°. Therefore, the total complex power de(cid:173)
`livered from the source is
`
`S = (PL + jQd - jQc
`= PL + j(QL - Qd
`Since the combined power factor is 0.95 (lagging),
`
`PL
`S = ypi + (QL - Qd2 = 0.95
`(QL - Qd ::; PL 1(_1_ - 1) = 328.7 VA (lagging)
`-y 0.952
`
`and therefore,
`
`Since
`
`Qc
`
`750 - 328.7 = 421.3 VA (leading)
`
`3-2-3-3 Three-Phase Circuits
`
`During a balanced steady-state operating condition, it is possible to analyze three-phase
`circuits such as that in Fig. 3-3a on a per-phase basis. The positive phase sequence is
`commonly assumed to be a-b-c. Using rms values to represent the magnitudes,
`
`•
`
`(3-16)
`
`and P phase
`
`VI cos <I>
`
`1.,e-j21r13 ::; le-j(<I>+21r/3)
`laei21r3 = le-j (<I>-21T/3)
`where I =:: VIZ. Assuming Z to be an inductive impedance with a positive value of </>, the
`phase voltage and current phasors are shown in Fig. 3-3b.
`It is possible to calculate the line-to-line voltages from the phase voltages, recogniz(cid:173)
`ing for example that vab = va - Vb' Figure 3-3c shows line-to-line voltage phasors where
`Vab = V uej1T16 leads Va by 30° and the line-to-line rms voltage magnitude is
`Vu V3V
`It is possible to calculate power on a per-phase basis as
`Sphase = VI
`Therefore, in a balanced system the total three-phase power can be expressed as
`S3-phase = 3Sphase = 3VI V3V ul
`
`(3-17)
`
`(3-18)
`
`(3-19)
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 022
`
`

`

`38
`
`CHAPTER 3 REVIEW OF BASIC ELECTRICAL AND MAGNETIC CIRCUIT CONCEPTS
`
`i" --
`
`(a)
`
`(b)
`
`Figure 3-3 Three-phase circuit.
`
`(c)
`
`and
`
`The above three-phase circuit operates on the same power factor as the per-phase
`power factor of cos <l>.
`It should be noted that even if a three-phase circuit is operating with nonsinusoidal
`voltages and currents, its total power can still be calculated on a per-phase basis provided
`the circuit is operating under a balanced, steady-state condition.
`
`(3-20)
`
`3-2-4 NON SINUSOIDAL WAVEFORMS IN STEADY STATE
`
`In power electronic circuits, dc or low frequency ac wavefonns are synthesized by using
`segments of an input wavefonn. The motor voltage produced by the power electronics
`inverter in an ac motor drive is shown in Fig. 3-40. Often, the line current drawn from the
`utility by the power electronic equipment is highly distorted, as shown in Fig. 3-4b. In
`steady state, such wavefonns repeat with a time period T and a frequency f (=w/2-rr) =
`liT. This repetition frequency is called the fundamental frequency, and it is usually
`designated by a subscript 1. In addition to a dominant component at Ibe fundamental
`
`Momentum Dynamics Corporation
`Exhibit 1018
`Page 023
`
`

`

`3-2 ELECTRIC CIRCUITS
`
`39
`
`1+ - - - - - - - - T= 1.. --------~
`r
`
`(a)
`
`(b)
`
`Figure 3-4 Nonsinusoidal