IPR2015-00762 - EXHIBIT 1008
`
`Zhongshan Broad Ocean Motor Co., Ltd., Petitioner
`
`Nidec Motor Corporation, Patent Owner
`
`1
`
`

`

`A JOHN WILEY 8: SONS, INC. PUBLICATION
`
`0-
`9 WILEY” I
`
`.
`
`INTERSCIENCE
`
`econd Edition
`
`PAUL C. KRAUSE
`OLEG WASYNCZUK
`SCOTT D. SUDHOFF
`
`Purdue University
`
`IEEE Power Engineering Society. Sponsor
`
`fV
`
`' :3mafia
`
`_
`SERIES
`°" POWER
`ENGINEERING
`
`IEEE Press Power Engineering Series
`Mohamed E. EI—Hawary. Series Editor
`
`

`

`13141516i7i819 20
`
`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
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`
`This book is primed on acid—free paper. @
`
`Copyright © 2002 by The Institute of Electrical and Electronics Engineers. Inc. All rights reserved.
`
`For ordering and customer service. call LEGO-CALL WILEY.
`
`Library of Congrats Cataloging-in-l’ublication is available.
`
`ISBN 0-471-14326—X
`
`Printed in the United States of America.
`
`

`

`aa:at»;ateatamaiémeeaarestartiéaartsaéai
`
`2.3 Voitage and Torque Equations I 76
`2.4 Basic Types of Direct-Current Machines I 78
`2.5 Dynamic Characteristics of Permanent—Magnet and Shunt dc Motors I 88
`2.6 Time-Domain Block Diagrams and State Equations '1 92
`2.7 Solution of Dynamic Characteristics by Lapiace Transformation I 98
`
`i
`1.2 Magnetically Coupled Circuits I
`1.3 Electromechanicai Energy Conversion I H
`
`1.4 Machine Windings and Air-Gap MMF I 35
`
`LS Winding lnductanees and Voltage Equations I 47
`References I 58
`
`Problems I 58
`
`Chapter 2 DIRECT-CURRENT MACHINES
`
`2.1
`
`Introduction I 67
`
`2.2 Elementary Direct—Current Machine I 68
`
`References I $104
`Problems I 105 ’
`
`PREFACE
`
`Chapter 1 BASIC PRINCIPLES FOR ELECTRIC
`MACHINE ANALYSIS
`
`Ll
`
`Introduction I
`
`l
`
`,:
`
`

`

`viii
`
`CONTENTS
`
`Chapter 3 REFERENCE—FRAME THEORY
`
`3.1
`
`3.2
`3.3
`
`3.4
`
`3.5
`
`3.6
`
`3.7
`
`3.8
`
`Introduction I 109
`
`_
`Background I 109
`Equations of Transformation: Changes of Variables I
`
`l 11
`
`Stationary Circuit Variables Transformed to the Arbitrary
`Reference Frame I 115
`
`Commonly Used Reference Frames I 123
`Transformation Between Reference Frames I 124
`
`Transformation of a Balanced Set I 126
`
`Balanced Steady-State Phasor Relationships I 127
`
`3.9
`
`Balanced Steady-State Voltage Equations I 130
`Variables Observed from Several Frames of Reference I 133
`3.10
`References I 137
`
`Problems I 138
`
`Chapter 4 SYMMETRICAL INDUCTION MACHINES
`
`4.1
`
`4.2
`
`4.3
`
`4.4
`
`4.5
`
`4.6
`4.7
`4.3
`
`4.9
`4.10
`4.11
`
`4.32
`
`4.13
`
`Torque Equation in Machine Variables I 197
`
`Torque Equation in Arbitrary Reference-Frame Variables I $53
`Commonly Used Reference Frames I 154
`Per Unit System I 155
`
`Introduction I 141
`
`Voltage Equations in Machine Variables I 142
`
`Torque Equation in Machine Variables I 146
`
`Equations of Transformation for Rotor Circuits I 147
`
`Voltage Equations in Arbitrary Reference-Frame Variables I
`
`l49
`
`Analysis of Steady-State Operation I 157
`Free Acceleration Characteristics I 165
`
`Free Acceleration Characteristics Viewed from Various
`Reference Frames I 172
`
`Dynamic Performance During Sudden Changes in Load Torque I 174
`Dynamic Performance During a B-Fhase Fault at the
`Machine Terminals I 181
`
`414
`
`Computer Simulation in the Arbitrary Reference Frame I 184
`References I 187
`Problems I 188
`
`Chapter 5 SYNCHRONOUS MACHINES
`
`Introduction I 191
`
`Voltage Equations in Machine Variables I 192
`
`5.1
`5.2
`
`5.3
`
`

`

`
`
`CONTENTS
`
`ix
`
`Stator Voltage Equations in Arbitrary Reference—Frame Variables I
`Voltage Equations in Rotor ReferenceFrame Variables:
`Park'3 Equations I 200
`Torque Equalions in Substitute Variables I 206
`Rotor Angie and Angle Between Rotors I 207
`
`:98
`
`_ Per Unit System I 209
`Analysis of Steady»State Operation I 210
`Dynamic Performance During aSudden Change in InputTorque I 219
`Dynamic Performance During a 3-Phase Fault at the
`Machine Terminals I 225
`
`Approximate Transient Torque Versus Rotor Angle Characteristics I 229
`
`Comparison ofA-ctual and Approximate Transient Torque—Angle
`Characteristics During a Sudden Change in input Torque:
`First Swing Transient Stability Limit I 232
`
`Comparison of Actual and Approximate Transient Torque—Angle
`Characteristics During a 3-Phase Fault at the Terminals: Critical
`Clearing Time I 239
`
`Derived Synchronous Machine Time Constants I 29|
`
`Voltage and Torque Equations in Machine Variables I 261
`62
`Voltage and Torque Equations in Rotor Reference-Frame Variables I 264
`6.3
`Analysis ofSteady-State Operation I 266
`6.4
`Dynamic Performance I 274
`6.5
`References I 281
`Problems I 281
`
`5.l5
`
`EqualwArea Criterion I 242
`
`5.16 Computer Simulation I 246
`References I 255
`
`Problems I 256
`
`Chapter 6 THEORY OF BRUSHLESS dc MACHINES
`
`6.I
`
`introduction I 26]
`
`Chapter 7 MACHINE EQUATIONS lN OPERATIONAL
`iMPEDANCES AND TIME CONSTANTS
`
`Introduction I283
`
`Park's Equations in Operational Form I 284
`
`Operational lmpedances and C(p) for a Synchronous Machine with
`Four Rotor Windings I 284
`Standard Synchronous Machine Reactances I 288
`
`Standard Synchronous Machine Time Constants I 290
`
`7.i
`
`7.2
`
`7.3
`
`7.4
`
`7.5
`7.6
`
`

`

`x
`
`CONTENTS
`
`7.7
`
`Parameters from Short—Circuit Characteristics I 294
`
`Parameters from Frethency-Response Characteristics I 301
`7.8
`References I 307
`
`Problems I 308
`
`Chapter 8 LINEARIZED MACHINE EQUATIONS
`
`8.i
`
`introduction I 311
`
`8.2 Machine Equations to Be Linearized I 312
`8.3
`Linearization of Machine Equations I 313
`
`Small—Displacement Stability: Eigenvalues I 323
`3.4
`Eigenvalues of Typical Induction Machines I 324
`8.5
`Eigenvalues of Typicai Synchronous Machines I 327
`8.6
`Transfer Function Formulation I 330
`8.7
`References I 335
`
`Problems I 335
`
`Chapter 9 REDUCED-ORDER MACHINE EQUATIONS
`
`
`
`Variables for Unsymmetrical 2-Phase Induction Machines I 373
`
`Induction Machine Largewfixcursion Behavior Predicted by
`Reduced-Order Equations I 343
`Synchronous Machine Large-Excursion Behavior Predicted
`by Reduced-Order Equations I 350
`Linearized Reduced-Order Equations I 354
`Eigenvalues Predicted by Linearized Reduced-Order Equations I 354
`Simulation of Reduced-Order Models I 355
`
`Introduction I 337
`
`Reduced-Order Equations I 338
`
`9.1
`
`9.2
`
`9.3
`
`9.4
`
`9.5
`9.6
`9.7
`
`Closing Comments and Guidelines I 358
`9.8
`References I 358
`Problems I 359
`
`Chapter 10 SYMMETRICAL AND UNSYMMETRICAL
`2-PHASE INDUCTION MACHINES
`
`10.1
`
`Introduction I 361
`
`10.2 Analysis of Symmetrical 2—Phase Induction Machines I 362
`10.3 Voltage and Torque Equations in Machine Variables for
`Unsymmetrical 2-Phase Induction Machines I 371
`10.4 Voltage and Torque Equations in Stationary ReferencewFrame
`
`

`

`.
`1
`
`CONTENTS
`
`xi
`
`Steady-State Operation of Unsymmetrical.
`uction Machines I 377
`.
`
`SEMICONTROLLED BRIDGE CONVERTERS
`
`I Introduction I 395
`Single»Phase Load‘Commutated Converter I 395
`3—Phase Load Contmutaled Converter I 406
`
`rencesr I 425
`oblems I 425
`
`Introduction I 427
`312,]
`Solid-State Converters for dc Drive Systems I 427
`12.2
`Steady-State and Dynamic Characteristics of acIdc Converter Drives I 431
`12.3
`12.4 One-Quadrantdc/dc Converter Drive I 443
`‘ 12.5 Two-Quadrant chdc Converter Drive I 460
`12.6
`Four-Quadrant chdc Converter Drive I 463
`12.7 Machine Control with Voltage-Controlled chdc Convener I 466
`12.8 Machine Control with Current-Controlled chdc Converter I 468
`References I 476
`
`13.] l Closedeoop Voltage and Current Controls I 516
`
`7
`
`Problems I 476
`
`Chapter 13
`
`FULLY CONTROLLED 3—PHASE BRIDGE
`CONVERTERS
`
`13.1
`
`13.2
`
`13.3
`13.4
`
`13.5
`
`13.6
`
`13.7
`13.8
`13.9
`
`lntroduction I 481
`
`The 3—Phase Bridge Converter I 481
`
`180° Voltage Source Operation I 487
`Pulse—Width Modulation I 494
`
`Sine-Triangle Modulation I 499
`
`Third-Harmonic Injection I 503
`
`Space-Vector Modulation I 506
`, Hysteresis Modulation I 510
`Delta Modulation I SEE
`
`13.10 Open-Loop Voflage and Current Control
`
`I 513
`
`

`

`Xii
`
`CONTENTS
`
`References I 520
`Problems I 521
`
`Chapter 14
`
`INDUCTION MOTOR DRIVES
`
`14.1
`
`14.2
`
`14.3
`14.4
`
`14.5
`
`14.6
`14.7
`
`Introduction I 525
`
`Volts-Per-Hertz Control
`
`I 525
`
`Constant Slip Current Control I 532
`Fie1d~0riented Control
`I 540
`
`Direct Rotor—Oriented Field-Oriented Control
`
`I 544
`
`Robust Direct Field-Oriented Control I 546
`Indirect Rotor Field-Oriented Control I 550
`
`Conclusions I 554
`14.8
`References I 554
`
`Problems I 555
`
`Conversion Factors, and Abbreviations
`
`15.13 Case Study: Current-Regulated Inverter-Based Speed Controller I 597
`References I 600
`Problems I 600
`
`Chapter ‘15 BHUSHLESS dc MOTOR DRIVES
`
`15.1
`
`15.2
`
`15.3
`15.4
`
`15.5
`15.6
`
`15.?
`
`E58
`15.9
`
`Introduction I 557
`
`Voltage-Source Inverter Drives I 558
`
`Equivalence of VSI Schemes to Idealized Source I 560
`Average-Value Analysis of VSI Drives I 568
`
`Steady-State Performance of V31 Drives I 57.1
`Transient and Dynamic Performance of V51 Drives I 574
`
`Consideration of Steady-State Harmonics I 573
`
`Case Study: Voltage-Source Inverter-Based Speed Control I 582
`Current-Regulated Inverter Drives I 586
`
`15.10 Voltage Limitations of Current—Source Inverter Drives I 590
`15.11 Current Command Synthesis I 591
`
`15.12 Average-Value Modeling of Current-Regulated Inverter Drives I 595
`
`Appendix A Trigonometric Relations, Constants and
`
`

`

`REFACE
`
`The first edition of this book was written by Paul C. Krause and published in 1986 by
`McGraw—Hill. Eight years tater the same book was republished by IEEE Press with.
`Oleg Wasynczuk and Scott D. Sudhoff added as coauthors. The lows of the first
`edition was the analysis ofelectric machines using reference frame theory, wherein
`the concept of the arbitrary reference frame was emphasized. Not only has this
`approach been embraced by the vast majority of electric machine analysts,
`it has
`also become the approach used in the analysis of electric drive systems. The use
`of reference—frame theory to analyze the complete drive system (machine, converter,
`and control) was not emphasized in the first edition. The goai ofthis edition is to fill
`this void and thereby meet the need of engineers whose job it is to analyze and
`design the complete drive system. For this reason the words “and Drive Systems”
`have been added to the title.
`
`xiii
`
`Although some of the material has been rearranged or revised, and in some cases
`eliminated, such as 3-phase symmetrical components, most of the material presented
`in the first ten chapters were taken from the original edition. For the most part, the
`material in Chapters li—lS on electric drive systems is new. In particular, the ana-
`lysis of converters used in electric drive systems is presented in Chapters 11 and 13
`while dc, induction. and brushless dc motor drives are analyzed in Chapters [2. i4,
`and 15, respectively.
`Central to the analysis used in this text is the transformation to the arbitrary refer~
`ence frame. All real and complex transformations used in machine and drive ana-
`lyses can be shown to be special cases of this general transformation. The modern
`electric machine and drive analyst must understand reference frame theory. For this
`reason, the complete performance of all electric machines and drives considered are
`illustrated by computer traces wherein variables are often portrayed in different
`
`

`

`xiv
`
`PREFACE
`
`frames of reference so that the student is able to appreciate the advantages and sig-
`nificance of the transformation used.
`The material presented in this text can be used most beneficially if the student has
`had an introductory course in electric machines. However, a senior would be com—
`fortable using this textbook as a first course. For this purpose, considerable time
`should be devoted to the basic principles discussed in Chapter 1, perhaps some of
`Chapter 2 covering basic dc machines. most of Chapter 3-covering reference frame
`theory, and the beginning sections of Chapters 4, 5. and 6 covering induction, syn—
`chronous. and brushless dc machines.
`Some of the material that wouid be of interest oniy to the electric power engineer
`has been reduced or eliminated from that given in the first edition. HoWever, the
`material found in the finai sections in Chapters 4 and 5 on induction and synchronous
`machines as well as operational impedances (Chapter 7), and reduced-order model-
`ing (Chapter 9) provide an excellent background for the power utility engineer.
`We would like to acknowledge the efforts and assistance of the reviewers. in par-
`ticular Mohamed E. El—Hawary, and the staff of IEEE Press and John Wiley & Sons.
`
`November 209!
`
`West bij'ayette. Indiana
`
`PAUL C. KRAUSE
`OLEG WASYNCZUK
`Scorr D. SUDHOFF
`
`

`

`EFERENCE-FRAME THEORY
`
`3.1
`
`INTRODUCTION
`
`109
`
`The voltage equations that describe the performance of induction and synchronous
`machines were established in Chapter 1. We found that some of the machine induc—
`ances are functions'of the rotor speed. whereupon the coefficients of the differential
`9equations (voltage equations) that describe. the behavior of these machines are time—
`arying except when the rotor is stalled. A change of variables is often used to reduce
`’ thecomplexity of these differential equations. There are several changes of variables
`that are used, and it was originally thought that each change ofvariables was different
`7 and therefore they were treated separately [1—43. it was later learned that 311 changes
`of variables used to transform real variables are contained in one [5.6]. This general
`transformation refers machine variables to a frame of reference that rotates at an
`
`arbitrary angular velocity. All known real transformations are obtained from this
`transformation by simply assigning the speed of the rotation of the reference frame.
`In this chapter this transformation is set forth and, because many of its properties
`can be studied without the complexities of the machine equations, it is applied to the
`equations that deseribe resistive, inductive. and capacitive circuit elements. By this
`approach, many ofthe basic concepts and interpretations of this general transforma—
`tion are readily and concisely established. Extending the material presented in this
`chapter to the analysis of ac machines is straightforward involving a minimum of
`trigonometric manipulations.
`
`3.2 BACKGROUND
`
`] introduced a new approach to electric machine ana-
`in the late 1920s, R. H. Park'l l
`lysis. He formulated a change of variables which, in effect, replaced the variables
`
`

`

`(voltages, currents. and flux linkages) associated with the stator windings of a syn;
`chronous machine with variables associated with fictitious windings rotating with
`the rotor. In other words, he transformed, or referred, the stator variables to a frame
`of reference fixed in the rotor. Park's transformation, which revolutionized electric
`machine analysis. has the unique property of eliminating all tlmaVflI'ylflg induct
`tances from the voltage equations of the synchronous machine which occur due to
`(i) electric circuits in relative motion and (2) electric circuits with varying magnetic
`reluctance.
`
`In the late 19305. H. C. Stanley {2] employed a change of variables in the
`analysis of induction machines. He showed that the time-varying inductances in ,
`the voltage equations of an induction machine due to electric circuits in relative
`motion could be eliminated by transforming the variables associated with the rotor
`windings (rotor variables) to variables assoeiated with fictitious static-nary windings.
`In this case the rotor variables are transformed to a frame reference fixed in the
`stator.
`
`110
`
`REFERENCE—FRAME THEORY
`
`machines.
`
`G. Kron {3] introduced a change of variables that eliminated the position or time-
`varying mutual
`inductances of a symmetrical induction machine by transforming
`both the stator variables and the rotor variables to a reference frame rotating in syn—
`chronism with the rotating magnetic field. This reference frame is commonly
`referred to as the synchronously rotating reference frame.
`D. S. Brereton et a1. [43 employed a change of variabies that also eliminated the ‘
`timeavarying inductances of a symmetrical induction machine by transforming the
`stator variables to a reference frame fixed in the rotor. This is essentially Park's
`transformation applied to induction machines.
`Park, Stanley. Kron, and Brereton et al. develoPed changes of variables, each
`of which appeared to be uniquely suited for a particular application. Consequently.
`each transformation was derived and treated separately in literature until
`it
`was noted in 1965 {5]
`that all known real
`transformations used in induction
`machine analysis are contained in one general transformation that eliminates all
`time—varying inductanCes by referring the stator and the rotor variables to a
`frame of reference that may rotate at any angular velocity or remain stationary.
`Ail known real
`transformations may then be obtained by simply assigning the
`appropriate speed of rotation, which may in fact be zero.
`to this so—caiied
`arbitrary reference frame.
`It
`is
`interesting to note that
`this transformation is
`sometimes referred to as the “generalized rotating real transformation," which
`may be somewhat misieading because the reference frame need not rotate. In
`any event, we will
`refer to it as the arbitrary reference frame as did the
`originators [5]. Later,
`it was noted that
`the stator variables of a synchronous
`machine could also be referred to the arbitrary reference frame [6]. However, we
`will
`find that
`the timevvarying inductances of a synchronous machine are
`eliminated only if
`the reference frame is
`fixed in the rotor (Park's transfor—
`mation); consequently the arbitrary reference frame does not offer the advantages
`in the analysis of the synchronous machines that it does in the case of induction
`
`

`

`Chapter 6
`
`THEORY OF BRUSHLESS dC'MACHINES
`
`1
`
`INTRODUCTION
`
`251
`
`The brushless dc motor is becoming widely used as a small horsepower control
`otor. This device has the physicat appearance of a 3-phase permanent magnet
`synchronous machine that is supplied from an inverter that converts a dc voltage
`3-phase altematingecurrent (ac) voltages with frequency corresponding instanta—
`usly to the rotor speed. The inverter—machine combination has the terminal and
`utput (Te vs. £11,) characteristics resembling those of a dc shunt machine during
`tor operation; hence the name fn'ushlessdc motor. In this brief chapter, equations
`derived which describe the Operation ofa brushless dc machine. The operation of
`:shless dc motor drives is considered in Chapter [5. ln this chapter We look at the
`forrnance of the brushtess dc motor with balanced 3—phase applied stator voltage
`,ith frequency corresponding to the rotor Speed. This allows as to become familiar
`31' the salient operating features of the inverter—motor combination without
`. ming involved with the harmonics of the phase voltages that occur due to the
`Wilching of the inverter.
`
`. VOLTAGE AND TORQUE EQUATIONS
`ACHINE VARIABLES
`
`POIG. brushless dc machine is depicted in Fig. 6.24. It has 3—phase, wye-con‘
`l3“Ed stator windings and a permanent magnet rotor. It
`is a synchronous machine.
`Stator windings are identical windings diSplaced 120°. each with Nr equivalent
`“-3 {It‘d resistance I}. For our analysis we will assume that the stator windings are
`
`

`

`252
`
`THEORY OF BHU SHLESS dc MACHINES
`
`Sensor
`
`Sinusoidally distributed. The three sensors shown in Fig. 6.2-1 are Hall eff'
`devices. When the north pole is under a sensor, its output is nonzero; with a so
`pole under the sensor, its output is zero In most applications the stator is suppil
`from an inverter that is switched at a frequency corresponding to the rotor speed. T
`states ofthe three sensors are used to determine the switching logic for the in»:e
`
`Two-pole, 3~phase brushless dc machine.
`
`Sensor
`
`c: axis .
`
`Figure 6.2-l
`
`

`

`263
`
`VOLTAGE AND TORQUE EQUATIONS lN MACHlNE VARIABLES
`
`Jury“?-
`
`11 effect'
`In the actual machine the sensors are not positioned over the rotor as shown in
`g
`l a south '
`Fig‘ 6.2-]. Instead they are piaced over a ring that is mounted on the shaft external
`£1-
`_'
`
`SUpplkd-Z
`to the stator windings and magnetized as the rotor. We wiil return to these sensors
`3
`
`eed. The _
`and the role they play later in this analysis. It is first necessary to establish the voltage
`E
`
`inverter, '
`and torque equations that can be used to describe the behavior of the permanent
`i
`:
`magnet synchronous machine.
`The voitage equations in machine variables are
`
`'
`
`,
`.
`
`where
`
`Vrtbtzr = rxlufxzc ipkaht's
`
`furl
`fbs
`(rtrbc'.r)T = ljrts
`r5=diag[rx i}
`
`r5]
`
`The iluxilinhages may bEWFitten
`
`S
`
`lube: = Lisiabhr 'l: 1;
`
`here L,- may be written from (LS—2341.547) and-(1.5-29)~(l.5-3l) or directly
`om (5.2-8). Also.
`
`it; a '2'‘H!
`
`stat},
`sin (0r — 3311)
`sin (0, + 2f)
`
`.
`
`.
`
`_
`
`.
`
`.
`
`.
`
`.
`
`_-, "' infirm- "“ trawler + 21531“) 511120:-
`
`.
`
`.
`
`-
`
`here 3:" is the amplitude of the flux linkages established by the permanent magnet
`s viewed from the stator phase windings.
`In other words the magnitude of pit;
`ottld be the opencircuit voltage induced in each stator phase winding. Damper
`indings are neglected because the permanent magnet is a poor electrical conductor
`rid the eddy currents that flow in the nonmagnetic materials securing the magnets
`resmali. Hence,
`large armature currents can be tolerated without significant
`emagnetization. We have assumed by (6.2-5) that the voltages induced in the stator
`indings by the permanent magnet are constant-amplitude sinusoidal voltages. For
`ansinusoidal induced voltages see references 1 and 2.
`The expression for the eEectromagnetic torque may be written in machine vari—
`CS from (5.3-4) by letting 2:” = anrjm. Thus
`
`
`
`P
`
`Lu g LIN
`3
`
`a) {M [(‘33 —
`
`.
`
`J5
`
`~%— ‘2— (if; - 2i”_.-I's.\ + 21'“_\1}_.-) cos20,]
`
`i
`
`m§z{1)cosflr + -,i- (in. — tm)stnt‘)r] }
`
`

`

`
`
`THEORY OF BRUSHLESS dc MACHINES
`
`254
`
`
`
`where LM and LM are defined by (5.2—11) and (5.242), respectively. The a
`
`expression for torque is positive for motor action. The torque and speed may;
`related as
`
`
`
`
`
`
`where J is kgml ; it is the inertia of the rotor and the connected load. Because:
`
`will be concerned primarily with motor action, the torque 21 is positive [Or a tor
`
`load. The constant B," is a damping coefficient associated with the rotational sysl
`
`of the machine and the mechanical load.
`It has the units Nvmvs per radian
`
`mechanical rotation. and it is generaliy small and often neglected.
`
`
`6.3 VOLTAGE AND TOFIOUE‘EQUATIONS IN ROTOR
`REFERENCE—FRAME VARIABLES
`
`
`The voltage equations in the rotor reference frame may be written directly fro
`
`
`Section 3.4 with w 2 car or from (5.5-! ) with the direction of positive stator CUFTEEES.
`into the machine.
`
`
`
`
`
`2.
`
`2 '
`
`.
`
`T2 = J (;)_pa), + 8,” (E5)mr +71},
`
`0
`L15
`
`1;: +1;
`lot-
`
`1
`0
`
`To be consistent with our previous notation, we have added the superscript r to R1”.
`expanded form we have
`
`,
`= mg, + twigs + p23;
`v9,r _ -r
`*r
`r
`1rd, — rad; ~— curx.“ + p20,;
`V0: = me. + Pia;
`
`where
`
`~r
`r 7
`it}: _ Llflqs
`tr
`r
`.r
`(I! = Ldlds + Am
`AD: 2 Lis‘l'fls
`
`
`
`
`
`
`
`
`where Lq = Ll; + Ling and LJ 5 Li's + Lama'-
`
`
`
`

`

`vOLTAGE AND TORQUE EOUATtONs IN ROTOR REFERENCEFHAME
`
`265
`
`The expression for electromagnetic torque may be written from (5.6—9) as
`
`Substituting (63 -7)(63--9} into (6 3——4)~(6.3—6) and because pal” = O, we can
`above
`in
`
`
`my be
`_
`Wile
`
`v;‘—= (r; + {71.4}th i mild?“ + to, am
`(6.310)
`
`.
`93.5: (r +1114)?Ir— cola-“f”
`(6.3-11)
`6‘2"?)
`
`905.2(1) + pL-‘sflflr
`(63712)
`156 We
`
`torque
`system
`lian of
`
`
`Tl“ :- (%)(%:)(A:£tirgas _ 197s:dc)
`(6'3'13)
`
`I
`
`
`
`substituting (6.3-?) and (6.3-8) into (6.3—13) yields
`
`(6.3-34)
`21,: (i) (P)a;gig + {artthe;35,51
`
`y from
`urrents "
`
`'
`
`I
`
`{6.3-1) ,
`
`{6.3-2
`
`(6.3-3
`
`.
`I'
`
`) 3'
`m'
`
`(6 3 4
`' 3
`—5
`(6.3 _
`
`The electromagnetic torque is positive for motor action.
`As pointed out in the previous section, the state of the sensors provides us with
`information regarding the position of the poles and thus the position of the q and d
`axes In other words when the machine15 supplied from an inverter. it is possible by
`' ontrolling the firing oftheinverter, to change the values of vf] and ugh. Recall that 0
`n the transformation equation to the rotor reference frame can be written as
`
`()0,
`“Jr = “a?
`
`(6.34 S)
`
`I For purposes of discussion, iet us assume that the applied stator voltages are
`Inusoidat so that
`
`'
`
`vm = 2v, cos 0“.
`2
`vi): = fiVJCOS Um "“ ”‘1:
`3
`2
`
`a“ = fivfios (0” + ~32)
`
`(6.3M 6)
`6.3-17
`
`(63-18)
`
`the stator voltages will have a
`When the machine is supplied from an inverter.
`lapped waveform. Nevertheless, [6.3-] 6)—(6.3-18) may he considered as the funda-
`Flifll components of these stepped phase voltages: Also,
`
`(3,:
`
`lid....
`{It
`
`

`

`(6444)
`
`Now, because co, r. we at all times, 45., is a constant during steady—state operation 0
`is changed by advancing or retarding the firing of the inverter relative to the roto
`position. in other words. we can, at any time, instantaneously adjust (12', by appropr
`ate firing of the inverter. thereby changing the phase relationship between the funda
`mental component of the 3ephase stator voltages and the rotor (permanent magnet
`In most applications. however div is fixed at zero so that as far as the fundament
`component is concerned “as is always zero and 12;, =\/§V_‘~
`In Section 2.6 the time—domain block diagrams and the state equations wer
`derived for a dc machine. By following the same procedure, the timedomain di
`gram and state equations can be established for the brushiess dc machine directl
`from the equations given above. Because this essentially is a repeat of the worki
`Section 2.6, the development is left as an exercise for the readers.
`
`265
`
`THEORY OF BRUSHLESS dc MACHENES
`
`The brushless dc machine is. by definition. a device where the frequency oflh
`fundamental component of the applied stator voltages corresponds to the speed 0
`the rotor, and we understand that this is accomplished by appropriately firing m
`inverter sapplying (driving) the machine. Hence, in a brushless dc machine, (on i
`(63—19) is 0),; and if (6.3-16)—(6.3—18) are substituted into Ki: we obtain
`
`' — fitgcosqbv
`
`— 2vjsin¢v
`
`95,: : Gait _ 6r
`
`6.4 ANALYSiS OF STEADY-STATE OPERATiON
`
`For steady—state operation with balanced, sinusoidai applied stator voltages, (6.3-10};
`and (6.3—11) may be written as
`
`"I
`V; = r51;I «1- (13,141; + 01,1”
`".
`Vdr = nil}, — erqlgi
`
`where uppercase letters denote steadystate (constant) quantities. It:5 clear that Ft[5'
`always constant The steady-state torque is expressed from (6.3- 14) with uppercase.
`letters as
`
`.
`
`(6.4-3)
`
`.3
`
`r 2 (2) (2) {121; + (L3 “ L4”; 5;.
`
`_
`
`It is possible to establish a phasor voltage equation from (6.4-! )and (6.4-2) Siml"
`lar to that for the synchronous machine. For this purpose let us write (6.342) as _
`
`0.... —-a + 4)..
`
`

`

`ANALYSIS OF STEADY-STATE OPERATION
`
`267
`
`For steady—state operation ¢,, is constant and represents the angular displacement
`between the peak value of the fundamental component of v,” and the q axis fixed
`in the rotor. if we reference the phasors to the q axis and letIt be aiong the positive
`real axis of the‘ stationary‘ phasor diagram. then (A, becomes the phase angle of V“
`and we can write
`
`Var —-V6’4" : Vcosq’il, wtjifisinr/
`
`(6.4—5)
`
`:Comparing (6.4-5) with (63-20) and (6.3~21), we see that
`
`WM: 2 V; we
`
`(6.46)
`
`if we go back and write equations for the 3-phase currents similar to (6.346)-
`E(6.3*18) in terms of ii“. then (6.3—22) would be in terms of 0,; and ‘i’t rather than
`i
`p and qbv; however, we would arrive at a similar relation for current as (6.4—6).
`particular.
`
`f1“: :15”J {is
`
`inm m (Iti‘jl1;,
`
`-- Substituting (6.4—1) and (6.4—2) into (6.4—6) and using (6.4-7) and (6.4-8) yields
`
`flat 3 (7:5 "i”jerqiim ‘i‘ Ea
`
`(64’9)
`
`”Wat + chime”
`
`(6.4-10)
`
`Chapter 5 the phasor diagram for the synchronous machine was referenced to the
`hosorof If,” (Vm). which was positioned along the real axis. Note that in the case of
`synchronous machine the phase angle of E", was 6 [see (59—18)]. whereas 5,. for
`brushless dc machine [see (6.4-10)] is along the reference (real) axis.
`
`oinmon Operating Mode
`
`rs
`
`m our earlier discussion we are aware that the values of V’ and are deter—
`by thatfiring of the driving inverter. However the condition with tbv—* 0,
`, effiupon V’: ft! and m—0 is used in most applications in this case,
`*2) may be solved for L,"In terms oi'i'r
`.
`-
`1,,
`sin“), ‘1;
`
`ror¢.,=o
`
`(sci-H)
`
`

`

`253
`
`THEORY OF BFIUSHLESS de MACHINES
`
`Substituting (6.4-1 1) into (6.4.2) yields
`
`.24 - 214
`
`
`Vt}? = (mm—W,“' m’ iL‘!) I; -l- rte/Li;
`I}
`
`for 4)” = 0
`
`'
`(6.44
`
`We now start to see a similarity between the voltage equation for the bmshl'
`dc machine operated in this mode (43‘, = 0) and the dc machine discussgd'j-
`Chapter 2 From (2.3-!) the steady-state armature voltage equation of a dc 3hr]:
`machine is
`
`Va“—“‘lat}, -i- GJILAFl'f
`
`if we neglect wEL,,Ld in (6.4—12) and if we assume the field current is constanti
`(6.4—13). then the two equations are identical in form. Let's note another similarity
`If we neglect the inductances, or set Lq = Lav, then the expression for the torqti
`given by (6.4-3) is identical in form to that of a dc shunt machine with a constan
`field current given by (2.375). We now see that the brushless dc motor is calle _
`brushiess dc motor, not because it has the same physical configuration as a (1
`machine but because its terminal characteristics may be made to resemble the
`of a dc machine. We must be careful, however, because in order for (64—12) an
`(64-13) to be identical in form, the term wELqLd must be significantly less tha
`:3. Let us see what effects this term has upon the torque versus speed characteristic
`If we solve (6.4 l2) forlqand we take that result along with (64-11) for 1;, an
`substitute these expressioqns into the expression for TE (6.43) and if we assum
`that Lg 2 L1,, we obtain the foilowing expression for torque:
`
`(6.445) abstitut
`
`fr
`
`TEE) (in) ’1'"
`
`r2 -l- (all?
`
`(V;was)
`
`rorqb..=0
`
`I
`
`(5.4-14)
`
`We have used LI for Lg and LL; because we have assumed that Lq = Lav.
`The steady- state torque—speed characteristics for a typical brushless dc motor are:
`Shown'tn Fig.2 6.4—1. Therein Lmq__ Ln”; and rfJU—— 0; hence Fig 6—4 1
`is a plot of
`(6.4- l4). If wELL~is neglected, then (6.4—14) yields a straight line T versus (1),. cha '
`acteristic for a constant VLF. Thus, if wELf could be neglected, the plot shown in
`Fig. 6.4—] would be a straight line. Although the Te versus at, is approximately linear
`over the region of motor operation where Tc 2 O and a), Z 0, it is not linear over the
`complete speed range. in fact, we see from Fig. 6.4-] that there appears to be a max—
`imum and minimum torque. Let us take the derivative of (64-14) with respect to a),
`and set the result to zero and solve for (1),. Thus, zero siope of the torque versus speed
`characteristics for L, 2 L, = LS, V6; = t/Z—WI. and Va: = 0 ((fiv = 0) occurs at
`
`...)I for (a, = 0
`
`

`

`Torque—speed characteristics of a brushless dc motor with Law =L,,,.;
`figure 6.4-1
`Lq =10, ti)» = 'v V; = five and
`5, : 0.
`
`Vperating Modes Achieuable by Phase Shifting the Applied Voltages
`
`Let us return to (6.4- i )—(6.4-3), and this time we wit! not restrict if), to zero. Hence.
`mm (6.4-2) we have
`
`
`
`ANALYSIS OF STEADY-STATE OPERATION
`
`259
`
`Due to the high reluctance of the magnetic material in the (1 axis, L; is less than L“.
`fer'some machine designs. in most cases. whether L,, > L0. or L, < L4. saiiency has
`tily secondary effects upon the torque—speed characteristics over the region of
`terest (7} 2 0 and w, 2 0). This is showrt in Fig. 6.4—2 when: Ln“, : 0.6L,” for
`e machine considered in Fig. 6.4—1.
`
`Figure 6.4-2 Same as Fig. 6.4—] with Lmq : 0.6L,,,.;.
`
`

`

`270
`
`THEORY OF BHUSHLESS dc MACHINES
`
`Let us again set L, =2 L; 2 L" which simplifies our work. in this case, T, and [r
`differ only by a constant multiplier. Solving (6.4-14) for 1;: yields
`9
`.
`ta
`'.
`L
`‘
`[41 "if; + £3):ng (V115
`rs his“)?
`Amwr')
`(6-448
`r __
`5
`r _w§. yr
`
`__ Mr
`
`We see from (63-20) and (63-21) that if we consider only the fundamental comp'
`nent of the 3-phase applied stator voltages, then we have
`
`Vras
`
`v5,
`
`x/ivxcosday
`
`afivssinqb‘.
`
`"
`
`ivfierein (priis—detei‘minEd by the switching of ihe inverter. Substituting-(61.419) amt
`(6.4—20) into (6.4—I 8) yields
`‘
`
`1' m9"
`
`firs Vs
`2
`2
`2‘
`r: + (0,1.if
`
`[cos gb,. + (1:; sin 47., —— falter}
`
`Here we are using time constants. or what appear to be time constants. in a steady-
`state equation. This is not normaily done; however, it allows us to work more Cffif
`cient