SECOND EDITION
`Pawl C. Krause
`OlegWa
`Seott li>
`
`

`
`ANALYSIS OF ELECTRIC
`MACHINERY
`AND DRIVE SYSTEMS
`
`Second Edition
`
`PAUL C. KRAUSE
`OLEG WASYNCZUK
`SCOTT D. SUDHOFF
`Purdue University
`
`Society, Sponsor
`IEEE Power Engineering
`
`IEEE
`PRESS
`SERIES
`
`OH POWER
`ENGINEERING
`
`IEEE Press Power Engineering Series
`
`Series Editor
`Mohamed E. EI-Hawary,
`
`A. IEEE
`v PRESS
`�WILEY­
`�INTERSCIENCE
`
`A JOHN WILEY & SONS, INC. PUBLICATION
`
`

`
`
`
`Copyright © 2002 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
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`stored in a retrieval
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`Printed in the United States of America.
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`ISBN 0-471-14326-X
`
`13 14 15 16 17 18 19 20
`
`

`
`CONTENTS
`
`PREFACE
`Chapter 1 BASIC PRINCIPLES FOR ELECTRIC
`MACHINE ANALYSIS
`I I
`1.1 Introduction
`1.2 Magnetically
`Coupled Circuits I
`1.3 Electromechanical
`Energy Conversion I II
`1.4 Machine Windings and Air-Gap MMF I 35
`1.5 Winding Inductances
`and Voltage Equations I 47
`References I 58
`Problems I 58
`
`Chapter 2 DIRECT-CURRENT MACHINES
`
`xi
`
`1
`
`67
`
`2.1 Introduction I 61
`
`2.2 Elementary
`Machine I 68
`Direct-Current
`2.3 Voltage and Torque Equations I 76
`Machines I 78
`2.4 Basic Types
`of Direct-CuiTent
`2.5 Dynamic Characteristics of
`and Shunt de Motors I 88
`Permanent-Magnet
`2.6 Time-Domain
`Block Diagrams and State Equations I 92
`2.7 Solution
`I 98
`
`of Dynamic Characteristics by Laplace Transformation
`References I I 04
`Problems I I 05 '
`
`vii
`
`

`
`viii CONTENTS
`
`Chapter 3 REFERENCE-FRAME THEORY
`
`1 09
`
`3.1 Imroduction
`I 109
`3.2 Background I 109
`3.3 Equations
`Changes of Variables I I ll
`of Transformation:
`3.4 Stationary
`to the Arbitrary
`Transformed
`Variables
`Circuit
`Frame I 1 1 5
`Reference
`3.5 Commonly Used Reference
`Frames I 123
`Frames I 124
`Between Reference
`3.6 Transformation
`Set I 1 26
`3.7 Transformation
`of a Balanced
`I 1 27
`Phasor Relationships
`Steady-State
`3.8 Balanced
`Voltage Equations I 130
`Steady-State
`3.9 Balanced
`from Several Frames of Reference I 133
`Observed
`3.10 Variables
`I 137
`References
`Problems I 138
`
`141
`
`Chapter 4 SYMMETRICAL INDUCTION MACHINES
`I ! 41 I
`4. 1
`Introduction
`I 142
`4.2
`in Machine Variables
`Voltage Equations
`I 146
`4.3
`in Machine Variables
`Torque Equation
`4.4
`for Rotor Circuits I 147
`of Transformation
`Equations
`I 149
`4.5
`Variables
`Reference-Frame
`in Arbitrary
`Voltage Equations
`Variables I 1 53
`4.6
`Reference-Frame
`in Arbitrary
`Torque Equation
`4.7
`Frames I 1 54
`Commonly Used Reference
`Per Unit System I !55
`4.8
`I 157
`4.9
`of Steady-State Operation
`Analysis
`I 165
`Characteristics
`Free Acceleration
`4.10
`4.11
`Viewed from Various
`Characteristics
`Free Acceleration
`Frames I 172
`Reference
`During Sudden Changes in Load Torque I 174
`4.1 2 Dynamic Performance
`4.13 Dynamic Performance
`During a 3-Phase Fault at the
`Machine Terminals I 181
`4.14 Computer Simulation
`Frame I 184
`in the Arbitrary Reference
`I 1 87
`References
`Problems I 188
`
`Chapter 5 SYNCHRONOUS MACHINES
`5 . 1 Introduction
`I 191
`I 192
`5.2 Voltage Equations
`in Machine Variables
`I 197
`5.3 Torque Equation
`in Machine Variables
`
`191
`
`

`
`19
`
`CONTENTS iX
`I 198
`Reference-Frame
`Variables
`in Arbitrary
`Stator Voltage Equations
`5.4
`Voltage Equations in Rotor Reference-Frame
`Variables:
`I 200
`5.5
`Park's Equations
`Variables I 206
`in Substitute
`Torque Equations
`5.6
`Rotor Angle and Angle Between Rotors I 207
`5.7
`Per Unit System I 209
`5.8
`I 210
`
`Analysis of Steady-State Operation
`5.9
`in Input Torque I 219
`
`Dynamic Performance During a Sudden Change
`5.10
`
`Dynamic Performance During a 3-Phase Fault at the
`5.11
`I 225
`Machine Terminals
`I 229
`Approximate Transient Torque Versus Rotor Angle Characteristics
`
`
`
`5.12
`
`
`
`Comparison of Actual and Approximate Transient Torque-Angle
`5.13
`Characteristics
`
`During a Sudden Change in Input Torque:
`Limit I 232
`
`First Swing Transient Stability
`Comparison of Actual and Approximate Transient Torque-Angle
`
`
`
`5.14
`
`
`Characteristics During a 3-Phase Fault at the Terminals: Critical
`Time I 239
`Clearing
`I 242
`Equal-Area Criterion
`5.15
`I 246
`Computer Simulation
`5.16
`I 255
`References
`Problems I 256
`Chapter 6 THEORY OF BRUSHLESS de MACHINES
`I 261
`6.1 Introduction
`I 261
`6.2 Voltage and Torque Equations in Machine Variables
`
`I 264
`6.3 Voltage and Torque Equations
`in Rotor Reference-Frame Variables
`I 266
`
`
`6.4 Analysis of Steady-State Operation
`6.5 Dynamic Performance
`I 274
`I 281
`References
`Problems I 281
`Chapter 7 MACHINE EQUATIONS IN OPERATIONAL
`
`261
`
`IMPEDANCES AND TIME CONSTANTS
`
`283
`
`with and G(p) for a Synchronous Machine 7.3 Operational Impedances
`
`I 283
`7 .I Introduction
`Form I 284
`7.2 Park's Equations in Operational
`
`
`
`
`Four Rotor W�ndings I 284
`I 288
`7.4 Standard
`
`Synchronous Machine Reactances
`Machine Time Constants I 290
`7.5 Standard Synchronous
`
`Machine Time Constants I 291
`7.6 Derived Synchronous
`
`

`
`X CONTENTS
`
`I 294
`7.7 Parameters from Short-Circuit Characteristics
`
`
`I 301
`
`7.8 Parameters from Frequency-Response
`Characteristics
`I 307
`References
`Problems I 308
`
`Chapter 8 LINEARIZED MACHINE EQUATIONS
`I 31 I
`8.1 Introduction
`I 312
`
`8.2 Machine Equations to Be Linearized
`I 313
`
`8.3 Linearization of Machine Equations
`I 323
`
`
`8.4 Small-Displacement Stability: Eigenvalues
`Machines I 324
`
`8.5 Eigenvalues of Typical Induction
`Machines I 327
`
`8.6 Eigenvalues of Typical Synchronous
`I 330
`
`Formulation 8. 7 Transfer Function
`
`I 335
`References
`Problems I 335
`
`311
`
`337
`
`
`
`9.3 Induction Machine Large-Excursion Behavior Predicted by
`
`Chapter 9 REDUCED-ORDER MACHINE EQUATIONS
`I 337
`9.1 Introduction
`I 338
`
`9.2 Reduced-Order Equations
`
`
`
`I 343
`
`Reduced-Order Equations
`Behavior Predicted 9.4 Synchronous Machine Large-Excursion
`
`
`
`I 350
`by Reduced-Order Equations
`I 354
`
`
`9.5 Linearized Reduced-Order Equations
`I 354
`
`
`
`
`Equations Reduced-Order 9.6 Eigenvalues Predicted by Linearized
`Models I 355
`
`9.7 Simulation of Reduced-Order
`I 358
`9.8 Closing Comments and Guidelines
`I 358
`References
`Problems I 359
`
`Chapter 1 0 SYMMETRICAL AND UNSYMMETRICAL
`2-PHASE INDUCTION MACHINES
`I 361
`10.1 Introduction
`Machines I 362
`I 0.3 Voltage and Torque Equations
`2-Phase Induction 10.2 Analysis of Symmetrical
`
`
`
`
`in Machine Variables for
`Machines I 371
`Unsymmetrical 2-Phase Induction
`
`
`Reference-Frame I 0.4 Voltage and Torque Equations in Stationary
`
`Machines I 373
`
`Variables for Unsymmetrical 2-Phase Induction
`
`361
`
`

`
`311
`
`337
`
`354
`
`361
`
`CONTENTS Xi
`
`395
`
`of Unsymmetric
`al
`Operation
`Analysis of Steady-Stat
`e
`2_phase Induction Machmes I 377
`_
`105
`Machines I 383
`Induction
`10_6 Single-Phase
`References I 393
`Problems I 393
`Chapter 11 SEMICONTROLLED BRIDGE CONVERTERS
`11.1 Introduction I 395
`I 395
`Converter
`Load·Commutated
`1 1.2 Single-Phase
`I 406
`Converter
`11.3 3-Phase Load Commutated
`I 425
`References
`Problems I 425
`Chapter 1 2 de MACHINE DRIVES
`I 427
`Introduction
`12.1
`for de Drive Systems I 427
`12.2 Solid-State Converters
`Drives I 431
`
`12.3 Steady-State and Dynamic Characteristics of ac/dc Converter
`I 443
`
`One-Quadrant de/de Converter Drive
`12.4
`Drive I 460
`
`Two-Quadrant de/de Converter
`12.5
`I 463
`12.6
`Four-Quadrant de/de Converter Drive
`I 466
`de/de Converter
`12.7 Machine Control with Voltage-Controlled
`I 468
`de/de Converter
`12.8 Machine Control with Current-Controlled
`I 476
`References
`Problems I 476
`
`427
`
`481
`
`Chapter 1 3 FULLY CONTROLLED 3-PHASE BRIDGE
`CONVERTERS
`I 481
`13.1 Introduction
`I 481
`13.2 The 3-Phase Bridge Converter
`I 487
`13.3 180° Voltage Source Operation
`Modulation I 494
`13.4 Pulse-Width
`I 499
`13.5 Sine-Triangle
`Modulation
`I 503
`13.6 Third-Harmonic Injection
`I 506
`13.7 Space-Vector Modulation
`I 510
`13.8 . Hysteresis
`Modulation
`I 512
`13.9 Delta Modulation
`13.10 Open-Loop Voltage and Current Control I 513
`I 516
`
`13.11 Closed-Loop Voltage and Current Controls
`
`

`
`Xii CONTENTS
`I 520
`I 521
`References
`Problems
`C hapter 14 INDUCTION MOTOR DRIVES
`I 525
`14.1 Introduction
`Hertz Control I 525
`14.2 Volts-Per-
`Slip Current Control I 532
`14.3 Constant
`Control I 540
`14.4 Field-Oriented
`Control I 544
`14.5 Direct Rotor-Oriented
`Field-Oriented
`Control I 546
`14.6 Robust Direct Field-Oriented
`Control I 550
`I 4. 7 Indirect
`Rotor Field-Oriented
`I 554
`14.8 Conclusions
`I 554
`References
`Problems I 555
`
`525
`
`557
`
`C hapter 1 5 BRUSHLESS de MOTOR DRIVES
`15.1 Introduction
`I 557
`Drives I 558
`15.2 Voltage-Source
`Inverter
`Source I 560
`15.3 Equivalence
`of VSI Schemes to Idealized
`of VSI Drives I 568
`15.4 Average-Value
`Analysis
`of VSI Drives I 571
`15.5 Steady-State
`Performance
`of VSI Drives I 574
`15.6 Transient
`and Dynamic Performance
`Harmonics I 578
`15.7 Consideration
`of Steady-State
`Speed Control I 582
`15.8 Case Study: Voltage-Source
`Inverter-Based
`Drives I 586
`15.9 Current-Regulated
`Inverter
`Drives I 590
`I 5.10 Voltage Limitations
`Inverter
`of Current-Source
`I 591
`15.1 1 Current Command Synthesis
`Drives I 595
`15.12 Average-Value Modeling of Current-Regulated
`Inverter
`15.13 Case Study: Current-Regulated
`I 597
`Speed Controller
`Inverter-Based
`I 600
`References
`Problems I 600
`
`Relations, Constants and
`Appendix A Trigonometric
`and Abbreviations
`Conversion Factors,
`
`INDEX
`
`603
`
`605
`
`

`
`5
`
`PREFACE
`
`in 1986 by
`The first edition of this book was written by Paul C. Krause and published
`
`
`
`
`McGraw-Hill. Eight years later the same book was republished by IEEE Press with
`
`Oleg Wasynczuk and Scott D. Sudhoff added as co-authors. The focus of the first
`
`
`
`
`
`was the analysis of electric machines using reference frame theory, wherein
`edition
`
`
`
`Not only has this the concept of the arbitrary reference frame was emphasized.
`
`
`
`
`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 goal of this 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.
`Although some of the material has been rearranged or revised, and in some cases
`
`
`
`
`
`
`
`
`
`
`eliminated, such as 3-phasc 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 11-15 on electric drive systems is new. In particular, the ana­
`
`
`
`
`lysis of conveners used in electric drive systems is presented in Chapters II and 13
`
`
`
`
`while de, induction, and brushless de motor drives are analyzed in Chapters 12, 14,
`
`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 comp�ter traces wherein variables
`
`are often portrayed in different
`
`xiii
`
`

`
`XiV PREFACE
`
`frames of reference so that the student is able to appreciate tl)e advantages and sig­
`
`
`
`
`
`
`
`
`nificance of the transformation used.
`
`
`has The material presented in this text can be used most beneficially if the student
`
`
`
`
`
`
`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 I, perhaps some of
`Chapter 2 covering
`most of Chapter 3 covering
`basic de machines,
`
`reference frame
`4, 5, and 6 covering
`
`
`
`theory, and the beginning sections of Chapters
`
`induction, syn­
`
`
`chronous, and brushless de machines.
`Some of the material that would be of interest only to the electric power engineer
`
`
`
`
`
`has been reduced or eliminated from that given in the first edition. However, the
`4 and 5 on induction
`
`
`material found in the final sections in Chapters
`and synchronous
`7), and reduced-order
`
`
`machines as well as operational impedances (Chapter 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
`and the staff of IEEE Press and John Wiley & Sons.
`
`ticular Mohamed E. El-Hawary,
`
`2001
`\Vest Lnfayerre, Indiana
`
`November
`
`PAUL c. KRAUSE
`OLEG WASYNCZUK
`Scorr D. SuoHoFF
`
`

`
`
`
`REFERENCE-FRAME THEORY
`
`3.1 INTRODUCTION
`The voltage equations that describe the performance of induction and synchronous
`
`
`
`
`
`
`
`machines were established in Chapter I. We found that some of the machine induc­
`
`
`
`tances are functions of the rotor speed, whereupon the coefficients of the differential
`
`
`
`
`
`
`equations (voltage equations) that describe the behavior of these machines are time­
`
`
`
`varying except when the rotor is stalled. A change of variables is often used to reduce
`
`
`
`
`the complexity of these differential equations. There are several changes of variables
`
`
`
`that are used, and it was originally thought that each change of variables was different
`[ 1-4 ]. It was later learned
`
`
`and therefore they were treated separately
`that all changes
`in one [5,6].
`
`
`
`of variables used to transform real variables are contained
`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 describe resistive, inductive, and capacitive circuit elements. By this
`
`
`
`
`approach, many of the 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
`
`In t11c late 1920s, R. H. Park'[ I] introduced
`
`
`
`
`
`a new approach to electric machine ana­
`
`
`
`
`
`lysis. He formulated a change of variables which, in effect, replaced the variables
`109
`
`

`
`11 0 REFERENCE-FRAME
`
`THEORY
`
`currents, and flux linkages) associated with the stator windings-of a syn­
`
`
`
`
`(voltages,
`3.3
`CH
`
`
`
`
`
`
`chronous machine with variables associated wit:1 fictitious windings rotating
`with
`
`
`
`
`
`
`the rotor. In other words, he transformed, or referred, the stator variables to a frame
`
`
`
`which revolutionized transformation, of reference fixed in the rotor. Park's
`electri
`c
`Altl
`
`
`
`
`
`machine analysis, has the unique property of eliminating all time-varying induc­
`tim
`
`
`
`
`
`from the voltage equations of the synchronous machine whi.ch occ·Jr due to
`tances
`vari
`
`
`
`
`
`
`(I) electric circuits in relative motion and (2) electric circuits with varying·;nagnetic
`ass•
`reluctance.
`use
`In the late 1930s, H. C. Stanley [2] employed a change �f vari::>.bles in the
`
`
`
`of1
`
`
`
`
`analysis of induction machines. He showed that the time-varying inductances
`in
`sen
`in relative
`
`
`
`
`
`the voltage equations of an induction machine due to electric circuits
`cia1
`
`
`
`motion could be eliminated by transforming the variables ass�ciated with the rotor
`var
`
`
`
`
`
`
`windings (rotor variables) to variables associated with fictitious statio:::ary windings.
`frru
`
`
`
`
`In this case the rotor variables are transformed to a frame reference fixed in the
`cia
`stator.
`ac-
`or time­
`
`
`
`G. Kron [3] introduced a change of variables that eliminated the position
`alSo
`
`
`
`
`
`varying mutual inductances of a symmetrica! induction machine by �ransforming
`
`
`
`
`
`both the stator variables and the rotor variables to a reference frame rotating in syn­
`ral
`
`
`
`
`chronism with the rotating magnetic field. This reference frame is commonly
`ma
`
`
`
`
`referred to as the synchronously rotating reference frame.
`mr
`D. S. Brereton
`et al. [4] employed a change of variables that also eliminated the
`
`
`
`all
`cir
`
`
`
`
`
`
`time-varying inductances of a symmetrical induction machine by transforming the
`the
`
`
`
`
`
`
`stator variables to a reference frame fixed in the rotor. This is essentially Park's
`
`
`transformation applied to induction machines.
`sta
`Park, Stanley, Kron, and Brereton et al. developed changes of variables, each
`
`
`
`
`
`. Consequently,
`
`
`
`
`of which appeared to be uniquely suited for a particular application
`
`
`
`
`
`each transformation was derived and treated separately in literz.ture 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
`wl
`
`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.
`
`
`
`All known real transformations may then be obtained by simp�y assigning the
`
`appropriate speed of rotation, which may in fact be zero, to this so-called
`
`
`arbitra1y reference frame.
`
`It is interesting to note that this transformation is
`
`
`
`
`
`
`
`sometimes referred to as the "generalized rotating real transformation," which
`In
`
`
`
`may be somewhat misleading because the reference frame need not rotate.
`
`
`
`any event, we will refer to it as the arbitrary reference frame as did the
`
`
`
`[5]. Later, it was noted that the stator variables of a synchronous
`originators
`
`
`
`
`machine could also be referred to the arbitrary reference frame [6]. However, we
`
`
`
`will find that the time-varying 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
`machines.
`
`It
`
`

`
`Chapter 6
`
`dc.MACHINES
`THEORY OF BRUSHLESS
`
`6.1 INTRODUCTION
`
`control
`a small horsepower
`used as
`widely
`de motor is becoming
`The brushless
`magnet
`permanent
`of a 3-phase
`appearance
`has the physical
`This device
`motor.
`a de voltage
`converts
`from an inverter that
`that is supplied
`machine
`synchronous
`instanta­
`corresponding
`with frequency
`alternating-current
`(ac) voltages
`to 3-phase
`has the terminal
`and
`combination
`achine
`The inverter-m
`to the rotor speed.
`neously
`(Te vs. w,) characteristics
`hence the name brush less de moror. In this brief chapter, equations
`motor operation;
`during
`those of a de shunt machine
`resembling
`output
`The operation
`of
`of a brush less de machine.
`the operation
`which describe
`are derived
`15. In this chapter
`we look at the
`in Chapter
`considered
`brush less de motor drives is
`voltage
`stator
`applied
`3-phase
`performance of the brush less de motor with balanced
`us to become familiar
`This allows
`speed.
`to the rotor
`corresponding
`With frequency
`with the salient
`without
`combination
`feature·s of the inverter-motor
`operating
`becoming involved
`that occur due to the
`of the phase voltages
`with the harmonics
`switching of the inverter.
`
`�ected stator
`
`6.2 VOLTAGE AND TORQUE EQUATIONS
`IN MACHINE VARIABLES
`A 2-polc, brushless
`is depicted
`in Fig. 6.2-1.
`de machine
`wye-con­
`It has 3-phase,
`It is a synchronous
`machine.
`and a permanent magnet
`rotor.
`dings
`wi
`he stator windings
`120°, each with Ns equivalent
`displaced
`windings
`are identical
`turns and resistance
`r.,.. For our analysis
`are
`windings
`we will assume that the stator
`261
`
`.n
`
`

`
`262 THEORY OF BRUSHLESS de MACHINES
`The three sensors shown in Fig. 6.2-1 are Hall effec
`distributed.
`sinusoidally
`its output is nonzero;
`When the north pole is under a sensor,
`devices.
`the stator is supplied
`its output is zero. In most applications
`pole under the sensor,
`to the rotor speed.lbe
`corresponding
`at a frequency
`is switched
`from an inverter that
`logic for the invene r.
`the switching
`are used to determine
`of the three sensors
`states
`
`with a sout�
`
`Sensor
`
`Sensor
`
`+
`
`N,
`r,
`
`Figure 6.2-1 Two-pole,
`de machine.
`brushless
`3-phase
`
`Wh(
`as 1 WOl
`win
`and
`are
`win
`non
`
`den
`
`ab\1
`
`1
`
`

`
`Jl effect
`1 a south
`supplied
`eed. The
`inverter.
`
`VOLTAGE AND TORQUE EQUATIONS IN MACHINE VARIABLES 263
`over the rotor as shown in
`not positioned
`In the actual machine the sensors are
`they are placed over a ring that is mounted on the shaft external
`Fig. 6.2-1.
`Instead
`We will return to these sensors
`as the rotor.
`and magnetized
`to the stator windings
`the voltage
`to establish
`It is first necessary
`and the role they play later in this analysis.
`of the permanent
`the behavior
`that can be used to describe
`and torque equations
`machine.
`magnet synchronous
`in machine variables
`are
`equations
`The voltage
`= r.,iahcs
`+ pJ..tlht'.<
`VtliJ<'X
`= [J,,, fbs fcs]
`(ft1bc.</
`r, = diag[ r_,
`1�,. 1'5 J
`
`(6.2-1)
`
`(6.2-2)
`(6.2-3)
`
`where
`
`may be written
`The flux linkages
`
`(6.2-4)
`and ( 1.5-29)-(
`where L .• may be written
`1.5-31)
`from ( 1.5-25)-( 1.5-27)
`or directly
`from (5.2-8).
`Also,
`
`(6.2-5)
`
`where;.;, is the amplitude
`
`of pl..�
`
`magnet
`by the permanent
`established
`of the flux linkages
`In other words the magnitude
`as viewed from the stator phase windings.
`Damper
`induced in each stator phase winding.
`voltage
`would be the open-circuit
`conductor
`magnet is a poor electrical
`because the permanent
`are neglected
`windings
`and the eddy currents
`the magnets
`securing
`materials
`that flow in the nonmagnetic
`without significant
`can be tolerated
`currents
`are small. Hence, large armature
`We have assumed by (6.2-5) that the voltages
`induced in the stator
`demagnetization.
`voltages.
`For
`litude sinusoidal
`magnet are constant-amp
`by the permanent
`windings
`I and 2.
`see references
`induced voltages
`nonsinusoidal
`The expression
`in machine vari­
`torque may be written
`electromagnetic
`for the
`by letting),;, = L,difi1• Thus
`T -(p) { ( L,J -L,") [ (
`·2 I ·2 I ·2 . . . . 2. . ) . 20
`ables from (5.3-4)
`'"·' -2'bs -2'n -ltl.rliJs
`e -2
`Sill r
`+ lb.<lcs
`-ltl.rln
`-T lh.,ln -'"·''"·
`1- v'3 ( '2 '2 2. . 2. . ) 20 ]
`' + lmlc.< COS r
`sin 0,]} (6.2-6)
`+ ;.;, [ C···
`-�h .. -� i,._,) cosO,+ .;; (h., -ic.,)
`
`3
`
`

`
`264 THEORY OF BRUSH LESS de MACHINES
`
`where Lmq and Lmd are defined by (5.2-11)
`and (5.2-12),
`respectively.
`The abov
`The torque and
`for motor action.
`for torque is positive
`expression
`related
`as
`
`speed may �
`
`load. Because we
`where J is kg· m2 ; it is the inertia
`load. It has the units N · m · s per radian of
`
`of the rotor and the connected
`for a torque
`the torque TL is positive
`primarily with motor
`action,
`will be concerned
`load. The constant Bm is a damping coefficient
`system
`with the rotational
`associated
`of the machine and the mechanical
`small and often neglected.
`and it is generally
`rotation,
`mechanical
`
`(6.2-7)
`
`6.3 VOLTAGE AND TORQUE EQUATIONS IN ROTOR
`REFERENCE-FRAME VARIABLES
`
`from
`directly
`frame may be written
`in the rotor reference
`The voltage equations
`stator
`positive
`with the direction of
`currents
`
`Section 3.4 with w = Wr or from (5.5-1)
`r •r � r + 1,.r
`+ Wr"-dq.f p qdOs
`= fs1qd0s
`VqdOs
`
`into the machine.
`
`( 6.3-1)
`
`where
`
`(6.3-2)
`
`(6.3-3)
`
`r to;.;
`we have added the superscript
`notation,
`with our previous
`To be consistent
`,. In
`form we have
`expanded
`
`j�sl [0]
`0 l [
`0 �ds + }.� I
`Lis lOs 0
`
`where
`
`(6.3-4)
`(6.3-5)
`(6.3-6)
`
`(6.3-7)
`(6.3-8)
`(6.3-9)
`
`V qs = rslqs WrAds PAqs
`r ·r + ,r + 1r
`r ·r 1 r _, 1 r
`v ds = rs1ds -WrAqs -,-P"'ds
`Vos = rsios
`+ PAOs
`Aqs = lf1qs
`1r L ·r
`).�s = Lt�i�s + ;.;;,
`}.Os = L,siOs
`where Lq = L1s + L,q and Ld = L1s + Lmd·
`
`

`
`VOLTAGE AND TORQUE EQUATIONS IN ROTOR REFERENCE·FRAME 265
`and because pA.:;, = 0, we can
`
`
`Substituting (6.3-7)-(6.3-9) into (6.3-4)-(6.3-6)
`write
`
`r ( L ) ·r L ·r
`Vds= rs+P d lds-Wr qlqs
`vo,· = (rs + pL,s)ios
`
`
`
`
`
`
`
`as torque may be written from (5.6-9) The expression for electromagnetic
`
`(6.3-10)
`( 6.3- 1 1)
`(6.3-12)
`
`above
`my be
`
`6.2-7)
`
`Jse we
`torque
`;ystem
`!ian of
`
`
`
`Substituting (6.3-7) and (6.3-8) into (6.3-13) yields
`
`
`
`
`
`
`
`(6.3-13
`)
`
`(6.3-14)
`
`(6.3-15)
`
`y from
`
`urrents
`
`(6.3-l )
`
`(6.3-3)
`
`In
`) A.�,.
`(6.3-4)
`(6.3-5)
`(6.3-6)
`
`The electromagnetic torque is positive for motor action.
`
`
`
`of the q and d
`As pointed out in the previous section, the state of the sensors provides us with
`
`
`
`
`
`informat
`
`
`ion regarding the position of the poles and thus the position
`
`
`
`
`axes. In other words, when the machine is supplied from an inverter, it is possible, by
`of v�s and v;i,. Recall
`that fJ,
`in the transformation
`
`
`
`controlling the firing of the inverter, to change the values
`
`
`
`equation to the rotor reference frame can be written as
`(6.3-2)
`dO,
`w,=-dt
`
`
`For purposes of discussion,
`
`sinusoidal
`so that
`Va, = hvscosO,,
`VIJs = vfi v,cos ( o.,.-2
`
`
`Vc.< = vfi vscos (oev + 2
`
`(6.3-16)
`
`(6.3-17
`)
`
`(6.3-18)
`
`37r)
`
`37r)
`
`When the machine is supplied from an inverter, the stator voltages will have a
`
`
`
`
`(6.3-7)
`
`
`
`
`
`stepped waveform. Nevertheless, (6.3-16)-(6.3-18) may be considered as the funda­
`(6.3-8)
`mental component
`.s of these stepped
`phase voltages:
`Also,
`(6.3-9)
`dO,,
`w�=- dt
`
`(6.3-19)
`
`
`
`
`
`let us assume that the applied stator voltages are
`
`

`
`266 THEORY OF BRUSHLESS de MACHINES
`
`to the speed of
`where the frequency of the
`a device
`is, by definition,
`The brush less de machine
`corresponds
`voltages
`stator
`fundamental component
`of the applied
`betWf
`w. in
`by appropriately
`that this is accomplished
`firing the
`and we understand
`the rotor,
`in the
`is wr; and if (6.3-16)-(6.
`de machine,
`Hence, in a brushless
`the machine.
`(driving)
`supplying
`inverter
`real a
`into K;, we obtain
`3-18) are substituted
`(6.3-19)
`v;s = ..fiv5cos</>v
`(6.3-20)
`V�s = -..fivs sin <l>v
`(6.3-21)
`
`and v
`
`Com
`
`where
`
`(6.3-22)
`
`If
`(6.3-
`
`Now, because Wr = We at all times, ¢v is a constant during steady-state operation or
`
`
`
`
`
`
`
`
`
`
`
`is changed by advancing or retarding the firing of the inverter relative to the rotor
`
`
`
`In other words, we can, at any time, instantaneously adjust ¢v by appropri­
`position.
`
`
`
`
`
`
`ate firing of the inverter, thereby changing the phase relationship between the funda­
`
`
`
`magnet). mental component of the 3-phase stator voltages and the rotor (permanent
`
`
`
`zero and v�s = .Jivs.
`
`
`
`In most applications, however, <l>v is fixed at zero so that as far as the fundamental
`v;j5 is always
`and
`
`component is concerned,
`In Section 2.6 the time-domain block diagrams and the state equations were
`
`
`
`
`
`
`
`
`derived for a de machine. By following the same procedure, the time-domain dia­
`
`
`
`
`gram and state equations can be established for the brushless de machine directly
`
`
`
`
`
`of the work in lly is a repeat this essentiafrom the equations given above. Because
`
`
`
`Section 2.6, the development is left as an exercise for the readers.
`
`0tv l In p:
`
`6.4 ANALYSIS OF STEADY-STATE OPERATION
`
`wh1
`
`
`
`
`For steady-state operation with balanced, sinusoidal applied stator voltages,
`(6.3-lO)
`
`
`
`
`
`
`
`and (6.3-11) may be written as
`
`v;s = rsl�s + WrLd(Js
`+ WrA:�
`Vd.r = rsl;s -WrLql;s
`
`(6.4-1)
`(6.4-2)
`
`In I
`
`ph:
`the
`the
`
`where uppercase lelters denote steady-state (constant) quantities. It is clear that;.;� is
`
`
`
`
`
`
`
`
`
`
`
`
`always constant. The steady-state torque is expressed from (6.3-14) with uppercase
`
`letters as
`
`It is possible to establish a phasor voltage equation from (6.4-1) and (6.4-2) simi­
`
`
`
`
`
`
`
`lar to that for the synchronous machine. For this purpose let us write (6.3-22) as
`
`( 6.4--4)
`
`(6.4--3)
`
`Cc Frc mi wl
`
`(6
`
`

`
`ANALYSIS OF STEADY-STATE
`
`OPERATION 267
`of V11.f and the q axis fixed
`the angular displacement and represems for steady-state operation ¢ .. is constant
`
`
`
`
`
`to the q axis and let it be along the positive
`between the peak value of the fundamental component
`
`
`the phasors in the rotor. If we reference
`
`the phase angle of V115
`
`
`
`real axis of the "stationary" phasor diagram, then ¢,.becomes
`and we can write
`
`
`
`
`
`and (6.3-21 ), we see that Comparing (6.4-5) with (6.3-20)
`
`
`
`( 6.4-5)
`
`(6.4-6)
`
`to (6.3-16)­currents similar [f we go back and write equations for the 3-phase
`
`
`
`
`would be in terms of Od and¢; rather
`than
`
`(6.3-18) in terms of 0�;, then (6.3-22)
`Btv and ¢v; however,
`
`
`
`
`we would arrive at a similar relation for current as (6.4-6).
`In particular,
`
`and
`
`(6.4-7)
`
`(6.4-8)
`
`
`
`Substituting (6.4-1) and (6.4-2) into (6.4-6) and using (6.4-7) and (6.4-8) yields
`
`
`
`
`
`
`
`
`
`where
`
`(6.4-9)
`
`(6.4-10)
`
`
`In Chapter 5 the phasor diagram for the synchronous machine was referenced to the
`
`
`
`
`of Vas (Vas), which was positioned
`phasor
`
`along the real axis. Note that in the case of
`the phase angle of Ea was {> [see (5.9-18)],
`Ea for
`the synchronous machine
`whereas
`
`
`
`
`
`the brushless de machine [see (6.4-1 0)] is along the reference (real) axis.
`
`Common Operating Mode
`Fr?m our earlier discussion we are aware that the values of V�s and V�s are deter­
`
`
`
`= v'2\lf and Vd.,
`
`
`
`
`Intoed by the firing of the driving inverter. However, the condition
`Whereupon v;,.
`
`= 0, is used in most applications. In this case,
` terms of l�s·
`for Ids i
`(6.4-2) may be solved
`for 4)v = 0
`
`with (P .. = 0,
`
`(6.4-11)
`
`_n
`
`

`
`268 THEORY OF BRUSHLESS de MACHINES
`
`
`
`
`
`
`(6.4-11) Substituting into (6.4-1) yields
`
`for ¢v = 0 (6.4-12)
`
`
`
`We now start to see a similarity between the voltage equation for the brushless
`in this mode ( ¢v = 0) and the de machine discussed in
`de machine operated
`
`
`
`Chapter 2. From (2.3-1) the steady-state armature voltage equation
`of a de shunt
`machine is
`
`( 6.4-13)
`
`Figure 6
`(l.v:::: L.t)
`
`Duel<
`for some
`only sec
`interest (
`the mad
`
`)
`
`(6.4-14
`)
`
`Substitut
`
`If we neglect w;LqLd
`in in (6.4-12) and if we assume the field current is constant
`
`
`
`(6.4-13),
`
`
`
`
`then the two equations are identical in form. Let's note another similarity.
`or set Lq = Ld. then the expression
`If we neglect the inductances,
`for the torque
`de motor is called a
`
`
`given by (6.4-3) is identical in form to that of a de shunt machine with a constant
`
`
`field current given by (2.3-5). We now see that the brushless
`brushless de motor, not because it has the same physical configuration as a de
`
`
`
`
`
`
`machine but because its terminal characteristics may be made to resemble those
`
`
`of a de machine. We must be careful, however, because in order for (6.4-12) and
`
`Operati
`in form, the term w;LqLd
`(6.4-13)
`to be identical
`
`must be significantly less than
`rs. Let us see what effects
`
`this term has upon the torque versus speed characteristics.
`Let us re
`for Ids and
`for l�s and we take that result along with (6.4-11)
`If we solve (6.4-12)
`from (6.•
`Te (6.4-3)
`
`
`
`substitute these expressions into the expression for and if we assume
`that Lq = LJ, we obtain the following
`
`expression for torque:
`(3) (p) r,,},:;.
`( , ).''
`Te = 2 2 r; + w;L; Vqs -w, "' for ¢v = 0
`We have used Ls for Lq and Ld because we have assumed that Lq = Ld.
`
`
`
`Therein Lmq = Lmd and <Pv = 0; hence, Fig. 6.4-l is a plot of
`shown in Fig. 6.4-l.
`line Te versus w, char­
`If w;L;
`is neglected, then (6.4-14) yields a straight
`(6.4-14).
`