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ADVANCED ELECTRIC DRIVES ADVANCED ELECTRIC DRIVES Analysis, Control, and Modeling Using MATLAB/Simulink® Ned Mohan

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ADVANCED ELECTRIC DRIVES

ADVANCED ELECTRIC DRIVES Analysis, Control, and Modeling Using MATLAB/Simulink®

Ned Mohan

Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See www.mathworks.com/trademarks for a list of additional trademarks. The MathWorks Publisher Logo identifies books that contain MATLAB® content. Used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book or in the software downloadable from http://www.wiley.com/WileyCDA/WileyTitle/productCd-047064477X.html and http://www.mathworks.com/ matlabcentral/fileexchange/?term=authorid%3A80973. The book’s or downloadable software’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular use of the MATLAB® software or related products. For MATLAB® and Simulink® product information, or information on other related products, please contact: The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 USA Tel 508-647-7000 Fax: 508-647-7001 E-mail: [email protected] Web: www.mathworks.com

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Mohan, Ned. Advanced electric drives : analysis, control, and modeling using MATLAB/Simulink® / Ned Mohan. pages cm Includes index. ISBN 978-1-118-48548-4 (hardback) 1. Electric driving–Computer simulation. 2. Electric motors–Mathematical models. 3. MATLAB. 4. SIMULINK. I. Title. TK4058.M5783 2014 621.460285'53–dc23 2014005496 Printed in the United States of America. 10 9 8 7 6 5 4 3 2 1

CONTENTS Preface

xiii

Notation

xv

1

Applications: Speed and Torque Control

1

1-1 1-2 1-3

History 1 Background 2 Types of ac Drives Discussed and the Simulation Software 2 1-4 Structure of this Textbook 3 1-5 “Test” Induction Motor 3 1-6 Summary 4 References 4 Problems 4 2

Induction Machine Equations in Phase Quantities: Assisted by Space Vectors 2-1 2-2

2-3

2-4

6

Introduction 6 Sinusoidally Distributed Stator Windings 6 2-2-1 Three-Phase, Sinusoidally Distributed Stator Windings 8 Stator Inductances (Rotor Open-Circuited) 9 2-3-1 Stator Single-Phase Magnetizing Inductance Lm,1-phase 9 2-3-2 Stator Mutual-Inductance Lmutual 11 2-3-3 Per-Phase Magnetizing-Inductance Lm 12 2-3-4 Stator-Inductance Ls 12 Equivalent Windings in a Squirrel-Cage Rotor 13 2-4-1 Rotor-Winding Inductances (Stator Open-Circuited) 13 v

vi CONTENTS

2-5

Mutual Inductances between the Stator and the Rotor Phase Windings 15 2-6 Review of Space Vectors 15 2-6-1 Relationship between Phasors and Space Vectors in Sinusoidal Steady State 17 2-7 Flux Linkages 18 2-7-1 Stator Flux Linkage (Rotor Open-Circuited) 18 2-7-2 Rotor Flux Linkage (Stator Open-Circuited) 19 2-7-3 Stator and Rotor Flux Linkages (Simultaneous Stator and Rotor Currents) 20 2-8 Stator and Rotor Voltage Equations in Terms of Space Vectors 21 2-9 Making the Case for a dq-Winding Analysis 22 2-10 Summary 25 Reference 25 Problems 26 3

Dynamic Analysis of Induction Machines in Terms of dq Windings 3-1 3-2

3-3

Introduction 28 dq Winding Representation 28 3-2-1 Stator dq Winding Representation 29 3-2-2 Rotor dq Windings (Along the Same dq-Axes as in the Stator) 31 3-2-3 Mutual Inductance between dq Windings on the Stator and the Rotor 32 Mathematical Relationships of the dq Windings (at an Arbitrary Speed ωd) 33 3-3-1 Relating dq Winding Variables to Phase Winding Variables 35 3-3-2 Flux Linkages of dq Windings in Terms of Their Currents 36 3-3-3 dq Winding Voltage Equations 37

28

CONTENTS vii

3-3-4

Obtaining Fluxes and Currents with Voltages as Inputs 40 3-4 Choice of the dq Winding Speed ωd 41 3-5 Electromagnetic Torque 42 3-5-1 Torque on the Rotor d-Axis Winding 42 3-5-2 Torque on the Rotor q-Axis Winding 43 3-5-3 Net Electromagnetic Torque Tem on the Rotor 44 3-6 Electrodynamics 44 3-7 d- and q-Axis Equivalent Circuits 45 3-8 Relationship between the dq Windings and the Per-Phase Phasor-Domain Equivalent Circuit in Balanced Sinusoidal Steady State 46 3-9 Computer Simulation 47 3-9-1 Calculation of Initial Conditions 48 3-10 Summary 56 Reference 56 Problems 57 4

Vector Control of Induction-Motor Drives: A Qualitative Examination 4-1 4-2

4-3

4-4 4-5

Introduction 59 Emulation of dc and Brushless dc Drive Performance 59 4-2-1 Vector Control of Induction-Motor Drives 61 Analogy to a Current-Excited Transformer with a Shorted Secondary 62 4-3-1 Using the Transformer Equivalent Circuit 65 d- and q-Axis Winding Representation 66 Vector Control with d-Axis Aligned with the Rotor Flux 67 4-5-1 Initial Flux Buildup Prior to t = 0− 67 4-5-2 Step Change in Torque at t = 0+ 68

59

viii CONTENTS

4-6

Torque, Speed, and Position Control 72 * (t ) 72 4-6-1 The Reference Current isq * 4-6-2 The Reference Current isd (t ) 73 4-6-3 Transformation and Inverse-Transformation of Stator Currents 73 4-6-4 The Estimated Motor Model for Vector Control 74 4-7 The Power-Processing Unit (PPU) 75 4-8 Summary 76 References 76 Problems 77 5

Mathematical Description of Vector Control in Induction Machines

79

5-1

Motor Model with the d-Axis Aligned Along the Rotor Flux Linkage λr -Axis 79 5-1-1 Calculation of ωdA 81 5-1-2 Calculation of Tem 81 5-1-3 d-Axis Rotor Flux Linkage Dynamics 82 5-1-4 Motor Model 82 5-2 Vector Control 84 5-2-1 Speed and Position Control Loops 86 5-2-2 Initial Startup 89 5-2-3 Calculating the Stator Voltages to Be Applied 89 5-2-4 Designing the PI Controllers 90 5-3 Summary 95 Reference 95 Problems 95 6

Detuning Effects in Induction Motor Vector Control 6-1 6-2

Effect of Detuning Due to Incorrect Rotor Time Constant τr 97 Steady-State Analysis 101 * 6-2-1 Steady-State isd /isd 104 * 6-2-2 Steady-State isq /isq 104

97

CONTENTS ix

6-2-3 Steady-State θerr 105 * 6-2-4 Steady-State Tem /Tem 106 6-3 Summary 107 References 107 Problems 108 7

Dynamic Analysis of Doubly Fed Induction Generators and Their Vector Control

109

7-1 Understanding DFIG Operation 110 7-2 Dynamic Analysis of DFIG 116 7-3 Vector Control of DFIG 116 7-4 Summary 117 References 117 Problems 117 8

Space Vector Pulse Width-Modulated (SV-PWM) Inverters 119 8-1 8-2 8-3 8-4

Introduction 119 Synthesis of Stator Voltage Space Vector vsa 119 Computer Simulation of SV-PWM Inverter 124 Limit on the Amplitude Vˆs of the Stator Voltage Space Vector vsa 125 Summary 128 References 128 Problems 129 9

Direct Torque Control (DTC) and Encoderless Operation of Induction Motor Drives 9-1 9-2 9-3 9-4

9-5

Introduction 130 System Overview 130 Principle of Encoderless DTC Operation 131 Calculation of λs, λr , Tem, and ωm 132 9-4-1 Calculation of the Stator Flux λ s 132 9-4-2 Calculation of the Rotor Flux λr 133 9-4-3 Calculation of the Electromagnetic Torque Tem 134 9-4-4 Calculation of the Rotor Speed ωm 135 Calculation of the Stator Voltage Space Vector 136

130

x CONTENTS

9-6 Direct Torque Control Using dq-Axes 139 9-7 Summary 139 References 139 Problems 139 Appendix 9-A 140 Derivation of Torque Expressions 140 10 Vector Control of Permanent-Magnet Synchronous Motor Drives

143

10-1 Introduction 143 10-2 d-q Analysis of Permanent Magnet (Nonsalient-Pole) Synchronous Machines 143 10-2-1 Flux Linkages 144 10-2-2 Stator dq Winding Voltages 144 10-2-3 Electromagnetic Torque 145 10-2-4 Electrodynamics 145 10-2-5 Relationship between the dq Circuits and the Per-Phase Phasor-Domain Equivalent Circuit in Balanced Sinusoidal Steady State 145 10-2-6 dq-Based Dynamic Controller for “Brushless DC” Drives 147 10-3 Salient-Pole Synchronous Machines 151 10-3-1 Inductances 152 10-3-2 Flux Linkages 153 10-3-3 Winding Voltages 153 10-3-4 Electromagnetic Torque 154 10-3-5 dq-Axis Equivalent Circuits 154 10-3-6 Space Vector Diagram in Steady State 154 10-4 Summary 156 References 156 Problems 156 11 Switched-Reluctance Motor (SRM) Drives 11-1 Introduction 157 11-2 Switched-Reluctance Motor 157 11-2-1 Electromagnetic Torque Tem 159 11-2-2 Induced Back-EMF ea 161

157

CONTENTS xi

11-3 11-4 11-5 11-6

Instantaneous Waveforms 162 Role of Magnetic Saturation 164 Power Processing Units for SRM Drives 165 Determining the Rotor Position for Encoderless Operation 166 11-7 Control in Motoring Mode 166 11-8 Summary 167 References 167 Problems 167 Index

169

PREFACE When I wrote the first version of this textbook in 2001, my opening paragraph was as follows: Why write a textbook for a course that has pretty much disappeared from the curriculum at many universities? The only possible answer is in hopes of reviving it (as we have been able to do at the University of Minnesota) because of enormous future opportunities that await us including biomedical applications such as heart pumps, harnessing of renewable energy resources such as wind, factory automation using robotics, and clean transportation in the form of hybrid-electric vehicles.

Here we are, more than a decade later, and unfortunately the situation is no different. It is hoped that the conditions would have changed when the time comes for the next revision of this book in a few years from now. This textbook follows the treatment of electric machines and drives in my earlier textbook, Electric Machines and Drives: A First Course, published by Wiley (http://www.wiley.com/college/mohan). My attempt in this book is to present the analysis, control, and modeling of electric machines as simply and concisely as possible, such that it can easily be covered in one semester graduate-level course. To do so, I have chosen a two-step approach: first, provide a “physical” picture without resorting to mathematical transformations for easy visualization, and then confirm this physics-based analysis mathematically. The “physical” picture mentioned above needs elaboration. Most research literature and textbooks in this field treat dq-axis transformation of a-b-c phase quantities on a purely mathematical basis, without relating this transformation to a set of windings, albeit hypothetical, that can be visualized. That is, we visualize a set of hypothetical dq windings along an orthogonal set of axes and then relate their currents and voltages to the a-b-c phase quantities. This discussion follows xiii

xiv PREFACE

seamlessly from the treatment of space vectors and the equivalent winding representations in steady state in the previous course and the textbook mentioned earlier. For discussion of all topics in this course, computer simulations are a necessity. For this purpose, I have chosen MATLAB/Simulink® for the following reasons: a student-version that is more than sufficient for our purposes is available at a very reasonable price, and it takes extremely short time to become proficient in its use. Moreover, this same software simplifies the development of a real-time controller of drives in the hardware laboratory for student experimentation—such a laboratory using 42-V machines is developed using digital control and promoted by the University of Minnesota. The MATLAB and Simulink files used in examples are included on the accompanying website to this textbook: www.wiley.com/go/advancedelectricdrives. As a final note, this textbook is not intended to cover power electronics and control theory. Rather, the purpose of this book is to analyze electric machines in a way that can be interfaced to well-known power electronic converters and controlled using any control scheme, the simplest being proportional-integral control, which is used in this textbook. Ned Mohan University of Minnesota

NOTATION 1. Variables that are functions of time v, i, λ Vˆ , Iˆ , λˆ 2. Peak values (of time-varying variables) V = Vˆ ∠θv, I = Iˆ∠θi 3. Phasors ˆ jθ, i (t ) = Ie ˆ jθ, λ (t ) = λˆe jθ 4. Space vectors H (t ), B(t ), F (t ), v(t ) = Ve For space vectors, the exponential notion is used where, e jθ = 1∠θ = cos θ + j sin θe jθ = 1∠θ = cos θ + j sin θ. Note that both phasors and space vectors, two distinct quantities, have their peak values indicated by “.”

SUBSCRIPTS Stator phases Rotor phases dq windings Stator Rotor Magnetizing Mechanical Mechanical Leakage

a, b, c A, B, C d, q s r m m (as in θm or ωm) mech (as in θmech or ωmech) ℓ

SUPERSCRIPTS Denotes the axis used as reference for defining a space vector (lack of superscript implies that the d-axis is used as the reference). *

Reference Value xv

xvi NOTATION

SYMBOLS p θ

ω ωmech θmech fl

Number of poles (p ≥ 2, even number) All angles, such as θm and the axes orientation (for example, ej2π/3), are in electrical radians (electrical radians equal p/2 times the mechanical radians). All speeds, such as ωsyn, ωd, ωdA, ωm, and ωslip (except for ωmech), are in electrical radians per second. The rotor speed is in actual (mechanical) radians per second: ωmech = (2/p)ωm. The rotor angle is in actual (mechanical) radians per second: θmech = (2/p)θm. Flux linkages are represented by fl in MATLAB and Simulink examples.

INDUCTION MOTOR PARAMETERS USED INTERCHANGEABLY Rr′ = Rr L′r = Lr

1

Applications: Speed and Torque Control

There are many electromechanical systems where it is important to precisely control their torque, speed, and position. Many of these, such as elevators in high-rise buildings, we use on daily basis. Many others operate behind the scene, such as mechanical robots in automated factories, which are crucial for industrial competitiveness. Even in general-purpose applications of adjustable-speed drives, such as pumps and compressors systems, it is possible to control adjustable-speed drives in a way to increase their energy efficiency. Advanced electric drives are also needed in wind-electric systems to generate electricity at variable speed, as described in Appendix 1-A in the accompanying website. Hybrid-electric and electric vehicles represent an important application of advanced electric drives in the immediate future. In most of these applications, increasing efficiency requires producing maximum torque per ampere, as will be explained in this book. It also requires controlling the electromagnetic toque, as quickly and as precisely as possible, illustrated in Fig. 1-1, where the load torque TLoad may take a step-jump in time, in response to which the electromagnetic torque produced by the machine Tem must also take a step-jump if the speed ωm of the load is to remain constant. 1-1 HISTORY In the past, many applications requiring precise motion control utilized dc motor drives. With the availability of fast signal processing capability, the role of dc motor drives is being replaced by ac motor drives. The Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

1

2 Applications: Speed and Torque Control TLoad;Tem

0

t

0

t

ωm

Fig. 1-1 Need for controlling the electromagnetic torque Tem.

use of dc motor drives in precise motion control has already been discussed in the introductory course using the textbook [1] especially designed for this purpose. Hence, our emphasis in this book for an advanced course (designed at a graduate level but that can be easily followed by undergraduates) will be entirely on ac motor drives. 1-2 BACKGROUND In the introductory course [1], we discussed electric drives in an integrative manner where the theory of electric machines was discussed using space vectors to represent sinusoidal field distribution in the air gap. This discussion included a brief introduction to power-processing units (PPUs) and feedback control systems. In this course, we build upon that discussion and discover that it is possible to understand advanced control of electric drives on a “physical” basis, which allows us to visualize the control process rather than leaving it shrouded in mathematical mystery. 1-3 TYPES OF AC DRIVES DISCUSSED AND THE SIMULATION SOFTWARE In this textbook, we will discuss all types of ac drives and their control in common use today. These include induction-motor drives, permanent-

“Test” Induction Motor 3

magnet ac drives and switched-reluctance drives. We will also discuss encoder-less operation of induction-motor drives. A simulation-based study is essential for discussing advanced electric drive systems. After a careful review of the available software, the author considers MATLAB/Simulink® to be ideal for this purpose—a student version that is more than sufficient for our purposes is available [2] at a very reasonable price, and it takes extremely short time to become proficient in its use. Moreover, the same software simplifies the development of a real-time controller of drives in the hardware laboratory for student experimentation—such a laboratory, using 42-V machines is being developed at the University of Minnesota using digital control. 1-4 STRUCTURE OF THIS TEXTBOOK Chapter 1 has introduced advanced electric drives. Chapter 2, Chapter 3, Chapter 4, Chapter 5, Chapter 6, Chapter 7 and Chapter 9 deal with induction-motor drives. Chapter 8 deals with the synthesis of stator voltage vector, supplied by the inverter of the PPU, using a digital signal processor. The permanent-magnet ac drives (ac servo drives) are discussed in Chapter 10 and the switched-reluctance motor drives are discussed in Chapter 11. A “test” motor is selected for discussing the design of controllers and for obtaining the performance by means of simulation examples for which the specifications are provided in the next section. In all chapters dealing with induction motor drives, the “test” induction motor used is described in the following section. The “test” motor for a permanentmagnet ac drive is described in Chapter 10. 1-5 “TEST” INDUCTION MOTOR For analyzing the performance of various control procedures, we will select a 1.5-MW induction machine as a “test” machine, for which the specifications are as follows: Power: Voltage:

1.5 MW 690 V (L-L, rms)

4 Applications: Speed and Torque Control

Frequency: 60 Hz Phases: 3 Number of Poles: 6 Full-Load Slip 1% Moment of Inertia 70 kg·m2 Per-Phase Circuit Parameters: Rs = 0.002 Ω Rr = 0.0015 Ω X s = 0.05 Ω X r = 0.047 Ω X m = 0.86 Ω. 1-6 SUMMARY This chapter describes the application of advanced ac motor drives and the background needed to undertake this study. The structure of this textbook is described in terms of chapters that cover all types of ac motor drives in common use. An absolute need for using a computer simulation program in a course like this is pointed out, and a case is made for using a general-purpose software, MATLAB/Simulink®. Finally, the parameters for a “test” induction machine are described— this machine is used in induction machine related chapters for analysis and simulation purposes. REFERENCES 1. N. Mohan, Electric Machines and Drives: A First Course, Wiley, Hoboken, NJ, 2011. http://www.wiley.com/college/mohan. 2. http://www.mathworks.com.

PROBLEMS 1-1 Read the report “Adaptive Torque Control of Variable Speed Wind Turbines” by Kathryn E. Johnson, National Renewable

Problems 5

Energy Laboratory (http://www.nrel.gov). Upon reading section 2.1, describe the Standard Region 2 Control and describe how it works in your own words. 1-2 Read the report “Final Report on Assessment of Motor Technologies for Traction Drives of Hybrid and Electric Vehicles” (http:// info.ornl.gov/sites/publications/files/pub28840.pdf) and answer the following questions for HEV/EV applications: (a) What are the types of machines considered? (b) What type of motor is the most popular choice? (c) What are the alternatives if NdFeB magnets are not available? (d) What are the advantages and disadvantages of SR motors? 1-3 Read the report “Evaluation of the 2010 Toyota Prius Hybrid Synergy Drive System” (http://info.ornl.gov/sites/publications/ files/Pub26762.pdf) and answer the following questions: (a) What are ECVT, PCU, and ICE? (b) What type of motor is used in this application?

2

Induction Machine Equations in Phase Quantities: Assisted by Space Vectors

2-1 INTRODUCTION In ac machines, the stator windings are intended to have a sinusoidally distributed conductor density in order to produce a sinusoidally distributed field distribution in the air gap. In the squirrel-cage rotor of induction machines, the bar density is uniform. Yet the currents in the rotor bars produce a magnetomotive force (mmf) that is sinsuoidally distributed. Therefore, it is possible to replace the squirrel-cage with an equivalent wound rotor with three sinsuoidally distributed windings. In this chapter, we will briefly review the sinusoidally distributed windings and then calculate their inductances for developing equations for induction machines in phase (a-b-c) quantities. The development of these equations is assisted by space vectors, which are briefly reviewed. The analysis in this chapter establishes the framework for the dq winding-based analysis of induction machines under dynamic conditions carried out in the next chapter.

2-2 SINUSOIDALLY DISTRIBUTED STATOR WINDINGS In the following analysis, we will also assume that the magnetic material in the stator and the rotor is operated in its linear region and has an infinite permeability.

Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

6

Sinusoidally Distributed Stator Windings 7 b-axis 3'

ib

4'

5'

ia 6'

2' a-axis ia

θ=π

ic

1'

7'

7

1 6

2 5

4

θ

θ=0

magnetic axis of phase a

3

c-axis (a)

(b)

Fig. 2-1 Stator windings.

In ac machines of Fig. 2-1a, windings for each phase ideally should produce a sinusoidally distributed radial field (F, H, and B) in the air gap. Theoretically, this requires a sinusoidally distributed winding in each phase. If each phase winding has a total of Ns turns (i.e., 2Ns conductors), the conductor density ns(θ) in phase-a of Fig. 2-1b can be defined as

ns (θ) =

Ns sin θ, 0 ≤ θ ≤ π. 2

(2-1)

The angle θ is measured in the counter-clockwise direction with respect to the phase-a magnetic axis. Rather than restricting the conductor density expression to a region 0 0, φm,i1 is a constant and therefore does not induce any voltage in Fig. 4-5b. Due to the voltage drop across R2, the current i2 declines, thus causing both φm,i2 and φℓ2 to decline (both of these are produced by i2). In accordance with the equivalent circuit of Fig. 4-5b, i2 decays exponentially, as shown in Fig. 4-5c

i2 (t ) = i2 (0+ )e−t τ2 ,

(4-11)

where τ2 is the time constant of winding 2:

τ2 =

L2 . R2

(4-12)

Theoretically, we can see that at t = 0+, a voltage impulse is necessary to make the current i1 jump because a finite amount of energy must be

ANALOGY TO A CURRENT-EXCITED TRANSFORMER 65

transferred instantaneously. This instantaneous energy increase (all of these associated energy levels were zero at t = 0−) is associated with: 1. Leakage flux of the primary (neglected in this discussion) 2. Leakage flux of the secondary in air 3. Slight increase of flux φ 2 (= φm,i1 − φm,i2 ) in the core. However, as argued in Reference [2], the volt-seconds needed to accomplish this instantaneous change in current are not excessive. After all, note that in a dc-motor drive, for a step change in torque, the armature current must be built up overcoming the inductive nature of the armature winding. A similar situation occurs in “brushless-dc” motor drives. 4-3-1 Using the Transformer Equivalent Circuit The earlier discussion can also be confirmed by considering the equivalent circuit of a two winding transformer with N1 = N2, shown in Fig. 4-6. For a step change in i1 at t = 0+, the instantaneous current division is based on inductances of the two parallel branches (resistance R2 will have a negligible effect): i2 (0+ ) =

Lm L i1 (0+ ) = m i1 (0+ ) Lm + L 2 L2

(4-13)

im (0+ ) =

L 2 L i1 (0+ ) = 2 i1 (0+ ). Lm + L 2 L2

(4-14)

and

R1

Ll1

Ll2

R2

i2 i2

i1 0

t

i1

Lm im

0

t

N1 = N2 = N

Fig. 4-6 Equivalent-circuit representation of the current-excited transformer with a short-circuited secondary.

66 VECTOR CONTROL OF INDUCTION-MOTOR DRIVES

Equation (4-13) shows the jump in i2 at t = 0+. Noting that di1/dt = 0 for t > 0, solving for i2 in the circuit in Fig. 4-6 confirms the decay in i2 due to R2

i2 (t ) = i2 (0+ )e−t τ2 .

(4-15)

4-4 d- AND q-AXIS WINDING REPRESENTATION A step change in torque requires a step change in the rotor current of a vector-controlled induction motor. We will make use of an orthogonal set of d- and q-axis windings, introduced in Chapter 3, producing the same mmf as three stator windings (each with Ns turns, sinusoidally distributed), with ia, ib, and ic flowing through them. In Fig. 4-7 at a time t, is (t ) and Fs (t ) are produced by ia(t), ib(t), and ic(t). The resulting mmf Fs (t ) = ( N s / p)is (t ) can be produced by the set of orthogonal stator windings shown in Fig. 4-7, each sinusoidally distributed with 3 / 2 N s turns: one winding along the d-axis, and the other along the q-axis. Note that this d–q axis set may be at any arbitrary angle with respect to the phase-a axis. In order to keep the mmf and the flux-density distributions the same as in the actual machine with three-phase windings, the currents in these two windings would have to be isd and isq, where, as shown

at t

q-axis ωd isq

3 2 Ns

irq

ωd 3 2 Ns

3 2 isq 3 2 irq

3 2 Ns

is ir

3 2isd ird θdA θm

θda

3 2ird

d-axis isd

ωm

A-axis rotor a-axis stator

Fig. 4-7 Stator and rotor mmf representation by equivalent dq winding currents.

VECTOR CONTROL WITH d-AXIS ALIGNED WITH ROTOR FLUX 67

in Fig. 4-7, these two current components are 2 / 3 times the projec tions of the is (t ) vector along the d-axis and q-axis.

4-5 VECTOR CONTROL WITH d-AXIS ALIGNED WITH THE ROTOR FLUX In the following analysis, we will assume that the d-axis is always aligned with the rotor flux-linkage space vector, that is, also aligned with Br (t ). 4-5-1 Initial Flux Buildup Prior to t = 0− We will apply the information of the last section to vector control of induction machines. As shown in Fig. 4-8, prior to t = 0−, the magnetizing currents are built up in three phases such that

ia (0− ) = Iˆm,rated

1 and ib (0− ) = ic (0− ) = − Iˆm,rated. 2

(4-16)

The current buildup prior to t = 0− may occur slowly over a long period of time and represents the buildup of the flux in the induction machine up to its rated value. These currents represent the rated magnetizing currents to bring the air gap flux density to its rated value. Note

q-axis t = 0−

isq = 0

φm,isd

is Br

d-axis isd

isd −∞ 0

t

Fig. 4-8 Currents and flux at t−.

68 VECTOR CONTROL OF INDUCTION-MOTOR DRIVES

that there will be no rotor currents at t = 0− (they decay out prior to t = 0−). Also, at t = 0−, the stator mmf can be represented by that produced by the d-axis winding (chosen to be along the a-axis) with a current isd, where

isd (0− ) =

2ˆ 2 3 ˆ 3ˆ I m,rated = I ms,rated = I m,rated 3 3 2 2

(4-17)

and

isq = 0.

(4-18)

We should note that the isd-produced stator leakage flux does not link the rotor, and hence it is of no concern in the following discussion. At t = 0−, the peak of the flux lines φm,isd linking the rotor is horizontally oriented. There is no rotor leakage flux because there are no currents flowing through the rotor bars. Only the flux φm,isd produced by the stator links the rotor. Therefore, Br (0− ), equal to Bms (0− ), is horizontally oriented along the d-axis (same as the a-axis at t = 0−). 4-5-2 Step Change in Torque at t = 0+ Next, we will see how this induction machine can produce a step change in torque. Initially, we will assume that the rotor is blocked from turning (ωmech = 0), a restriction that will soon be removed. At t = 0+, the three stator currents are changed as a step in order to produce a step change in the q-axis current isq, without changing isd, as shown in Fig. 4-9a. The current isq in the stator q winding produces the flux lines φm,isq that cross the air gap and link the rotor. The leakage flux produced by isq can be safely neglected from the discussion here (because it does not link the shorted rotor cage), similar to neglecting the leakage flux produced by the primary winding of the transformer in the previous analogy. Turning our attention to the rotor at t = 0+, we note that the rotor is a short-circuited cage, so its flux linkage cannot change instantaneously. To oppose the flux lines produced by isq, currents are instantaneously induced in the rotor bars by the transformer action, as shown in Fig. 4-9a.

VECTOR CONTROL WITH d-AXIS ALIGNED WITH ROTOR FLUX 69 ωslip isq

t = 0+

isq

ωmech = 0

φm,isq

0

φm,ir φlr

ωslip

y y' x

a-axis

Br isd

x'

isd

net = 0 −∞ 0

t

(a)

(b) +

Fig. 4-9 Currents at t = 0 .

This current distribution in the rotor bars is sinusoidal, as justified below using Fig. 4-9b: To justify the sinusoidal distribution of current in the rotor bars, assume that the bars x − x′ constitute one short-circuited coil, and the bars y − y′ the other coil. The density of flux lines produced by isq is sinusoidally distributed in the air gap. The coil x − x′ links most of the flux lines produced by isq. But the coil y − y′ links far fewer flux lines. Therefore, the current in this coil will be relatively smaller than the current in x − x′.

These rotor currents in Fig. 4-9a produce two flux components with peak densities along the q-axis and of the direction shown: 1. The magnetizing flux φm,ir that crosses the air gap and links the stator. 2. The leakage flux φℓr that does not cross the air gap and links only the rotor. By the theorem of constant flux linkage, at t = 0+, the net flux linking the short-circuited rotor in the q-axis must remain zero. Therefore, at t = 0+, for the condition that φrq,net = 0 (taking flux directions into account):

70 VECTOR CONTROL OF INDUCTION-MOTOR DRIVES Bms

Blr

θr

t = 0+ a-axis

Br

Fig. 4-10 Flux densities at t = 0+.

φm,isq (o+ ) = φm,ir (o+ ) + φr (o+ ).

(4-19)

Since isd and the d-axis rotor flux linkage have not changed, the net flux, Br , linking the rotor remains the same at t = 0+ as it was at t = 0−. The space vectors at t = 0+ are shown in Fig. 4-10. No change in the net flux linking the rotor implies that Br has not changed; its peak is still horizontal along the a-axis and of the same magnitude as before. The rotor currents produced instantaneously by the transformer action at t = 0+, as shown in Fig. 4-9a, result in a torque Tem(0+). This torque will be proportional to Bˆ r and isq (slightly less than isq by a factor of Lm/Lr due to the rotor leakage flux, where Lr equals Lm + L′r in the per-phase equivalent circuit of an induction machine):

L Tem = k1Bˆ r m isq , Lr

(4-20)

where k1 is a constant. If no action is taken beyond t = 0+, the rotor currents will decay and so will the force on the rotor bars. This current decay would be like in a transformer of Fig. 4-6 with a short-circuited secondary and with the primary excited with a step of current source. In the transformer case of Fig. 4-6, decay of i2 could be prevented by injecting a voltage equal to R2i2(0+) beyond t = 0+ to overcome the voltage drop across R2. In the case of an induction machine, beyond t = 0+, as shown in Fig. 4-11, we will equivalently rotate both the d-axis and the q-axis stator windings at an appropriate slip speed ωslip in order to maintain Br (t ) completely along the d-axis with a constant amplitude of Bˆ r , and to maintain the same rotor-bar current distribution along the q-axis. This corresponds to the beginning of a new steady state. Therefore, the steady-state analysis of induction machine applies. As the d-axis and the q-axis windings rotate at the appropriate value of ωslip (notice that the rotor is still blocked from turning in Fig. 4-11),

VECTOR CONTROL WITH d-AXIS ALIGNED WITH ROTOR FLUX 71 q-axis ωslip

t>0

isq

φm,isq

isq

φm,ir φlr

Br

ωslip d-axis

isd a-axis

net = 0

Fig. 4-11 Current and fluxes at some time t > 0, with the rotor blocked.

there is no net rotor flux linkage along the q-axis. The flux linkage along the d-axis remains constant with a flux density Bˆ r “cutting” the rotor bars and inducing the bar voltages to cancel the iRbar voltage drops. Therefore, the entire distribution rotates with time, as shown in Fig. 4-11 at any arbitrary time t > 0. For the relative distribution and hence the torque produced to remain the same as at t = 0+, the two windings must rotate at an exact ωslip, which depends linearly on both the rotor resistance Rr′ and isq (slightly less by the factor Lm/Lr due to the rotor leakage flux), and inversely on Bˆ r

ωslip = k2

Rr′(Lm / Lr )isq , Bˆ r

(4-21)

where k2 is a constant. Now we can remove the restriction of ωmech = 0. If we need to produce a step change in torque while the rotor is turning at some speed ωmech, then the d-axis and the q-axis windings should be equivalently rotated at the appropriate slip speed ωslip relative to the rotor speed ωm (= ( p / 2)ωmech ) in electrical rad/s, that is, at the synchronous speed ωyn = ωm + ωslip, as shown in Fig. 4-12.

72 VECTOR CONTROL OF INDUCTION-MOTOR DRIVES q-axis ωsyn

t>0

isq

isq

φm,isq φ m,ir φlr

ωm

Br

ωsyn

d-axis

isd a-axis

net = 0

Fig. 4-12 Vector-controlled condition with the rotor speed ωm (= ( p / 2)ωmech ) electrical rad/s.

4-6 TORQUE, SPEED, AND POSITION CONTROL In vector control of induction-motor drives, the stator phase currents ia(t), ib(t), and ic(t) are controlled in such a manner that isq(t) delivers the desired electromagnetic torque while isd(t) maintains the peak * * (t ) and isd rotor-flux density at its rated value. The reference values isq (t ) are generated by the torque, speed, and position control loops, as discussed in the following section. * (t ) 4-6-1 The Reference Current isq * (t ) depends on the desired torque, which is calThe reference value isq culated within the cascade control of Fig. 4-13, where the position loop is the outermost loop and the torque loop is the innermost loop. The loop bandwidths increase from the outermost to the innermost loop. * The error between the reference (desired) position, θmech (t ), and the measured position θmech(t) is amplified by a proportional (P) amplifier * (t ) . The error between the to generate the speed reference signal ωmech * reference speed ωmech (t ) and the measured speed ωmech(t) is amplified

Torque, Speed, and Position Control 73

Br ωmech (measured) ωmech

Br (calculated)

Tem

B*r

Br θBr

Σ

PI

ωmech (measured) motor estimated model

isd calculations

isq * isd

* θmech

ω*mech

Σ θmech (measured)

P

Σ ωmech (measured)

PI

* Tem

* isq

Σ

ia*

dq ib* to abc ic*

abc to dq

current regulated PPU

ia ib ic

Motor

PI

Tem (calculated)

ωmech d / dt (measured)

θmech encoder

Fig. 4-13 Vector-controlled induction motor drive with a current-regulated PPU. In a multipole machine, measured speed should be converted into electrical radians per second.

by a proportional-integral (PI) amplifier to generate the torque refer* * ence Tem (t ). Finally, the error between Tem (t ) and the calculated torque Tem(t) is amplified by another PI amplifier to generate the reference * (t ). value isq * 4-6-2 The Reference Current isd (t )

For measured speed values below the rated speed of the motor, the rotor flux-density peak Bˆ r is maintained at its rated value as shown by the speed versus flux-density block in Fig. 4-13. Above the rated speed, the flux density is reduced in the flux-weakening mode, as discussed in the previous course. The error between Bˆ r* and the calculated flux-density peak Bˆ r is amplified by a PI amplifier to generate the refer* ence value isd (t ). 4-6-3 Transformation and Inverse-Transformation of Stator Currents Fig. 4-13 shows the angle θBr (t ) of the d-axis, with respect to the stationary a-axis, to which the rotor flux-density space vector Br (t ) is aligned. The angle θBr is the same as θda in Chapter 3 if the d-axis is aligned with the rotor flux λr at all times such that λrq = 0. This angle is computed by the vector-controlled motor model, which is described in the next

74 VECTOR CONTROL OF INDUCTION-MOTOR DRIVES * section. Using the d-axis angle θBr (t ), the reference current signals isd (t ) * and isq (t ) are transformed into the stator current reference signals ia* (t ), ib* (t ), and ic* (t ), as shown in Fig. 4-13 by the transform block (dq − to − abc), same as [Ts]dq→abc in Equation (3-18) of the previous chapter. The current-regulated PPU uses these reference signals to supply the desired currents to the motor (details of how it can be accomplished are discussed briefly in Section 4-7). The stator currents are measured and the d-axis angle θBr (t ) is used to transform them using a matrix same as [Ts]abc→dq in Equation (3-12) of the previous chapter into the signals isd(t) and isq(t), as shown by the inverse transform block (abc − to − dq) in Fig. 4-13.

4-6-4 The Estimated Motor Model for Vector Control The estimated motor model in Fig. 4-13 has the following measured inputs: the three stator phase currents ia(t), ib(t), and ic(t), and the measured rotor speed ωmech(t). The motor model also needs accurate estimation of the rotor parameters Lm, L′r , and Rr′ . The following parameters are calculated in the motor model for internal use and also as outputs: the angle θBr (with respect to the stationary phase-a axis) to which the d-axis is aligned, the peak of the rotor flux density Bˆ r (t ), and the electromagnetic torque Tem(t). In the estimated motor model, Bˆ r (t ) is computed by considering the dynamics along the d-axis, which is valid in the flux-weakening mode, where Bˆ r (t ) is decreased to allow operation at higher than rated speed. The electromagnetic torque Tem(t) is computed based on Eq. (4-20) (the complete torque expression will be derived in the next chapter). The angle θBr (t ) is computed by first calculating the slip speed ωslip(t) based on Eq. (4-21) (the complete expression will be derived in the next chapter). This slip speed is added to the measured rotor speed ωm (= ( p / 2)ωmech ) to yield the instantaneous synchronous speed of the d- and the q-axes:

ωsyn (t ) = ωm (t ) + ωslip (t ).

(4-22)

With θBr = 0 at starting by initially aligning the rotor flux-density space vector along the a-axis, integrating the instantaneous synchronous speed results in the d-axis angle as follows:

The Power-Processing Unit (PPU) 75 t

θBr (t ) = 0 +

∫ω

syn

( τ ) ⋅ dτ ,

(4-23)

0

where τ is a variable of integration. Based on these physical principles, the mathematical expressions are clearly and concisely developed in the next chapter.

4-7 THE POWER-PROCESSING UNIT (PPU) The task of the PPU in Fig. 4-13 is to supply the desired currents based on the reference signals to the induction motor. This PPU is further illustrated in Fig. 4-14a, where phases b and c are omitted for simplification. One of the easiest ways to ensure that the motor is supplied the desired currents is to use hysteresis control similar to that discussed in Chapter 7 for ECM drives and Chapter 10 for PMAC drives in the previous course [1]. The measured phase current is compared with its

phase a

Vd

ia* (t) qA (t) ia (t) (a) actual current reference current 0

(b)

Fig. 4-14 (a) Block diagram representation of hysteresis current control; (b) current waveform.

76 VECTOR CONTROL OF INDUCTION-MOTOR DRIVES

reference value in the hysteresis comparator, whose output determines the switch state (up or down), resulting in a current waveform as shown in Fig. 4-14b. In spite of the simplicity of the hysteresis control, one perceived drawback of this controller is that its switching frequency changes as a function of the back-emf waveform. For this reason, constant switching frequency PPU are used as described in Chapter 8 of this book.

4-8 SUMMARY In this chapter, we have qualitatively examined how it is possible to produce a step in torque in a squirrel-cage induction machine. This analysis is aided by the steady-state analysis of induction machines using space vectors in the previous course, which clearly shows the orthogonal relationship between the rotor flux-linkage space vector and the rotor mmf space vector. In vector control, we keep the rotor flux linkage constant in amplitude (which can be decreased in the flux-weakening mode). Controlling the rotor flux linkage requires a dq winding analysis, where the current in the equivalent d winding of the stator is kept constant in order to keep the rotor flux along the d-axis constant. A step change in the stator q winding current suddenly induces currents in the rotor equivalent q-axis winding while keeping its flux linkage zero. Therefore, the rotor flux linkage remains unchanged in amplitude. Sudden appearance of currents along the rotor q-axis, in the presence of d-axis flux, results in a step change in torque. To maintain the induced rotor q winding current from decaying, the dq winding set must be rotated at an appropriate slip speed with respect to the rotor.

REFERENCES 1. N. Mohan, Electric Machines and Drives: A First Course, Wiley, Hoboken, NJ, 2011. http://www.wiley.com/college/mohan. 2. A. Hughes, J. Corda, and D. Andrade, “Vector Control of Cage Induction Motors: A Physical Insight,” IEE Proceedings: Electric Power Applications, vol. 143, no. 1, Jan. 1996, pp. 59–68.

Problems 77

PROBLEMS 4-1 Draw the dynamic dq-axis equivalent circuits under the condition that the rotor flux is aligned with d-axis, such that λrq and dλrq/dt are zero at all times. Apply a step change of current in the q-axis circuit and explain what happens. 4-2 In an induction motor described with the following nameplate data, establishing the rated air gap flux density requires Îm = 2.54 A. To build up to this rated flux, calculate the three-phase currents at t = 0−. Nameplate Data Power: 3 HP/2.4 kW Voltage: 460 V (L-L, rms) Frequency: 60 Hz Phases: 3 Full Load Current: 4 A (rms) Full-Load Speed: 1750 rpm Full-Load Efficiency: 88.5% Power Factor: 80.0% Number of Poles: 4 Per-Phase Motor Circuit Parameters: Rs = 1.77 Ω Rr = 1.34 Ω X s = 5.25 Ω (at 60 Hz) X r = 4.57 Ω (at 60 Hz) X m = 139.0 Ω (at 60 Hz) Full-Load Slip = 1.72% The iron losses are specified as 78 W and the mechanical (friction and windage) losses are specified as 24 W. The inertia of the machine is given. Assuming that the reflected load inertia is approximately the same as the motor inertia, the total equivalent inertia of the system is Jeq = 0.025 kg · m2.

78 VECTOR CONTROL OF INDUCTION-MOTOR DRIVES

4-3 In the machine of Problem 4-2, a desired step-torque at t = 0+ requires step change in isq = 2 A from its initial zero value. Calculate the phase currents at t = 0+, which result in the desired step change in q-axis current while maintaining the rated flux density in the air gap. 4-4 In the machine of Problems 4-1 and 4-2, the slip speed at which the equivalent d-axis and the q-axis windings need to be rotated is ωslip = 2.34 electrical rad/s. Assuming that the rotor is blocked from turning, calculate the phase currents at t = 8 ms. 4-5 Repeat Problem 4-4, if the rotor is turning and the speed can be assumed constant at 1100 rpm even after the step change in torque at t = 0+.

5

Mathematical Description of Vector Control in Induction Machines

In vector control described qualitatively in Chapter 4, the d-axis is aligned with the rotor flux linkage space vector such that the rotor flux linkage in the q-axis is zero. With this as the motivation, we will first develop a model of the induction machine where this condition is always met. Such a model of the machine would be valid regardless if the machine is vector controlled, or if the voltages and currents are applied as under a general-purpose operation (line-fed or in adjustable speed drives described in the previous course). After developing the motor model, we will study vector control of induction-motor drives, assuming that the exact motor parameters are known—effects of errors in parameter estimates are discussed in the next chapter. We will first use an idealized current-regulated PWM (CR-PWM) inverter to supply motor currents calculated by the controller. As the last step in this chapter, we will use an idealized space vector pulse width-modulated inverter (discussed in detail in Chapter 7) to supply motor voltages that result in the desired currents calculated by the controller. 5-1 MOTOR MODEL WITH THE d-AXIS ALIGNED ALONG THE ROTOR FLUX LINKAGE λr -AXIS As noted in the qualitative description of vector control, we will align the d-axis (common to both the stator and the rotor) to be along the rotor flux linkage λr ( = λˆr e j 0 ), as shown in Fig. 5-1. Therefore, Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

79

80 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

q-axis

at t ωd 3 2 Ns

isq

ωd 3 2 Ns

irq

3 2 isq 3 2 irq

3 2 Ns

is 3 2isd

ir

3 2ird

θm

isd

ωm

ird

θdA

d-axis

A-axis rotor a-axis stator

θda

Fig. 5-1 Stator and rotor mmf representation by equivalend dq winding currents. The d-axis is aligned with λˆr . ω λ

λ

λ

ω λ

λ

λ

ω λ

Fig. 5-2 Dynamic circuits with the d-axis aligned with λr .

λrq (t ) = 0.

(5-1)

Equating λrq in Eq. (3-22) to zero,

irq = −

Lm isq. Lr

(5-2)

The condition that the d-axis is always aligned with λr such that λrq = 0 also results in dλrq/dt to be zero. Using λrq = 0 and dλrq/dt = 0 in the d- and the q-axis dynamic circuits, we can obtain the simplified circuits shown in Fig. 5-2a and b. Note that Eq. (5-2) is consistent with the equivalent circuit of Fig. 5-2b, where Lr = Lℓr + Lm.

MOTOR MODEL WITH THE d-AXIS ALIGNED 81

Next, we will calculate the slip speed ωdA and the electromagnetic torque Tem in this new motor model in terms of the rotor flux λrd and the stator current component isq in the q-winding (under vector control conditions, λrd would be kept constant except in the field-weakening mode and the toque production will be controlled by isq). We will also establish the dynamics of the rotor flux λrd in the rotor d-winding (λrd varies during flux buildup at startup and when the motor is made to go into the flux weakening mode of operation).

5-1-1 Calculation of ωdA As discussed earlier, under the condition that the d-axis is always aligned with the rotor flux, the q-axis rotor flux linkage is zero, as well as dλrq/dt = 0. Therefore, in a squirrel-cage rotor with vrq = 0, Eq. (3-32) results in

ωdA = −Rr

irq , λrd

(5-3)

which is consistent with the equivalent circuit of Fig. 5-2b. In the rotor circuit, the time-constant τr, called the rotor time-constant, is

τr =

Lr . Rr

(5-4)

Substituting for irq from Eq. (5-2), in terms of τr, the slip speed can be expressed as

ωdA =

Lm isq. τ rλrd

(5-5)

5-1-2 Calculation of Tem Since the flux linkage in the q-axis of the rotor is zero, the electromagnetic torque is produced only by the d-axis flux in the rotor acting on the rotor q-axis winding. Therefore, from Eq. (3-46),

p Tem = − λrd irq. 2

(5-6)

82 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

λ

Fig. 5-3 The d-axis circuit simplified with a current excitation.

In Eq. (5-6), substituting for irq from Eq. (5-2) Tem =

p Lm isq . λrd 2 Lr

(5-7)

5-1-3 d-Axis Rotor Flux Linkage Dynamics To obtain the dependence of λrd on isd, we will make use of the equivalent circuit in Fig. 5-2a, and redraw it as in Fig. 5-3 with a current excitation by isd. From Fig. 5-3, in terms of Laplace domain variables, ird ( s) = −

sLm isd ( s). Rr + sLr

(5-8)

In the rotor d-axis winding, from Eq. (3-21),

λrd = Lr ird + Lmisd.

(5-9)

Substituting for ird from Eq. (5-8) into Eq. (5-9), and using τr from Eq. (5-4),

λrd ( s) =

Lm isd ( s). (1 + sτ r )

(5-10)

In time domain, the rotor flux linkage dynamics expressed by Eq. (5-10) is as follows:

d L λ λrd + rd = m isd. dt τr τr

(5-11)

5-1-4 Motor Model Based on above equations, a block diagram of an induction-motor model, where the d-axis is aligned with the rotor flux linkage, is shown

MOTOR MODEL WITH THE d-AXIS ALIGNED 83

mech

Fig. 5-4 Motor model with d-axis aligned with λr .

in Fig. 5-4. The currents isd and isq are the inputs, and λrd, θda, and Tem are the outputs. Note that ωd (= ωdA + ωm) is the speed of the rotor field, and therefore, the rotor-field angle with respect to the stator a-axis (see Fig. 5-1) is t

θda (t ) = 0 +

∫ ω (τ )dτ, d

(5-12)

0

where τ is the variable of integration, and the initial value of θda is assumed to be zero at t = 0. EXAMPLE 5-1 The motor model developed earlier, with the d-axis aligned with λr , can be used to model induction machines where vector control is not the objective. To illustrate this, we will repeat the simulation of Example 3-3 of a line-fed motor using this new motor model (which is much simpler) and compare simulation results of these two examples. Solution We need to recalculate initial flux values because now the rotor flux linkage is completely along the d-axis. This is done in a MATLAB (Continued)

84 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

file EX5_1calc.m on the accompanying website. The initial part in this file is the same as in EX3_1.m (used in Example 3-3), in which the initial values of the angles thetar and thetas for λr and λs are calculated with respect to the d-axis aligned to the stator a-axis with θda(0) = 0. In the present model, with the d-axis aligned with λr , the rotor flux linkage angle is zero, and the stator flux linkage angle with respect to the d-axis equals (thetas—thetar) in terms of their values in EX3.1m. The Simulink schematic for this example is called EX5_1.mdl (included on the accompanying website) and its top-level diagram is shown in Fig. 5-5. The resulting torque and speed plots due to a load torque disturbance in this line-fed machine are plotted in Fig. 5-6, which are identical to the results obtained in Example 3-3.

5-2 VECTOR CONTROL One of the vector control methods is discussed in this section. It is called indirect vector control in the rotor flux reference frame. For many other possible methods and their pros and cons, readers are urged to look at several books on vector control and the IEEE transactions and conference proceedings of its various societies.

Fig. 5-5 Simulation of Example 5-1.

Vector Control 85

Fig. 5-6 Results of Example 5-1.

Tem λrd θda

* isd * isq

Transformations

Motor Model (Fig. 5-4)

dq to abc

ia* ib* ic*

isd

isq

abc to dq

Current − Regulated PPU

ia ib ic

Motor

Transformations

ωmech (measured)

θmech

d dt

Encoder

Fig. 5-7 Vector-controlled induction motor with a CR-PWM inverter.

A partial block diagram of a vector-controlled induction motor drive is shown in Fig. 5-7, with the two reference (or command) currents * indicated by “*” as inputs. The d-winding reference current isd controls * the rotor flux linkage λrd, whereas the q-winding current isq controls the electromagnetic torque Tem developed by the motor. The reference dq winding currents (the outputs of the proportional-integral PI controllers described in the next section) are converted into the reference phase currents ia* (t ), ib* (t ), and ic* (t ). A current-regulated switch-mode converter (the power-processing unit, PPU) can deliver the desired

86 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

currents to the induction motor, using a tolerance-band control described in the previous chapter. However, in such a current-regulation scheme, the switching frequency within the PPU does not remain constant. If it is important to keep this switching frequency constant, then an alternative is described in a Section 5-3 using a space-vector pulsewidth-modulation scheme, which is discussed in detail in Chapter 8. 5-2-1 Speed and Position Control Loops * * The current references isd and isq (inputs in the block diagram of Fig. 5-7) are generated by the cascaded torque, speed, and position control * loops shown in the block diagram of Fig. 5-8, where θmech is the position reference input. The actual position θmech and the rotor speed ωmech (where ωm = p/2 × ωmech) are measured, and the rotor flux linkage λrd is calculated as shown in the block diagram of Fig. 5-8 (same as Fig. 4-13 of the previous chapter). For operation in an extended speed range beyond the rated speed, the flux weakening is implemented as a function of rotor speed in computing the reference for the rotor flux linkage.

EXAMPLE 5-2 In this example, we will consider the drive system of Example 5-1 under vector control described earlier. The initial conditions in the motor are identical to that in the previous example. We will neglect the torque loop in this example, where all the motor parameter estimates are assumed to be perfect. (We will see the effect of estimate errors in the motor parameters in the next chapter.) The objective of the speed loop is to keep the speed at its initial value, in spite of the load torque disturbance at t = 0.1 second. We will design the speed loop with a bandwidth of 25 rad/s and a phase margin of 60°, using the same procedure as in Reference [1]. Solution Initial flux values are the same as in Example 5-1. These calculations are repeated in a MATLAB file EX5_2calc.m on the accompanying website. To design the speed loop (without the torque loop), the * torque expression is derived as follows at the rated value of isd : In

87

* θmech

ωmech

P

Σ

ωmech (measured)

Σ

* ωmech

ωmech

* λrd

PI

Σ

Tem (calculated)

* Tem

PI

PI

Tem λrd θda

ia* dp ib* to abc ic*

ωmech d / dt (measured)

i*sq

* isd

θda

Fig. 5-4

abc to dq

ia current i regulated b ic PPU

θmech

isd isq

Fig. 5-8 Vector controlled induction motor drive with a current-regulated PPU.

θmech (measured)

Σ

* λrd

λrd (calculated)

ωmech Estimated Motor Model

Motor

88 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

steady state under vector control, ird = 0 in Fig. 5-3. Therefore, in Eq. (5-9)

λrd = Lmisd

(under vector control in steady state).

(5-13)

Substituting for λrd from Eq. (5-13) and for irq from Eq. (5-2) into the * torque expression of Eq. (5-7) at the rated isd , Tem =

p L2m * isd isq 2L r

(under vector control in steady statte),

(5-14)

k

where k is a constant. The speed loop diagram is shown in Fig. 5-9 where the PI controller constants are calculated in EX5_2calc.m on the basis that the crossover frequency of the open loop is 25 rad/s and the phase margin is 60°. The simulation diagram of the file EX5_2.mdl (included on the accompanying website) is shown in Fig. 5-10, and the torque and speed are plotted in Fig. 5-11.

* ωmech

Σ

kp

ki s

isq (s)

k

Tem

1 sJeq

Fig. 5-9 Design of the speed-loop controller.

Fig. 5-10 Simulation of Example 5-2.

ωmech

Vector Control 89

Fig. 5-11 Simulation results of Example 5-2.

5-2-2 Initial Startup Unlike the example above where the system was operating in steady state initially, the system must be started from standstill conditions. Initially, the flux is built up to its rated value, keeping the torque to be * * zero. Therefore, initially isq is zero. The reference value λrd of the rotor flux at zero speed is calculated in the block diagram of Fig. 5-8. The value of the rotor-field angle θda is assumed to be zero. The division by zero in the block diagram of Fig. 5-4 is prevented until λrd takes on some finite (nonzero) value. This way, three stator currents build up to their steady state dc magnetizing values. The rotor flux builds up entirely along the a-axis. Once the dynamics of the flux build-up is completed, the drive is ready to follow the torque, speed and position commands. 5-2-3 Calculating the Stator Voltages to Be Applied It is usually desirable to keep the switching frequency within the switchmode converter (PPU) constant. Therefore, it is a common practice to calculate the required stator voltages that the PPU must supply to the motor, in order to make the stator currents equal to their reference values.

90 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

We will first define a unitless term called the leakage factor σ of the induction machine as: σ = 1−

L2m . Ls Lr

(5-15)

Substituting for ird from Eq. (5-9) into Eq. (3-19) for λsd,

λsd = σLs isd +

Lm λrd. Lr

(5-16)

From Eq. (3-20) for λsq, using Eq. (5-2) under vector-controlled conditions λsq = σ Ls isq.

(5-17)

Substituting these into Eq. (3-28) and Eq. (3-29) for vsd and vsq,

d L d vsd = Rs isd + σ Ls isd + m λrd − ωd σ Ls isq dt r dt L

(5-18)

d L vsq = Rs isq + σ Ls isq + ωd m λrd + ωdσ Ls isd . dt Lr

(5-19)

′ vsd

vsd,comp

and

′ vsq

vsq,comp

5-2-4 Designing the PI Controllers In the d-axis voltage equation of Eq. (5-18), on the right side only the first two terms are due to the d-axis current isd and disd/dt. The other terms due to λrd and isq can be considered as disturbances. Similarly in Eq. (5-19), the terms due to λrd and isd can be considered as disturbances. Therefore, we can rewrite these equations as:

′ = Rsisd + σLs vsd

d isd dt

(5-20)

′ = Rsisq + σ Ls vsq

d isq, dt

(5-21)

and

Vector Control 91

isd

Tem λrd θda

vsd,comp

isq

va*

v'sd

vb* * vsq

* isq

isd Transformations

θda

* vsd

* isd

Motor Model

vc*

v'sq

Space Vector Modulated PPU

Motor

ia ib ic

Transformations

isq

vsq,comp θmech (measured)

θmech

Encoder

Fig. 5-12 Vector control with applied voltages.

where the compensation terms are

Lm d λrd − ωdσ Ls isq Lr dt

(5-22)

L vsq,comp = ωd m λrd + σ Ls isd . Lr

(5-23)

vsd ,comp =

and

As shown in the block diagram of Fig. 5-12, we can generate the * * * * reference voltages vsd and vsq from given isd and isq , and using the calculated values of λrd, isd, and isq, and the chosen value of ωd. Using the calculated value of θda in the block diagram of Fig. 5-12, the reference values va* , vb* , and vc* for the phase voltages are calculated. The actual stator voltages va, vb, and vc are supplied by the power electronics converter, using the stator voltage space vector modulation technique discussed in Chapter 8. ′ signals in Fig. 5-12, we will employ PI controllers ′ and vsq To obtain vsd in the current loops. To compute the gains of the proportional and the integral portions of the PI controllers, we will assume that the compensation is perfect. Hence, each channel results in a block diagram of Fig. 5-13 (shown for d-axis), where the “motor-load plant” can be represented by the transfer functions below, based on Eq. (5-20) and Eq. (5-21):

92 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

Fig. 5-13 Design of the current-loop controller.

isd ( s) =

1 ′ ( s) vsd Rs + sσ Ls

(5-24)

isq ( s) =

1 ′ ( s). vsq Rs + sσ Ls

(5-25)

and

Now, the gain constants of the PI controller in Fig. 5-13 (same in the q-winding) can be calculated using the procedure illustrated in the following example.

EXAMPLE 5-3 Repeat the vector control of Example 5-2 by replacing the CR-PWM inverter by a space vector pulse width-modulated inverter, which is assumed to be ideal. The speed loop specifications are the same as in Example 5-2. The current (torque) loop to generate reference voltage has 10 times the bandwidth of the speed loop and the same phase margin of 60°. Solution Calculations for the initial conditions are repeated in the MATLAB file EX5_3calc.m, which is included on the accompanying website. It also shows the procedure for calculating the gain constants of the PI controller of the current loop. The simulation diagram of the SIMULINK file EX5_3.mdl (included on the accompanying website) is shown in Fig. 5-14, and the simulation results are plotted in Fig. 5-15.

93

Fig. 5-14 Simulation of Example 5-3.

94 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

Fig. 5-15 Simulation results of Example 5-3.

EXAMPLE 5-4 Consider the “test” machine described in Chapter 1. This machine is operating in steady state under its rated conditions, supplying its rated torque. At t = 1 second, the load torque suddenly goes to onehalf of its initial value and stays there. The objective is to maintain the speed of this machine at its initial value. Design a vector control scheme with the d-axis aligned to the rotor flux. Design the speed loop to have a bandwidth of 25 rad/s and the phase margin of 60°. Design the torque (current) loop to have a bandwidth of 250 rad/second and a phase margin of 60°. Simulate the system with and without the compensation terms and plot various quantities as functions of time. Solution See the complete solution on the accompanying website.

Problems 95

5-3 SUMMARY In this chapter, we first developed a model of the induction machine where the d-axis is always aligned with the rotor flux linkage space vector. Such a model of the machine is valid regardless if the machine is vector controlled, or if the voltages and currents to it are applied as under a general-purpose operation as discussed in Chapter 3. This is illustrated by Example 5-1. After developing the earlier-mentioned motor model, we studied vector control of induction motor drives, assuming that the exact motor parameters are known—effects of errors in parameter estimates are discussed in the next chapter. We first used an idealized CR-PWM inverter to supply motor currents calculated by the controller. This vector control is illustrated by means of Example 5-2. As the last step in this chapter, we used an idealized space vector pulse-width-modulated inverter (discussed in detail in Chapter 8) to supply motor voltages that result in the desired currents calculated by the controller. This is illustrated by means of Example 5-3.

REFERENCE 1. N. Mohan, Electric Machines and Drives: A First Course, Wiley, Hoboken, NJ, 2011. http://www.wiley.com/college/mohan.

PROBLEMS 5-1 In Example 5-1, comment how isd, isq, and λˆr vary under the dynamic condition caused by the change in load torque. Plot and comment on ωd under steady state, as well as under dynamic conditions. 5-2 Modify the simulation of Example 5-1 for a line start from standstill at t = 0, with the rated load torque. 5-3 In Example 5-2, plot the stator dq winding currents, the phase currents, ωd, and ωdA.

96 MATHEMATICAL DESCRIPTION OF VECTOR CONTROL

5-4 Add the blocks necessary in the simulation of Example 5-2 to plot phase voltages. 5-5 In the simulation of Example 5-2, include the torque loop, assuming its bandwidth to be 10 times larger than the speed loop bandwidth of 25 rad/s (keeping the phase margin in both loops at 60°). Compare results with those in Example 5-2. 5-6 Include flux weakening in the simulation of Example 5-2 by modifying the simulation as follows: initially in the steady-state operating condition, the load torque is one-half the rated torque of the motor. Instead of the load disturbance at t = 0.1 second, the speed reference is ramped linearly to reach 1.5 times the full-load motor speed in 2 seconds. 5-7 Plot phase voltages in Example 5-3. 5-8 Add the compensation terms in the simulation of Example 5-3. Compare results with those of Example 5-3. 5-9 Repeat Problem 5-6 in the simulation of Example 5-3 by including flux weakening, 0.1 second, the speed reference.

6

Detuning Effects in Induction Motor Vector Control

In vector control described in Chapters 4 and 5, we assumed that the induction machine parameters were known exactly. In practice, the estimated parameters may be off by a significant amount. This is particularly true of the rotor time constant τr (=Lr/Rr), which depends on the rotor resistance that increases significantly as the rotor heats up. In this chapter, we will calculate the steady-state error due to the incorrect estimate of the rotor resistance and also look at its effect on the dynamic response of vector-controlled drives [1,2].

6-1 EFFECT OF DETUNING DUE TO INCORRECT ROTOR TIME CONSTANT τr We will define a detuning factor to be the ratio of the actual and the estimated rotor time constants as

kτ =

τr , τ r ,est

(6-1)

where the estimated quantities are indicated by the subscript “est.” To analytically study the sensitivity of the vector control to kτ, we will simplify our system by assuming that the rotor of the induction machine is blocked from turning, that is, ωmech = 0. Also, we will assume an open* * loop system, where the command (reference) currents are isd and isq . As shown in Fig. 6-1 at t = 0−, the stator a-axis, the rotor A-axis, and the d-axis are all aligned with λr , which is built up to its rated value. Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

97

98 Detuning Effects in Induction Motor Vector Control q-axis

isq = 0

t = 0−

λr isd

a-axis d-axis

Fig. 6-1 dq windings at t = 0−.

qest-axis t> 0

q-axis * isq isq

is_dq

d-axis

i*sq

isd

isq

isd * θda isd

dest-axis i*sd

θda,est a-axis

Fig. 6-2 dq windings at t > 0; drawn for kτ imd Qr = vrqird = ωdAλrd ird = − vrd = 0 vrq = ωdAλrd = − (Continued)

116 DYNAMIC ANALYSIS OF DOUBLY FED INDUCTION GENERATORS

cos(θdA ) − sin(θdA ) vA (t ) 2 4π 4π vrd vB (t ) = cos θdA + − sin θdA + 3 3 3 vrq vC (t ) 2π 2π cos θ − sin θ + + dA dA 3 3 Various space vectors are shown in Fig. 7-5. q-axis

vs_dq = jvsq ir_dq

irq

isd

imd ird

d-axis

vr_dq = jvrq

is_dq

isq

Fig. 7-5 Space vector diagram for Example 7-2.

7-2 DYNAMIC ANALYSIS OF DFIG Equations for DFIG in terms of dq windings are the same as described in Chapter 3, where it is assumed that the rotor windings have the same number of turns as the stator windings, that is, Nr = Ns However, for n = (Nr/Ns), these equations can be rewritten, left as homework problems. 7-3 VECTOR CONTROL OF DFIG In Chapter 5, vector control was described by aligning the d-axis with the rotor flux. However, in controlling DFIG, it is common to align the d-axis with the stator voltage vector since stator voltages are easy to

Problems 117

measure [2]. With this choice of the reference frame, the d-axis stator current contributes to the real power P, and the q-axis stator current contributes to the reactive power of the DFIG. As discussed earlier, these are controlled by controlling the rotor currents ird and irq.

EXAMPLE 7-3 Consider a DFIG as a “test” machine, described in Chapter 1. Design the controller and show the output results. Solution A detailed controller design procedure and the results are on the accompanying website.

7-4 SUMMARY Doubly fed induction generators (DFIGs) are used in harnessing wind energy. In this chapter, the principle of operation of doubly fed induction machines is described mathematically in order to apply vector control.

REFERENCES 1. N. Mohan, Electric Machines and Drives, Wiley, Hoboken, NJ, 2012. http:// www.wiley.com/college/mohan. 2. T. Brekken, “A Novel Control Scheme for a Doubly-Fed Wind Generator under Unbalanced Grid Voltage Conditions,” PhD thesis, University of Minnesota, July 2005.

PROBLEMS 7-1 A DFIG is operating in the motoring mode at a subsynchronous speed at a leading power factor (supplying Qs from the grid).

118 DYNAMIC ANALYSIS OF DOUBLY FED INDUCTION GENERATORS

Calculate the signs of various quantities in this mode of operation. 7-2 A DFIG is operating in the generator mode at a subsynchronous speed at a leading power factor (supplying Qs to the grid). Calculate the signs of various quantities in this mode of operation. 7-3 Equations for DFIG in terms of dq windings are the same as described in Chapter 3, where it is assumed that the rotor windings have the same number of turns as the stator windings, that is, Nr = Ns. However, write these equations for n = (Nr/Ns) and draw dq winding equivalent circuits (a) “Seen” from the stator-side. (b) “Seen” from the rotor-side.

8

Space Vector Pulse WidthModulated (SV-PWM) Inverters

8-1 INTRODUCTION In Chapter 5, we briefly discussed current-regulated pulse widthmodulated (PWM) inverters using current-hysteresis control, in which the switching frequency fs does not remain constant. The desired currents can also be supplied to the motor by calculating and then applying appropriate voltages, which can be generated based on the sinusoidal pulse-width-modulation principles discussed in basic courses in electric drives and power electronics [1]. However, the availability of digital signal processors in control of electric drives provides an opportunity to improve upon this sinusoidal pulse-width modulation by a procedure described in this chapter [2,3], which is termed space vector pulse-width modulation (SV-PWM). We will simulate such an inverter using Simulink for use in ac drives. 8-2 SYNTHESIS OF STATOR VOLTAGE SPACE VECTOR vsa In terms of the instantaneous stator phase voltages, the stator space voltage vector is (8-1) vsa (t ) = va (t )e j 0 + vb (t )e j 2 π / 3 + vc (t )e j 4 π / 3. In the circuit of Fig. 8-1, in terms of the inverter output voltages with respect to the negative dc bus and hypothetically assuming the stator neutral as a reference ground Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

119

120 SPACE VECTOR PULSE WIDTH-MODULATED INVERTERS +

ia

a

ib

b

Vd

c

+

vb vc − v + a ic +

− N qa

qb

qc

Fig. 8-1 Switch-mode inverter.

va = vaN + vN ; vb = vbN + vN ; vc = vcN + vN .

(8-2)

Substituting Eq. (8-2) into Eq. (8-1) and recognizing that

e j 0 + e j 2 π / 3 + e j 4 π / 3 = 0,

(8-3)

the instantaneous stator voltage space vector can be written in terms of the inverter output voltages as

vsa (t ) = vaN e j 0 + vbN e j 2 π / 3 + vcN e j 4 π / 3.

(8-4)

A switch in an inverter pole of Fig. 8-1 is in the “up” position if the pole switching function q = 1, otherwise in the “down” position if q = 0. In terms of the switching functions, the instantaneous voltage space vector can be written as

vsa (t ) = Vd (qa e j 0 + qbe j 2 π / 3 + qc e j 4 π / 3 ).

(8-5)

With three poles, eight switch-status combinations are possible. In Eq. (8-5), the stator voltage vector vsa (t ) can take on one of the following seven distinct instantaneous values, where in a digital representation, phase “a” represents the least significant digit and phase “c” the most significant digit (e.g., the resulting voltage vector due to the switch status combination 011 is represented as v3 ): ( =3 )

SYNTHESIS OF STATOR VOLTAGE SPACE VECTOR vsa 121

vsa (000) = v0 = 0 vsa (001) = v1 = Vd e j 0 vsa (010) = v2 = Vd e j 2 π / 3 vsa (011) = v3 = Vd e jπ / 3 vsa (100) = v4 = Vd e j 4 π / 3 vsa (101) = v5 = Vd e j 5π / 3 vsa (110) = v6 = Vd e jπ vsa (111) = v7 = 0.

(8-6)

In Eq. (8-6), v0 and v7 are the zero vectors because of their values. The resulting instantaneous stator voltage vectors, which we will call the “basic vectors,” are plotted in Fig. 8-2. The basic vectors form six sectors, as shown in Fig. 8-2. The objective of the SV-PWM control of the inverter switches is to synthesize the desired reference stator voltage space vector in an optimum manner with the following objectives: • • •

A constant switching frequency fs Smallest instantaneous deviation from its reference value Maximum utilization of the available dc-bus voltages

v2 (010)

v3 (011)

sector 2 sector 1

vs

sector 3

v6 (110)

v1 (001)

sector 4

a-axis

sector 6 sector 5

v4 (100)

v5 (101)

Fig. 8-2 Basic voltage vectors ( v0 and v7 not shown).

122 SPACE VECTOR PULSE WIDTH-MODULATED INVERTERS v3 = Vd e j π / 3

vs = Vs e j θs yv3

θs

v1 = Vd e j 0

xv1

Fig. 8-3 Voltage vector in sector 1.

• •

Lowest ripple in the motor current, and Minimum switching loss in the inverter.

The above conditions are generally met if the average voltage vector is synthesized by means of the two instantaneous basic nonzero voltage vectors that form the sector (in which the average voltage vector to be synthesized lies) and both the zero voltage vectors, such that each transition causes change of only one switch status to minimize the inverter switching loss. In the following analysis, we will focus on the average voltage vector in sector 1 with the aim of generalizing the discussion to all sectors. To synthesize an average voltage vector vsa ( =Vˆs e jθs ) over a time period Ts in Fig. 8-3, the adjoining basic vectors v1 and v3 are applied for intervals xTs and yTs, respectively, and the zero vectors v0 and v7 are applied for a total duration of zTs. In terms of the basic voltage vectors, the average voltage vector can be expressed as

1 vsa = [ xTsv1 + yTsv3 + zTs ⋅ 0] Ts

(8-7)

vsa = xv1 + yv3,

(8-8)

x + y + z = 1.

(8-9)

or where

In Eq. (8-8), expressing voltage vectors in terms of their amplitude and phase angles results in

SYNTHESIS OF STATOR VOLTAGE SPACE VECTOR vsa 123

Vˆse jθs = xVd e j 0 + yVd e jπ / 3.

(8-10)

By equating real and imaginary terms on both sides of Eq. (8-10), we can solve for x and y (in terms the given values of Vˆs, θs, and Vd) to synthesize the desired average space vector in sector 1 (see Prob lem 8-1). Having determined the durations for the adjoining basic vectors and the two zero vectors, the next task is to relate the earlier discussion to the actual poles (a, b, and c). Note in Fig. 8-2 that in any sector, the adjoining basic vectors differ in one position; for example, in sector 1 with the basic vectors v1 (001) and v3 (011), only the pole “b” differs in the switch position. For sector 1, the switching pattern in Fig. 8-4 shows that pole-a is in “up” position during the sum of xTs, yTs, and z7Ts intervals, and hence for the longest interval of the three poles. Next in the length of duration in the “up” position is pole-b for the sum of yTs, and z7Ts intervals. The smallest in the length of duration is pole-c for only z7Ts interval. Each transition requires a change in switch state in only one of the poles, as shown in Fig. 8-4. Similar switching

vtri

vcontrol,a vcontrol,b

0

vcontrol,c

Vd 0

vaN x/ 2 vbN

0 z0 / 2

x /2

V y/ 2 vcN

0

y/ 2

Vd

z0 / 2 z7

Ts / 2 Ts

Fig. 8-4 Waveforms in sector 1; z = z0 + z7.

124 SPACE VECTOR PULSE WIDTH-MODULATED INVERTERS

patterns for the three poles can be generated for any other sector (see Problem 8-2).

8-3 COMPUTER SIMULATION OF SV-PWM INVERTER In computer simulations, for example, using Simulink, as well as in hardware implementation using rapid prototyping tools such as from DSPACE [4], the earlier described pulse-width modulation of the stator voltage space vector can be carried out by comparing control voltages with a triangular waveform signal at the switching frequency to generate switching functions. It is similar to the sinusoidal PWM approach only to the extent of comparing control voltages with a triangular waveform signal. However, in SV-PWM, the control voltages do not have a purely sinusoidal nature as those in the sinusoidal PWM. In an induction machine with an isolated neutral, the three-phase voltages sum to zero (see Problem 8-3) va (t ) + vb (t ) + vc (t ) = 0. (8-11) a To synthesize an average space vector vs with phase components va, vb, and vc (the dc-bus voltage Vd is specified), the control voltages can be written in terms of the phase voltages as follows, expressed as a ratio of Vˆtri (the amplitude of the constant switching frequency triangular signal vtri used for comparison with these control voltages):

vcontrol,a va − vk = Vd / 2 Vˆtri vcontrol,b vb − vk = Vd / 2 Vˆtri

(8-12)

vcontrol,c vc − vk . = Vd / 2 Vˆtri where

vk =

max(va , vb , vc ) + min(va , vb , vc ) . 2

(8-13)

Deriving Eq. (8-13) is left as a homework problem (Problem 8-5).

LIMIT ON THE AMPLITUDE Vˆs 125

EXAMPLE 8-1 In a three-phase inverter, the dc bus voltage Vd = 700 V. Using the space vector modulation principles, calculate and plot the control voltages in steady state to synthesize a 60-Hz output with a line-line rms value of 460 V. Assume that Vˆtri = 5 V and the switching frequency fs = 10 kHz. Solution Fig. 8-5 shows the block diagram in Simulink, which is included on the accompanying website, to synthesize the ac output voltages. The results are plotted in Fig. 8-6.

8-4 LIMIT ON THE AMPLITUDE Vˆs OF THE STATOR VOLTAGE SPACE VECTOR vsa First, we will establish the absolute limit on the amplitude Vˆs of the average stator voltage space vector at various angles. The limit on the amplitude equals Vd (the dc-bus voltage) if the average voltage vector lies along a nonzero basic voltage vector. In between the basic vectors, the limit on the average voltage vector amplitude is that its tip can lie on the straight lines shown in Fig. 8-7, forming a hexagon (see Problem 8-6). However, the maximum amplitude of the output voltage vsa should be limited to the circle within the hexagon in Fig. 8-7 to prevent distortion in the resulting currents. This can be easily concluded from the fact that in a balanced sinusoidal steady state, the voltage vector vsa rotates at the synchronous speed with its constant amplitude. At its maximum amplitude,

vsa,max (t ) = Vˆs,max e jωsynt.

(8-14)

Therefore, the maximum value that Vˆs can attain is

600 3 Vˆs,max = Vd cos = Vd. 2 2

(8-15)

126 Fig. 8-5 Simulation of Example 8-1.

LIMIT ON THE AMPLITUDE Vˆs 127

Fig. 8-6 Simulation results of Example 8-1.

Vd Vˆs,max 30° Vd

Fig. 8-7 Limit on amplitude Vˆs .

128 SPACE VECTOR PULSE WIDTH-MODULATED INVERTERS

From Eq. (8-15), the corresponding limits on the phase voltage and the line–line voltages are as follows: 2 V Vˆphase,max = Vˆs,max = d 3 3

(8-16)

and

VLL,max (rms) = 3

Vˆphase,max 2

=

Vd = 0.707Vd. 2

(8-17)

The sinusoidal pulse-width modulation in the linear range discussed in the previous course on electric drives and power electronics results in a maximum voltage

VLL,max (rms) =

3 Vd = 0.612Vd 2 2

(sinusoidal PWM). (8-18)

Comparison of Eq. (8-17) and Eq. (8-18) shows that the SV-PWM discussed in this chapter better utilizes the dc bus voltage and results in a higher limit on the available output voltage by a factor of (2 / 3 ), or by approximately 15%higher, compared with the sinusoidal PWM.

SUMMARY In this chapter, an approach called SV-PWM is discussed, which is better than the sinusoidal PWM approach in utilizing the available dc-bus voltage. Its modeling using Simulink is described.

REFERENCES 1. N. Mohan, Electric Machines and Drives: A First Course, Wiley, Hoboken, NJ, 2011. http://www.wiley.com/college/mohan. 2. H.W. van der Broek et al., “Analysis and Realization of a Pulse Width Modulator Based on Voltage Space Vectors,” IEEE Industry Applications Society Proceedings, 1986, pp. 244–251.

Problems 129

3. J. Holtz, “Pulse Width Modulation for Electric Power Converters,” chapter 4 in Power Electronics and Variable Frequency Drives, ed. B.K. Bose, IEEE Press, New York, 1997. 4. http://www.dspace.de.

PROBLEMS 8-1 In a converter, Vd = 700 V. To synthesize an average stator voltage vector vsa = 563.38e j 0.44 V, calculate x, y, and z. 8-2 Repeat if vsa = 563.38e j 2.53 V. Plot results similar to those in Fig. 8-4. 8-3 Show that in an induction machine with isolated neutral, at any instant of time, va(t) + vb(t) + vc(t) = 0. 8-4 Given that vsa = 563.38e j 0.44 V, calculate the phase voltage components. 8-5 Derive Eq. (8-12). 8-6 Derive that the maximum limit on the amplitude of the space vector forms the hexagonal trajectory shown in Fig. 8-7.

9

Direct Torque Control (DTC) and Encoderless Operation of Induction Motor Drives

9-1 INTRODUCTION Unlike vector-control techniques described in previous chapters, in the direct-control (DTC) scheme, no dq-axis transformation is needed, and the electromagnetic torque and the stator flux are estimated and directly controlled by applying the appropriate stator voltage vector [1–3]. It is possible to estimate the rotor speed, thus eliminating the need for rotor speed encoder.

9-2 SYSTEM OVERVIEW Figure 9-1 shows the block diagram of the overall system, which includes the speed and the torque feedback loops, without a speed encoder. The estimated speed ωmech,est is subtracted from the reference (desired) * speed ωmech , and the error between the two acts on a PI-controller to * generate the torque reference signal Tem . The estimated speed generates the reference signal for the stator flux linkage λˆs* (thus allowing flux weakening for extended range of speed operation), which is compared with the estimated stator flux linkage λˆ s,est. The errors in the electromagnetic torque and the stator flux, combined with the angular position ∠θs of the stator flux linkage space vector, determine the stator voltage space vector vs that is applied to the motor during each sampling interval ΔT, for example, equal to 25 μs. Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

130

Principle of Encoderless DTC Operation 131

+

* Tem PI Σ Σ + − + − ω mech Tem

* ω mech

λˆs*

+

Σ

− λˆ

s

Selection of vs

< λs

Vd

−

qa

qb qc

Estimator

ic

ib

ia

IM

Fig. 9-1 Block diagram of DTC.

Estimating the electromagnetic torque and the stator flux linkage vector requires measuring the stator currents and the stator phase voltages—the latter, as shown in Fig. 9-1, are indirectly calculated by measuring the dc-bus voltage and knowing within the digital controller the status of the inverter switches. 9-3 PRINCIPLE OF ENCODERLESS DTC OPERATION Prior to detailed derivations, we can enumerate the various steps in the estimator block of Fig. 9-1 as follows, where all space vectors are implicitly expressed in electrical radians with respect to the stator a-axis as the reference axis (unless explicitly mentioned otherwise): 1. From the measured stator voltages and currents, calculate the stator flux linkage space vector λs : λs (t ) = λs (t −∆T ) +

t

∫

t−∆T

(vs − Rs is ) ⋅ dτ = λˆs e jθs .

132 DIRECT TORQUE CONTROL AND ENCODERLESS OPERATION

2. From λs and is , calculate the rotor flux space vector λr and hence the speed of the rotor flux linkage vector, where ΔTω is a sampling time for speed calculation: θ (t ) − θr (t −∆Tω ) L d λr = r (λs − σ Ls is ) = λˆr e jθr and ωr = θr = r . ∆Tω Lm dt 3. From λs and is , calculate the estimated electromagnetic torque Tem: 2 p Tem = Im(λsconjis ). 3 2

4. From λr and Tem,est, estimate the slip speed ωslip and the rotor speed ωm: ωslip =

2 3 Tem Rr ˆ 2 and ωm = ωr − ωslip. p 2 λr

In the stator voltage selection block of Fig. 9-1, an appropriate stator voltage vector is calculated to be applied for the next sampling interval ΔT based on the errors in the torque and the stator flux, in order to keep them within a hysteretic band. 9-4 CALCULATION OF λ s , λr , Tem, AND ωm 9-4-1 Calculation of the Stator Flux λ s The stator voltage equation with the stator a-axis as the reference is

d vs = Rs is + λs. dt

(9-1)

From Eq. (9-1), the stator flux linkage space vector at time t can be calculated in terms of the flux linkage at the previous sampling time as

λs (t ) = λs (t −∆T ) +

t

∫

(vs − Rs is ) ⋅ dτ = λˆs e jθs ,

(9-2)

t−∆T

where τ is the variable of integration, the applied stator voltage remains constant during the sampling interval ΔT, and the stator current value is that measured at the previous time step.

CALCULATION OF λ s , λ r, Tem, AND ωm 133

9-4-2 Calculation of the Rotor Flux λr From Chapter 3,

λs = Ls is + Lm ir

(9-3)

λr = Lr ir + Lm is.

(9-4)

L λs ir = − s is, Lm Lm

(9-5)

and

Calculating ir from Eq. (9-3),

and substituting it into Eq. (9-4),

L LL λr = r λs − s r is + Lm is Lm Lm 2 Lr L = λs − Ls is 1 − m , Lm L s Lr (=σ )

(9-6)

where the leakage factor σ is defined as (similar to Eq. 5-15 in Chapter 5)

σ = 1−

L2m . Ls Lr

(9-7)

Therefore, the rotor flux linkage space vector in Eq. (9-6) can be written as

L λr = r (λs − σLs is ) = λˆr e jθr . Lm

(9-8)

We should note that similar to Eq. (9-2), for the stator flux linkage vector, the rotor flux linkage space vector can be expressed as follows, recognizing that the rotor voltage in a squirrel-cage rotor is zero

λrA (t ) = λrA (t −∆T ) +

t

∫

t−∆T

A ( − Rr ir A ) ⋅ dτ = λˆr e jθr ,

(9-9)

134 DIRECT TORQUE CONTROL AND ENCODERLESS OPERATION λs (t)

vs •∆T λs (t − ∆T) λr (t) ≈ λr (t − ∆T)

θs θr

θrA θm

Rotor A-axis a-axis

Fig. 9-2 Changing the position of stator flux-linkage vector.

where the space vectors and angles (in electrical radians) are expressed with respect to the rotor A-axis shown in Fig. 9-2. The above equation shows that the rotor flux changes very slowly with time (in amplitude and in phase angle θrA with respect to the rotor A-axis) only due to a small voltage drop across the rotor resistance. 9-4-3 Calculation of the Electromagnetic Torque Tem The electromagnetic torque developed by the motor can be estimated in terms of the stator flux and the stator current, or in terms of the stator flux and the rotor flux. We will derive both expressions in Appendix 9-A; however, the final expressions that we need are given below. Torque depends on the magnitude of the stator and the rotor fluxes, and the angle between the two space vectors. As derived in Appendix 9-A, in terms of the machine leakage inductance Lσ (also defined in Appendix 9-A)

2 p Lm ˆ ˆ Tem = λ λ sin θsr, 3 2 L2σ s r

(9-10a)

θsr = θs − θr.

(9-10b)

where

The angles in Eq. (9-10) are expressed in electrical radians with respect the stator a-axis, as shown in Fig. 9-2. Equation (9-2) and Fig. 9-2 show that the torque can be controlled quickly by rapidly changing the position of the stator flux linkage space vector (i.e., θs, hence θsr) by applying an appropriate voltage space

CALCULATION OF λ s , λ r, Tem, AND ωm 135

vector during the sampling interval ΔT, while the rotor flux space vector position θr (= θrA + θm ) changes relatively slowly. Thus, in accordance with Eq. (9-10a), a change in θsr results in the desired change in torque. For torque estimation, it is better to use the expression below (derived in Appendix 9-A) in terms of the estimated stator flux linkage and the measured stator currents,

2 p Tem = Im(λsconjis ), 3 2

(9-11)

which, unlike the expression in Eq. (9-10a), does not depend on the rotor flux linkage (note that the rotor flux linkage in Eq. (9-8) depends on correct estimates of Ls, Lr, and Lm). 9-4-4 Calculation of the Rotor Speed ωm A much slower sampling rate with a sampling interval ΔTω, for example, equal to 1 ms, may be used for estimating the rotor speed. The speed of the rotor flux in electrical radians per second (rad/s) is calculated from the phase angle of the rotor flux space vector in Eq. (9-8) as follows:

ωr =

θ (t ) − θr (t − ∆Tω ) d θr = r . dt ∆Tω

(9-12)

The slip speed is calculated as follows: In Chapter 5, the torque and the speed expressions are given by Eq. (5-7) and Eq. (5-5), where in the motor model, the d-axis is aligned with the rotor flux linkage space vector. These equations are repeated below:

Tem =

p Lm isq λrd 2 Lr

(9-13)

and

ωslip = Rr

1 Lm isq , λrd Lr

(9-14)

where ωslip is the slip speed, the same as ωdA in Eq. (5-5) of Chapter 5. Calculating isq from Eq. (9-13) and substituting it into Eq. (9-14) (and

136 DIRECT TORQUE CONTROL AND ENCODERLESS OPERATION

recognizing that in the model with the d-axis aligned with the rotor flux linkage, λrd = 2 / 3λˆr ), the slip speed in electrical radians per second is

ωslip =

2 3 Tem Rr ˆ 2 . p 2 λr

(9-15)

Therefore, the rotor speed can be estimated from Eq. (9-12) and Eq. (9-15) as

ωm = ωr − ωslip,

(9-16)

where all speeds are in electrical radians per second. In a multipole machine with p ≥ 2,

ωmech = (2 / p)ωm.

(9-17)

9-5 CALCULATION OF THE STATOR VOLTAGE SPACE VECTOR A common technique in DTC is to control the torque and the stator flux amplitude with a hysteretic band around their desired values. Therefore, at a sampling time (with a sampling interval of ΔT), the decision to change the voltage space vector is implemented only if the torque and/or the stator flux amplitude are outside their range. Selection of the new voltage vector depends on the signs of the torque and the flux errors and the sector in which the stator flux linkage vector lies, as explained later. The plane of the stator voltage space vector is divided into six sectors, as shown in Fig. 9-3. We should note that these sectors are different than those defined for the stator voltage space vector-PWM in Chapter 8. The central vectors for each sector, which lie in the middle of a sector, are the basic inverter vectors, as shown in Fig. 9-3. The choice of the voltage space vector for sector 1 is explained later with the help of Fig. 9-4 and Eq. (9-10). Assuming that the stator flux linkage space vector is along the central vector, the roles of various voltage vectors can be tabulated in Table 9-1. There are some additional observations: The voltage vectors would have the same effects as tabulated earlier, provided the stator fluxlinkage space vector is anywhere in sector 1. The use of voltage vectors

Calculation of the Stator Voltage Space Vector 137 b-axis

v3(011)

v2(010) 3

2

v6(110)

v1(001) a-axis

1

4

6

5

v5(101)

v4(100) c-axis

Fig. 9-3 Inverter basic vectors and sectors.

v2

v3

λs

sector 1

v1

v6 v4

v5

Fig. 9-4 Stator voltage vector selection in sector 1. TABLE 9-1 Effect of Voltage Vector on the Stator Flux-Linkage Vector in Sector 1 vs Tem λˆs v3 Increase Increase v2 Increase Decrease v4 Decrease Decrease v5 Decrease Increase

v1 and v6 is avoided because their effect depends on where the stator flux-linkage vector is in sector 1. A similar table can be generated for all other sectors. Use of zero vectors v0 (000) and v7 (111) results in the stator flux linkage vector essentially unchanged in amplitude and in the angular position θs. In the torque expression of Eq. (9-10b), for small values of θsr in electrical radians,

138 DIRECT TORQUE CONTROL AND ENCODERLESS OPERATION

sin θsr ≈ (θs − θr ).

(9-18)

With a zero voltage vector applied, assuming that the amplitudes of the stator and the rotor flux linkage vectors remain constant, Tem k(θs − θr ),

(9-19)

where k is a constant. With the zero voltage vector applied, the position of the stator flux-linkage vector remains essentially constant, thus Δθs 0. Similarly, the position of the rotor flux-linkage vector, with respect to the rotor A-axis, remains essentially constant, that is, ΔθrA 0. However, as can be observed from Fig. 9-2, ∆θr = ∆θm + ∆θrA. Therefore the position of the rotor flux-linkage vector changes, albeit slowly, and the change in torque in Eq. (9-19) can be expressed as

∆Tem −k(∆θm ) (with zero voltage vector applied).

(9-20)

Equation (9-20) shows that applying a zero stator voltage space vector causes change in torque in a direction opposite to that of ωm. Therefore, with the rotor rotating in a positive (counter-clockwise) direction, for example, it may be preferable to apply a zero voltage vector to decrease torque in order to keep it within a hysteretic band. In literature, there is no uniformity on the logic of space vector selection to keep the stator flux amplitude and the electromagnetic torque within their respective hysteretic bands. One choice of space vectors is illustrated by means of Example 9-1.

EXAMPLE 9-1 The “test” induction motor described in Chapter 1 is operated using encoderless DTC for speed control, as described in a file Ex9_1.pdf (which can be printed) on the website accompanying this textbook. Model this system using Simulink and plot the desired results. Solution Various subsystems and the simulation results are included on the website associated with this textbook.

Problems 139

9-6 DIRECT TORQUE CONTROL USING dq-AXES It is possible to perform the same type of control by aligning the d-axis with the stator flux-linkage vector. The amplitude of the stator fluxlinkage vector is controlled by applying vsd along the d-axis, and the torque is controlled by applying vsq along the q-axis. The advantage of this type of control over the hysteretic control described earlier is that it results in a constant switching frequency. This is described by Example 9-2 in accompanying website.

9-7 SUMMARY This chapter discusses the direct torque control (DTC) scheme, where, unlike the vector control, no dq-axis transformation is needed and the electromagnetic torque and the stator flux are estimated and directly controlled by applying the appropriate stator voltage vector. It is possible to estimate the rotor speed, thus eliminating the need for rotor speed encoder.

REFERENCES 1. M. Depenbrock, “Direct Self Control (DSC) of Inverter-Fed Induction Machines,” IEEE Transactions on Power Electronics, 1988, pp. 420–429. 2. I. Takahashi and T. Noguchi, “A New Quick Response and High Efficiency Strategy of an Induction Motor,” IEEE Transactions on Industry Applications, vol. 22, no. 7, 1986, pp. 820–827. 3. P. Tiitinen and M. Surendra, “The Next Generation Motor Control Method, DTC Direct Torque Control,” Proceedings of the International Conference on Power Electronics, Drives and Energy Systems, PEDES’96 New Delhi (India), pp. 37–43.

PROBLEMS 9-1 Using the parameters of the “test” induction machine described in Chapter 1, show that it is much faster to change electromagnetic

140 DIRECT TORQUE CONTROL AND ENCODERLESS OPERATION

torque by changing the position of the stator flux-linkage vector, rather than by changing its amplitude. Assume that the machine is operating under rated conditions. 9-2 Obtain the stator voltage vectors needed in other sectors, similar to what has been done in Table 9-1 for sector 1. 9-3 Assuming that the “test” machine is operating under the rated conditions, compute the effect of applying a zero voltage space vector on the flux linkage space vectors and on the electromagnetic torque produced. 9-4 In the system of Example 9-1, how can the modeling be simplified if the speed is never required to reverse? 9-5 Experiment with other schemes for selecting voltage space vector and compare results with that in Example 9-1. 9-6 In Example 9-1, include the field-weakening mode of operation.

APPENDIX 9-A Derivation of Torque Expressions The electromagnetic torque in terms of the stator flux and the stator current can be expressed as follows: Tem =

2p Im(λsconj is ) 32

(9A-1)

To derive the above expression, it is easiest to assume it be correct and to substitute the components to prove it. Taking the complex conjugate on both sides of Eq. 9-3,

λsconj = Ls isconj + Lm ir conj

(9A-2)

Substituting in Eq. 9A-1, Tem =

conj conj conj 2p 2p {Ls Im( i i i L is ) (9A-3) s s ) + Lm Im( ir s )} = m Im( ir 32 32 (=0 )

Appendix 9-A 141

Even though the dq transformation is not used in DTC, we can make use of dq transformations to prove our expressions. Therefore, in terms of an arbitrary dq reference set and the corresponding components substituted in Eq. 9A-3,

Tem =

2p 3 3 p Lm Im{ (ird − jirq ) (isd + jisq )} = Lm (isqird − isd irq ) 32 2 2 2 (9A-4)

which is identical to Eq. 3-47 of Chapter 3, thus proving the torque expression of Eq. 9A-1 to be correct. Another torque expression, which we will not use directly but which is the basis on which the selection of the stator voltage vector is made, is as follows: Tem =

2 p Lm conj Im( λ ) sλr 3 2 L2σ

(9A-5)

where the machine leakage inductance is defined as

Lσ = Ls Lr − L2m

(9A-6)

Again assuming the above expression in Eq. 9A-5 to be correct and substituting the expressions for the fluxes from Eqs. 9-3 and 9-4, 2 3 2 = 3

Tem =

conj conj p Lm Im{( L i + L i )( L i + L i )} s s m r r r m s 2 L2σ conj conj 2 p Lm 2 p Lm L L i i L i Im( ) + Im( ). s r s r m r is 3 2 L2σ 2 L2σ

(9A-7)

Note that Im( ir isconj ) = − Im( is ir conj ). Therefore, in Eq. 9A-7, Tem =

conj conj 2 p Lm 2p 2 L i ( L L − L )Im( i i ) = Im( ) s r m s r m s ir 32 3 2 L2σ 2 ( =Lσ )

p Lm Im{(isd + jisq )(ird − jirq )} 2 p = Lm (isqird − isd irq ), 2 =

(9A-8)

142 DIRECT TORQUE CONTROL AND ENCODERLESS OPERATION

which is identical to Eq. 3-47 of Chapter 3, thus proving the torque expression of Eq. 9A-5 to be correct. In Eq. 9A-5, expressing flux linkages in their polar form, 2 3 2 = 3

Tem =

2 p Lm ˆ ˆ p Lm Im(λˆs e jθs ⋅ λˆr e− jθr ) = λsλr Im(e jθsr ) 2 2 Lσ 3 2 L2σ p Lm ˆ ˆ λsλr sin θsr 2 L2σ

(9A-9)

θsr = θs − θr

(9A-10)

where

is the angle between the two flux-linkage space vectors.

10

Vector Control of Permanent-Magnet Synchronous Motor Drives

10-1 INTRODUCTION In the previous course [1], we looked at permanent-magnet synchronous motor drives, also known as “brushless-dc motor” drives in steady state, where without the help of dq analysis, it was not possible to discuss dynamic control of such drives. In this chapter, we will make use of the dq-analysis of induction machines, which is easily extended to analyze and control synchronous machines.

10-2 d-q ANALYSIS OF PERMANENT MAGNET (NONSALIENT-POLE) SYNCHRONOUS MACHINES In synchronous motors with surface-mounted permanent magnets, the rotor can be considered magnetically round (non-salient) that has the same reluctance along any axis through the center of the machine. A simplified representation of the rotor magnets is shown in Fig. 10-1a. The three-phase stator windings are sinusoidally distributed in space, like in an induction machine, with Ns number of turns per phase. In Fig. 10-1b, d-axis is always aligned with the rotor magnetic axis, with the q-axis 90° ahead in the direction of rotation, assumed to be counter-clockwise. The stator three-phase windings are represented by equivalent d- and q-axis windings; each winding has 3 / 2 N s turns, which are sinusoidally distributed. Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

143

144 VECTOR CONTROL OF PERMANENT-MAGNET DRIVES b-axis

b-axis q-axis

ib

ib

Br a' N

θm

N ia

S

a-axis ic

c-axis

θm ia a-axis

S

ic

d-axis

a

c-axis

(a)

(b)

Fig. 10-1 Permanent-magnet synchronous machine (shown with p = 2).

10-2-1 Flux Linkages The stator d and q winding flux linkages can be expressed as follows:

λsd = Ls isd + λfd

(10-1)

λsq = Ls isq,

(10-2)

and

where in Eq. (10-1) and Eq. (10-2), Ls = Lℓs + Lm, and λfd is the flux linkage of the stator d winding due to flux produced by the rotor magnets (recognizing that the d-axis is always aligned with the rotor magnetic axis). 10-2-2 Stator dq Winding Voltages Using Eq. (3-28) and Eq. (3-29), developed for induction machines in Chapter 3, in dq windings,

vsd = Rs isd +

d λsd − ωmλsq dt

(10-3)

vsq = Rs isq +

d λsq + ωmλsd, dt

(10-4)

and

d-q ANALYSIS 145

where the speed of the equivalent dq windings is ωd = ωm (in electrical rad/s) in order to keep the d-axis always aligned with the rotor magnetic axis [2]. The speed ωm is related to the actual rotor speed ωmech as

ωm =

p ωmech. 2

(10-5)

10-2-3 Electromagnetic Torque Using the analysis for induction machines in Chapter 3 and Eq. (3-46) and Eq. (3-47), we can derive the following equation (see Problem 3-9a in Chapter 3), which is also valid for synchronous machines:

Tem =

p (λsd isq − λsqisd ). 2

(10-6)

Substituting for flux linkages in the above equation for a nonsalientpole machine,

Tem =

p p [(Ls isd + λfd )isq − Ls isqisd ] = λfd isq (nonsalient). 2 2

(10-7)

10-2-4 Electrodynamics The acceleration is determined by the difference of the electromagnetic torque and the load torque (including friction torque) acting on Jeq, the combined inertia of the load and the motor:

d T −TL ωmech = em , dt J eq

(10-8)

where ωmech is in rad/s and is related to ωm as shown in Eq. (10-5). 10-2-5 Relationship between the dq Circuits and the Per-Phase Phasor-Domain Equivalent Circuit in Balanced Sinusoidal Steady State In this section, we will see that under a balanced sinusoidal steady state condition, the two dq winding equivalent circuits combine to result in the per-phase equivalent circuit of a synchronous machine that we have derived in the previous course. Note that in a synchronous motor used

146 VECTOR CONTROL OF PERMANENT-MAGNET DRIVES

in a “brush-less dc” drive, the synchronous speed equals the rotor speed on an instantaneous basis, therefore our choice of ωd = ωm also results in ωd = ωm = ωsyn. Under a balanced sinusoidal steady-state condition, dq winding quantities are dc and their time derivatives are zero. In Eq. (10-3) and Eq. (10-4), for stator voltages, substituting flux linkages from Eq. (10-1) and Eq. (10-2) results in vsd = Rs isd −ωm Ls isq

(10-9)

vsq = Rs isq + ωm Ls isd + ωmλfd.

(10-10)

and

Multiplying both sides of Eq. (10-10) by (j) and adding to Eq. (10-9) (and multiplying both sides of the resulting equation by 3 / 2 ) leads to the following space vector equation, with the d-axis as the reference axis: vs = Rs is + jωm Ls is + j 3 / 2ωmλfd , (10-11) e fs

noting that vs = 3 / 2 (vsd + jvsq ) and so on. Dividing both sides of the above space vector equation by 3/2, we obtain the following phasor equation for phase a in a balanced sinusoidal steady state: 2 Va = Rs I a + jωm Ls I a + jωm λfd . 3

(10-12)

E fa

The above equation corresponds to the per-phase equivalent circuit of Fig. 10-2 that was derived in the previous course under a balanced sinusoidal steady-state condition.

Ia +

jωmLs Rs

jωmLls

jωmLm + Efa (Eˆfa = kEωm)

Va −

−

Fig. 10-2 Per-phase equivalent circuit in steady state (ωm in electrical rad/s).

d-q ANALYSIS 147

Relationship between kE and λfd From Eq. (10-12),

2 λ fd ωm = kE ωm, Eˆ fa = 3

(10-13)

kE

Therefore,

kE =

2 λfd. 3

(10-14)

10-2-6 dq-Based Dynamic Controller for “Brushless DC” Drives In the previous course, in the absence of the dq analysis, a hysteretic converter was used, where the switching frequency does not remain constant. In this section, we will see that it is possible to use a converter with a constant switching frequency. The block diagram of such a control system is shown in Fig. 10-3. In Eq. (10-3) and Eq. (10-4), using the flux linkages of Eq. (10-1) and Eq. (10-2), voltages can be expressed as follows, recognizing that the time-derivative of the rotor-produced flux λfd is zero:

vsd = Rs isd + Ls

d isd + (−ωm Ls isq ) dt comp

(10-15)

d

mech

Fig. 10-3 Controller in the dq reference frame.

148 VECTOR CONTROL OF PERMANENT-MAGNET DRIVES

and

vsq = Rs isq + Ls

d isq + ωm (Ls isd + λfd ). dt comp

(10-16)

q

In Fig. 10-3, the PI controllers in both channels are designed assuming that the inverter is ideal and the compensation (decoupling) terms in Eq. (10-15) and Eq. (10-16) are utilized, to result in the desired phase margin at the chosen open-loop crossover frequency. Flux Weakening In the normal speed range below the rated speed, the reference for the d winding current is kept zero (ids = 0). Beyond the rated speed, a negative current in the d winding causes flux weakening (a phenomenon similar to that in brush-type dc machines and induction machines), thus keeping the back-emf from exceeding the rated voltage of the motor. A negative value of isd in Eq. (10-10) causes vsq to decrease. To operate synchronous machines with surface-mounted permanent magnets at above the rated speed requires a substantial negative d winding current to keep the terminal voltage from exceeding its rated value. Note that the total current into the stator cannot exceed its rated value in steady state. Therefore, the higher the magnitude of the d winding current, the lower the magnitude of the q winding current has to be, since 3ˆ 2 2 isd + isq ≤ Iˆdq,rated = I a,rated . 2

(10-17)

EXAMPLE 10-1 For analyzing performance of the dynamic control procedure, a motor from a commercial vendor catalog [3] is selected, whose specifications are as follows: Nameplate Data Continuous Stall Torque: 3.2 Nm Continuous Current: 8.74 A

d-q ANALYSIS 149

Peak Torque: Peak Current: Rated Voltage: Rated Speed: Phases: Number of Poles:

12.8 Nm 31.5 A 200 V 6000 rpm 3 4

Per-Phase Motor Circuit Parameters Rs = 0.416 Ω Ls = 1.365 mH Voltage Constant kE (as in Eq. 10-13 and Fig. 10-2): 0.0957 V/ (electrical rad/s) The total equivalent inertia of the system (motor–load combination) is Jeq = 3.4×10−4 kg ⋅ m 2. Initially, the drive is operating in steady state at its rated speed, supplying its rated torque of 3.2 Nm to the mechanical load connected to its shaft. At time t = 0.1 second, a load-torque disturbance occurs, which causes it to suddenly decrease by 50% (there is no change in load inertia). The feedback control objective is to keep the shaft speed at its initial steady-state value subsequent to the load–torque disturbance. Design the speed feedback controller with the open-loop crossover frequency of 2500 rad/s and a phase margin of 60 degrees. The open-loop crossover frequency of the internal current feedback loop is ten times higher than that of the speed loop and the phase margin is 60°. Solution The simulation block diagram is shown in Fig. 10-4 and the Simulink file EX9_1.mdl is included in the accompanying website to this textbook. The simulation results are shown in Fig. 10-5.

150 Fig. 10-4 Simulation of Example 10-1.

Salient-Pole Synchronous Machines 151

Fig. 10-5 Simulation results of Example 10-1.

q-axis

b-axis d-axis q-axis isq

• a-axis

d-axis irq

ird ird ifd

+

a-axis (b)

c-axis (a)

Fig. 10-6 Salient-pole machine.

10-3 SALIENT-POLE SYNCHRONOUS MACHINES Synchronous machines with interior permanent magnets result in unequal reluctance along the d- and the q-axis. In this section, we will go a step further and assume a salient-pole rotor structure as shown in Fig. 10-6a with a rotor field excitation and nonidentical damper

152 VECTOR CONTROL OF PERMANENT-MAGNET DRIVES

windings along the d- and the q-axis. In such a machine, we have the inductances described in the next section. 10-3-1 Inductances In the stator dq windings,

Lsd = Lmd + Ls

(10-18)

Lsq = Lmq + Ls,

(10-19)

and

where the magnetizing inductance of the d winding is not equal to that of the q winding (Lmd ≠ Lmq) due to the nonsalient nature of the rotor. However, both windings have the same leakage inductance Lℓs, which is not affected by the rotor structure. As shown in Fig. 10-6b, we will replace the actual field winding of Nf turns in the rotor by an equivalent field winding with 3 / 2 N s turns (where Ns equals the number of turns in each phase of the stator windings), supplied by an equivalent field-winding current ifd. This procedure results in the equivalent field winding having the same magnetizing inductance Lmd as the stator d winding, hence the equivalent field winding inductance can be written as

Lfd = Lmd + Lfd,

(10-20)

where Lℓfd is the leakage inductance of the equivalent field winding. Similarly in Fig. 10-6b, replacing the actual damper windings with equivalent damper windings, each with 3 / 2 N s turns, we can write the following equations for the inductances of the equivalent rotor damper windings (with a subscript “r”):

Lrd = Lmd + Lrd

(10-21)

Lrq = Lmq + Lrq,

(10-22)

and

where Lℓrd and Lℓrq are the leakage inductance of the equivalent rotor damper windings.

Salient-Pole Synchronous Machines 153

10-3-2 Flux Linkages In terms of these inductances, the flux linkages of the various windings can be expressed as follows: Stator dq Winding Flux Linkages

λsd = Lsd isd + Lmd ird + Lmd i fd

(10-23)

λsq = Lsq isq + Lmq irq.

(10-24)

and

Rotor dq Winding Flux Linkages

λrd = Lrd ird + Lmd isd + Lmd i fd

(d-axis damper)

(10-25)

λrq = Lrqisq + Lmqisq (q-axis damper)

(10-26)

λfd = Lfd id + Lmd isd + Lmd ird.

(10-27)

and

10-3-3 Winding Voltages In terms of the above flux linkages, winding voltages can be written as follows, assuming that ωd = ωm in order to keep the d-axis aligned with the rotor magnetic axis. Stator dq Winding Voltages

vsd = Rs isd +

d λsd − ωmλsq dt

(10-28)

vsq = Rs isq +

d λsq + ωmλsd. dt

(10-29)

and

Rotor dq Winding Voltages

d v λrd rd = Rrd ird + dt

(10-30)

d vrq = Rrqirq + λrq dt

(10-31)

( =0 )

(=0 )

154 VECTOR CONTROL OF PERMANENT-MAGNET DRIVES

and

vfd = Rfd i fd +

d λfd. dt

(10-32)

10-3-4 Electromagnetic Torque Substituting for flux linkages from above in Eq. (10-6):

Tem =

p [Lmd (i fd + ird )isq + (Lsd − Lsq )isd isq − Lmq irq isd ]. (10-33) 2 field + damper in d -axis saliency

10-3-5 dq-Axis Equivalent Circuits Following the procedure used in Chapter 3 for deriving the dq-axis equivalent circuits for induction machines, we can draw the equivalent circuits shown in Fig. 10-7 for the d- and q-axis windings, respectively. 10-3-6 Space Vector Diagram in Steady State In a balanced sinusoidal steady state with ωd = ωm, the damper winding currents in the rotor are zero, as well as all the time derivatives of

Fig. 10-7 Equivalent circuits for a salient-pole machine.

Salient-Pole Synchronous Machines 155

currents and flux linkages in dq windings. Therefore, in Eq. (10-23) and Eq. (10-24),

λsd = Lsd isd + Lmd i fd

(10-34)

λsq = Lsq isq.

(10-35)

and

From Eq. (10-28) and Eq. (10-29), using Eq. (10-34) and Eq. (10-35) vsd = Rs isd −ωm Lsqisq

(10-36)

vsq = Rs isq + ωm Lsd isd + ωm Lmd i fd.

(10-37)

and

Multiplying Eq. (10-37) by (j) and adding to Eq. (10-36),

vsd + jvsq = Rs isd + jRs isq + jωm Lsd isd + jωm Lmd i fd − ωm Lsqisq, (10-38)

which is represented by a space vector diagram in Fig. 10-8a, where

vsd + jvsq =

2 vs 3

(10-39)

isd + jisq =

2 is. 3

(10-40)

and

The corresponding phasor diagram is shown in Fig. 10-8b.

Im Va

2 / 3vs

vsd

2 / 3is vsq

(a)

Ia

q-axis

Re

(b)

d-axis

Fig. 10-8 Space vector and phasor diagrams.

156 VECTOR CONTROL OF PERMANENT-MAGNET DRIVES

10-4 SUMMARY In this chapter, we have extended the dq-analysis of induction machines to analyze and control synchronous machines.

REFERENCES 1. N. Mohan, Electric Machines and Drives: A First Course, Wiley, Hoboken, NJ, 2011. http://www.wiley.com/college/mohan. 2. E.W. Kimbark, Power System Stability: Synchronous Machines, IEEE Press, New York, 1995. 3. http://www.baldor.com.

PROBLEMS 10-1 In the simulation of Example 10-1, replace the ideal inverter by an appropriate SV-PWM inverter, similar to that described in Chapter 8. 10-2 Implement flux-weakening in Example 10-1 for extended speed operation. 10-3 Derive the torque expression in Eq. (10-33) for a salient-pole synchronous motor.

11

Switched-Reluctance Motor (SRM) Drives

11-1 INTRODUCTION In the previous course [1], we have studied variable reluctance stepper motors, whose construction requires a salient stator and a salient rotor. Stepper motors are generally used for position control in an open-loop manner, where by counting the number of electrical pulses supplied and knowing the step angle of the motor, it is possible to rotate the shaft by a desired angle without any feedback. In contrast, switchedreluctance motor (SRM) drives, also doubly salient in construction, are intended to provide continuous rotation and compete with induction motor and brushless dc motor drives in certain applications, such as washing machines and automobiles, with many more applications being contemplated. In this chapter, we will briefly look at the basic principles of SRM operation and how it is possible to control them in an encoderless manner. 11-2 SWITCHED-RELUCTANCE MOTOR Cross-section of a four-phase SRM is shown in Fig. 11-1, which looks identical to a variable-reluctance stepper motor. It has a four-phase winding on the stator. In order to achieve a continuous rotation, each phase winding is excited by an appropriate current at an appropriate rotor angle, as well as de-excited at a proper angle. For rotating it in the counterclockwise direction, the excitation sequence is a-b-c-d. Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

157

158 SWITCHED-RELUCTANCE MOTOR DRIVES θmech = 0o

θmech

a d

b

c

c

d

b a

Fig. 11-1 Cross-section of a four-phase 8/6 switched reluctance machine. θmech = 0o = θal

θmech

θmech = 0o

θmech

a

a d

b

d

c

c

d

b

θun

b

c

c

b

d

a

a

(a)

(b)

Fig. 11-2 (a) Aligned position for phase a; (b) unaligned position for phase a.

An SRM must be designed to operate with magnetic saturation and the reason to do so will be discussed later on in this chapter. Fig. 11-2 shows the aligned and the unaligned rotor positions for phase a. For phase a, the flux linkage λa as a function of phase current ia is plotted in Fig. 11-3 for various values of the rotor position. In the unaligned position where the rotor pole is midway between two stator poles (see Fig. 11-2b, where θmech equals θun), the flux path includes a large air gap, thus the reluctance is high. Low flux density keeps the magnetic structure in its linear region, and the phase inductance has a small value.

Switched-Reluctance Motor 159

Fig. 11-3 Typical flux linkage characteristics of an SRM.

λa λ2 λ1

2 1

θ1 + ∆θmech θ1

0

I1

ia

Fig. 11-4 Calculation of torque.

As the rotor moves toward the aligned position of Fig. 11-2a (where θmech equals zero), the characteristics become progressively more saturated at higher current values. 11-2-1 Electromagnetic Torque Tem With the current built up to a value I1, as shown in Fig. 11-4, holding the rotor at a position θ1 between the unaligned and the aligned

160 SWITCHED-RELUCTANCE MOTOR DRIVES

positions, the instantaneous electromagnetic torque can be calculated as follows: Allowing the rotor to move incrementally under the influence of the electromagnetic torque from position θ1 to θ1 + Δθmech, keeping the current constant at I1, the incremental mechanical work done is ∆Wmech = Tem∆θmech.

(11-1)

The increment of energy supplied by the electrical source is

∆Welec = area (1 − λ1 − λ2 − 2 − 1),

(11-2)

and the incremental increase in energy storage associated with the phase-a winding is

∆Wstorage = area (0 − 2 − λ2 − 0) − area (0 − 1 − λ1 − 0).

(11-3)

The mechanical work performed is the difference of the energy supplied by the electrical source minus the increase in energy storage

∆Wmech = ∆Welec − ∆Wstorage.

(11-4)

Therefore in Eq. (11-4),

Tem∆θ = area (1 − λ1 − λ2 − 2 − 1) − {area (0 − 2 − λ2 − 0) − area (0 − 1 − λ1 − 0)} = {a rea (1 − λ1 − λ2 − 2 − 1) + area (0 − 1 − λ1 − 0)} (11-5) area ( 0−1−2−λ2 −0 )

− area (0 − 2 − λ2 − 0) = area (0 − 1 − 2 − 0), which is shown shaded in Fig. 11-4. Therefore,

Tem =

area (0 − 1 − 2 − 0) , ∆θmech

(11-6)

which is in the direction to increase this area. The area between the λ − i characteristic and the horizontal current axis is usually defined as

Switched-Reluctance Motor 161

the co-energy W′. Therefore, the area in Eq. (11-6) shown shaded in Fig. 11-4 represents an increase in co-energy. Thus, on a differential basis, we can express the instantaneous electromagnetic torque developed by this motor as the partial derivative of co-energy with respect to the rotor angle, keeping the current constant ∂W ′ ∂θmech

Tem =

.

(11-7)

ia =constant

11-2-2 Induced Back-EMF ea With phase a excited by ia, the movement of the rotor results in a backemf ea, and the voltage across the phase-a terminals includes the voltage drop across the resistance of the phase winding:

va = Ria + ea

(11-8)

and

ea =

d λa (ia, θmech ), dt

(11-9)

where the phase winding flux linkage is a function of the phase current and the rotor position, as shown in Fig. 11-3. In terms of partial derivatives, we can rewrite the back-emf in Eq. (11-9) as: ea =

∂λa ∂ia

θmech

d ∂λa ia + dt ∂θmech

ia

d θmech, dt

(11-10)

where it is important to recognize that a partial derivative with respect to one variable is obtained by keeping the other variable constant. In Fig. 11-3, the movement of the rotor by an angle Δθmech, keeping the current constant results in a back-emf, which from Eq. (11-10) can be written as:

ea =

∂λa ∂θmech

d ∂λa θmech = dt ia ∂θmech ωmech

d ωmech ia = 0, dt ia

(11-11)

162 SWITCHED-RELUCTANCE MOTOR DRIVES

where ωmech is the instantaneous rotor speed. However, the instantaneous power (eaia) is not equal to the instantaneous mechanical output due to the change in stored energy in the phase winding.

11-3 INSTANTANEOUS WAVEFORMS For clear understanding, we will initially assume an idealized condition where it is possible to supply the phase winding with a current ia that has a rectangular waveform as a function of θmech, as shown in Fig. 11-5. The current is assumed to be built up instantaneously (this will require infinite voltage) at the unaligned position θun and instantaneously goes to zero at the aligned position θal. The corresponding waveforms for the electromagnetic torque Tem,a and the induced back-emf ea are also plotted by means of Eq. (11-7) and Eq. (11-11) respectively, with the current held constant. 20 ia (A) 10 0 −40 1

θun

θal

30

θun

θal

30

λa (V-sec) 0.5 0 −40 40 Tem,a 20 (Nm) 0 −40 300 ea (V)

30

200 100 0 −40

θmech (deg)

30

Fig. 11-5 Performance assuming idealized current waveform.

Instantaneous Waveforms 163

The objectives in selecting these two rotor positions θun and θal for current flow are twofold: (1) to maximize the average torque per ampere, and (2) to build up the current to its desired level while the back-emf is small. We can appreciate that with the current flow prior to the unaligned position and after the aligned position, the instantaneous torque would be negative, which would be counter to our objective of maximizing the average torque per ampere. At the unaligned position, the winding inductance is the lowest, and it is easier to build up current in that position compared with other rotor positions. To achieve instantaneous build-up and decay of phase current assumed in the plots of Fig. 11-5 would require that an infinite phase voltage (positive and negative) is available. In reality with a finite voltage available from the power processing unit to the motor, the phase current waveform for a four-phase motor may look as shown in Fig. 11-6, with the corresponding flux linkage and torque waveforms. 20 ia (A) 10 0 θon θun 1

θoff θal

λa 0.5 (V-sec) 0 θon θun 40

θoff θal

Tem,a 20 (Nm) 0 θon θun

θoff θal

40 (Nm) 20 0 θon θun

Tem Tem,a

Tem,b

Tem,c

Tem,d

θoff θal

Fig. 11-6 Performance with a power-processing unit.

164 SWITCHED-RELUCTANCE MOTOR DRIVES

The phase current build-up is started at an angle θon (prior to the unaligned position), and the current decay is started at an angle θoff (prior to the aligned position). In order to reduce torque ripple, there is generally overlapping of phases where during a short duration, two of the phases are simultaneously excited. The bottom part in Fig. 11-6 shows the resultant electromagnetic torque Tem by summing the torque developed by each of the four phases.

11-4 ROLE OF MAGNETIC SATURATION [2] Magnetic saturation plays an important role in SRM drives. During each excitation cycle, a large ratio of the energy supplied to a phase winding should be converted into mechanical work, rather than returned to the electrical source at the end of the cycle. We will call this an energy conversion factor. In Fig. 11-7, assuming magnetic saturation and a finite voltage available from the power-processing unit, this factor is as follows:

Energy Conversion Factor =

Wem . Wem + Wf

(11-12)

As can be seen from Fig. 11-7, this factor would be clearly higher in the idealized case where an instantaneous build-up and decay of

aligned position λa θoff

Wf Wem

unaligned position 0

ia

Fig. 11-7 Flux linkage trajectory during motoring.

Power Processing Units for SRM Drives 165

phase current is assumed. However, without saturation this energy conversion factor is limited to a value of nearly 50%. Magnetic saturation also keeps the rating of the power processor unit from becoming unacceptable. It should be noted that the Energy Conversion factor is not the same as energy efficiency of the motor, although there is a correlation—a lower energy conversion factor means that a larger fraction of energy sloshes back and forth between the power-processing unit and the machine, resulting in power losses in the form of heat and thus in a lower energy efficiency.

11-5 POWER PROCESSING UNITS FOR SRM DRIVES A large number of topologies for SRM converters have been proposed in the literature. Fig. 11-8 shows a topology that is most versatile. For current build-up, both transistors are turned on simultaneously. (This also shows the robustness of the SRM drive power processing unit, where turning on both transistors simultaneously is normal, which can be catastrophic in other drives.) To maintain the current within a hysteretic band around the reference value, either one of the transistors is turned off, thus making the current freewheel through the opposite diode, or both transistors are tuned off, in which case the current flows into the dc bus and decreases in magnitude. The later condition is also used to quickly de-energize a phase winding.

Da2

Sa2

Db2

Sb2

Db1

Dc2

Sc2

Sd1 Dc1

Dd2

Phase d

−

Da1

Sc1

Phase c

Vdc

Sb1

Phase b

+

Phase a

Sa1

Dd1

Sd2

Fig. 11-8 Power converter for a four-phase switched reluctance drive.

166 SWITCHED-RELUCTANCE MOTOR DRIVES v

+

∑

− iR

λ θ(λ, i)

θmech

i

Fig. 11-9 Estimation of rotor position.

11-6 DETERMINING THE ROTOR POSITION FOR ENCODERLESS OPERATION It is necessary to determine the rotor position so that the current build-up and decay can be started at rotor positions θon and θoff, respectively, for each phase. There are various methods proposed in the literature to determine the rotor position. One of the easiest methods to explain is shown by means of Fig. 11-3 and Fig. 11-9. The motor can be characterized to achieve the family of curves shown in Fig. 11-3, where the flux linkage of a phase is plotted as a function of the phase current for various values of the rotor position. From this, information, knowing the flux linkage and the current allows the determination of the rotor position. In Fig. 11-9, the flux linkage of an excited phase is computed by integrating the difference of the applied phase voltage and the voltage drop across the winding resistance (see Eq. 11-8). The combination of the measured phase current and the estimated flux linkage then determines the rotor position, using the information of Fig. 11-3.

11-7 CONTROL IN MOTORING MODE A simple block diagram for speed control is shown in Fig. 11-10, where the actual rotor position is either sensed or estimated using the method described in the previous section or some other technique. The speed error between the reference speed and the actual speed is amplified by means of a PI (proportional-integral) controller to generate a current reference. The rotor angle determines which phases are to be excited, and their current is controlled to equal the reference current as much

Problems 167

ω*

mech

Iref

PI

∑

+

current

power

controller signals

−

λa

gate converter

ωmech s

position

Fig. 11-10 Control block diagram for motoring.

as possible in view of the limited dc-bus voltage of the power-processing unit shown in Fig. 11-8.

11-8 SUMMARY This chapter discusses SRM drives, doubly salient in construction, which are intended to provide continuous rotation and compete with induction motor and brushless dc motor drives in certain applications, such as washing machines and automobiles, with many more applications being contemplated. In this chapter, we briefly looked at the basic principles of SRM operation and how it is possible to control them in an encoderless manner.

REFERENCES 1. N. Mohan, Electric Machines and Drives: A First Course, Wiley, Hoboken, NJ, 2011. http://www.wiley.com/college/mohan. 2. T.J.E. Miller, ed., Electronic Control of Switched Reluctance Machines, Newnes, Oxford, 2001.

PROBLEMS 11-1 Show that without magnetic saturation, the energy conversion factor in Fig. 11-7 would be limited to 50%.

168 SWITCHED-RELUCTANCE MOTOR DRIVES

11-2 What would the plot of the phase inductance be as a function of the rotor angle (between the unaligned and the aligned rotor positions) for various values of the phase current. 11-3 Although only the motoring mode is discussed in this chapter, SRM drives (like all other drives) can also be operated in a generator mode. Explain how this mode of operation is possible in SRM drives.

INDEX Note: Page numbers in italics refer to figures. a-b-c phase winding variables, relating dq winding variables to, 35–36 αβ windings, derivation of dq winding voltages, 37–40, 37, 39 Acceleration determination of, 44 in permanent-magnet synchronous motor drives, 145 ac motor drives, 1–2 types of, 2–3 Adjustable-speed drives, 1 Air gap dq winding in, 38, 39, 41 flux density in, 67, 69 flux density space vectors in, 15, 20 sinusoidal flux-density distribution in, 11, 15–16 sinusoidally distributed stator windings and, 7–8, 30 in switched-reluctance motor drives, 158 Amplitude, of stator voltage space vector, 125–128, 127 Angle error, 105, 107

Angles, 7 d-axis, 34, 35, 39, 73–75, 98 d-q axis, 66 rotor, 28, 83, 84, 89, 157, 161, 164 symbols for, xvi voltage vector, 122–123, 125, 134 Applications electric drive, 1 wind energy, 109 Brushless dc motor drives dq-based dynamic controller for, 147–151, 147 performance, 59–62 See also Permanent-magnet synchronous motor drives Circuits d- and q-axis equivalent, 45–46, 45 dq, 145–147 dq-axis equivalent, 154, 154 dq-winding equivalent, 45–46, 45 transformer equivalent, 65–66, 65 Co-energy, switched-reluctance motor drives and, 161

Advanced Electric Drives: Analysis, Control, and Modeling Using MATLAB/Simulink®, First Edition. Ned Mohan. © 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

169

170 INDEX Collinear stator space vectors, 19, 19 Computer simulation dq windings, 47–56 SV-PWM inverter, 124–125, 126–127 See also Matlab; Simulink Conductor density, 7 Control. See Vector control CR-PWM. See Current-regulated PWM (CR-PWM) inverter Current-controlled brushless dc (BLDC) motor drive, 60 Current-excited transformer with secondary shortcircuited, 62–66, 62, 64 transformer equivalent circuit, 65–66, 65 Current-loop controller, 90–92, 92 Current-regulated power-processing unit, 72–75, 73, 85–86, 87 Current-regulated PWM (CR-PWM) inverter, 79, 85, 99, 100 Currents dq-winding, from flux linkages, 47–48 d-winding reference, 85 flux linkages of dq windings in terms of, 36 obtaining, with voltages as inputs, 40–41, 41 q-winding, 85 reference, 72–73 Current waveform, 75 Damper windings, 152 d-axis aligned along rotor flux linkage λr -axis, 79–84, 80 aligned with rotor flux, vector control with, 67–72, 67–72

aligned with stator voltage vector, 116–117 flux linkages and currents in doubly fed induction generators and, 112 rotor flux linkage dynamics and, 82 d-axis equivalent circuits, 45–46, 45 d-axis voltage equations, 90–91 d-axis winding representation, 66–67, 66 dc bus voltage, 128 dc motor drives, 1–2, 60 performance of, 59–62 Detuning effects in induction motor vector control, 97–107 due to incorrect rotor time constant, 97–101 steady-state analysis of, 101–107 Detuning factor, defined, 97 DFIGs. See Doubly fed induction generators (DFIGs) Digital signal processors, 119 Direct torque control (DTC), 107, 130–139 electromagnetic torque, calculation of, 134–135 encoderless DTC operation, principle of, 131–132 rotor flux, calculation of, 133–134 rotor speed, calculation of, 135–136 stator flux, calculation of, 132 stator voltage space vector, calculation of, 136–138 system overview, 130–131, 131 using dq-axes, 139 Doubly fed induction generators (DFIGs), 109–117, 110 benefits in wind applications, 109 dynamic analysis of, 116

INDEX 171

electromagnetic torque and, 113 flux linkages and currents and, 112 operation of, 110–116 stator and rotor power inputs, 112–113 stator voltages and, 111 vector control of, 116–117 dq axis(axes) direct torque control using, 139 in doubly fed induction generators, 111–117 dq-axis equivalent circuits, in salient-pole synchronous machines, 154, 154 dq-axis transformation, xiii–xiv, 22 dq-based dynamic controller, for brushless dc drives, 147–151, 147 dq circuits, per-phase phasor-domain equivalent circuit and, in balanced sinusoidal steady state, 145–147 dq transformation, xiii, 22, 141 dq winding analysis case for, 22–25 of induction machines operating under dynamic conditions (see Induction machines, dynamic analysis of in terms of dq windings) of permanent magnet synchronous machines, 143–151 dq winding currents, 85 stator and rotor representation by equivalent, 32–33, 33 dq-winding equivalent circuits, 45–46, 45 dq winding flux linkages and currents, 40–41, 41

dq winding quantities, transformation of phase quantities into, 34–35, 34 dq windings detuning effect due to incorrect rotor time constant and, 97–101, 98 dynamic analysis of induction machines in terms of (see Induction machines, dynamic analysis of in terms of dq windings) flux linkages of, in term of their currents, 36 induction motor model in terms of, 47–56, 49 mathematical relationships of, at arbitrary speed, 33–41 per-phase phasor-domain equivalent circuit, in balanced sinusoidal steady state, 46–47, 47 representation, 28–33, 29 subscripts, xv dq winding speed, choice of, 41–42 dq winding variables, relating to phase winding variables, 35–36 dq winding voltages, 33 equations, 37–40 DSPACE, 124 DTC. See Direct torque control (DTC) d-winding reference current, 85 Dynamic analysis of doubly fed induction generators, 116 of induction machines operating under dynamic conditions (see Induction machines, dynamic analysis of in terms of dq windings)

172 INDEX Dynamic controller, dq-based, 147–151, 147 Electrical Machines and Drives: A First Course (Mohan), xiii Electrical radians, 7 Electric vehicles, 1 Electrodynamics of induction machines, 44–45 of permanent-magnet synchronous motor drives, 145 Electromagnetic torque, 42–44 acceleration and, 44 calculating, 131, 132, 134–135 controlling, 1, 2 in dc-motor drive, 60–61 in doubly fed induction generators, 113 inductances and, 44 in motor model, 74 in motor model with d-axis aligned along rotor flux linkage λr -axis, 81–82 net, on the rotor, 44 in permanent-magnet synchronous motor drives, 145 plotting, 53, 54 on rotor d-axis winding, 42–43, 42 on rotor q-axis winding, 43, 44 in salient-pole synchronous machines, 154 space vector equations and, 17 speed, position control, and, 72–75 steady-state, 106 in switched-reluctance motor drives, 159–161, 159, 162, 164 See also Torque

Encoderless operation of induction-motor drives, 3 (see also Direct torque control (DTC)) of switched-reluctance motor drives, 166 Energy conversion factor, 165 Factory automation, 59 Feedback control systems, 2 Field distributions, space vectors representing, 16–17, 16 Flux dq winding analysis and, 23, 30 obtaining, with voltages as inputs, 40–41, 41 See also Magnetic flux Flux currents, in doubly fed induction generators, 112 Flux density, 10 rotor, 61, 84 Flux-density distributions, 8, 11 Flux linkages, 18–21 calculating, 50–51, 55 in current-excited transformer with secondary short-circuited, 62–63 in doubly fed induction generators, 112 dq-winding currents from, 47–48 of dq windings in terms of their currents, 36 magnetic, 11 in permanent-magnet synchronous motor drives, 144 of phase-a, 12 in salient-pole synchronous machines, 153 in switched-reluctance motor drives, 158, 159, 164, 166

INDEX 173

symbol, xvi See also Rotor flux linkage; Stator flux linkage Flux weakening,, dq-based dynamic controller and, 148 Four-phase switched reluctance drive, power converter for, 165, 165 Friction torque, 44 Hybrid-electric vehicles, 1 Hysteresis control, 75–76, 75 Hysteretic band, direct torque control and, 136, 139 Hysteretic converter, 147 Indirect vector control, in rotor flux reference frame, 84–94, 85 initial start-up, 89 PI controllers, designing, 90–92 speed and position control loops and, 86–89 stator voltages, calculating, 89–90 Induced back-EMF, for switchedreluctance motor drives, 161–162 Inductance electromagnetic torque and, 44 in salient-pole synchronous machines, 152 single-phase magnetizing, 9–10, 10, 12 See also Magnetizing inductance; Mutual inductance Induction machine equations in phase quantities, 6–25 dq-winding analysis of induction machines, 22–25 equivalent windings in a squirrelcage rotor, 13–14 flux linkages, 18–21

mutual inductances between the stator and the rotor phase windings, 15 sinusoidally distributed stator windings, 6–9 space vectors, 15–18 stator and rotor voltage equations in terms of space vectors, 21 stator inductances (rotor open-circuited), 9–12 Induction machines, dynamic analysis of in terms of dq windings, 28–56 choice of dq winding speed, 41–42 computer simulation, 47–56 d- and q-axis equivalent circuits, 45–46 dq winding representation, 28–33 electrodynamics, 44–45 electromagnetic torque and, 42–44 mathematical relationships of dq windings, 33–41 relationship between dq windings and per-phase phasor-domain equivalent circuit in balanced sinusoidal steady state, 46–47 Induction motor, 2, 3 encoderless operation of (see Direct torque control (DTC)) parameters, xvi specifications for “test,” 3–4 vector control of (see Vector control of induction-motor drives) Induction motor model computer simulation, 47–56, 49 initial conditions, calculating, 48–56 phasor analysis, 49–56

174 INDEX Instantaneous waveforms, switchedreluctance motor drives and, 162–164, 162 Inverse-transformation, of stator currents, 73–74 Inverters current-regulated pulse-width modulated, 79, 85, 99, 100 pulse-width modulated, 121 switch-modulated, 120, 120 See also Space vector pulse width-modulated (SV-PWM) inverters Kirchhoff’s current law, 18 Leakage factor, 90 Leakage flux in current-excited transformer with secondary shortcircuited, 63–64, 65 phase-a, 9 step change in torque and, 69 Leakage inductance, 9–10, 10, 152 of equivalent windings, 31, 32 of rotor phase, 13–14 Leakage subscript, xv Line-line voltages, limits on, 128 Load torque acceleration and, 44 calculating, 50–53, 56 Magnetic flux linkage, 11 Magnetic saturation, switchedreluctance motor drives and, 158, 164–165, 164 Magnetizing flux, 9, 11 in current-excited transformer with secondary short-circuited, 63 step change in torque and, 69

Magnetizing inductance per-phase, 12, 13 of rotor phase, 13 single-phase, 9–10, 10, 12 Magnetizing subscript, xv Magnetomotive force (mmf) space vectors, 15–16, 15, 29, 61 Mathematical description of vector control indirect vector control in rotor flux reference frame, 84–94 motor model with the d-axis aligned along the rotor flux linkage λr -axis, 79–95, 80, 83 Mathematical relationships of dq windings, at arbitrary speed, 33–41 MATLAB, xiv, 3 Mechanical subscript, xv mmf. See Magnetomotive force (mmf) space vectors Motor model with d-axis aligned along rotor flux linkage λr -axis, 79–84, 80, 83 for vector control, 74–75 Multi-pole machine, peak conductor density in, 7 Mutual inductance, 11, 11 between dq windings on stator and rotor, 32–33, 33 between stator and rotor phase windings, 15 Nonsalient-pole synchronous machines, 143–151 Park’s transformation, 28 Peak values, symbols for, xv

INDEX 175

Permanent-magnet ac drives, 2–3 Permanent-magnet synchronous machine, 144 Permanent-magnet synchronous motor drives, vector control of, 143–156 d-q analysis of permanent magnet synchronous machines, 143–151 salient-pole synchronous machines, 151–155, 151 Per-phase magnetizing inductance, 12, 13 Per-phase phasor-domain equivalent circuit dq circuits and, in balanced sinusoidal steady state, 145–147, 146 dq windings and, in balanced sinusoidal steady state, 46–47, 47 Phase a flux linkage of, 12 sinusoidally distributed stator windings for, 6–8, 7 switched-reluctance motor drive positions for, 158 Phase b, sinusoidally distributed stator windings for, 8–9 Phase c, sinusoidally distributed stator windings for, 8–9 Phase currents, dq windings for, 29 Phase quantities analysis of induction machine in, 23–25 transformation into dq winding quantities, 34–35, 34 Phase voltage, limits on, 128 Phasor analysis, induction motor model simulation, 49–56

Phasor diagram, for salient-pole synchronous machines, 155 Phasors space vectors and, in sinusoidal steady state, 17–18, 17 symbols, xv PI. See Proportional-integral (PI) amplifier; Proportional-integral (PI) controller PMAC drives, 75 Position control, 1–4 torque, speed, and, 72–75 Position control loops, 72, 86–89 Position of rotor field, 61 Power electronics converter, 91 Power-processing unit (PPU), 2, 75–76 current-regulated, 85–86, 87 induction motor drive with current-regulated, 72–75, 73 switched-reluctance motor drives and, 163, 163, 165, 165 switching frequency and, 86, 89 vector control and, 60–61 Proportional-integral (PI) amplifier, 72–73 Proportional-integral (PI) controller, 148 designing, 90–92, 92 direct torque control and, 130 switched-reluctance motor drives and, 166 PSpice, 3 Pulse-width modulated (PWM) inverter current-regulated, 79, 85, 99, 100 objective of, 121 See also Space vector pulse width-modulated (SV-PWM) inverters

176 INDEX q-axis, flux linkages and currents in doubly fed induction generators, 112 q-axis equivalent circuits, 45–46, 45 q-axis winding representation, 66–67, 66 q-winding current, 85 Radial field distribution, 8 Reference current, 72–73 Reference phase currents, 85 Reference voltages, 91 Resistance of equivalent windings, 14, 31, 32 phase winding, 161, 166 rotor, 71, 97, 100, 106–107, 111, 134 secondary winding, 63 Robotics, 59 Rotor in doubly fed induction generator, 109 mutual inductance between dq windings on, 32–33, 33 net torque on, 44 subscripts, xv in surface-mounted permanent magnets, 143 See also Squirrel-cage rotor Rotor αβ windings, dq winding voltages and, 39–40, 39 Rotor angle, 28, 83, 84, 89, 157, 161, 164 Rotor angle symbol, xvi Rotor axis-A, 17 Rotor current space vector, 21, 31–32 Rotor d-axis, torque on, 42–43, 42 Rotor dq winding, 31–32 Rotor dq winding flux linkages, in salient-pole synchronous machines, 153

Rotor dq winding voltages, in salient-pole synchronous machines, 153 Rotor-field angle, 89 Rotor flux, 89 steady state analysis of, 103–104 Rotor flux density, 61, 84 Rotor flux linkage, 19–21, 20 d-axis aligned along, 79–84, 80 dq-winding currents from, 47–48 Rotor flux linkage space vector, 138 calculating, 132, 133–134 calculation of electromagnetic torque and, 135 vector control with d-axis aligned with the, 67–72, 67–72 Rotor mmf space vector, 31–32, 32 Rotor phase subscripts, xv Rotor phase windings, mutual inductances between, 15 Rotor positions, for switchedreluctance motor drives, 158, 163, 166, 166 Rotor power inputs, in doubly fed induction generators, 113 Rotor q-axis, torque on, 43, 44 Rotor real and reactive powers, in doubly fed induction generators, 113 Rotor resistance, 71, 97, 100, 106–107, 111, 134 Rotor resistance, steady-state error and, 97, 101–102, 106–107 Rotor speed calculation of, 132, 135–136 plotting, 53, 54 symbol for, xvi Rotor time constant, detuning effects due to incorrect, 97–101 Rotor voltage equations, in terms of space vectors, 21, 22

INDEX 177

Rotor voltages in doubly fed induction generators, 112 Rotor windings, 36 Rotor-windings inductances (stator open-circuited), 13–14, 14 Salient-pole synchronous machines, 151–155, 151 Secondary winding current, 63 Secondary winding leakage flux, 63–64 Secondary winding resistance, 63 Self-inductance of stator phase winding, 9 Simulation software. See Matlab; Simulink Simulink®, xiv, xvi, 3, 4, 25, 51, 53, 83, 86, 92, 106, 124 Single-phase magnetizing inductance, 9–10, 10, 12 Sinusoidally distributed stator windings, 6–9, 7 Sinusoidal steady state dq circuits and per-phase phasordomain equivalent circuit in balanced, 145–147, 146 dq windings and per-phase phasor-domain equivalent circuit in, 46–47, 47 phasors and space vectors in, 17–18, 17 space vector diagram in, 154–155, 155 Slip speed calculation of, 132, 135–136 in motor model with d-axis aligned along rotor flux linkage λr -axis, 81 steady state estimated value and, 101–103

Space vector diagram in steady state, salient-pole synchronous machines and, 154–155, 155 Space vector pulse width-modulated (SV-PWM) inverters, 92, 93–94, 95, 119–128 computer simulation of, 124–125, 126–127 limit on amplitude of stator voltage space vector, 125–128, 128 synthesis of stator voltage space vector, 119–124 Space vectors magnetomotive force, 15–16, 15, 29, 61 phasors and, in sinusoidal steady state, 17–18, 17 review of, 15–18 rotor current, 21, 31–32 stator and rotor voltage equations in terms of, 21 stator current, 16–17, 16, 21, 29–30 superscripts and, 33–34 symbols, xv See also Rotor flux linkage space vector; Stator flux linkage space vector; Stator voltage space vector Speed symbols for, xvi torque, position control, and, 72–75 See also Rotor speed; Slip speed Speed control, 1–4 of switched-reluctance motor drives, 166–167, 167 Speed control loops, 86–89, 88 Squirrel-cage rotor equivalent phase voltages in, 21 equivalent windings in, 6, 13–14

178 INDEX Squirrel-cage rotor (cont’d) rotor voltage in, 133 slip speed in, 81 SRM. See Switched-reluctance motor (SRM) drives Stator in doubly fed induction generator, 109 mutual inductance between dq windings on, 32–33, 33 subscripts, xv Stator αβ windings, dq winding voltages and, 37–39, 37 Stator currents, transformation and inverse-transformation of, 73–74 Stator current space vector, 16–17, 16, 21, 29–30 Stator dq current vector, 103 Stator dq winding flux linkages, in salient-pole synchronous machines, 153 Stator dq windings inductance and, 152 representation, 29–31 Stator dq winding voltages in permanent-magnet synchronous motor drives, 144–145 in salient-pole synchronous machines, 153 Stator flux amplitude, direct torque control and, 136 Stator flux linkage, 18–19, 19, 20–21 calculation of electromagnetic torque and, 134–135 dq-winding currents from, 47–48 Stator flux linkage space vector, 137–138

calculating, 131, 132, 134 calculation of electromagnetic torque and, 134 d-axis aligned with, 139 Stator inductances, 9–12 Stator mutual-inductance, 11, 11 Stator phase subscripts, xv Stator phase windings, mutual inductances between, 15 Stator power inputs, in doubly fed induction generators, 112 Stator real and reactive powers, in doubly fed induction generators, 113 Stator single-phase magnetizing inductance, 9–10, 10, 12 Stator voltage equations, in terms of space vectors, 21, 22 Stator voltages calculating, 89–90 in doubly fed induction generators, 111, 116–117 Stator voltage space vector calculation of, 136–138 limit on amplitude of, 125–128, 127 sectors, 136–137, 137, 137(table) synthesis of, 119–124 Stator windings, 36, 66, 66 sinusoidally distributed, 6–9, 7 Steady-state analysis, 101–107 Stepper motors, 157 Subscripts, xv Superscripts, xv space vectors and, 33–34 SV-PWM. See Space vector pulse width-modulated (SV-PWM) inverters Switched-reluctance drives, 3

INDEX 179

Switched-reluctance motor (SRM) drives, 157–167 control in motoring mode, 166– 167, 167 electromagnetic torque and, 159– 161, 159 four-phase 8/6 switched reluctance machine, 157, 158, 163, 165 induced back-EMF and, 161–162 instantaneous waveforms and, 162–164, 162 power-processing units for, 165, 165 role of magnetic saturation in, 164–165, 164 rotor position for encoderless operation in, 166, 166 switched-reluctance motor, 157– 162, 158 Switching frequency direct torque control and constant, 139 in power-processing unit, 86, 89 Switch-mode inverter, 120, 120 Symbols, xv–xvi Test induction motor, 138 specifications for, 3–4 Theorem of constant flux linkage, 69 Three-phase, sinusoidally distributed stator windings, 8–9, 8 Tolerance-band control, 86 Torque dq winding analysis and, 23, 30 friction, 44 speed and position control and, 59

step change in, 68–72, 69 See also Electromagnetic torque; Load torque Torque control, 1–4 Torque factor, 104, 105 Torque loop, 72 Torque per ampere, maximum, 1 Torque reference, 73 Torque reference signal, 130 Transformation dq, xiii, 22, 141 Park’s, 28 of stator currents, 34, 73–74 Transformation matrix, 34, 35, 36 Transformer equivalent circuit, in current-excited transformer with secondary shortcircuited, 65–66, 65 Two coupled-coil system, 23–24 Variables that are functions of time, symbols for, xv Vector control detuning effects in, 97–107 of doubly fed induction generators, 109–117 indirect (see Indirect vector control, in rotor flux reference frame) principles of, 28 Vector control in induction machines, mathematical description of, 79–95 indirect vector control in rotor flux reference frame, 84–94 motor model with the d-axis aligned along the rotor flux linkage λr -axis, 79–84, 80, 83

180 INDEX Vector control of induction-motor drives, 59–76, 61 current-excited transformer with shorted secondary and, 62–66, 62 with current-regulated powerprocessing unit, 72–75, 73 d- and q-axis winding representation, 66–67 with d-axis aligned with rotor flux, 67–72 emulation of dc and brushless dc drive performance, 59–62 estimated motor model, 74–75 power-processing unit, 75–76 torque, speed, and position control and, 72–75 Vector control of permanent-magnet synchronous motor drives, 143–156

d-q analysis of permanent magnet synchronous machines, 143–151 salient-pole synchronous machines, 151–155, 151 Voltage impulse, current-excited transformer with secondary short-circuited and, 64–65 Voltages rotor, 112 stator, 89–90, 111, 116–117 Voltage vectors, 121, 121, 122–123, 122. See also Stator voltage space vector Wind-electric systems, 1 Wind energy, doubly fed induction generators and, 109 Winding voltages, in salient-pole synchronous machines, 153–154 Zero vectors, 121–123, 137–138

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