Permanent Magnet Synchronous and Brushless DC Motor Drives © 2010 by Taylor and Francis Group, LLC Permanent Magnet Synchronous and Brushless DC Motor Drives R. Krishnan Electrical and Computer Engineering Department Virginia Tech Blacksburg, Virginia, U.S.A. Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business © 2010 by Taylor and Francis Group, LLC MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft- ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Krishnan, R. (Ramu) Permanent magnet synchronous and brushless DC motor drives / author, Ramu Krishnan. p. cm. “A CRC title.” Includes bibliographical references and index. ISBN 978-0-8247-5384-9 (hardcover : alk. paper) 1. Electric motors, Synchronous. 2. Brushless direct current electric motors. I. Title. TK2787.K75 2010 621.46--dc22 2009029824 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2010 by Taylor and Francis Group, LLC To My brothers: Govindan and Elango My sisters: Jothi, Parvathi, and Saya © 2010 by Taylor and Francis Group, LLC Contents Preface....................................................................................................................xvii Acknowledgments ...................................................................................................xxi Author ..................................................................................................................xxiii List of Symbols ......................................................................................................xxv PART I Introduction to Permanent Magnets and Machines and Converters and Control Chapter 1 Permanent Magnets and Machines ......................................................3 1.1 Permanent Magnets ...................................................................3 1.1.1 Demagnetization Characteristics .................................5 1.1.2 Operating Point and Air Gap Line ...............................7 1.1.3 Energy Density ...........................................................10 1.1.4 Energy Stored in the Magnet......................................12 1.1.5 Magnet Volume ..........................................................13 1.1.6 Effect of External Magnetic Field Intensity ...............13 1.1.6.1 Analytical Approach ...................................14 1.1.6.2 Graphical Approach ....................................17 1.2 Arrangement of PMs ...............................................................21 1.3 Magnetization of PMs .............................................................23 1.3.1 Radial and Parallel Magnetizations ...........................23 1.3.2 Halbach Array ............................................................25 1.4 PM ac Machines ......................................................................31 1.4.1 Machine Confi gurations .............................................31 1.4.2 PM Rotor Types ..........................................................34 1.4.2.1 Surface-Mounted PMSM ............................34 1.4.2.2 Surface-Inset PMSM ..................................34 1.4.2.3 Interior PMSM ............................................35 1.4.2.4 Line-Start PMSM .......................................37 1.4.3 Hybrid Machines ........................................................38 1.4.3.1 Flux Reversal PMSM ..................................39 1.4.3.2 Flux Switching Machine .............................42 1.4.3.3 PM SRM or Doubly Salient PM Machine ......................................................43 1.4.4 Concentrated Coil-Wound PMSMs ............................46 1.4.5 Types of PMSMs ........................................................48 1.5 Fundamentals of Synchronous Machines ................................51 1.5.1 Principle of Operation ................................................51 1.5.2 MMF of a Coil ...........................................................52 vii © 2010 by Taylor and Francis Group, LLC viii Contents 1.5.3 Sinusoidal MMF Distribution ....................................53 1.5.3.1 Concentric Winding ....................................53 1.5.3.2 Distributed Winding ...................................53 1.5.4 Induced EMF ..............................................................54 1.5.4.1 Winding Distribution Factor .......................57 1.5.4.2 Winding Pitch Factor ..................................59 1.5.4.3 Skew Factor ................................................61 1.5.4.4 Winding Factor ...........................................63 1.5.5 Types of Windings......................................................64 1.5.5.1 Single-Layer Winding .................................64 1.5.5.2 Double-Layer Winding ...............................64 1.5.6 Rotating Magnetic Field .............................................66 1.5.6.1 Sinusoidal MMF Distribution .....................67 1.5.6.2 Rectangular MMF Distribution ..................71 1.6 Fundamental Synchronous Machine Relationships ................71 1.6.1 Effective Air Gap .......................................................72 1.6.2 Effect of Magnet on Induced EMF ............................74 1.6.3 Electromagnetic Power and Torque ............................75 1.6.4 Fundamental Representation of Electromagnetic Torque .........................................................................76 1.6.5 Machine Output Equation ..........................................78 1.6.6 Equivalent Magnet Current ........................................81 1.6.7 Safe Peak Stator Current ............................................82 1.6.8 Inductances .................................................................84 1.6.8.1 Self-Inductance per Phase...........................87 1.6.8.2 Magnetizing Inductance .............................87 1.6.8.3 Synchronous Inductance .............................88 1.6.8.4 d- and q-Axes Inductances .........................88 1.6.9 Effect of Stator Excitation on Air Gap Flux Density ...............................................................89 1.7 Core Losses .............................................................................92 1.7.1 Stator Core Losses ......................................................92 1.7.2 Eddy Current Losses ..................................................95 1.7.2.1 Eddy Current Loss in Tooth .......................96 1.7.2.2 Eddy Current Loss in Yoke ........................99 1.7.3 Evaluation of Peak Tooth and Yoke Flux Densities ...................................................................100 1.7.4 Hysteresis Losses .....................................................101 1.7.5 Core Loss Measurement in the Machine..................102 1.8 Resistive Losses .....................................................................103 1.9 Initial Machine Design ..........................................................105 1.10 Cogging Torque .....................................................................107 1.10.1 Cause and Magnitude ...............................................107 1.10.2 Basics of Cogging Torque ........................................107 1.10.2.1 General Expression for Cogging Torque .......................................................109 © 2010 by Taylor and Francis Group, LLC Contents ix 1.10.3 Analysis and Computation .......................................110 1.10.4 Factors Affecting Cogging Torque ...........................111 1.10.5 Mitigation Methods ..................................................111 1.10.5.1 Skewing ....................................................112 1.10.5.2 Varying Magnet Width .............................113 1.10.5.3 Varying Slot Width ...................................114 1.10.5.4 Shifting Alternate Pair of Poles ................114 1.10.5.5 Notching of Teeth .....................................114 1.11 Basic Types of PMSMs Based on Flux Paths ........................115 1.12 Vibration and Noise ...............................................................122 References ........................................................................................123 Chapter 2 Introduction to Inverters and Their Control .....................................135 2.1 Power Device .........................................................................136 2.1.1 Power Devices and Switching ..................................136 2.1.1.1 Power Diodes ............................................136 2.1.1.2 MOSFET ...................................................137 2.1.1.3 Insulated Gate Bipolar Transistor .............138 2.1.2 Switching of Power Devices .....................................139 2.1.3 Device Losses ...........................................................142 2.1.3.1 Conduction Loss .......................................142 2.1.3.2 Switching Losses ......................................143 2.2 DC Input Source ....................................................................146 2.2.1 DC Input Source with Diode Bridge Rectifi er .........147 2.3 DC to ac Power Conversion ...................................................150 2.3.1 Single-Phase Half-Wave Inverter .............................150 2.3.2 Single-Phase Full-Wave Inverter ..............................153 2.3.3 Three-Phase Inverter ................................................155 2.4 Real Power .............................................................................161 2.5 Reactive Power ......................................................................161 2.6 Need for Inverter Control ......................................................162 2.7 Pulse Width Modulation ........................................................164 2.7.1 PWM Implementation ..............................................167 2.7.1.1 Reference Signal Generation ....................168 2.7.1.2 Sampling of Reference Signals .................169 2.7.1.3 Transfer Characteristics of PWM- Controlled Inverters ..................................171 2.7.1.4 Off-Line Optimized PWM .......................172 2.8 Hysteresis Current Control ....................................................173 2.9 Space Vector Modulation ......................................................176 2.9.1 Switching States of the Inverter ...............................177 2.9.2 Principle of Space Vector Modulation .....................177 2.9.2.1 Transfer Characteristics of SVM- Controlled Inverters ..................................183 2.9.3 Space Vector Modulator Implementation .................188 © 2010 by Taylor and Francis Group, LLC x Contents 2.9.4 Switching Ripple in SVM ........................................188 2.9.4.1 Hybrid PWM Controller ...........................192 2.10 Inverter Switching Delay .......................................................195 2.10.1 Control Modeling of the Three-Phase Inverter ........207 2.11 Input Power Factor Correction Circuit ..................................208 2.11.1 Single-Phase Power Factor Correction Circuit .........208 2.11.2 Three-Phase Power Factor Correction Circuit .........210 2.12 Four-Quadrant Operation ......................................................211 2.13 Converter Requirements ........................................................214 References ........................................................................................215 PART II Permanent Magnet Synchronous Machines and Their Control Chapter 3 Dynamic Modeling of Permanent Magnet Synchronous Machines ..........................................................................................225 3.1 Real-Time Model of a Two-Phase PMSM .............................226 3.2 Transformation to Rotor Reference Frames ..........................231 3.3 Three-Phase to Two-Phase Transformation ..........................236 3.3.1 Unbalanced Operation ..............................................238 3.4 Zero Sequence Inductance Derivation ..................................239 3.5 Power Equivalence ................................................................241 3.6 Electromagnetic Torque .........................................................242 3.7 Steady-State Torque Characteristics ......................................244 3.8 Models in Flux Linkages .......................................................250 3.9 Equivalent Circuits ................................................................251 3.10 Per Unit Model ......................................................................254 3.11 Dynamic Simulation ..............................................................257 3.12 Small-Signal Equations of the PMSM ..................................262 3.12.1 Derivation .................................................................262 3.13 Evaluation of Control Characteristics of the PMSM .............264 3.13.1 Transfer Functions and Frequency Responses .........264 3.14 Computation of Time Responses ...........................................266 3.15 Space Phasor Model ..............................................................270 3.15.1 Principle ...................................................................270 3.15.2 Model Derivation .....................................................270 References ........................................................................................276 Chapter 4 Control Strategies for a Permanent Magnet Synchronous Machine ............................................................................................279 4.1 Vector Control .......................................................................279 4.2 Derivation of Vector Control .................................................280 4.2.1 Electromagnetic Torque ...........................................282 © 2010 by Taylor and Francis Group, LLC Contents xi 4.2.2 Stator d- and q-Axis Currents in Stator Reference Frames .....................................................283 4.2.3 Mutual Flux Linkages ..............................................283 4.2.4 Role of a Torque Angle in Machine Operations .......284 4.2.5 Key Results ...............................................................284 4.3 Drive System Schematic ........................................................285 4.3.1 Torque-Controlled Drive System .............................285 4.3.1.1 Hysteresis Current Controller ...................287 4.3.2 Simulation and Results of a Torque-Controlled Drive System ............................................................288 4.3.3 Speed-Controlled Drive System ...............................293 4.3.3.1 Mutual Flux Programming .......................294 4.3.3.2 Torque Programming in the Flux-Weakening Region ...........................294 4.3.4 Simulation and Results of a Speed-Controlled Drive System ............................................................296 4.4 Control Strategies ..................................................................303 4.4.1 Constant (δ = 90°) Torque Angle Control .................304 4.4.2 Unity Power Factor Control ......................................308 4.4.3 Constant Mutual Flux Linkages Control ..................312 4.4.4 Angle Control of Air Gap Flux and Current Phasors .....................................................................314 4.4.5 Optimum Torque per Unit Current Control .............317 4.4.6 Constant Power Loss Control ...................................320 4.4.7 Maximum Effi ciency Control ..................................325 References ........................................................................................328 Chapter 5 Flux-Weakening Operation ..............................................................331 5.1 Maximum Speed ...................................................................332 5.2 Flux-Weakening Algorithm ...................................................333 5.2.1 Indirect Control Scheme ..........................................336 5.2.2 Constant Torque Mode Controller ............................336 5.2.3 Flux-Weakening Controller ......................................338 5.2.4 System Performance .................................................339 5.3 Direct Flux Weakening .........................................................343 5.3.1 Maximum Permissible Torque Limit .......................344 5.3.2 Speed Control Scheme .............................................344 5.3.3 Implementation Strategy ..........................................345 5.3.3.1 Lookup Tables Realization .......................346 5.3.4 System Performance .................................................348 5.4 Parameter Sensitivity .............................................................349 5.4.1 Stator Resistance Variation ......................................349 5.4.2 Rotor Flux Linkage Variation ..................................350 5.4.3 q-Axis Inductance Variation ....................................350 5.5 Model-Free (Parameter-Insensitive) Flux-Weakening Method ...................................................................................351 © 2010 by Taylor and Francis Group, LLC xii Contents 5.6 Six-Step Voltage and Constant Back EMF Control Strategies for PMSM .............................................................353 5.6.1 Constant Back EMF Control Strategy ......................354 5.6.1.1 The Basics .................................................354 5.6.1.2 Maximum Current in the Flux-Weakening Region ...........................355 5.6.1.3 Operational Boundary ..............................355 5.6.1.4 Maximum Speed in the Flux-Weakening Region ...........................356 5.6.2 Six-Step Voltage Control Strategy............................356 5.6.2.1 Fundamental Analysis ..............................357 5.6.2.2 Steady-State Current in SSV Mode ..........359 5.6.2.3 Operational Boundary for the SSV Control Strategy ........................................359 5.6.2.4 Comparison ...............................................360 5.7 Direct Steady-State Evaluation ..............................................361 5.7.1 Input Voltages ...........................................................362 5.7.2 Machine Equations in State Space Form .................365 5.7.3 Boundary Matching Condition and Solution ...........366 5.7.4 MATLAB® Program ................................................369 5.8 Flux Weakening in SMPM and IPM Synchronous Machines ...............................................................................371 5.8.1 Flux-Weakening Equations in New Base .................371 References ........................................................................................376 Chapter 6 Design of Current and Speed Controllers ........................................379 6.1 Current Controller .................................................................380 6.1.1 Rotor Reference Frame Current Controllers ............381 6.1.2 Stator Reference Frames Current Controller ............382 6.1.3 Deadbeat Current Controllers ..................................385 6.1.3.1 Deadbeat Controller ..................................385 6.1.3.2 Predictive Deadbeat Controller ................386 6.1.3.3 Improved Predictive Deadbeat Controller ..................................................388 6.2 Speed Controller ....................................................................389 6.2.1 Derivation of Block Diagram ...................................390 6.2.2 Simplifi ed Current Loop Transfer Function .............391 6.2.3 Speed Controller .......................................................393 6.2.3.1 Smoothing .................................................396 References ........................................................................................398 Chapter 7 Parameter Sensitivity and Compensation .........................................401 7.1 Introduction ...........................................................................401 7.1.1 Ratio of Torque to Its Reference ...............................402 7.1.2 Ratio of Mutual Flux Linkages to Its Reference ......403 © 2010 by Taylor and Francis Group, LLC