Design of Brushless
Permanent-Magnet
Machines
J.R. Hendershot Jr.
T.J.E. Miller
Contents
1 GENERAL INTRODUCTION l
1.1 D e f i n i t i o n s a n d t y p e s of b r u s h l e s s m o t o r 1
1.2 Commutation ,. 4
1.3 O p e r a t i o n of 3 - p h a s e b r u s h l e s s DC m o t o r 5
1.3.1 EMF waveform 7
1.3.2 Torque and EMF constants 10
1.3.3 Speed/torque characteristic 11
1.4 Sinewave motors and generators , 16
1.4.1 Phasor representation 19
1.4.2 Voltage 22
1.5 Practical considerations 23
2 MACHINE TYPES a n d APPLICATIONS 25
2.1 Machine configuration , ... 25
2.1.1 Reasons for variety 25
2.1.2 Classification 27
2.2 Radial-flux m a c h i n e s 30
2.2.1 Interior-rotor surface-magnet machines 30
2.2.2 Interior-rotor interior-magnet machines (IPM) 32
2.2.3 Exterior-rotor machines . . . , , . . , , , , 35
2.3 Axial-flux, l i n e a r a n d o t h e r m a c h i n e s 37
2.4 Gallery 43
xiv CONTENTS
3 BASIC DESIGN CHOICES 65
3.1 Machine and drive configuration 67
3.1.1 Squarewave and sinewave drives 67
3.1.1.1 Squarewave drive 67
3.1.1.2 Sinewave drive 68
3.1.2 Salient-pole and nonsalient-pole machines 68
3.1.2.1 Nonsalient-pole machines 68
3.1.2.2 Salient-pole machines 69
3.2 N u m b e r of p h a s e s , p o l e s a n d s l o t s 71
3.2.1 Number of phases 71
3.2.1.1 Practical considerations 71
3.2.1.2 Number of phases in electrical systems 72
3.2.1.3 Number of phases in electrical machines 75
3.2.1.4 Distribution of coils between phases 77
3.2.1.5 Number of phases in inverters and rectifiers 80
3.2.2 Numbers of slots and poles 82
3.3 S i z i n g t h e ABC of e l e c t r i c m a c h i n e d e s i g n 87
3.3.1 The output equation 87
3.4 Rotor design 92
3.4.1 Length/diameter ratio 92
3.4.2 Airgap length 92
3.4.3 First estimate of magnet dimensions 93
3.4.4 Exploratory selection of magnet grade 94
3.4.5 Magnet overhang 95
3.4.6 Rotor yoke dimensions 96
3.5 Stator design 97
3.5.1 Cutting the laminations 97
3.5.2 Choice of core plate 97
3.5.3 Stacking 98
3.5.4 Insulating the slots 100
3.5.5 Slot-fill factor 101
3.5.6 Winding and inserting the phase coils 103
3.5.7 Varnishing 104
3.5.8 Winding with multiple-strand conductors 104
CONTENTS XV
3.5.9 N u m b e r of stator slots , , ... 105
3.5.10 Stator core d i m e n s i o n s 105
3.5.11 Stator tooth-tips 106
3.5.12 Cogging a n d s k e w 107
3.5.13 M a n a g e m e n t of end-turns , 109
3.6 Electrical design of windings 110
3.6.1 Definitions 110
3.6.2 Integral-slot w i n d i n g s 111
3.6.3 Windings for squarewave drive 115
3.6.4 Fractional-slot w i n d i n g s 118
3.6.4.1 A rule and two examples . . . . 118
3.6.4.2 The 12/10 motor; alternative windings 124
3.6.4.3 Pitch factor 128
3.6.4.4 Sinewave and squarewave motors 130
3.6.5 Irregular s l o t t i n g 131
3.6.6 Systematic analysis of slot/pole ratio and w i n d i n g s 133
3.6.7 Winding r e s i s t a n c e 139
3.6.7.1 Resistance calculation 139
3.6.7.2 Relationship between resistance and copper weight 140
3.6.7.3 Variation of resistance with temperature 140
3.6.7.4 AC resistance 143
3.7 Magnet r e t e n t i o n . . . 153
4 FLUX, EMF, A N D TORQUE 157
4.1 Permanent magnets and magnetic circuits 157
4.1.1 Magnetic e q u i v a l e n t circuits 158
4.1.1.1 Airgap flux distribution 164
4.1.1.2 Clearance gap and equivalent magnet 165
4.1.1.3 Magnet divided by thin bracing bridges 167
4.1.2 Direct solution of Laplace / P o i s s o n e q u a t i o n s 169
4.1.3 Finite-element method 174
4.2 EMF 178
4.2.1 Formula 179
4.2.1.1 EMF constant of squarewave motors 179
4.2.1.2 EMF constant of sinewave motors 180
xvi CONTENTS
4.2.2 BLV waveform method 181
4.2.3 Toothflux waveform method 183
4.3 Torque 185
4.3.1 Torque constants 186
4.3.1.1 Three-phase squarewave motor 186
4.3.1.2 Sinewave motors 187
4.4 Torque a n d i n d u c t a n c e 190
4.4.1 Salient-pole machines in phase variables 190
4.4.2 Salient-pole machines in dq axes 193
4.5 i-psi l o o p 197
4.6 P r o p e r t i e s of t h e elliptical i-psi l o o p 203
5 INDUCTANCE 209
5.1 D e f i n i t i o n of i n d u c t a n c e a n d flux-linkage 210
5.1.1 Alternative definitions 211
5.1.1.1 di/dt 211
5.1.1.2 Flux times turns 211
5.1.2 Other necessary laws of electromagnetism 211
5.1.3 Turns squared 212
5.2 Important practical effects of i n d u c t a n c e 213
5.3 Inductance components 214
5.4 Airgap i n d u c t a n c e of surface-magnet m a c h i n e s 215
5.4.1 Airgap Self 215
5.4.2 Airgap mutual 217
5.4.3 Examples of airgap inductance calculation 217
5.4.4 General case of airgap inductance 221
5.5 Slot-leakage i n d u c t a n c e 226
5.6 End-winding leakage inductance 233
CONTENTS xvii
5.7 I n d u c t a n c e s of s l o t l e s s ( a i r g a p ) w i n d i n g s 238
5.7.1 Helical windings 241
5.7.2 Lawrenson's method 241
5.8 Equivalent sine-distributed windings 242
5.9 Synchronous inductance . 243
5.9.1 Static measurement of synchronous inductance 246
5.10 I n d u c t a n c e s of s a l i e n t - p o l e m a c h i n e s . . . . 247
5.10.1 dq-axis inductances from Park's transform 248
5.10.2 Synchronous inductance coefficients 252
5.10.3 Direct calculation of synchronous inductance 253
5.10.4 Differential leakage inductance 258
5.10.5 Static measurement again 260
5.11 I n d u c t a n c e f r o m f i n i t e - e l e m e n t c a l c u l a t i o n s 262
5.12 M a g n e t i z a t i o n c u r v e s b e y o n d i n d u c t a n c e 263
5.12.1 Magnetization curves in dqr-axes 266
5.13 S a t u r a t i o n i n t h e dq-axis model 267
5.14 D e m a g n e t i z a t i o n 268
6 SQUAREWAVE DRIVE 273
Introduction 273
6.1 Three-phase bipolar drives ....... 274
6.1.1 Waveforms and commutation sequences 274
6.1.2 Current regulation 279
6.1.3 Commutation 282
6.1.4 3-phase squarewave control strategies 286
6.1.5 Accumulations for mean and RMS currents 288
6.1.6 Selection of appropriate switching strategy 289
xviii CONTENTS
6.2 T r a n s i e n t analysis of 3-phase d r i v e s , 291
6.2.0.1 Wye connection .,. 293
6.2.0.2 Delta connection 296
6.2.0.3 Regeneration (over-running); no-load speed 301
6.2.0.4 Phase advance 304
6.2.0.5 Dwell control 306
6.2.1 Salient-pole machines with squarewave drive 309
6.2.2 Back-EMF sensing 312
6.3 1- a n d 2 - p h a s e u n i p o l a r d r i v e s 315
6.4 Controller architecture ... 321
7 SINEWAVE DRIVE 325
Introduction , 325
7.1 The phasor diagram motor operation 327
7.1.1 Torque/angle curves 332
7.1.2 The voltage locus diagram 336
7.1.3 T h e c i r c l e a n d ellipse d i a g r a m s 338
7.1.4 C a l c u l a t i o n of t h e t o r q u e / s p e e d c h a r a c t e r i s t i c 349
7.1.5 The synchronous reluctance motor 361
7.1.6 Summary calculated characteristics 367
7.2 Electronic control 368
7.2.1 T h e n e e d for c u r r e n t r e g u l a t i o n 369
7.2.2 Historical development 371
7.2.3 O v e r v i e w of c o n t r o l l e r s 373
7.2.4 Switching representation by voltage vectors 374
7.2.5 Six-step 375
7.2.6 Hysteresis-band current regulator 377
7.2.7 dq_W_CR 381
7.2.8 Sine/triangle r a m p comparison . 383
7.2.9 V o l t a g e PWM ( s i n e / t r i a n g l e ) 385
7.2.10 The synchronous regulator 389
7.2.11 Space-vector controller 391
7.2.12 D i r e c t t o r q u e control (DTC) 396
7.2.13 S u m m a r y of v o l t a g e c a p a b i l i t i e s 404
CONTENTS xix
8 k T A N D kE, A N D FIGURES-OF-MERIT 405
8.1 Introduction 405
8.2 k T & k E of s q u a r e w a v e a n d s i n e w a v e m o t o r / d r i v e s 407
8.2.1 DC commutator motor and drive 407
8.2.2 3-phase squarewave motor and drive 411
8.2.3 3-phase sinewave motor and drive 415
8.2.4 3-phase sinewave motor with squarewave drive 417
8.2.5 3-phase squarewave motor with sinewave drive 419
8.2.6 3-phase squarewave & sinewave systems compared 422
8.2.7 Example calculations (3-phase) 424
8.2.8 2-phase squarewave motor and drive 426
8.2.9 2-phase sinewave motor and drive 428
8.2.10 2-phase sinewave motor with squarewave drive 430
8.2.11 2-phase squarewave motor with sinewave drive 432
8.2.12 2-phase squarewave & sinewave systems compared 435
8.3 F i g u r e s of m e r i t 436
8.3.1 kT and kE 436
8.3.2 Efficiency and power factor , 436
8.3.3 Torque/Inertia ratio 437
8.3.4 Power r a t e 437
8.3.5 Speed r a t e and mechanical time-constant 439
8.3.6 Motor constant 440
8.4 The brushless PM motor in control systems 442
8.4.1 Classical transfer function between voltage & speed 443
8.4.2 Brushless DC motor model including inductance 445
8.4.3 Closed-loop feedback system 446
8.4.4 Response of generic second-order system 448
8.4.5 Dynamic braking 449
xx CONTENTS
9 GENERATING 451
9.1 Introduction ,,, 451
9.2 Configurations a n d loads 454
9.2.1 No-load (open-circuit) 455
9.2.2 Steady-state short-circuit 456
9.2.3 Passive impedance load 457
9.2.4 Voltage regulation curves 459
9.2.5 Connection to an infinite bus 462
9.2.6 Diode rectifier load 464
9.2.7 Active rectification , . . . 467
9.3 Short-circuit faults 468
9.3.1 Classical analysis 468
9.3.2 Transient Magnetic Field by Fourier Transform 472
10 MULTIPLE-PHASE MACHINES 475
Introduction . . . 475
10.1 P o l y p h a s e m a c h i n e s ...... 475
10.2 Multiplex w i n d i n g s 478
10.2.1 Reasons for using multiplex windings 479
10.2.2 Fault-tolerant machines 480
10.3 Analysis of m u l t i p l e x w i n d i n g s 481
10.3.1 Balance 484
10.4 Matrix analysis of t h e i n d u c t a n c e s 485
10.5 Torque 491
10.6 Steady-state o p e r a t i o n : p h a s o r diagram 493
10.7 S o l u t i o n m e t h o d t r a n s i e n t .. 495
10.8 F i n i t e - e l e m e n t analysis 496
CONTENTS xxi
11 LINE-START MOTORS 497
11.1 I n t r o d u c t i o n 497
11.2 H i s t o r y 500
11.3 A n a l y s i s of p o l y p h a s e line-start m o t o r s ... 503
11.3.1 Steady state 503
11.3.2 A s y n c h r o n o u s o p e r a t i o n a n d s t a r t i n g 506
11.3.3 Analysis of s y n c h r o n i z a t i o n 510
11.4 A n a l y s i s of s i n g l e - p h a s e l i n e - s t a r t m o t o r s 517
11.4.1 S t e a d y state; n o rotor cage 517
11.4.2 Symmetrical c o m p o n e n t s ... 519
11.4.3 A s y n c h r o n o u s a n d starting performance 537
11.5 A d v a n c e d t o p i c s 542
11.5.1 Winding h a r m o n i c s 542
11.5.2 Bar-pair-by-bar-pair m o d e l of t h e rotor cage 543
11.5.3 C o n n e c t i o n circuits 550
12 LOSSES a n d COOLING . 553
12.1 I n t r o d u c t i o n 553
12.2 J o u l e l o s s e s i n s t a t o r c o n d u c t o r s 554
12.3 Core l o s s e s . .., 555
12.3.1 The n a t u r e of core losses .... 555
12.3.2 Core loss properties of practical materials 556
12.3.3 Calculation of core losses 559
12.4 R o t o r e d d y - c u r r e n t l o s s e s 561
12.4.1 Causes of rotor loss 561
12.4.1.1 Loss mechanisms in the magnets themselves 563
12.4.1.2 Resistance-or inductance-limited eddy-currents? 564
12.4.1.3 Hysteresis loss in magnets 566
12.4.2 H a r m o n i c l o s s e s i n surface-magnet m a c h i n e s 568
12.4.2.1 Solution of the Complex Diffusion Equation 570
xxii CONTENTS
12.4.2.2 Exterior-rotor machine; 2-region model 574
12.4.2.3 Evaluation of the Exciting Harmonic Current Sheets 580
12.4.2.4 Balanced operation of 3-phase machines 586
12.4.2.5 Unbalanced operation of 3-phase machines 589
12.4.3 S e g m e n t e d m a g n e t s a n d finite-length effects 602
12.4.3.1 Circumferential segmentation ., 604
12.4.3.2 Simplified analysis of double segmentation 610
12.4.3.3 End-effect; segmentation in the axial direction 611
12.4.3.4 Russell and Norsworthy's method 616
12.4.3.5 Alternative analysis of segmented magnets 618
12.4.4 Slot ripple 620
12.4.4.1 Flux-dip-sweeping analysis of losses in thin can 624
12.4.4.2 Rotor can losses 626
12.4.5 Harmonic l o s s e s i n t h e IPM 628
12.4.5.1 Losses caused by time-harmonics in the current 628
12.4.5.2 Losses caused by flux-pulsations (slotting) 629
12.4.6 Subtransient i n d u c t a n c e and time-constant 631
12.4.6.1 Effect of segmentation on subtransient reactance 635
12.4.6.2 Coupling coefficient of the IPM 638
12.4.6.3 Rotor time-constant 642
12.4.7 F i n i t e - e l e m e n t calculation of losses 644
12.5 Windage, friction and bearing losses 647
12.6 Thermal analysis and cooling 648
12.6.1 The n e e d for c o o l i n g 648
12.6.2 Cooling a n d efficiency 649
12.6.3 Responsibility for t e m p e r a t u r e rise 650
12.6.4 Heat removal 650
12.6.5 D e t a i l e d analysis of c o o l i n g 652
12.6.5.1 Conduction 652
12.6.5.2 Radiation 653
12.6.5.3 Convection 654
12.6.5.4 Some rules of thumb 655
12.6.5.5 Internal temperature distribution 656
12.6.5.6 Thermal equivalent circuit 657
12.6.5.7 Some useful tables 658
12.6.6 I n t e r m i t t e n t o p e r a t i o n 660
CONTENTS xxi
13 TESTING 667
13.1 I n t r o d u c t i o n 667
13.2 O b j e c t i v e s of t e s t i n g 667
13.3 B a s i c t e s t s a n d m e a s u r e m e n t s 668
13.3.1 Inertia 668
13.4 R e s i s t a n c e 669
13.5 E M F T e s t i n g 670
13.6 G e n e r a t o r l o a d t e s t i n g 671
13.7 M o t o r l o a d t e s t i n g 672
13.8 T o r q u e T e s t i n g 672
13.8.1 Torque constant kT 672
13.8.2 Cogging torque 673
13.8.3 On-line estimation of torque using the i-psi loop 674
13.9 T h e r m a l T e s t i n g 675
13.9.1 Thermal equivalent-circuit parameters 675
13.10 I n d u c t a n c e T e s t i n g 676
14 APPENDIX 681
14.1 F r e q u e n t l y a s k e d q u e s t i o n s 681
14.1.1 Machine Design Questions ,,. 681
14.1.1.1 How do I decide the shape and size of the machine? 681
14.1.1.2 How do I choose the number of slots and poles? 682
14.1.1.3 How do I design the stator teeth and slots? 682
14.1.1.4 How do I decide the number of turns? 684
14.1.1.5 How do I decide the type of stator winding? 685
14.1.1.6 How can I get a fractional number of turns/coil? 685
14.1.1.7 How can I reduce the wire size? 685
14.1.1.8 How can I reduce the inductance? 686
14.1.1.9 How can I increase the inductance? 686
14.1.1.10 How do I choose between SPM and IPM? 686
xxiv CONTENTS
14.1.1.11 How do I choose between exterior or interior rotor? 688
14.1.1.12 When should I consider an axial-flux machine? 688
14.1.1.13 How do I decide the rotor geometry? 689
14.1.1.14 How can I reduce the inertia? 691
14.1.1.15 How can I improve the torque linearity? 692
14.1.1.16 How can I reduce torque ripple? 692
14.1.1.17 How do I design a PM synchronous generator? 692
14.1.1.18 How do I test a PM synchronous machine? 692
14.1.1.19 Why isn't my measured kE equal to kT? 692
14.1.1.20 How do I calculate the machine temperature? 692
14.1.1.21 What are the main effects of temperature? 693
14.1.1.22 How can I prevent demagnetization? 694
14.1.1.23 How can I reduce the noise level? 695
14.1.1.24 How can I reduce the motor cost? 695
14.1.1.25 How about EMF ripple? 696
14.1.1.26 How about a sine-EMF motor with squarewave drive? 6 9 6
14.1.2 P e r f o r m a n c e a n d Control Questions 697
14.1.2.1 How can I increase efficiency? 697
14.1.2.2 How can I increase power-factor? 698
14.1.2.3 How can I get smooth rotation at low speed? 698
14.1.2.4 How can I make the motor go faster? 699
14.1.2.5 How can I get a more sinusoidal EMF waveform? 700
14.1.2.6 How can I get a more sinusoidal current waveform? 700
14.1.2.7 How do I avoid first-turn insulation failure? 700
14.1.2.8 How do I avoid bearing currents? 702
14.1.2.9 What causes machines to fail? 702
14.2 S a l i e n c y 703
14.3 Half t u r n s 706
14.4 S e r i e s a n d parallel i n d u c t a n c e s 709
14.5 Gearing 714
14.6 U n i t s of i n e r t i a 716
14.7 Calculation of i n e r t i a 721
Symbols, Abbreviations, and Explanatory Notes 723
Bibliography 737
Index 755
