Electrical Machine Fundamentals with Numerical Simulation using MATLAB / SIMULINK
Description
Electrical Machine Fundamentals with Numerical Simulation using MATLAB/Simulink provides readers with a basic understanding of all key concepts related to electrical machines (including working principles, equivalent circuit, and analysis). It elaborates the fundamentals and offers numerical problems for students to work through. Uniquely, this text includes simulation models of every type of machine described in the book, enabling students to design and analyse machines on their own.
Unlike other books on the subject, this book meets all the needs of students in electrical machine courses. It balances analytical treatment, physical explanation, and hands-on examples and models with a range of difficulty levels. The authors present complex ideas in simple, easy-to-understand language, allowing students in all engineering disciplines to build a solid foundation in the principles of electrical machines. This book:
* Includes clear elaboration of fundamental concepts in the area of electrical machines, using simple language for optimal and enhanced learning
* Provides wide coverage of topics, aligning with the electrical machines syllabi of most international universities
* Contains extensive numerical problems and offers MATLAB/Simulink simulation models for the covered machine types
* Describes MATLAB/Simulink modelling procedure and introduces the modelling environment to novices
* Covers magnetic circuits, transformers, rotating machines, DC machines, electric vehicle motors, multiphase machine concept, winding design and details, finite element analysis, and more
Electrical Machine Fundamentals with Numerical Simulation using MATLAB/Simulink is a well-balanced textbook perfect for undergraduate students in all engineering majors. Additionally, its comprehensive treatment of electrical machines makes it suitable as a reference for researchers in the field.
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Persons
Dr. Atif Iqbal is a Full Professor in the Department of Electrical Engineering, Qatar University. He is an IET Fellow (UK), IE Fellow (India), and Senior Member of the IEEE, as well as Associate Editor, IEEE Trans. on Industrial Electronics.
Dr. Shaikh Moinoddin is a Senior Member of the IEEE, India. He is former Assistant Professor in Electrical Engineering at the University of Polytechnic, Aligarh Muslim University, Aligarh, India.
Dr. Bhimireddy Prathap Reddy is currently working as a Post-doc at the Department of Electrical Engineering, Qatar University and is a member of the IEEE.
Content
Preface xxi
Acknowledgements xxiii
1 Fundamentals of Electrical Machines 1
1.1 Preliminary Remarks 1
1.2 Basic Laws of Electrical Engineering 1
1.2.1 Ohm's Law 1
1.2.2 Generalization of Ohm's Law 2
1.2.2.1 Derivation of Eq. (1.6) 2
1.2.3 Ohm's Law for Magnetic Circuits 3
1.2.4 Kirchhoff's Laws for Magnetic Circuits 3
1.2.5 Lorentz Force Law 5
1.2.6 Biot-Savart Law 6
1.2.7 Ampere Circuital Law 17
1.2.8 Faraday's Law 20
1.2.8.1 Motional emf 24
1.2.9 Flux Linkages and Induced Voltages 29
1.2.10 Induced Voltages 29
1.2.11 Induced Electric Fields 30
1.2.12 Reformulation of Faraday's Law 31
1.3 Inductance 38
1.3.1 Application of Ampere's Law to Find B in a Solenoid 39
1.3.2 Magnetic Field of a Toroid 40
1.3.3 The Inductance of Circular Air-Cored Toroid 40
1.3.4 Mutual Inductance 44
1.4 Energy 47
1.5 Overview of Electric Machines 49
1.6 Summary 58
Problems 58
References 67
2 Magnetic Circuits 69
2.1 Preliminary Remarks 69
2.2 Permeability 69
2.3 Classification of Magnetic Materials 70
2.3.1 Uniform Magnetic Field 72
2.3.2 Magnetic-Field Intensity 72
2.4 Hysteresis Loop 74
2.4.1 Hysteresis Loop for Soft Iron and Steel 76
2.5 Eddy-Current and Core Losses 78
2.6 Magnetic Circuits 82
2.6.1 The Magnetic Circuit Concept 82
2.6.2 Magnetic Circuits Terminology 82
2.6.2.1 Limitations of the Analogy Between Electric and Magnetic Circuits 86
2.6.3 Effect of Air Gaps 86
2.6.3.1 Magnetic Circuit with an Air Gap 86
2.6.3.2 Magnetic Forces Exerted by Electromagnets 89
2.7 Field Energy 100
2.7.1 Energy Stored in a Magnetic Field 100
2.7.1.1 The Magnetic Energy in Terms of the Magnetic Induction B 101
2.7.1.2 The Magnetic Energy in Terms of the Current Density J and the Vector Potential A 102
2.7.1.3 The Magnetic Energy in Terms of the Current I and of the Flux ¿¿¿¿m 103
2.7.1.4 The Magnetic Energy in Terms of the Currents and Inductances 103
2.8 The Magnetic Energy for a Solenoid Carrying a Current I 104
2.9 Energy Flow Diagram 106
2.9.1 Power Flow Diagram of DC Generator and DC Motor 106
2.9.1.1 Power Flow Diagram and Losses of Induction Motor 108
2.9.1.2 Rotational Losses 109
2.10 Multiple Excited Systems 110
2.11 Doubly Excited Systems 113
2.11.1 Torque Developed 116
2.11.1.1 Excitation Torque 117
2.11.1.2 Reluctance Torque 122
2.12 Concept of Rotating Magnetic Field 126
2.12.1 Rotating Magnetic Field due to Three-Phase Currents 126
2.12.1.1 Speed of Rotating Magnetic Field 130
2.12.1.2 Direction of Rotating Magnetic Field 131
2.12.2 Alternate Mathematical Analysis for Rotating Magnetic Field 131
2.13 Summary 134
Problems 135
References 144
3 Single-Phase and Three-Phase Transformers 147
3.1 Preliminary Remarks 147
3.2 Classification of Transformers 149
3.2.1 Classification Based on Number of Phases 149
3.2.1.1 Single-Phase Transformers 149
3.2.1.2 Three-Phase Transformers 149
3.2.1.3 Multi-Phase Transformers 150
3.2.2 Classification Based on Operation 150
3.2.2.1 Step-Up Transformers 150
3.2.2.2 Step-Down Transformers 151
3.2.3 Classification Based on Construction 151
3.2.3.1 Core-Type Transformers 151
3.2.3.2 Shell-Type Transformers 151
3.2.4 Classification Based on Number of Windings 153
3.2.4.1 Single-Winding Transformer 153
3.2.4.2 Two-Winding Transformer 153
3.2.4.3 Three-Winding Transformer 153
3.2.5 Classification Based on Use 153
3.2.5.1 Power Transformer 153
3.2.5.2 Distribution Transformer 154
3.3 Principle of Operation of the Transformer 154
3.3.1 Ideal Transformer 154
3.4 Impedance Transformation 157
3.5 DOT Convention 158
3.6 Real/Practical Transformer 158
3.7 Equivalent Circuit of a Single-Phase Transformer 160
3.8 Phasor Diagrams Under Load Condition 166
3.9 Testing of Transformer 170
3.9.1 Open-Circuit Test 171
3.9.2 Short-Circuit Test 172
3.10 Performance Measures of a Transformer 175
3.10.1 Voltage Regulation 175
3.10.1.1 Condition for Maximum Voltage Regulation 177
3.10.1.2 Condition for Zero Voltage Regulation 177
3.10.2 Efficiency of Transformer 180
3.10.3 Maximum Efficiency Condition 181
3.11 All-Day Efficiency or Energy Efficiency 185
3.12 Autotransformer 186
3.13 Three-Phase Transformer 190
3.13.1 Input (Y), Output (¿) 192
3.13.2 Input Delta (¿), Output Star (Y) 194
3.13.3 Input Delta (¿), Output Delta (¿) 195
3.13.4 Input Star (Y), Output Star (Y) 196
3.14 Single-Phase Equivalent Circuit of Three-Phase Transformer 197
3.15 Open-Delta Connection or V Connection 200
3.16 Harmonics in a Single-Phase Transformer 205
3.16.1 Excitation Phenomena in a Single-Phase Transformer 208
3.16.2 Harmonics in a Three-Phase Transformer 210
3.16.2.1 Star-Delta Connection with Grounded Neutral 213
3.16.2.2 Star-Delta Connection without Grounded Neutral 214
3.16.3 Summary 214
3.16.4 Star-Star with Isolated Neutral 214
3.17 Disadvantages of Harmonics in Transformer 215
3.17.1 Effect of Harmonic Currents 215
3.17.2 Electromagnetic Interference 215
3.17.3 Effect of Harmonic Voltages 215
3.17.4 Summary 216
3.17.5 Oscillating Neutral Phenomena 216
3.18 Open Circuit and Short-Circuit Conditions in a Three-Phase Transformer 217
3.19 Matlab/Simulink Model of a Single-Phase Transformer 219
3.20 Matlab/Simulink Model of Testing of Transformer 222
3.21 Matlab/Simulink Model of Autotransformer 223
3.22 Matlab/Simulink Model of Three-Phase Transformer 223
3.23 Supplementary Solved Problems 232
3.24 Summary 249
3.25 Problems 249
References 255
4 Fundamentals of Rotating Electrical Machines and Machine Windings 257
4.1 Preliminary Remarks 257
4.2 Generator Principle 257
4.2.1 Simple Loop Generator 257
4.2.2 Action of Commutator 259
4.2.3 Force on a Conductor 260
4.2.3.1 DC Motor Principle 260
4.2.3.2 Motor Action 261
4.3 Machine Windings 261
4.3.1 Coil Construction 261
4.3.1.1 Coil Construction: Distributed Winding 261
4.3.1.2 Coil Construction: Concentrated Winding 262
4.3.1.3 Coil Construction: Conductor Bar 262
4.3.2 Revolving (Rotor) Winding 262
4.3.3 Stationary (Stator) Winding 262
4.3.4 DC ArmatureWindings 262
4.3.4.1 Pole Pitch (Yp) 263
4.3.4.2 Coil Pitch or Coil Span (Ycs) 263
4.3.4.3 Back Pitch (Yb) 263
4.3.4.4 Front Pitch (Yf) 264
4.3.4.5 Resultant Pitch (Y) 264
4.3.4.6 Commutator Pitch (a) 264
4.3.5 Lap Winding 265
4.3.5.1 Lap Multiple or Parallel Windings 265
4.3.5.2 Formulas for Lap Winding 266
4.3.5.3 Multiplex, Single, Double, and Triple Windings 267
4.3.5.4 Meaning of the Term Re-entrant 268
4.3.5.5 Multiplex Lap Windings 268
4.3.6 WaveWinding 279
4.3.6.1 Formulas forWave Winding 281
4.3.6.2 MultiplexWave or Series-ParallelWinding 282
4.3.6.3 Formulas for Series-Parallel Winding 283
4.3.7 Symmetrical Windings 284
4.3.7.1 Possible SymmetricalWindings for DC Machines of a Different Number of Poles 284
4.3.8 Equipotential Connectors (Equalizing Rings) 284
4.3.9 Applications of Lap andWave Windings 286
4.3.10 Dummy or Idle Coils 310
4.3.10.1 Dummy Coils 310
4.3.11 Whole-CoilWinding and Half-CoilWinding 311
4.3.12 Concentrated Winding 312
4.3.13 Distributed Winding 312
4.4 Electromotive Force (emf) Equation 313
4.4.1 emf Equation of an Alternator [1] 313
4.4.1.1 Winding Factor (Coil Pitch and Distributed Windings) 313
4.4.2 Winding Factors 313
4.4.2.1 Pitch Factor or Coil Pitch (Pitch Factor (Kp) or Coil Span Factor [Kc]) 314
4.4.3 Distribution Factor (Breadth Factor (Kb) or Distribution Factor (Kd)) 315
4.4.3.1 Distribution Factor (Kd) 315
4.5 Magnetomotive Force (mmf) of ACWindings 316
4.5.1 mmf and Flux in Rotating Machine 316
4.5.2 Main Air-Gap Flux (Field Flux) 316
4.5.3 mmf of a Coil [5] 316
4.5.3.1 mmf 316
4.5.3.2 mmf of Distributed Windings 317
4.5.3.3 mmf SpaceWave of a Single Coil 317
4.5.3.4 mmf SpaceWave of One Phase of a Distributed Winding [6] 319
4.6 Harmonic Effect [7] 322
4.6.1 The Form Factor and the emf per Conductor 322
4.6.2 TheWave Form 323
4.6.3 Problem Due to Harmonics 324
4.6.4 Elimination or Suppression of Harmonics 324
4.6.4.1 Shape of Pole Face 324
4.6.4.2 Use of Several Slots per Phase per Pole 324
4.6.4.3 Use of Short-Pitch Windings 325
4.6.4.4 Effect of the Y- and ¿ -Connection on Harmonics 327
4.6.4.5 Harmonics Produced by Armature Slots 328
4.7 Basic Principles of Electric Machines 330
4.7.1 AC Rotating Machines 331
4.7.1.1 The Rotating Magnetic Field 331
4.7.1.2 The Relationship between Electrical Frequency and the Speed of Magnetic Field Rotation 333
4.7.1.3 Reversing the Direction of the Magnetic Field Rotation 335
4.7.1.4 The Induced Voltage in AC Machines 335
4.7.1.5 The Induced Voltage in a Coil on a Two-Pole Stator 335
4.7.1.6 The Induced Voltage in a Three-Phase Set of Coils 337
4.7.1.7 The rms Voltage in a Three-Phase Stator 338
4.7.2 The Induced Torque in an AC Machine 338
4.8 Summary 339
Problems 339
References 340
5 DC Machines 341
5.1 Preliminary Remarks 341
5.2 Construction and Types of DC Generator 342
5.2.1 Construction of DC Machine 342
5.2.2 Types of DC Generator 343
5.3 Principle of Operation of DC Generator 345
5.3.1 Voltage Build-Up in a DC Generator 346
5.3.2 Function of Commutator 347
5.4 Commutation Problem and Solution 349
5.4.1 Brush Shifting 349
5.4.2 Commutating Poles 350
5.4.3 Compensating Windings 350
5.5 Types of Windings 351
5.6 emf Equations in a DC Generator 351
5.7 Brush Placement in a DC Machine 353
5.8 Equivalent Circuit of DC Generator 354
5.9 Losses of DC Generator 354
5.10 Armature Reaction 360
5.10.1 No-Load Operation 361
5.10.2 Loaded Operation 361
5.11 Principle of Operation of a DC Motor 362
5.11.1 Equivalent Circuit of a DC Motor 363
5.12 emf and Torque Equations of DC Motor 364
5.13 Types of DC Motor 364
5.13.1 Separately Excited DC Motor 364
5.13.2 Self-Excited DC Motor 365
5.13.2.1 Shunt DC Motor 365
5.13.2.2 Series DC Motor 366
5.14 Characteristics of DC Motors 367
5.14.1 Separately Excited and DC Shunt Motor 368
5.14.2 DC Series Motor 369
5.14.3 Compound Motor 370
5.15 Starting of a DC Motor 371
5.15.1 Design of a Starter for a DC Motor 372
5.15.2 Types of Starters 373
5.15.2.1 Three-Point Starter 373
5.15.2.2 Four-Point Starter 374
5.16 Speed Control of a DC Motor 374
5.16.1 Separately Excited and DC Shunt Motor 375
5.16.2 DC Series Motor 376
5.17 Solved Examples 378
5.18 Matlab/Simulink Model of a DC Machine 387
5.18.1 Matlab/Simulink Model of a Separately/ Shunt DC Motor 387
5.18.2 Matlab/Simulink Model of a DC Series Motor 387
5.18.3 Matlab/Simulink Model of a Compound DC Motor 388
5.19 Summary 392
Problems 392
Reference 399
6 Three-Phase Induction Machine 401
6.1 Preliminary Remarks 401
6.2 Construction of a Three-Phase Induction Machine 402
6.2.1 Stator 402
6.2.2 Stator Frame 403
6.2.3 Rotor 403
6.3 Principle Operation of a Three-Phase Induction Motor 404
6.3.1 Slip in an Induction Motor 406
6.3.2 Frequency of Rotor Voltage and Current 407
6.3.3 Induction Machine and Transformer 408
6.4 Per-phase Equivalent Circuit of a Three-Phase Induction Machine 408
6.5 Power Flow Diagram in a Three-Phase Induction Motor 415
6.6 Power Relations in a Three-Phase Induction Motor 416
6.7 Steps to Find Powers and Efficiency 417
6.8 Per-Phase Equivalent Circuit Considering Stray-Load Losses 420
6.9 Torque and Power using Thevenin's Equivalent Circuit 421
6.10 Torque-Speed Characteristics 424
6.10.1 Condition for Maximum Torque 427
6.10.2 Condition for Maximum Torque at Starting 429
6.10.3 Approximate Equations 429
6.11 Losses in a Three-Phase Induction Machine 433
6.11.1 Copper Losses or Resistive Losses 433
6.11.2 Magnetic Losses 434
6.11.3 Mechanical Losses 434
6.11.4 Stray-Load Losses 434
6.12 Testing of a Three-Phase Induction Motor 435
6.12.1 No-Load Test 435
6.12.2 Blocked Rotor Test 436
6.12.3 DC Test 437
6.12.4 Load Test 438
6.12.5 International Standards for Efficiency of Induction Machines 441
6.12.6 International Standards for the Evaluation of Induction Motor Efficiency 442
6.13 Starting of a Three-Phase Induction Motor 443
6.13.1 Direct-on-Line Start 446
6.13.2 Line Resistance Start 447
6.13.3 Star-Delta Starter 448
6.13.4 Autotransformer Starter 449
6.14 Speed Control of Induction Machine 451
6.14.1 By Varying the Frequency of the Supply 451
6.14.2 Pole Changing Method 452
6.14.2.1 Multiple Numbers of Windings 453
6.14.2.2 Consequent Pole Method 453
6.14.3 Stator Voltage Control 454
6.14.3.1 Voltage/Frequency = Constant Control 455
6.14.3.2 Rotor Resistance Variation 456
6.14.3.3 Rotor Voltage Injection Method 456
6.14.3.4 Cascade Connection of Induction Machines 456
6.14.3.5 Pole-Phase Modulation for Speed Control 458
6.15 Matlab/Simulink Modelling of the Three-Phase Induction Motor 461
6.15.1 Plotting Torque-Speed Curve under Steady-State Condition 464
6.15.2 Dynamic Simulation of Induction Machine 464
6.16 Practice Examples 469
6.17 Summary 482
Problems 482
References 489
7 Synchronous Machines 491
7.1 Preliminary Remarks 491
7.2 Synchronous Machine Structures 492
7.2.1 Stator and Rotor 492
7.3 Working Principle of the Synchronous Generator 496
7.3.1 The Synchronous Generator under No-Load 498
7.3.2 The Synchronous Generator under Load 498
7.4 Working Principle of the Synchronous Motor 501
7.5 Starting of the Synchronous Motor 502
7.5.1 Starting by External Motor 502
7.5.2 Starting by using Damper Winding 503
7.5.3 Starting by Variable Frequency Stator Supply 503
7.6 Armature Reaction in Synchronous Motor 503
7.7 Equivalent Circuit and Phasor Diagram of the Synchronous Machine 506
7.7.1 Phasor Diagram of the Synchronous Generator 508
7.7.2 Phasor Diagram of the Synchronous Motor 510
7.8 Open-Circuit and Short-Circuit Characteristics 514
7.8.1 Open-Circuit Curve 514
7.8.2 Short-Circuit Curve 516
7.8.3 The Unsaturated Synchronous Reactance 517
7.8.4 The Saturated Synchronous Reactance 517
7.8.5 Short-Circuit Ratio 518
7.9 Voltage Regulation 520
7.9.1 Emf or Synchronous Method 521
7.9.2 The Ampere-Turn or mmf Method 522
7.9.3 Zero-Power Factor Method or Potier Triangle Method 526
7.9.3.1 Steps for Drawing Potier Triangles 526
7.9.3.2 Procedure to Obtain Voltage Regulation using the Potier Triangle Method 526
7.10 Efficiency of the Synchronous Machine 529
7.11 Torque and Power Curves 533
7.11.1 Real/Active Output Power of the Synchronous Generator 534
7.11.2 Reactive Output Power of the Synchronous Generator 535
7.11.3 Complex Input Power to the Synchronous Generator 536
7.11.4 Real/Active Input Power to the Synchronous Generator 536
7.11.5 Reactive Input Power to the Synchronous Generator 537
7.12 Maximum Power Output of the Synchronous Generator 537
7.13 Capability Curve of the Synchronous Machine 541
7.14 Salient Pole Machine 545
7.14.1 Phasor Diagram of a Salient Pole Synchronous Generator 547
7.14.2 Power Delivered by a Salient Pole Synchronous Generator 552
7.14.3 Maximum Active and Reactive Power Delivered by a Salient Pole Synchronous Generator 555
7.14.3.1 Active Power 555
7.14.3.2 Reactive Power 555
7.15 Synchronization of an Alternator with a Bus-Bar 558
7.15.1 Process of Synchronization 560
7.16 Operation of a Synchronous Machine Connected to an Infinite Bus-Bar (Constant Vt and f ) 562
7.16.1 Motor Operation of Change in Excitation at Fixed Shaft Power 562
7.16.2 Generator Operation for Change in Output Power at Fixed Excitation 565
7.17 Hunting in the Synchronous Motor 570
7.17.1 Role of the DamperWinding 572
7.18 Parallel Operation of Synchronous Generators 572
7.18.1 The Synchronous Generator Operating in Parallel with the Infinite Bus Bar 574
7.19 Matlab/Simulink Model of a Salient Pole Synchronous Machine 581
7.19.1 Results Motoring Mode 585
7.19.2 Results Generator Mode 585
7.20 Summary 586
Problems 587
Reference 591
8 Single-Phase and Special Machines 593
8.1 Preliminary Remarks 593
8.2 Single-phase Induction Machine 593
8.2.1 Field System in a Single-phase Machine 594
8.3 Equivalent Circuit of Single-phase Machines 597
8.3.1 Equivalent Circuit Analysis 599
8.3.1.1 Approximate Equivalent Circuit 600
8.3.1.2 Thevenin's Equivalent Circuit 601
8.4 How to Make a Single-phase Induction Motor Self Starting 602
8.5 Testing of an Induction Machine 608
8.5.1 DC Test 609
8.5.2 No-load Test 609
8.5.3 Blocked-Rotor Test 610
8.6 Types of Single-Phase Induction Motors 612
8.6.1 Split-Phase Induction Motor 612
8.6.2 Capacitor-Start Induction Motor 612
8.6.3 Capacitor-Start Capacitor-Run Induction Motor (Two-Value Capacitor Method) 613
8.7 Single-Phase Induction Motor Winding Design 614
8.7.1 Split-Phase Induction Motor 617
8.7.2 Capacitor-Start Motors 618
8.8 Permanent Split-Capacitor (PSC) Motor 621
8.9 Shaded-Pole Induction Motor 622
8.10 Universal Motor 622
8.11 Switched-Reluctance Motor (SRM) 624
8.12 Permanent Magnet Synchronous Machines 624
8.13 Brushless DC Motor 625
8.14 Mathematical Model of the Single-phase Induction Motor 626
8.15 Simulink Model of a Single-Phase Induction Motor 627
8.16 Summary 633
Problems 633
Reference 637
9 Motors for Electric Vehicles and Renewable Energy Systems 639
9.1 Introduction 639
9.2 Components of Electric Vehicles 641
9.2.1 Types of EVs 641
9.2.1.1 Battery-Based EVs 642
9.2.1.2 Hybrid EVs 643
9.2.1.3 Fuel-Cell EVs 646
9.2.2 Significant Components of EVs 649
9.2.2.1 Battery Bank 649
9.2.2.2 DC-DC Converters 661
9.2.2.3 Power Inverter 662
9.2.2.4 Electric Motor 663
9.2.2.5 Transmission System or Gear Box 663
9.2.2.6 Other Components 663
9.3 Challenges and Requirements of Electric Machines for EVs 663
9.3.1 Challenges of Electric Machines for EVs 664
9.3.2 Requirements of Electric Machines for EVs 664
9.4 Commercially Available Electric Machines for EVs 667
9.4.1 DC Motors 667
9.4.2 Induction Motor 667
9.4.3 Permanent Magnet Synchronous Motors (PMSM) 668
9.4.4 Brushless DC Motors 668
9.4.5 Switched Reluctance Motors (SRMs) 669
9.5 Challenges and Requirements of Electric Machines for RES 669
9.6 Commercially Available Electric Machines for RES 671
9.6.1 DC Machine 671
9.6.2 Induction Machines 671
9.6.3 Synchronous Machines 674
9.6.4 Advanced Machines for Renewable Energy 675
9.7 Summary 676
References 677
10 Multiphase (More than Three-Phase) Machines Concepts and Characteristics 679
10.1 Preliminary Remarks 679
10.2 Necessity of Multiphase Machines 679
10.2.1 Evolution of Multiphase Machines 680
10.2.2 Advantages of Multiphase Machines 683
10.2.2.1 Better Space Harmonics Profile 683
10.2.2.2 Better Torque Ripple Profile 684
10.2.2.3 Improved Efficiency 686
10.2.2.4 Fault Tolerant Capability 686
10.2.2.5 Reduced Ratings of Semiconductor Switches and Better Power/Torque Distribution 688
10.2.2.6 Torque Enhancement by Injecting Lower-Order Harmonics into Stator Currents 688
10.2.3 Applications of Multiphase Machines 689
10.3 Working Principle 691
10.3.1 Multiphase Induction Machine 691
10.3.2 Multiphase Synchronous Machine 691
10.4 Stator-Winding Design 692
10.4.1 Three-PhaseWindings 695
10.4.1.1 Single-Layer Full-Pitch Winding 695
10.4.1.2 Single-Layer Short-Pitch Winding 698
10.4.1.3 Double-Layer Full-PitchWinding 699
10.4.1.4 Double-Layer Short-Pitch Winding 699
10.4.1.5 Fractional-Slot Winding 701
10.4.2 Five-PhaseWindings 701
10.4.3 Six-Phase Windings 706
10.4.3.1 Symmetrical Winding of Six-Phase Machine 707
10.4.3.2 Asymmetrical Winding 710
10.4.4 Nine-PhaseWindings 710
10.5 Mathematical Modelling of Multiphase Machines 715
10.5.1 Mathematical Modelling of Multiphase Induction Machines in Original Phase-Variable Domain 715
10.5.2 Transformation Matrix for Multiphase Machines 718
10.5.3 Modelling of Multiphase Induction Machines in Arbitrary Reference Frames 720
10.5.4 Commonly used Reference Frames 722
10.5.5 Modelling of a Multiphase Synchronous Machine 723
10.6 Vector Control Techniques for Multiphase Machines 725
10.6.1 Indirect Field-Oriented Control or Vector-Control Techniques for Multiphase Induction Machines 726
10.6.2 Vector Control for Multiphase Synchronous Machines 730
10.7 Matlab/Simulink Model of Multiphase Machines 731
10.7.1 Dynamic Model of the Nine-Phase Induction Machine 731
10.7.2 Dynamic Model of the Nine-Phase Synchronous Machine 734
10.8 Summary 741
Problems 741
References 742
11 Numerical Simulation of Electrical Machines using the Finite Element Method 745
11.1 Introduction 745
11.2 Methods of Solving EM Analysis 747
11.2.1 Analytical Techniques 749
11.2.2 Numerical Techniques 750
11.2.2.1 Finite Difference Method 752
11.2.2.2 Finite Element Method 753
11.2.2.3 Solution of Laplace Equation Using the Finite Element Method 753
11.3 Formulation of 2-Dimensional and 3-Dimensional Analysis 758
11.3.1 Maxwell Equations 759
11.3.1.1 Gauss Law 759
11.3.1.2 Gauss Law of Magnetism 760
11.3.1.3 Ampere's Integral Law 761
11.3.1.4 Faraday's Integral Law 761
11.3.1.5 Differential Form of Maxwell Equations 761
11.3.2 FEM Adaptive Meshing 763
11.3.3 FEM Variation Principle 764
11.4 Analysis and Implementation of FEM Machine Models 765
11.4.1 RMxprt Design to Implement a Maxwell Model of Machine 765
11.4.2 Power Converter Design in Simplorer 776
11.4.3 Integration of Power Converter with a Maxwell Model for Testing Drive 776
11.5 Example Model of Three-Phase IM in Ansys Maxwell 2D 778
11.6 Summary 793
References 793
Index 795
1
Fundamentals of Electrical Machines
1.1 Preliminary Remarks
The study of the construction of electrical machines, along with the principles and laws on which they operate, gives the knowledge to predict behaviour and characteristics of the machines. Therefore, the study of electric machines requires an understanding of basic electromagnetic fields. The purpose of this chapter is to give some familiarity with the principles and laws that help understand the principle of Electric Machinery.
1.2 Basic Laws of Electrical Engineering
1.2.1 Ohm's Law
The current through an ideal resistance element is directly proportional to the voltage drop across it, as shown in Figure 1.1.
Ideal resistance element here refers to the element without any parasitic inductance or capacitance.
The relationship between voltage and current in an ideal resistor is given in Eq. (1.1) in which v is in volts, i is in amps, and the constant of proportionality is resistance R measured in ohms (O).This simple formula is known as Ohm's law in honour of the German physicist, Georg Ohm, whose original experiments led to this incredibly useful and important relationship.
(1.1)Notice that voltage v is measured across the resistor. That is, it is the voltage at point A with respect to the voltage at point B. When the current is in the direction shown, the voltage at A with respect to B is positive, so it is quite common to say that there is a voltage drop across the resistor.
An equivalent relationship for a resistor is given in Eq. (1.2), where the current is given in terms of voltage and the proportionality constant is conductance G with units of Siemens (S). In older literature, the unit of conductance was mhos.
(1.2)Figure 1.1 (a) An ideal resistor symbol. (b) voltage-current relationship.
Magnetic field problems involving components such as current coils, ferromagnetic cores, and air gaps can be solved as magnetic circuits according to the analogous behaviour of the magnetic quantities to the corresponding electric quantities in an electric circuit.
1.2.2 Generalization of Ohm's Law
Generalization of Ohm's law means expressing the law in terms of E (electric field in V/m) and J (current density in Amp/m2), rather than V and I. Consider a length l of a conductor of uniform cross-sectional area A through which a current I flows. In general, the electrical resistance of the conductor is proportional to its length, l, and inversely proportional to its cross-sectional area, A. Thus, resistance can be expressed as
(1.3)Here, the constants ? and s are called the resistivity and conductivity of the conducting medium, respectively, and are measured in units of ohm-meters and (ohm-meter)-1. Therefore, Ohm's law is stated as
(1.4)But, I/A = Jz (assuming that the conductor is aligned along the z-axis) and V/l = Ez, hence, Eq. (1.4) modifies to
(1.5)Equation (1.5) can be generalized as
(1.6a)or
(1.6b)where s is the conductivity of the medium. Equations 1.6a and (1.6b) are the vector form of Ohm's law.
1.2.2.1 Derivation of Eq. (1.6)
Let a unit volume of conductor contains n number of free electrons. In the presence of electric field E, the free electrons accelerate with drift velocity
(1.7)where q, t, and me are charged in coulomb, relaxation time in seconds and mass of electron respectively. The total charge of electrons (-nq) is moving with velocity (vd). Therefore, the current density
(1.8)Thus comparing Eqs. (1.6b) and (1.8), the conductivity can be defined as
(1.9)Let P be the power dissipated per unit volume inside the conducting medium. P is given as
(1.10)1.2.3 Ohm's Law for Magnetic Circuits
This law is also called as Hopkinson's law, after John Hopkinson, but was formulated earlier by Henry Augustus Rowland in 1873. It states that
(1.11)where , f, and R are magneto motive force (MMF) in Amp-turn (A.t), flux in Wb and reluctance in A.t/Wb, respectively.
1.2.4 Kirchhoff's Laws for Magnetic Circuits
As in electric circuits, Kirchhoff's laws can be applied in magnetic circuits because Kirchhoff's voltage and Kirchhoff's current laws are analogous to Ampere's law and Gauss Law, respectively.
- Magnetic circuits obey other laws of electric circuits such as reluctances in series and parallel
A space containing a quasi-stationary electric or magnetic field may be partitioned into flux tubes: geometrical figures in which all lines of flux are perpendicular to their bases and no lines of flux cut their sides; see Figure 1.2.
Lines of the equal magnetic scalar potential, u, are perpendicular to lines of flux, f; therefore, the bases of a flux tube are equipotential planes. The magnetic scalar potential difference between the bases equals the MMF drop. Generally, the ratio of potential difference at the ends of the flux tube that contains no current and the flux through it, is a function of flux tube geometry and the characteristics of the medium. Mathematically, this ratio is equal to:
(1.12)Figure 1.2 A flux tube in a field [1].
where l is the total flux tube length, A is its cross-sectional area, and c is a function of material properties. The quantity R is defined as reluctance in magnetic fields and is analogous to resistance in electric fields and the inverse of capacitance in electrostatic fields. R is a function of field quantities and also a function of geometry. The quantity c is equal to the flux tube material permeability, µ, for magnetic fields. For electric fields, c is equal to the flux tube conductivity, s, while for electrostatic fields c is the permittivity, e. For the three field types, one can write:
(1.13) (1.14) (1.15)The analogy between the three types of fields is almost complete, except in one detail: to maintain a certain energy level in a magnetic and an electrostatic field, no support from outside the field is needed. On the contrary, in an electric field, there must be a current from a source to cover losses. Electrostatic and magnetic fields can store energy, whereas in an electric field, the complete energy is irreversibly converted into heat. If all quantities in a field are constant, then stored energy in a magnetic or an electrostatic field is constant. Under the same circumstances, the energy lost in an electric field increases in proportion to time. A comparison of the field is tabulated in Table 1.1.
By comparing expressions for stored energy in electrostatic and magnetic fields, one can answer the question of why the magnetic field is chosen as an electromechanical energy conversion medium. Assuming the same level of field strength in both types of fields in free space, one can store µo/eo, or approximately 1.42 × 105, more energy per volume in a magnetic than in an electrostatic field. Stored energy per volume is representative of electro-mechanical energy conversion capability and, therefore, the choice of the magnetic field as a medium is obvious.
Table 1.1 Analogies between different types of fields [1].
Type of field Magnetic Electrostatic Electric Material Constant µ e S Flux or current density Field strength Reluctance, capacitance, resistance Potential difference Flux, charge, current Energy ?VdQ ?IVdtFurther comparison between electric and magnetic fields shows that magnetic circuits almost always operate in a saturated, i.e. nonlinear mode, while most elements in electric circuits are linear. There is also a large difference in the ratio of material constants between non-conducting and conducting media in magnetic and electric fields. The electrical conductivity of air is typically more than six orders of magnitude less than the conductivity of a current-carrying medium while, in a...
