ELECTRIC MACHINES Steady State, Transients, and Design with MATLAB ION BOLDEA LUCIAN TUTELEA Lop) CRC Press ^ ^ J Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business
Contents Preface xvii Part I Steady State 1 Introduction 3 1.1 Electric Energy and Electric Machines 3 1.2 Basic Types of Transformers and Electric Machines 6 1.3 Losses and Efficiency 13 1.4 Physical Limitations and Ratings 17 1.5 Nameplate Ratings 19 1.6 Methods of Analysis 21 1.7 State of the Art and Perspective 22 1.8 Summary 25 1.9 Proposed Problems 26 References 27 2 Electric Transformers 31 2.1 AC Coil with Magnetic Core and Transformer Principles 32 2.2 Magnetic Materials in EMs and Their Losses 38 2.2.1 Magnetization Curve and Hysteresis Cycle 38 2.2.2 Permanent Magnets 41 2.2.3 Losses in Soft Magnetic Materials 42 2.3 Electric Conductors and Their Skin Effects 45 2.4 Components of Single- and 3-Phase Transformers 51 2.4.1 Cores 52 2.4.2 Windings 54 2.5 Flux Linkages and Inductances of Single-Phase Transformers 59 2.5.1 Leakage Inductances of Cylindrical Windings 61 2.5.2 Leakage Inductances of Alternate Windings 62 2.6 Circuit Equations of Single-Phase Transformers with Core Losses 64 2.7 Steady State and Equivalent Circuit 65 2.8 No-Load Steady State (I 2 = 0)/Lab 2.1 67 2.8.1 Magnetic Saturation under No Load 70 v
vi Contents 2.9 Steady-State Short-Circuit Mode /Lab 2.2 71 2.10 Single-Phase Transformers: Steady-State Operation on Load/Lab 2.3 74 2.11 Three-Phase Transformers: Phase Connections 78 2.12 Particulars of 3-Phase Transformers on No Load 82 2.12.1 No-Load Current Asymmetry 82 2.12.2 Y Primary Connection for the 3-Limb Core 83 2.13 General Equations of 3-Phase Transformers 84 2.13.1 Inductance Measurement/Lab 2.4 86 2.14 Unbalanced Load Steady State in 3-Phase Transformers/Lab 2.5 87 2.15 Paralleling 3-Phase Transformers 91 2.16 Transients in Transformers 94 2.16.1 Electromagnetic (R,L) Transients 94 2.16.2 Inrush Current Transients/Lab 2.6 95 2.16.3 Sudden Short Circuit from No Load (Vi, =0)/Lab 2.7 96 2.16.4 Forces at Peak Short-Circuit Current 97 2.16.5 Electrostatic (C,R) Ultrafast Transients 99 2.16.6 Protection Measures of Anti-Overvoltage Electrostatic Transients 102 2.17 Instrument Transformers 102 2.18 Autotransformers. 103 2.19 Transformers and Inductances for Power Electronics 106 2.20 Preliminary Transformer Design (Sizing) by Example 108 2.20.1 Specifications 108 2.20.2 Deliverables 109 2.20.3 Magnetic Circuit Sizing 109 2.20.4 Windings Sizing 110 2.20.5 Losses and Efficiency 112 2.20.6 No-Load Current 112 2.20.7 Active Material Weight 113 2.20.8 Equivalent Circuit 113 2.21 Summary 114 2.22 Proposed Problems 115 References 118 3 Energy Conversion and Types of Electric Machines 121 3.1 Energy Conversion in Electric Machines 121 3.2 Electromagnetic Torque 122 3.2.1 Cogging Torque (PM Torque at Zero Current) 123 3.3 Passive Rotor Electric Machines. 124 3.4 Active Rotor Electric Machines 127 3.4.1 DC Rotor and AC Stator Currents 128 3.4.2 AC Currents in the Rotor and the Stator 128 3.4.3 DC (PM) Stator and AC Rotor 129
Contents vii 3.5 Fix Magnetic Field (Brush-Commutator) Electric Machines 131 3.6 Traveling Field Electric Machines 132 3.7 Types of Linear Electric Machines 134 3.8 Summary 139 3.9 Proposed Problem 140 References 141 Brush-Commutator Machines: Steady State 143 4.1 Introduction 143 4.1.1 Stator and Rotor Construction Elements 143 4.2 Brush-Commutator Armature Windings 146 4.2.1 Simple Lap Windings by Example: N s = 16, 2 Pl = 4 148 4.2.2 Simple Wave Windings by Example: N s = 9,2p 1 =2 150 4.3 Brush-Commutator 152 4.4 Airgap Flux Density of Stator Excitation MMF 154 4.5 No-Load Magnetization Curve by Example 155 4.6 PM Airgap Flux Density and Armature Reaction by Example 160 4.7 Commutation Process 164 4.7.1 AC Excitation Brush-Commutation Winding 166 4.8 EMF. 168 4.9 Equivalent Circuit and Excitation Connection 170 4.10 DC Brush Motor/Generator with Separate (or PM) Excitation/Lab 4.1 171 4.11 DC Brush PM Motor Steady-State and Speed Control Methods /Lab 4.2 173 4.11.1 Speed Control Methods 175 4.12 DC Brush Series Motor/Lab 4.3 181 4.12.1 Starting and Speed Control 183 4.13 AC Brush Series Universal Motor 184 4.14 Testing Brush-Commutator Machines/Lab 4.4 187 4.14.1 DC Brush PM Motor Losses, Efficiency, and Cogging Torque 188 4.15 Preliminary Design of a DC Brush PM Automotive Motor by Example 191 4.15.1 PM Stator Geometry 192 4.15.2 Rotor Slot and Winding Design 193 4.16 Summary 195 4.17 Proposed Problems 197 References 201
Vlll Contents 5 Induction Machines: Steady State 203 5.1 Introduction: Applications and Topologies 203 5.2 Construction Elements 204 5.3 AC Distributed Windings 206 5.3.1 Traveling MMF of AC Distributed Windings 206 5.3.2 Primitive Single-Layer Distributed Windings (q > 1, Integer) 209 5.3.3 Primitive Two-Layer 3-Phase Distributed Windings (q = Integer) 211 5.3.4 MMF Space Harmonics for Integer q (Slots/Pole/Phase) 212 5.3.5 Practical One-Layer AC 3-Phase Distributed Windings 216 5.3.6 Pole Count Changing AC 3-Phase Distributed Windings 220 5.3.7 Two-Phase AC Windings 221 5.3.8 Cage Rotor Windings 222 5.4 Induction Machine Inductances 225 5.4.1 Main Inductance 225 5.4.2 Leakage Inductance 227 5.5 Rotor Cage Reduction to the Stator 229 5.6 Wound Rotor Reduction to the Stator 230 5.7 Three-Phase Induction Machine Circuit Equations 230 5.8 Symmetric Steady State of 3-Phase IMs 234 5.9 Ideal No-Load Operation/Lab 5.1 236 5.10 Zero Speed Operation (S = 1)/Lab 5.2 238 5.11 No-Load Motor Operation (Free Shaft)/Lab 5.3 241 5.12 Motor Operation on Load (1 > S > 0)/Lab5.4 243 5.13 Generating at Power Grid (n >f 1 /p l,s < 0)/Lab 5.5 244 5.14 Autonomous Generator Mode (S < 0)/Lab 5.6 245 5.15 Electromagnetic Torque and Motor Characteristics 246 5.16 Deep-Bar and Dual-Cage Rotors 252 5.17 Parasitic (Space Harmonics) Torques 253 5.18 Starting Methods 256 5.18.1 Direct Starting (Cage Rotor) 256 5.18.2 Reduced Stator Voltages 257 5.18.3 Additional Rotor Resistance Starting 258 5.19 Speed Control Methods 259 5.19.1 Wound Rotor IM Speed Control 261 5.20 Unbalanced Supply Voltages 264 5.21 One Stator Phase Open by Example 265 5.22 One Rotor Phase Open 269 5.23 Capacitor Split-Phase Induction Motors 270 5.24 Linear Induction Motors 275 5.24.1 End and Edge Effects in LIMs 277
I Contents ix 5.25 Regenerative and Virtual Load Testing of IMs/Lab 5.7... 280 5.26 Preliminary Electromagnetic IM Design by Example 282 5.26.1 Magnetic Circuit 283 5.26.2 Electric Circuit 287 5.26.3 Parameters 287 5.26.4 Starting Current and Torque 290 5.26.5 Breakdown Slip and Torque 291 5.26.6 Magnetization Reactance, X m, and Core Losses, piron 291 5.26.7 No-Load and Rated Currents, 1Q and I 293 5.26.8 Efficiency and Power Factor 294 5.26.9 Final Remarks 294 5.27 Summary 295 5.28 Proposed Problems 298 References 300 Synchronous Machines: Steady State 303 6.1 Introduction: Applications and Topologies 303 6.2 Stator (Armature) Windings for SMs 306 6.2.1 Nonoverlapping (Concentrated) Coil SM Armature Windings 307 6.3 SM Rotors: Airgap Flux Density Distribution andemf 314 6.3.1 PM Rotor Airgap Flux Density 317 6.4 Two-Reaction Principle via Generator Mode 318 6.5 Armature Reaction and Magnetization Reactances, X dm and X qm 320 6.6 Symmetric Steady-State Equations and Phasor Diagram... 323 6.7 Autonomous Synchronous Generators 325 6.7.1 No-Load Saturation Curve/Lab 6.1 325 6.7.2 Short-Circuit Curve: (I sc (J F ))/Lab 6.2 326 6.7.3 Load Curve: V s (I s )/Lab 6.3 326 6.8 Synchronous Generators at Power Grid/Lab 6.4 331 6.8.1 Active Power/Angle Curves: P e (5 V ) 332 6.8.2 V-Shaped Curves 333 6.8.3 Reactive Power Capability Curves 334 6.9 Basic Static- and Dynamic-Stability Concepts 335 6.10 Unbalanced Load Steady State of SGs/Lab 6.5 339 6.10.1 Measuring X d, X q, Z_, andx 0 /Lab 340 6.11 Large Synchronous Motors 344 6.11.1 Power Balance 345 6.12 PM Synchronous Motors: Steady State 346 6.13 Load Torque Pulsations Handling by Synchronous Motors /Generators 349
x Contents 6.14 Asynchronous Starting of SMs and Their Self-Synchronization to Power Grid 352 6.15 Single-Phase and Split-Phase Capacitor PM Synchronous Motors 353 6.15.1 Steady State of Single-Phase Cageless-Rotor PMSMs 354 6.16 Preliminary Design Methodology of a 3-Phase PMSM by Example 357 6.17 Summary 363 6.18 Proposed Problems 366 References 370 Part II Transients 7 Advanced Models for Electric Machines 375 7.1 Introduction 375 7.2 Orthogonal (dq) Physical Model 376 7.3 Pulsational and Motion-Induced Voltages in dq Models 378 7.4 dq Model of DC Brush PM Motor (cu b = 0) 379 7.5 Basic dq Model of Synchronous Machines (cut, = ш г) 380 7.6 Basic dq Model of Induction Machines (шь = 0,iu r,a>i) 382 7.7 Magnetic Saturation in dq Models 383 7.8 Frequency (Skin) Effect Consideration in dq Models 386 7.9 Equivalence between dq Models and AC Machines 387 7.10 Space Phasor (Complex Variable) Model 389 7.11 High-Frequency Models for Electric Machines 392 7.12 Summary 393 7.13 Proposed Problems 396 References 398 8 Transients of Brush-Commutator DC Machines 401 8.1 Introduction 401 8.2 Orthogonal (dq) Model of DC Brush Machines with Separate Excitation 401 8.3 Electromagnetic (Fast) Transients 404 8.4 Electromechanical Transients 406 8.4.1 Constant Excitation (PM) Flux, dr 406 8.4.2 Variable Flux Transients 410 8.4.3 DC Brush Series Motor Transients 411 8.5 Basic Closed-Loop Control of DC Brush PM Motor 413 8.6 DC-DC Converter-Fed DC Brush PM Motor 413 8.7 Parameters from Test Data/Lab 8.1 415 8.8 Summary 417
Contents xi 8.9 Proposed Problems 418 References 419 9 Synchronous Machine Transients 421 9.1 Introduction 421 9.2 Phase Inductances of SMs 422 9.3 Phase Coordinate Model 423 9.4 dqo Model Relationships of 3-Phase SM Parameters 425 9.5 Structural Diagram of the SM dqo Model 427 9.6 pu dqo Model of SMs 430 9.7 Balanced Steady State via the dqo Model 432 9.8 Laplace Parameters for Electromagnetic Transients 436 9.9 Electromagnetic Transients at Constant Speed 437 9.10 Sudden 3-Phase Short Circuit from a Generator at No Load/Lab 9.1 439 9.11 Asynchronous Running of SMs at a Given Speed 442 9.12 Reduced-Order dqo Models for Electromechanical Transients 446 9.12.1 Neglecting Fast Stator Electrical Transients 446 9.12.2 Neglecting Stator and Rotor Cage Transients 447 9.12.3 Simplified (Third-Order) dq Model Adaptation for SM Voltage Control 447 9.13 Small-Deviation Electromechanical Transients (in PU)... 449 9.14 Large-Deviation Electromechanical Transients 453 9.14.1 Asynchronous Starting and Self-Synchronization of DC-Excited SMs/Lab 9.2 453 9.14.2 Asynchronous Self-Starting of PMSMs to Power Grid 455 9.14.3 Line-to-Line and Line-to-Neutral Faults 455 9.15 Transients for Controlled Flux and Sinusoidal Current SMs 456 9.15.1 Constant d-axis (i ^) Flux Transients in Cageless SMs 457 9.15.2 Vector Control of PMSMs at Constant л1> ао (ко = const) 460 9.15.3 Constant Stator Flux Transients in Cageless SMs at cosi[>i = 1 461 9.15.4 Vector Control of SMs with Constant Flux (i ) s ) and cos cp s = 1 464 9.16 Transients for Controlled Flux and Rectangular Current SMs 465 9.16.1 Model of Brushless DC Motor Transients 465 9.16.2 DC-Excited Cage Rotor SM Model for Rectangular Current Control 468
xii Contents 9.17 Switched Reluctance Machine Modeling for Transients 469 9.18 Split-Phase Cage Rotor SMs 475 9.19 Standstill Testing for SM Parameters/Lab 9.3 477 9.19.1 Saturated Steady-State Parameters, L^m and Lq m, from Current Decay Tests at Standstill 478 9.19.2 Single Frequency Test for Subtransient Inductances, I/j and Ц 481 9.19.3 Standstill Frequency Response Tests 481 9.20 Linear Synchronous Motor Transients 483 9.21 Summary 486 9.22 Proposed Problems 489 References 492 10 Transients of Induction Machines 495 10.1 Three-Phase Variable Model 495 10.2 dq (Space Phasor) Model of IMs 497 10.3 Three-Phase IM-dq Model Relationships 499 10.4 Magnetic Saturation and Skin Effects in the dq Model... 500 10.5 Space Phasor Model Steady State: Cage and Wound Rotor IMs 501 10.6 Electromagnetic Transients 507 10.7 Three-Phase Sudden Short Circuit/Lab 10.1 509 10.7.1 Transient Current at Zero Speed 512 10.8 Small-Deviation Electromechanical Transients 512 10.9 Large-Deviation Electromechanical Transients/Lab 10.2.. 514 10.10 Reduced-Order dq Model in Multimachine Transients... 516 10.10.1 Other Severe Transients 518 10.11 m/n r Actual Winding Modeling of IMs with Cage Faults 518 10.12 Transients for Controlled Magnetic Flux and Variable Frequency 522 10.12.1 Complex Eigenvalues of IM Space Phasor Model 522 10.13 Cage Rotor Constant Stator Flux Transients and Vector Control Basics 524 10.13.1 Cage-Rotor Constant Rotor Stator Flux Transients and Vector Control Basics 529 10.13.2 Constant Rotor Flux Transients and Vector Control Principles of Doubly Fed IMs 532 10.14 Doubly Fed IM as a Brushless Exciter for SMs 533 10.15 Parameter Estimation in Standstill Tests/Lab 10.3 537 10.15.1 Standstill Flux Decay for Magnetization Curve Identification: W^ (I m ) 538
Contents xiii 10.15.2 Identification of Resistances and Leakage Inductances for Standstill Flux Decay Tests 540 10.15.3 Standstill Frequency Response Tests 541 10.16 Split-Phase Capacitor IM Transients/Lab 10.4 542 10.16.1 Phase Variable Model 543 10.16.2 dq Model 544 10.17 Linear Induction Motor Transients 545 10.18 Summary 549 10.19 Proposed Problems 553 References 557 Part III FEM Analysis and Optimal Design 11 Essentials of Finite Element Method in Electromagnetics 561 11.1 Vectorial Fields 561 11.1.1 Coordinate Systems 561 11.1.2 Operations with Vectors 563 11.1.3 Line and Surface (Flux) Integrals of a Vectorial Field 564 11.1.4 Differential Operations 565 11.1.5 Integral Identities 567 11.1.6 Differential Identities 568 11.2 Electromagnetic Fields 569 11.2.1 Electrostatic Fields 569 11.2.2 Fields of Current Densities 570 11.2.3 Magnetic Fields 571 11.2.4 Electromagnetic Fields: Maxwell Equations 572 11.3 Visualization of Fields 573 11.4 Boundary Conditions 576 11.4.1 Dirichlet's Boundary Conditions 576 11.4.2 Neumann's Boundary Conditions 576 11.4.3 Mixed Robin's Boundary Conditions 577 11.4.4 Periodic Boundary Conditions 577 11.4.5 Open Boundaries 577 11.4.5.1 Problem Truncation 577 11.4.5.2 Asymptotical Boundary Conditions 577 11.4.5.3 Kelvin Transform 579 11.5 Finite Element Method 581 11.5.1 Residuum (Galerkin's) Method 582 11.5.2 Variational (Rayleigh-Ritz) Method 583 11.5.3 Stages in Finite Element Method Application 583 11.5.3.1 Domain Discretization 583 11.5.3.2 Choosing Interpolation Functions 584
xiv Contents 11.5.3.3 Formulation of Algebraic System Equations 584 11.5.3.4 Solving Algebraic Equations 584 11.6 2D FEM 584 11.7 Analysis with FEM 586 11.7.1 Electromagnetic Forces 589 11.7.1.1 Integration of Lorenz Force 589 11.7.1.2 Maxwell Tensor Method 590 11.7.1.3 Virtual Work Method 590 11.7.2 Loss Computation 591 11.7.2.1 Iron Losses 591 References 592 12 FEM in Electric Machines: Electromagnetic Analysis 595 12.1 Single-Phase Linear PM Motors 595 12.1.1 Preprocessor Stage 597 12.1.2 Postprocessor Stage... 602 12.1.3 Summary 609 12.2 Rotary PMSMs (6/4) 609 12.2.1 BLDC: Preprocessor Stage 610 12.2.2 BLDC Motor Analysis: Postprocessor Stage 616 12.2.3 Summary 632 12.3 The 3-Phase Induction Machines 632 12.3.1 Induction Machines: Ideal No Load 636 12.3.2 Rotor Bar Skin Effect 642 12.3.3 Summary 649 References 650 13 Optimal Design of Electric Machines: The Basics 651 13.1 Electric Machine Design Problem 651 13.2 Optimization Methods 653 13.3 Optimum Current Control 659 13.4 Modified Hooke-Jeeves Optimization Algorithm 664 13.5 Electric Machine Design Using Genetic Algorithms 670 References 674 14 Optimization Design of Surface PMSMs 675 14.1 Design Theme 675 14.2 Electric and Magnetic Loadings 675 14.3 Choosing a Few Dimensioning Factors 677 14.4 A Few Technological Constraints 678 14.5 Choosing Magnetic Materials 678 14.6 Dimensioning Methodology 681 14.6.1 Rotor Sizing 685 14.6.2 PM Flux Computation 686
Contents XV 14.6.3 Weights of Active Materials 692 14.6.4 Losses 693 14.6.5 Thermal Verification 694 14.6.6 Machine Characteristics 694 14.7 Optimal Design with Genetic Algorithms 694 14.7.1 Objective (Fitting) Function 696 14.7.2 PMSM Optimization Design Using Genetic Algorithms: A Case Study 697 14.8 Optimal Design of PMSMs Using Hooke-Jeeves Method... 709 14.9 Conclusion 710 References 715 15 Optimization Design of Induction Machines 717 15.1 Realistic Analytical Model for Induction Machine Design... 717 15.1.1 Design Theme 717 15.1.2 Design Variables 718 15.1.3 Induction Machine Dimensioning 719 15.1.3.1 Rotor Design 721 15.1.3.2 Stator Slot Dimensions 724 15.1.3.3 Winding End-Connection Length 724 15.1.4 Induction Machine Parameters 725 15.2 Induction Motor Optimal Design Using Genetic Algorithms 729 15.3 Induction Motor Optimal Design Using Hooke-Jeeves Algorithm 739 15.4 Machine Performance 742 15.5 Conclusion 750 References 751 Index 753