Modeling and Application of Electromagnetic and Thermal Field in Electrical Engineering (eBook)

Foreword 5
Preface 10
Motivation 10
Outline of This Book 11
Co-authorship and Edition 11
The Authors’ Expectations 12
Acknowledgements 13
Contents 14
About the Editors 26
Engineering Electromagnetic and Thermal Field Problems and FEM Fundamentals 28
1 General Survey of Engineering Electromagnetic and Thermal Field Problems 29
Abstract 29
1.1 Overview of Engineering Electromagnetic and Thermal Field Modeling 30
1.2 New Challenges Posed by UHV Transformer Engineering 31
1.3 Some Key Research Projects 35
1.3.1 Accurate Analysis of Total Core Loss 35
1.3.2 Efficient Solution of Transformer Winding Loss 37
1.3.3 Modeling and Control of Stray-Field Loss in Structural Parts 37
1.3.4 Numerical Prediction and Measurement of Electromagnetic and Thermal Fields 38
1.4 Realization of Accurate Modeling and Simulation of Electromagnetic and Thermal Performance 39
1.5 Overall Composition of the Book 42
References 43
2 Low-Frequency Electromagnetic Fields and Finite Element Method 47
Abstract 47
2.1 Introduction 48
2.2 Maxwell’s Equations 49
2.3 Governing Equations for Analysis of Low-Frequency Eddy Current Problems 51
2.4 Ar-V-Ar-Based Method 53
2.5 Scalar and Vector Galerkin Weight Function 54
2.6 Discussion on Edge Elements 55
2.7 Comparison of Basic Equations and Galerkin Residuals of Nodal Elements and Edge Elements 57
2.8 Comparison of Nonzero Entries and Total Unknowns in Coefficient Matrix 57
2.8.1 Unknowns and Number of Nonzero Entries in Matrix 59
2.8.1.1 Nodal Element 59
2.8.1.2 Edge Element 60
2.8.2 Comparison of Nonzero Entries and Total Unknowns in Matrix 61
2.9 Concluding Remarks 63
Acknowledgements 63
Appendix: Formulation of A-V-A and Galerkin Weighted Residual Processing 64
Basic Model of A-V-A 64
Governing Equation of A-V-A 66
Eddy Current Region 66
Non-Eddy Current Region 66
Galerkin Weighted Residual Processing 66
Galerkin Residuals 66
Residuals Processing 67
Eddy Current Region 67
Non-Eddy Current Region 72
References 73
Engineering Electromagnetic and Thermal Field Modeling 76
3 Some Key Techniques in Electromagnetic and Thermal Field Modeling 77
Abstract 77
3.1 Introduction 77
3.2 Special Elements 78
3.2.1 What Is Special Element? 78
3.2.2 Distribution of Potentials in Special Elements 79
3.2.3 Finite Element Formulation 81
3.2.4 Some Examples 81
3.3 Voltage-Driven Analysis 85
3.3.1 FEM Considering Voltage Source 85
3.3.2 An Example 88
3.4 Optimal Design Method 89
3.4.1 Various Optimization Methods 89
3.4.2 Experimental Design Method (EDM) 90
3.4.3 Rosenbrock’s Method (RBM) 92
3.4.4 Evolution Strategy (ES) 94
3.4.5 ON/OFF Method 95
3.4.6 Example of Application 99
3.5 Magneto-Thermal Coupled Analysis 110
3.5.1 Thermal Analysis 110
3.5.2 Magneto-Thermal Analysis 112
3.5.3 Magneto-Thermal-Fluid Analysis 114
3.6 Summary 123
References 123
4 Solution of Coupled Electromagnetic and Thermal Fields 125
Abstract 125
4.1 Simulation as a Design and Analysis Tool 126
4.2 Modeling 127
4.2.1 Geometry 128
4.2.2 Coils and Sources 129
4.2.3 Circuits 130
4.2.4 Material Properties 131
4.2.5 Material Modeling 132
4.2.5.1 Permeability 132
4.2.5.2 Core Loss 133
4.2.5.3 Electric Conductivity, Thermal Conductivity, Specific Heat Capacity, and Mass Density 136
4.2.6 Boundary Conditions 136
4.2.7 Accuracy Considerations 138
4.2.7.1 Mesh Controls 140
4.2.7.2 Adaption 141
4.2.7.3 Polynomial Order 142
4.2.7.4 Time Steps 146
4.2.7.5 Solver Options 146
4.3 Result Evaluation 147
4.3.1 Fields 148
4.3.2 Global Quantities 149
4.3.3 Scripting 150
4.4 Electromagnetic Field Computation 150
4.4.1 Solving the Electromagnetic Field Problem 150
4.4.1.1 Conducting Components 151
4.4.1.2 Non-conducting Components and Stranded Coil Regions 151
4.4.1.3 Conductors with Holes 152
4.4.2 Boundary Conditions 152
4.4.3 Problem Size 152
4.4.4 Surface Impedance Modeling 153
4.4.5 Skin Depth Modeling 155
4.5 Temperature Field Computation 155
4.5.1 Solving the Thermal Field Problem 155
4.5.2 Problem Sizes 156
4.6 Mechanism of Coupling Electromagnetic and Thermal Field Solutions 156
4.6.1 Sources of Heat Generation 156
4.6.2 Solving the Coupled Electromagnetic–Thermal Problem 157
4.6.3 Coupled Solution Controls 158
4.6.4 Coupled Electromagnetic–Thermal–Flow Simulation 159
4.7 Concluding Remarks 159
References 160
5 Development of Customized Scripts 162
Abstract 162
5.1 Introduction 162
5.2 Basics of the Script 164
5.2.1 Definition and Role of the Script 164
5.2.2 Classification of the Script 164
5.2.3 Concise Basic Syntax of VBScript 165
5.2.3.1 Basic Format 165
5.2.3.2 Special Symbols 165
5.2.3.3 Data Type of VBScript 168
5.2.3.4 VBScript Constants 168
5.2.3.5 VBScript Variables 169
5.2.3.6 VBScript Array Variables 170
5.2.3.7 Operators of VBScript 171
5.2.3.8 Control Statements 172
5.2.3.9 VBScript Procedures 176
5.3 Script Development in Simcenter MAGNET 178
5.3.1 Automatic Modeling 178
5.3.1.1 Script File 178
5.3.1.2 Script Form 179
5.3.2 Recording Script File 182
5.3.3 Interoperability 182
5.3.4 Export the Field Data 186
5.4 Development of a Script for Transformer Winding Parameters Calculation 187
5.4.1 Requirements for Script to Calculate Transformer Winding Parameters 187
5.4.2 Goal of the Script Used to Calculate Transformer Winding Parameters 188
5.4.3 FEM Method to Calculate the Eddy Loss of Windings 189
5.4.4 Implementation Process 190
5.5 Summary 196
Acknowledgements 197
References 197
6 Harmonic-Balanced Finite Element Method and Its Application 198
Abstract 198
6.1 Development of HBFEM 198
6.1.1 Basic Theory of HBFEM 199
6.1.2 Coupling Between Electric Circuits and the Magnetic Field 200
6.1.3 Epstein Frame-like Core Model Under DC-Biased Magnetization 201
6.1.4 Simulation and Analysis 203
6.2 The Fixed-Point Harmonic-Balanced Method 214
6.2.1 The Fixed-Point Technique 215
6.2.2 Fixed-Point Harmonic-Balanced Equation 215
6.2.3 Electromagnetic Coupling 217
6.2.4 Validation and Discussion 220
6.3 Decomposed Harmonic-Balanced Method 222
6.3.1 The Fixed-Point Reluctivity 222
6.3.2 Linearization and Decomposition 222
6.3.3 Force Computation of a Gapped Reactor Core Model 225
Acknowledgements 232
References 232
Measurement and Modeling of Magnetic Material and Component Properties 234
7 Fundamentals of Magnetic Material Modeling 235
Abstract 235
7.1 Introduction 236
7.2 Modeling of B–H Curve 236
7.2.1 Relationship Between B and H 236
7.2.2 Sectional Polynomial Approximation 237
7.2.3 Approximation of B–H Curve at High Flux Density 239
7.3 Modeling of Magnetic Anisotropy 240
7.3.1 Problem of Two B–H Curves Model 240
7.3.2 Multi-B–H Curve Model 242
7.3.3 E& SS Model
7.4 Hysteresis Modeling 251
7.4.1 Various Hysteresis Models 251
7.4.2 Interpolation Method 252
7.4.3 Preisach Model 254
7.4.4 Jiles–Atherton Model 259
7.4.5 Stoner–Wohlfarth Model 261
7.4.6 Effect of Hysteresis on Flux Distribution of Single-Phase Transformer 265
7.5 Estimation of Iron Loss 266
7.5.1 Iron Loss Under Alternating Flux 266
7.5.2 Iron Loss Under Rotating Flux 269
7.6 Modeling of Laminated Core 272
7.6.1 Laminated Core and Various Modeling Methods 272
7.6.2 Homogenization Method 272
7.6.3 Two-Zone Method 273
7.7 Factors Affecting Magnetic Properties of Electrical Steel 279
7.7.1 Residual Stress by Cutting 279
7.7.2 Compressive Stress 281
7.7.3 Effect of Press and Shrink Fitting on Iron Loss of Motor Core 282
7.7.4 Iron Loss Under Rotating Flux Excitation 284
7.7.5 Iron Loss Under DC Bias Excitation 286
7.8 Summary 289
References 290
8 Magnetic Measurement Based on Epstein Combination and Multi-angle Sampling 293
Abstract 293
8.1 Introduction 293
8.2 Magnetic Properties Under Rotating Flux Conditions 295
8.3 Application and Improvement of Epstein Frame Measurement 297
8.3.1 Epstein Frame 297
8.3.2 Epstein Combination and Loss Data-Based Weighted Processing Method 299
8.3.2.1 On the Ploss-Based Weighted Method 299
8.3.2.2 Typical Results of Path Length 301
8.3.2.3 Remarks on the Improved Epstein Measurement 302
8.4 Magnetic Measurement Based on Multi-angle Sampling 303
8.4.1 Multi-direction Magnetic Measurement 303
8.4.2 Multi-angle Sampling 303
8.5 Measurement Results and Discussions 305
8.5.1 Bm–Hm Curve Before Annealing (30P120) 305
8.5.2 Bm–Wt Curve Before Annealing (30P120) 307
8.5.3 Comparison of Bm–Hm Curves Measured Before and After Annealing 307
8.5.4 Comparison of Bm–Wt Curves Measured Before and After Annealing 308
8.6 Measuring Record of Voltage Starting Distortion in Magnetic Measurement Before and After Sample-Annealing 315
8.7 Concluding Remarks 315
Acknowledgements 318
References 318
9 Electromagnetic Property Modeling Based on Product-Level Core Models 320
Abstract 320
9.1 Introduction 321
9.2 Measurement of Magnetic Properties of Product-Level Core Model 322
9.2.1 Two-Laminated Core Models 322
9.2.2 Experimental Equipment 322
9.2.3 Experimental Content and Circuit 323
9.2.4 Measurement Procedure and Key Points 323
9.3 Measurement Results of Magnetic Properties of Product-Level Core Model 329
9.3.1 Waveforms of Exciting Current and Voltage 329
9.3.2 Experimental Data of Core Models 329
9.3.3 Magnetic Properties of Core Models and Comparison with Material Properties 332
9.3.3.1 Magnetic Properties of Core Models 332
9.3.3.2 Magnetic Properties of Silicon Steel 332
9.3.3.3 Comparison of Magnetic Properties 332
9.4 Separation of Exciting Power and Active Power Loss in the Joint Area and Middle Uniform Area of the Core 333
9.4.1 Separation of Exciting Power 333
9.4.2 Separation of the Active Power Loss 338
9.5 Specific Total Loss Calculation of the Middle Uniform Area with Two-Core Method 340
9.6 Determination of Building Factor of Core Model 342
9.7 Research on Magnetic Measurement of Transformer Core at Different Ambient Temperatures 345
9.7.1 Experimental Setup and Process 346
9.7.2 Measurement Results and Analysis 347
9.7.2.1 Material Properties Under Different Temperatures 347
9.7.2.2 Magnetic Properties of Core Model Measured Under Different Temperatures 349
9.8 Magnetic Properties Modeling Based on Ring Cores Before and After Annealing 355
9.8.1 Ring Core 355
9.8.2 Annealing Conditions 356
9.8.3 Experimental Result 356
9.8.3.1 Bm–Hb Curves of Ring Core Before and After Annealing 358
9.8.3.2 Bm–Ploss Curves of Ring Core Before and After Annealing 359
9.8.3.3 Exciting Power (Bm–Se) Curves of Ring Core Before and After Annealing 359
9.9 Concluding Remarks 360
9.9.1 Separation of Exciting Power and Magnetic Loss Based on Laminated Core Models 360
9.9.2 Effect of Temperature on the Magnetic Properties 360
Acknowledgements 361
Appendix 9.1: Magnetic Property Curves of the Ring Core Before and After Annealing 361
References 364
10 Rotational Magnetic Properties Measurement and Modeling 366
Abstract 366
10.1 Development of Rotational Magnetic Properties Measurement 366
10.1.1 Measurement Methods 367
10.1.2 Measurement Apparatus in Field-Metric Method 368
10.1.3 Techniques for Measuring B and H in Field-Metric Method 369
10.1.4 3-D Magnetic Testing System 371
10.1.4.1 3-D Excitation Structure 371
3-D Magnetization Structure 371
10.1.4.2 Modeling of the 3-D Magnetization Circuit 371
10.1.4.3 Shaping of the Core Poles and Magnetic Concentration 373
10.1.4.4 Parameters of the Excitation Circuit 374
10.1.4.5 Excitation Model of the 3-D Magnetic Testing 374
10.1.5 B–H Combined Sensing Structure 377
10.1.6 Calibration and Compensation of the 3-D Tester 378
10.2 Measurement and Analysis of the Rotational Magnetic Properties 381
10.2.1 Magnetic Properties of the Soft Magnetic Composite Materials 381
10.2.1.1 Magnetic Properties of SMC Material Under Circular Rotating Excitations 381
10.2.1.2 Magnetic Properties of SMC Material Under Elliptical Rotating Excitations 382
10.2.1.3 Magnetic Properties of SMC Material Under Spherical Rotating Excitations 383
10.2.1.4 Harmonic Analysis Under Rotational Excitation 385
10.2.1.5 Measurement and Calculation of 3-D Rotational Core Loss 386
10.2.2 Magnetic Properties of the Silicon Steels 387
10.2.2.1 Alternating Magnetic Properties Along Different Directions 388
10.2.2.2 Rotational Magnetic Properties of the Silicon Steels 389
10.2.2.3 Core Loss Analysis 390
10.3 Vector Hysteresis Model 392
10.3.1 Definition of the Vector Hysteron 393
10.3.1.1 Magnetic Particles and Equipotential Lines 393
10.3.1.2 Definition of the Vector Hysteron 394
10.3.1.3 Magnetization Direction of 2-D Vector Hysteron 395
10.3.1.4 Influence of Interaction Field Between Hysterons 396
10.3.2 Modeling of the Vector Hysteresis Characteristics 397
10.3.2.1 Magnetization Process of 2-D Hysterons 397
10.3.2.2 Modeling of the Vector Hysteresis Characteristics 399
10.3.3 Magnetic Properties Prediction and Validation 401
10.3.3.1 Alternating Magnetic Properties Prediction and Validation 401
10.3.3.2 Rotational Magnetic Properties’ Prediction and Validation 403
10.4 Summary 406
Acknowledgements 406
References 406
11 Measurement and Prediction of Magnetic Property of GO Silicon Steel Under Non-standard Excitation Conditions 409
Abstract 409
11.1 Introduction 410
11.2 1-D Magnetic Measurement Under Non-standard Conditions 410
11.2.1 Measurement of Magnetic Loss Under Harmonic or DC-Bias Condition 411
11.2.2 Measurement of Magnetic Loss Under Harmonic and DC-Bias Condition 413
11.2.3 Measurement of Magnetization Property 417
11.2.4 Measurement of B–H Loop Under Harmonic and DC-Bias 418
11.3 Magnetic Measurement Under Non-standard Conditions Based on an Integrated Magnetic Measure-Bench 418
11.3.1 Magnetic Property Measure-Bench and Two Core Model Schemes 418
11.3.2 Harmonic Excitation 421
11.3.3 Results and Discussions 422
11.3.3.1 Measurement of Specific Total Losses (Model C70) 422
11.3.3.2 Measurement of Specific Total Loss (Model C50) 428
11.3.4 Specific Total Loss (Model (C70–C50)) 428
11.3.5 Comparisons Among Specific Total Losses Measured by Two Core Models 429
11.3.6 Comparison of Specific Total Loss Results (Using Core Models and Epstein Frame) 432
11.3.7 Exciting Power Inside Laminated Core 432
11.3.8 Remarks 1 437
11.4 Measurement and Numerical Analysis of Magnetic Loss Under AC–DC Hybrid Excitation 438
11.4.1 Core Model Used for Magnetic Measurement Under Harmonic and DC-Bias Excitations 438
11.4.2 Feasibility of Magnetic Measurement Based on the New Core Model 441
11.4.2.1 Effect of Lamination L? on Magnetic Loss in Square Laminated Frame 441
11.4.2.2 Typical Measurement Results Under AC–DC Hybrid Excitation 442
11.4.3 Numerical Calculation and Validation of Magnetic Loss Inside Square Laminated Frame Under AC–DC Hybrid Excitation Conditions 443
11.4.3.1 DC Magnetic Field 443
11.4.3.2 Zoning Scheme 444
11.4.4 Remarks 2 446
11.5 Concluding Remarks 446
Acknowledgements 446
Appendix 1 Non-standard Magnetic Measurement Results 447
1.1 Magnetic Property of GO Silicon Steel Measured Under Non-standard Conditions 447
1.2 Magnetic Properties Measured Under Non-standard Conditions Based on Laminated Core Models 454
References 466
Validation Based on a Well-Established Benchmarking System 468
12 Establishment and Development of Benchmark Family (P21) 469
Abstract 469
12.1 Introduction 470
12.2 Development of TEAM Problem 21 472
12.2.1 Modeling and Prediction of Stray-Field Loss 472
12.2.2 Proposal and Updates to Problem 21 473
12.3 Definition of Problem 21 Benchmark Family 476
12.3.1 Benchmark Models 477
12.3.2 Benchmark Family Data 486
12.3.3 Field Quantities to Be Calculated 488
12.4 Numerical Analysis and Measurement 489
12.4.1 On Eddy Current Analysis Method 489
12.4.2 Measurement of Magnetic Flux Density and Interlinkage Flux at Specified Positions 491
12.4.3 Indirect Determination of Loss in Conducting Components 494
12.4.4 Determination of Upper and Lower Bounds of Losses 495
12.4.5 Eddy Current Losses in Exciting Coils 496
12.5 Typical Calculated and Measured Results 497
12.5.1 Problem 210 (P210-A and P210-B) 497
12.5.2 Problem 21a 499
12.5.3 Problem 21b 503
12.5.4 Problem 21c 507
12.5.5 Loss Spectrum of Problem 21 Benchmark Family 509
12.6 Problem 21 in Magnetic Saturation 512
12.6.1 Nonlinear Iterative Convergence Process Under Different Excitation Conditions 512
12.6.2 Introduction of Magnetic Saturation Factor 513
12.6.3 Analysis of Quasi-saturation 515
12.7 Further Co-research for Problem 21 Family 519
12.7.1 New Proposal of Problem 21 Family 519
12.7.2 Improved Method to Determine Stray-Field Loss 520
12.8 Summary and Outlook 522
12.8.1 Summary on Problem 21 Family 522
12.8.2 Outlook on the Future Co-research 523
Acknowledgements 524
Appendix 12.1: Characteristics of Magnetic Steel Plates Used in Problem 21 Family 524
Appendix 12.2: Characteristics of Silicon Steel Sheets Used in Problem 21 Family 525
Appendix 12.3: Reference Data of Problem 21 Family 529
References 531
13 Analysis and Validation of Additional Iron Loss Based on Benchmark Models 535
Abstract 535
13.1 Introduction 535
13.2 Model Structure and Design Data 536
13.2.1 Structure and Dimension of the Models 536
13.2.2 Locations of the Search Coil 537
13.3 Experimental Method and Targets 539
13.3.1 Experimental Circuit 539
13.3.2 Measurement Procedure 540
13.4 Measurement Results 542
13.4.1 Measured Loss Results of P21d-M 542
13.4.2 Measured Loss Results of P21d-M2 542
13.4.3 Flux Waveforms Obtained by Search Coils in Models P21d-M and P21d-M2 544
13.4.4 Average Flux Density Waveforms in the Laminated Sheets of Models P21d-M and P21d-M2 544
13.4.5 Determination of Maximum Values of Flux Density Based on P21d-M and P21d-M2 546
13.4.6 Remarks on the Measured Results 549
13.4.7 3-D Finite Element Computation of Additional Iron Loss 553
13.4.7.1 3-D Finite Element Model 553
13.4.8 Measured and Calculated Results of Iron Loss and Magnetic Flux 554
13.4.9 Comparison Between Waveforms of Measured and Calculated Flux 557
13.4.10 Measured and Calculated Flux Densities at Specified Positions 561
13.4.11 Question and Discussion 562
13.5 Concluding Remarks 563
Acknowledgements 563
Appendix 13.1 Magnetic Loss and Flux Under Different Excitation Patterns 564
13.1.1 Benchmarking Results Based on P21d-M 564
13.1.2 Benchmarking Results Based on P210-B 565
13.1.3 Remarks 567
References 567
Transformer-Related Electromagnetic and Thermal Modeling and Application 568
14 Electromagnetic and Thermal Modeling Based on Large Power Transformers 569
Abstract 569
14.1 Introduction 569
14.2 Measurement of Electromagnetic and Thermal Properties of Commonly Used Metal Materials 570
14.2.1 Conductivity Measurement 571
14.2.1.1 Test Specimen 571
14.2.1.2 Measurement Method 571
14.2.1.3 Test Temperature Control 572
14.2.1.4 DC Resistance Tester 572
14.2.1.5 Measured Conductivity 572
14.2.2 Measurement of DC Magnetization Curve of Magnetic Steel Plate 573
14.2.2.1 Ring-Shaped Specimens 573
14.2.2.2 Measured Results of DC Magnetization Curve of Magnetic Steel Plate 574
14.2.3 Surface Heat Transfer Coefficient of Steel Plate 574
14.2.3.1 Models for Measuring Surface Temperature Property of the Steel Plate 576
14.2.3.2 Measurement Methods and Instruments 576
14.2.3.3 Measurement Results 577
14.3 Validation of Modeling and Simulation of Loss and Surface Hot-Spot Temperature of Steel Plate 578
14.3.1 Test Model 580
14.3.2 Measuring System 580
14.3.3 Results and Validation 580
14.4 3-D FE Model for Simulating Transformer Component Loss 585
14.4.1 3-D Mesh for Magnetic Steel Plate 585
14.4.1.1 Determining the Standard Loss for Simulation of Magnetic Steel Specimen 585
14.4.1.2 Analysis of Simulated Losses and Errors Obtained by Different Mesh Approaches 585
14.4.2 3-D Mesh for Non-magnetic Steel Plate 589
14.5 Engineering Application of Electromagnetic and Thermal Simulation 592
14.5.1 Large Single-Phase Autotransformer (700 MVA/750 KV) 593
14.5.1.1 The Specifications 593
14.5.1.2 Magnetic Field Simulation of Winding 593
14.5.2 Modeling and Simulation of Preliminary Structural Design 594
14.5.3 Thermal Field Simulation of Components in the Active Part 595
14.5.4 Modeling and Simulation of Optimized Structures 597
14.5.5 Discussion 598
14.6 Summary 601
Acknowledgements 601
References 602
15 Engineering-Oriented Modeling and Experimental Research on DC-Biased Transformers 603
Abstract 603
15.1 Introduction 603
15.1.1 DC Bias Phenomenon on Power Transformers 604
15.1.2 Brief Overview of Investigation on DC-Biased Problem 607
15.1.3 Key Research Projects 609
15.2 Magnetic Properties of Product-Level Laminated Core Under DC Bias Condition 610
15.2.1 ?–I Curve and B?H Curve of Transformer Core 610
15.2.2 Bm – W Curve of Transformer Core 621
15.3 Calculation of the Exciting Current Under DC Bias Condition 622
15.3.1 Principle of Simple Iteration to Determine DC Flux in Transformer Core 622
15.3.2 Validation of Simple Iteration Method 624
15.3.2.1 No-Load Exciting Current Without DC Bias (Idc?=?0 A) 624
15.3.2.2 No-Load Exciting Current Under DC Bias Condition (Idc?=?1.26 A) 628
15.3.2.3 No-Load Exciting Current Under DC Bias Condition (Idc?=?2.53 A) 628
15.3.2.4 No-Load Exciting Current Under DC Bias Condition (Idc?=?3.2 A) 629
15.3.3 Harmonic Analysis of Exciting Current [22] 629
15.4 Modeling and Computation of Magnetic Field and Loss Under DC Bias Condition 632
15.4.1 Some Key Factors in Modeling Under DC Bias Condition 632
15.4.2 Computation of Magnetic Field and Loss Under No-Load and DC Bias Condition 634
15.4.2.1 Results Under No-Load Condition with Idc?=?0 A 634
15.4.2.2 Results Under No-Load Condition with Idc?=?3.2 A 636
15.4.3 Computation of Magnetic Field and Loss Under Load and DC Bias Condition 643
15.4.3.1 Results Under Load Condition with Idc?=?0 A 643
15.4.3.2 Results Under Load Condition with Idc?=?3.2 A 647
15.4.4 Influence of DC Bias on Loss 650
15.4.4.1 Results Under No-Load Condition 650
15.4.4.2 Results Under Load Condition 650
15.4.4.3 Influence of Bias on No-Load and Load Loss 651
15.5 The Experimental Research on the DC-Biased 500 KV Autotransformer 652
15.5.1 No-Load Loss Measurement Under DC Bias Condition 653
15.5.2 Harmonics Analysis of Exciting Current 655
15.5.3 Measurement of Sound Level Under DC Bias Condition 666
15.6 On the Ability to Withstand DC Bias for Power Transformers 669
15.7 Summary and Outlook 673
Acknowledgements 675
Appendix: Magnetic Property Data Under DC Bias Conditions 675
References 678
16 Modeling and Validation of Thermal-Fluid Field of Transformer Winding Based on a Product-Level Heating and Cooling Model 681
Abstract 681
16.1 Introduction 682
16.2 Test Model 683
16.3 Experiment Instruments and the Performance 686
16.4 Measurement Methodology 686
16.5 Numerical Modeling and Simulation of Thermal-Fluid Field in Transformer Winding 688
16.6 Results and Discussions 692
16.7 Summary 700
Acknowledgements 700
References 700