Transformer Design Principles - With Applications to Core-Form Power Transformers

Cover......Page 1
Transformer Design Principles: With Applications to Core-Form Power Transformers......Page 2
Transformer Design Principles: With Applications to Core-Form Power Transformers......Page 3
Contents......Page 5
Preface......Page 13
1.1 Historical Background......Page 14
1.2 Uses in Power Systems......Page 15
1.3 Core- Form and Shell- Form Transformers......Page 21
1.4 Stacked and Wound Core Construction......Page 22
1.5 Transformer Cooling......Page 23
1.6 Winding Types......Page 25
1.7 Insulation Structures......Page 27
1.8 Structural Elements......Page 31
1.9 Modern Trends......Page 34
2.1 Introduction......Page 36
2.2 Basic Magnetism......Page 38
2.3 Hysteresis......Page 41
2.4 Magnetic Circuits......Page 43
2.5 Inrush Current......Page 48
2.6 Distinguishing Inrush from Fault Current......Page 51
2.7 Optimal Core Stacking......Page 59
3.2 Circuit Model of the Core......Page 62
3.3 Two- Winding Transformer Circuit Model with Core......Page 66
3.4 Approximate Two- Winding Transformer Circuit Model without Core......Page 69
3.5 Vector Diagram of a Loaded Transformer with Core......Page 74
3.6 Per- Unit System......Page 75
3.7 Voltage Regulation......Page 77
4.1 Introduction......Page 81
4.2 General Method for Determining Inductances and Mutual Inductances......Page 82
4.2.1 Energy by Magnetic Field Methods......Page 83
4.2.2 Energy from Electric Circuit Methods......Page 85
4.3 Two- Winding Leakage Reactance Formula......Page 88
4.4 Ideal Two-, Three-, and Multiwinding Transformers......Page 92
4.4.1 I deal Autotransformers......Page 95
4.5 Leakage Reactance for Two- Winding Transformers Based on Circuit Parameters......Page 96
4.6 Leakage Reactances for Three- Winding Transformers......Page 100
4.6.1 L eakage Reactance for an Autotransformer with a Tertiary Winding......Page 104
4.6.2 L eakage Reactance between Two Windings Connected in Series and a Third Winding......Page 109
4.6.3 L eakage Reactance of a Two- Winding Autotransformer with X- Line Taps......Page 110
4.6.4 More General Leakage Reactance Calculations......Page 113
5.1 Phasors......Page 114
5.2 Wye and Delta Three- Phase Connections......Page 118
5.3 Zig- Zag Connection......Page 123
5.4 Scott Connection......Page 125
5.5 Symmetrical Components......Page 128
6.1 Introduction......Page 134
6.2 Fault Current Analysis on Three- Phase Systems......Page 135
6.2.2 Single- Phase Line- to- Ground Fault......Page 138
6.2.3 L ine- to- Line Fault......Page 139
6.2.4 Double Line- to- Ground Fault......Page 140
6.3 Fault Currents for Transformers with Two Terminals per Phase......Page 141
6.3.2 Single- Phase Line- to- Ground Fault......Page 144
6.3.3 L ine- to- Line Fault......Page 145
6.3.4 Double Line- to- Ground Fault......Page 146
6.3.5 Zero- Sequence Circuits......Page 147
6.3.6 Numerical Example for a Single Line- to- Ground Fault......Page 149
6.4 Fault Currents for Transformers with Three Terminals per Phase......Page 150
6.4.1 Three- Phase Line- to- Ground Fault......Page 153
6.4.2 Single- Phase Line- to- Ground Fault......Page 154
6.4.3 Line- to- Line Fault......Page 156
6.4.4 Double Line- to- Ground Fault......Page 158
6.4.6 Numerical Examples......Page 160
6.5 Asymmetry Factor......Page 164
7.1 Introduction......Page 165
7.2 Basic Principles......Page 167
7.3 Squashed Delta Phase- Shifting Transformer......Page 169
7.3.1 Zero- Sequence Circuit Model......Page 173
7.4 Standard Delta Phase- Shifting Transformer......Page 175
7.4.1 Zero- Sequence Circuit Model......Page 179
7.5 Two- Core Phase- Shifting Transformer......Page 180
7.5.1 Zero- Sequence Circuit Model......Page 184
7.6 Regulation Effects......Page 185
7.7 Fault Current Analysis......Page 187
7.7.1 Squashed Delta Fault Currents......Page 189
7.7.2 Standard Delta Fault Currents......Page 191
7.7.3 Two- Core Phase- Shifting Transformer Fault Currents......Page 192
7.8 Zig- Zag Transformer......Page 194
7.8.1 Calculation of Electrical Characteristics......Page 195
7.8.2 Per- Unit Formulas......Page 199
7.8.3 Zero- Sequence Impedance......Page 200
7.8.4 Fault Current Analysis......Page 202
8.1 Introduction......Page 204
8.2.1 Two- Winding Leakage Inductance......Page 205
8.2.2 Multiwinding Transformers......Page 206
8.2.3 Transformer Loading......Page 210
8.3.1 Two Secondary Windings in Series......Page 211
8.3.2 Primary Winding in Series with a Secondary Winding......Page 212
8.3.3 A utotransformer......Page 213
8.4 Multiphase Transformers......Page 214
8.4.1 Delta Connection......Page 216
8.4.2 Zig- Zag Connection......Page 218
8.5 Generalizing the Model......Page 220
8.6 Regulation and Terminal Impedances......Page 222
8.7 Multiterminal Transformer Model for Balanced and Unbalanced Load Conditions......Page 224
8.7.1 Theory......Page 225
8.7.2 A dmittance Representation......Page 227
8.7.2.1 Delta Winding Connection......Page 229
8.7.3 Impedance Representation......Page 231
8.7.3.1 Ungrounded Y Connection......Page 232
8.7.3.2 Series- Connected Windings from the Same Phase......Page 234
8.7.3.3 Zig- Zag Winding Connection......Page 236
8.7.3.5 Three Windings Joined......Page 237
8.7.4 Terminal Loading......Page 238
8.7.5.2 Winding Currents and Voltages......Page 239
8.7.6.1 Autotransformer with Buried Delta Tertiary and Fault on Low- Voltage Terminal......Page 240
8.7.6.2 Power Transformer with Fault on Delta Tertiary......Page 241
8.7.6.3 Power Transformer with Fault on Ungrounded Y Secondary......Page 242
8.7.7.1 Standard Delta Phase- Shifting Transformer......Page 243
8.7.8 Discussion......Page 245
9.1 Introduction......Page 247
9.2 Theory......Page 248
9.3.1 Rabins\' Method Applied to Calculate the Leakage Reactance between Two Windings That Occupy Different Radial Positions......Page 261
9.3.2 Rabins\' Method Applied to Calculate the Leakage R eactance between Two Axially Stacked Windings......Page 265
9.3.3 Rabins\' Method Applied to Calculate the Leakage Reactance for a Collection of Windings......Page 267
9.4 Application of Rabins\' Method to Calculate the ­Self-Inductance of and Mutual Inductance between Coil Sections......Page 268
9.5 Determining the B- Field......Page 271
9.6 Determination of Winding Forces......Page 272
9.7 Numerical Considerations......Page 275
10.1 Introduction......Page 276
10.2 Force Calculations......Page 278
10.3 Stress Analysis......Page 280
10.3.1 Compressive Stress in the Key Spacers......Page 282
10.3.2 A xial Bending Stress per Strand......Page 283
10.3.3 Tilting Strength......Page 286
10.3.4 Stress in the Tie Bars......Page 290
10.3.5 Stress in the Pressure Ring......Page 296
10.3.6 Hoop Stress......Page 297
10.3.7 R adial Bending Stress......Page 298
10.4 Radial Buckling Strength......Page 305
10.4.1 Free Unsupported Buckling......Page 306
10.4.2 Constrained Buckling......Page 309
10.4.3 E xperiment to Determine Buckling Strength......Page 310
10.5 Stress Distribution in a Composite Wire– Paper Winding Section......Page 314
10.6 Additional Mechanical Considerations......Page 319
11.1.1 Planar Geometry......Page 321
11.1.2 Cylindrical Geometry......Page 324
11.1.3 Spherical Geometry......Page 327
11.1.4 Cylinder– Plane Geometry......Page 328
11.2.1 Physical Basis......Page 333
11.2.2 Conformal Mapping......Page 334
11.2.3 Schwarz– Christoffel Transformation......Page 338
11.2.4 Conformal Map for the Electrostatic Field Problem......Page 340
11.2.4.1 Electric Potential and Field Values......Page 344
11.2.4.2 Calculations and Comparison with a Finite Element Solution......Page 352
11.2.4.3 Estimating Enhancement Factors......Page 356
11.3 Finite Element Electric Field Calculations......Page 358
12.1 Introduction......Page 363
12.2 Distributive Capacitance along a Winding or Disk......Page 364
12.3 Stein\'s Disk Capacitance Formula......Page 369
12.4 General Disk Capacitance Formula......Page 377
12.5 Coil Grounded at One End with Grounded Cylinders on Either Side......Page 378
12.6 Static Ring on One Side of a Disk......Page 380
12.7 Terminal Disk without a Static Ring......Page 381
12.8 Capacitance Matrix......Page 382
12.9 Two Static Rings......Page 384
12.11 Winding Disk Capacitances with Wound- in Shields......Page 388
12.11.1 Analytic Formula......Page 389
12.11.2 Circuit Model......Page 392
12.11.3 Experimental Methods......Page 397
12.11.4 Results......Page 398
12.12 Multistart Winding Capacitance......Page 402
13.1 Introduction......Page 405
13.2 Principles of Voltage Breakdown......Page 406
13.2.1 Breakdown in Solid Insulation......Page 411
13.2.2 Breakdown in Transformer Oil......Page 413
13.3 Geometric Dependence of Transformer- Oil Breakdown......Page 416
13.3.2 Planar Geometry......Page 418
13.3.3 Cylindrical Geometry......Page 421
13.3.4 Spherical Geometry......Page 422
13.3.5 Comparison with Experiment......Page 424
13.3.6 Generalization......Page 425
13.3.6.2 Breakdown for the Disk– Disk- to- Ground Plane Geometry......Page 427
13.3.7 Discussion......Page 430
13.4 Insulation Coordination......Page 431
13.5 Continuum Model of Winding Used to Obtain the Impulse– Voltage Distribution......Page 435
13.5.1 Uniform Capacitance Model......Page 436
13.5.2 Traveling Wave Theory......Page 440
13.6.1 Circuit Description......Page 441
13.6.3 Capacitance Calculations......Page 444
13.6.4 Impulse– Voltage Calculations and Experimental Comparisons......Page 445
13.6.5 Sensitivity Studies......Page 450
14.1 Introduction......Page 452
14.2 No- Load or Core Losses......Page 453
14.2.2 I nterlaminar Losses......Page 457
14.3.1 I2R Losses......Page 460
14.3.2 Stray Losses......Page 461
14.3.2.1 Eddy Current Losses in the Coils......Page 462
14.3.2.2 Tieplate Losses......Page 466
14.3.2.3 Tieplate and Core Losses Due to Unbalanced Currents......Page 474
14.3.2.4 Tank and Clamp Losses......Page 480
14.3.3 Stray Losses Obtained from 3D Finite Element Analyses......Page 487
14.3.3.1 Shunts on the Clamps......Page 488
14.3.3.2 Shunts on the Tank Wall......Page 491
14.3.3.3 Effects of Three- Phase Currents on Losses......Page 492
14.3.3.4 Stray Losses from the 3D Analysis versus Analytical and Test Losses......Page 494
14.4 Tank and Shield Losses Due to Nearby Busbars......Page 495
14.4.2 Losses Obtained Analytically......Page 496
14.4.2.1 Current Sheet......Page 497
14.4.2.2 Delta Function Current......Page 498
14.4.2.3 Collection of Delta Function Currents......Page 500
14.4.2.4 Model Studies......Page 503
14.5 Tank Losses Associated with the Bushings......Page 505
14.5.1 Comparison with a 3D Finite Element Calculation......Page 510
15.1 Introduction......Page 512
15.2 Thermal Model of a Disk Coil with Directed Oil Flow......Page 513
15.2.1 Oil Pressures and Velocities......Page 515
15.2.2 Oil Nodal Temperatures and Path Temperature Rises......Page 519
15.2.3 Disk Temperatures......Page 521
15.3 Thermal Model for Coils without Directed Oil Flow......Page 522
15.4 Radiator Thermal Model......Page 523
15.5 Tank Cooling......Page 527
15.6 Oil Mixing in the Tank......Page 528
15.7 Time Dependence......Page 531
15.8 Pumped Flow......Page 532
15.9 Comparison with Test Results......Page 533
15.10 Determining m and n Exponents......Page 538
15.11 Loss of Life Calculation......Page 540
15.12 Cable and Lead Temperature Calculation......Page 543
15.13 Tank Wall Temperature Calculation......Page 547
15.14 Tieplate Temperature......Page 549
15.15 Core Steel Temperature Calculation......Page 552
16.2 General Description of Load Tap Changers......Page 554
16.3 Types of Regulation......Page 555
16.4.1 Resistive Switching......Page 557
16.4.2 Reactive Switching with Preventive Autotransformers......Page 558
16.5.1 Power Transformers......Page 560
16.5.2 Autotransformers......Page 561
16.5.3 Use of Auxiliary Transformers......Page 566
16.6 General Maintenance......Page 567
17.1 Setting the Impulse Test Generator to Achieve Close to Ideal Waveshapes......Page 570
17.1.1 Impulse Generator Circuit Model......Page 572
17.1.2 Transformer Circuit Model......Page 574
17.1.3 Determining the Generator Settings for Approximating the Ideal Waveform......Page 577
17.1.4 Practical Implementation......Page 581
17.2.1 Lumped Parameter Model......Page 584
17.2.1.1 Numerical Example......Page 586
17.2.2 Traveling Wave Theory......Page 587
17.3 Air Core Inductance......Page 589
17.4 Electrical Contacts......Page 592
17.4.1 Contact Resistance......Page 593
17.4.2 Thermal Considerations......Page 594
17.4.3 Practical Considerations......Page 597
References......Page 598