Transformer Design Principles: With Applications to Core-Form Power Transformers, Second Edition

602 Pages·2010·8.06 MB·English

by  Robert M. Del Vecchio, Bertrand Poulin, Pierre T. Feghali, Dilipkumar M. Shah, Rajendra Ahuja

Transformer
Design
Principles
With Applications to
Core-Form Power Transformers
Second Edition
Transformer
Design
Principles
With Applications to
Core-Form Power Transformers
Second Edition
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Library of Congress Cataloging-in-Publication Data
Transformer design principles : with applications to core-form power transformers /
authors, Robert M. Del Vecchio … [et al.]. -- 2nd ed.
p. cm.
“A CRC title.”
Includes bibliographical references and index.
ISBN 978-1-4398-0582-4 (alk. paper)
1. Electric transformers--Design and construction. I. Del Vecchio, Robert M. II. Title.
TK2551.T765 2010
621.31’4--dc22 2009050603
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Contents
Preface ................................................................................................................... xiii
1. Introduction .....................................................................................................1
1.1 Historical Background .........................................................................1
1.2 Uses in Power Systems .........................................................................2
1.3 Core-Form and Shell-Form Transformers .........................................8
1.4 Stacked and Wound Core Construction ............................................9
1.5 Transformer Cooling .......................................................................... 10
1.6 Winding Types .................................................................................... 12
1.7 Insulation Structures .......................................................................... 14
1.8 Structural Elements ............................................................................ 18
1.9 Modern Trends .................................................................................... 21
2. Magnetism and Related Core Issues ........................................................23
2.1 Introduction .........................................................................................23
2.2 Basic Magnetism .................................................................................25
2.3 Hysteresis .............................................................................................28
2.4 Magnetic Circuits ................................................................................30
2.5 Inrush Current .....................................................................................35
2.6 Distinguishing Inrush from Fault Current .....................................38
2.7 Optimal Core Stacking .......................................................................46
3. Circuit Model of a Two-Winding Transformer with Core ................... 49
3.1 Introduction ......................................................................................... 49
3.2 Circuit Model of the Core .................................................................. 49
3.3 Two-Winding Transformer Circuit Model with Core ....................53
3.4 Approximate Two-Winding Transformer Circuit
Model without Core ............................................................................56
3.5 Vector Diagram of a Loaded Transformer with Core .................... 61
3.6 Per-Unit System ................................................................................... 62
3.7 Voltage Regulation ..............................................................................64
4. Reactance and Leakage Reactance Calculations .................................... 69
4.1 Introduction ......................................................................................... 69
4.2 General Method for Determining Inductances and
Mutual Inductances ............................................................................ 70
4.2.1 Energy by Magnetic Field Methods .................................... 71
4.2.2 Energy from Electric Circuit Methods................................ 73
4.3 Two-Winding Leakage Reactance Formula .................................... 76
v
vi Contents
4.4 Ideal Two-, Three-, and Multiwinding Transformers. ...................80
4.4.1 Ideal Autotransformers. ........................................................83
4.5 Leakage Reactance for Two-Winding Transformers Based
on Circuit Parameters. ........................................................................84
4.5.1 Leakage Reactance for a Two-Winding
Autotransformer ....................................................................88
4.6 Leakage Reactances for Three-Winding Transformers .................88
4.6.1 Leakage Reactance for an Autotransformer with a
Tertiary Winding ................................................................... 92
4.6.2 Leakage Reactance between Two Windings Connected
in Series and a Third Winding ............................................ 97
4.6.3 Leakage Reactance of a Two-Winding
Autotransformer with X-Line Taps ..................................... 98
4.6.4 More General Leakage Reactance Calculations .............. 101
5. Phasors, Three-Phase Connections, and Symmetrical
Components ................................................................................................. 103
5.1 Phasors ................................................................................................ 103
5.2 Wye and Delta Three-Phase Connections ..................................... 107
5.3 Zig-Zag Connection .......................................................................... 112
5.4 Scott Connection ............................................................................... 114
5.5 Symmetrical Components ............................................................... 117
6. Fault Current Analysis .............................................................................. 123
6.1 Introduction ....................................................................................... 123
6.2 Fault Current Analysis on Three-Phase Systems ......................... 124
6.2.1 Three-Phase Line-to-Ground Fault ................................... 127
6.2.2 Single-Phase Line-to-Ground Fault ................................... 127
6.2.3 Line-to-Line Fault ................................................................ 128
6.2.4 Double Line-to-Ground Fault ............................................ 129
6.3 Fault Currents for Transformers with Two Terminals per
Phase ................................................................................................... 130
6.3.1 Three-Phase Line-to-Ground Fault ................................... 133
6.3.2 Single-Phase Line-to-Ground Fault ................................... 133
6.3.3 Line-to-Line Fault ................................................................ 134
6.3.4 Double Line-to-Ground Fault ............................................ 135
6.3.5 Zero-Sequence Circuits ....................................................... 136
6.3.6 Numerical Example for a Single Line-to-Ground
Fault ....................................................................................... 138
6.4 Fault Currents for Transformers with Three Terminals per
Phase ................................................................................................... 139
6.4.1 Three-Phase Line-to-Ground Fault ................................... 142
6.4.2 Single-Phase Line-to-Ground Fault ................................... 143
6.4.3 Line-to-Line Fault ................................................................ 145
6.4.4 Double Line-to-Ground Fault ............................................ 147
Contents vii
6.4.5 Zero-Sequence Circuit.s. .................................................1..4.9.
6.4.6 Numerical Examples ........................................................... 149
6.5 Asymmetry Factor ............................................................................ 153
7. Phase-Shifting and Zig-Zag Transformers ........................................... 155
7.1 Introduction ....................................................................................... 155
7.2 Basic Principles .................................................................................. 157
7.3 Squashed Delta Phase-Shifting Transformer ................................ 159
7.3.1 Zero-Sequence Circuit Model ............................................ 163
7.4 Standard Delta Phase-Shifting Transformer ................................. 165
7.4.1 Zero-Sequence Circuit Model ............................................ 169
7.5 Two-Core Phase-Shifting Transformer .......................................... 170
7.5.1 Zero-Sequence Circuit Model ............................................ 174
7.6 Regulation Effects ............................................................................. 175
7.7 Fault Current Analysis ..................................................................... 177
7.7.1 Squashed Delta Fault Currents .......................................... 179
7.7.2 Standard Delta Fault Currents ........................................... 181
7.7.3 Two-Core Phase-Shifting Transformer Fault
Currents ................................................................................. 182
7.8 Zig-Zag Transformer .........................................................................184
7.8.1 Calculation of Electrical Characteristics ...........................185
7.8.2 Per-Unit Formulas ...............................................................189
7.8.3 Zero-Sequence Impedance .................................................190
7.8.4 Fault Current Analysis ........................................................192
8. Multiterminal Three-Phase Transformer Model ..................................195
8.1 Introduction .......................................................................................195
8.2 Theory .................................................................................................196
8.2.1 Two-Winding Leakage Inductance ....................................196
8.2.2 Multiwinding Transformers ...............................................197
8.2.3 Transformer Loading ...........................................................201
8.3 Transformers with Winding Connections within a Phase ..........202
8.3.1 Two Secondary Windings in Series ...................................202
8.3.2 Primary Winding in Series with a Secondary
Winding .................................................................................203
8.3.3 Autotransformer ..................................................................204
8.4 Multiphase Transformers .................................................................205
8.4.1 Delta Connection..................................................................207
8.4.2 Zig-Zag Connection .............................................................209
8.5 Generalizing the Model .................................................................... 211
8.6 Regulation and Terminal Impedances ...........................................213
8.7 Multiterminal Transformer Model for Balanced and
Unbalanced Load Conditions ..........................................................215
8.7.1 Theory ....................................................................................216
8.7.2 Admittance Representation ................................................218
viii Contents
8.7.2.1 Delta Winding Connection .................................220
8.7.3 Impedance Representation .................................................222
8.7.3.1 Ungrounded Y Connection .................................223
8.7.3.2 Series-Connected Windings from the
Same Phase ............................................................225
8.7.3.3 Zig-Zag Winding Connection .............................227
8.7.3.4 Autoconnection ....................................................228
8.7.3.5 Three Windings Joined .......................................228
8.7.4 Terminal Loading ................................................................229
8.7.5 Solution Process ...................................................................230
8.7.5.1 Terminal Currents and Voltages ........................230
8.7.5.2 Winding Currents and Voltages .........................230
8.7.6 Unbalanced Loading Examples ......................................... 231
8.7.6.1 Autotransformer with Buried Delta Tertiary
and Fault on Low-Voltage Terminal ..................231
8.7.6.2 Power Transformer with Fault on Delta
Tertiary...................................................................232
8.7.6.3 Power Transformer with Fault on
Ungrounded Y Secondary ..................................233
8.7.7 Balanced Loading Example ................................................234
8.7.7.1 Standard Delta Phase-Shifting Transformer ....234
8.7.8 Discussion .............................................................................236
9. Rabins’ Method for Calculating Leakage Fields, Leakage
Inductances, and Forces in Transformers .............................................. 239
9.1 Introduction ....................................................................................... 239
9.2 Theory ................................................................................................. 240
9.3 Rabins’ Formula for Leakage Reactance ........................................253
9.3.1 Rabins’ Method Applied to Calculate the Leakage
Reactance between Two Windings That Occupy
Different Radial Positions ...................................................253
9.3.2 Rabins’ Method Applied to Calculate the Leakage
Reactance between Two Axially Stacked Windings ....... 257
9.3.3 Rabins’ Method Applied to Calculate the Leakage
Reactance for a Collection of Windings............................ 259
9.4 Application of Rabins’ Method to Calculate the
S elf-Inductance of and Mutual Inductance between Coil
Sections ............................................................................................... 260
9.5 Determining the B-Field .................................................................. 263
9.6 Determination of Winding Forces ..................................................264
9.7 Numerical Considerations ............................................................... 267
10. Mechanical Design ..................................................................................... 269
10.1 Introduction .......................................................................................269
10.2 Force Calculations .............................................................................271
Contents ix
10.3 Stress Analysis.. ................................................................................. 273
10.3.1 Compressive Stress in the Key Spacer..s.. ........................ 275
10.3.2 Axial Bending Stress per Stran.d.. .................................... 276
10.3.3 Tilting Strength.. .................................................................. 279
10.3.4 Stress in the Tie Bars.. ......................................................... 283
10.3.5 Stress in the Pressure Rin.g.. .............................................. 289
10.3.6 Hoop Stress. .......................................................................... 290
10.3.7 Radial Bending Stress.. ....................................................... 291
10.4 Radial Buckling Strength.. ............................................................... 298
10.4.1 Free Unsupported Buckling... ............................................ 299
10.4.2 Constrained Buckling.. ........................................................ 302
10.4.3 Experiment to Determine Buckling Streng.t..h.. .............. 303
10.5 Stress Distribution in a Composite Wire–Paper
Winding Section. ............................................................................... 307
10.6 Additional Mechanical Consideration..s.. ..................................... 312
11. Electric Field Calculation..s. ...................................................................... 315
11.1 Simple Geometries.. .......................................................................... 315
11.1.1 Planar Geometry.. ................................................................ 315
11.1.2 Cylindrical Geometry.. ........................................................ 318
11.1.3 Spherical Geometry.. ........................................................... 321
11.1.4 Cylinder–Plane Geometry.. ................................................ 322
11.2 Electric Field Calculations Using Conformal Mapping............... 327
11.2.1 Physical Basis.. ...................................................................... 327
11.2.2 Conformal Mapping.. .......................................................... 328
11.2.3 Schwarz–Christoffel Transformatio.n... ............................ 332
11.2.4 Conformal Map for the Electrostatic Field
Problem ................................................................................. 334
11.2.4.1 Electric Potential and Field Value..s. ................. 338
11.2.4.2 Calculations and Comparison with a Finite
Element Solution.. ................................................ 346
11.2.4.3 Estimating Enhancement Facto.r.s.. ................... 350
11.3 Finite Element Electric Field Calculatio.n..s.. ................................ 352
12. Capacitance Calculation.s. ........................................................................ 357
12.1 Introduction. ...................................................................................... 357
12.2 Distributive Capacitance along a Winding or Di.s..k.. ................ 358
12.3 Stein’s Disk Capacitance Formul..a. ............................................... 363
12.4 General Disk Capacitance Formu.l.a.. ............................................ 371
12.5 Coil Grounded at One End with Grounded Cylinders on
Either Side. ......................................................................................... 372
12.6 Static Ring on One Side of a Dis.k.. ................................................ 374
12.7 Terminal Disk without a Static Ring............................................... 375
12.8 Capacitance Matrix.. ......................................................................... 376
12.9 Two Static Rings.. .............................................................................. 378
x Contents
12.10 Static Ring between the First Two Disks. .......................................382
12.11 Winding Disk Capacitances with Wound-in Shields ................... 382
12.11.1 Analytic Formula .................................................................383
12.11.2 Circuit Model ........................................................................386
12.11.3 Experimental Methods........................................................ 391
12.11.4 Results ................................................................................... 392
12.12 Multistart Winding Capacitance..................................................... 396
13. Voltage Breakdown and High-Voltage Design..................................... 399
13.1 Introduction .......................................................................................399
13.2 Principles of Voltage Breakdown ....................................................400
13.2.1 Breakdown in Solid Insulation ..........................................405
13.2.2 Breakdown in Transformer Oil ..........................................407
13.3 Geometric Dependence of Transformer-Oil
Breakdown .........................................................................................410
13.3.1 Theory ....................................................................................412
13.3.2 Planar Geometry ..................................................................412
13.3.3 Cylindrical Geometry ..........................................................415
13.3.4 Spherical Geometry .............................................................416
13.3.5 Comparison with Experiment ............................................418
13.3.6 Generalization ......................................................................419
13.3.6.1 Breakdown for the Cylinder–Plane
Geometry ...............................................................421
13.3.6.2 Breakdown for the Disk–Disk-to-Ground
Plane Geometry ....................................................421
13.3.7 Discussion .............................................................................424
13.4 Insulation Coordination ...................................................................425
13.5 Continuum Model of Winding Used to Obtain the
Impulse–Voltage Distribution .........................................................429
13.5.1 Uniform Capacitance Model ..............................................430
13.5.2 Traveling Wave Theory .......................................................434
13.6 Lumped-Parameter Model for Transient Voltage
Distribution ........................................................................................435
13.6.1 Circuit Description ...............................................................435
13.6.2 Mutual and Self-Inductance Calculations ........................438
13.6.3 Capacitance Calculations ....................................................438
13.6.4 Impulse–Voltage Calculations and Experimental
Comparisons .........................................................................439
13.6.5 Sensitivity Studies ................................................................444
14. Losses ............................................................................................................447
14.1 Introduction .......................................................................................447
14.2 No-Load or Core Losses ...................................................................448
14.2.1 Building Factor ..................................................................... 452
14.2.2 Interlaminar Losses ............................................................. 452