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

595 Pages·2010·14.93 MB·English

by  Dilipkumar M. Shah, Rajendra Ahuja

TRANSFORMER
DESIGN PRINCIPLES
© 2002 by CRC Press
TRANSFORMER
DESIGN PRINCIPLES
With Applications to
Core-Form Power Transformers
Robert M.Del Vecchio
Bertrand Poulin
Pierre T.Feghali
Dilipkumar M.Shah
Rajendra Ahuja
CRC PRESS
Boca Raton London New York Washington, D.C.
© 2002 by CRC Press
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© 2002 by CRC Press
CONTENTS
Preface xi
1 Introduction to Transformers 1
1.1 Historical Background 1
1.2 Uses in Power Systems 3
1.3 Core-Form and Shell-Form Transformers 9
1.4 Stacked and Wound Core Construction 10
1.5 Transformer Cooling 15
1.6 Winding Types 17
1.7 Insulation Structures 20
1.8 Structural Elements 24
1.9 Three-Phase Connections 29
1.10 Modern Trends 36
2 Transformer Circuit Models, Including Magnetic Core
Characteristics and Applications 39
2.1 Introduction 39
2.2 Basic Magnetism 42
2.3 Hysteresis 46
2.4 Magnetic Circuits 49
2.5 Inrush Current 54
2.6 Transformer Circuit Model with Core 66
2.7 Per Unit System 80
2.8 Voltage Regulation 82
3 Reactance Calculations 87
3.1 Introduction 87
3.2 Ideal Transformers 88
3.2.1 Ideal Autotransformer 92
3.3 Leakage Impedance for 2-Winding Transformers 94
3.3.1 Leakage Impedance for a 2-Winding
Autotransformer 98
3.4 Leakage Impedances for 3-Winding Transformers 99
3.4.1 Leakage Impedances for an
Autotransformer with Tertiary 104
© 2002 by CRC Press
vi CONTENTS
3.4.2 Leakage Impedance between 2 Windings
Connected in Series and a Third Winding 109
3.4.3 Leakage Impedance of a 2-Winding
Autotransformer with X-Line Taps 110
3.4.4 More General Leakage Impedance Calculations 113
3.5 Two Winding Leakage Reactance Formula 114
4 Fault Current Calculations 119
4.1 Introduction 119
4.2 Symmetrical Components 121
4.3 Fault Analysis on 3-Phase Systems 127
4.3.1 3-Phase Line to Ground Fault 129
4.3.2 Single Phase Line to Ground Fault 130
4.3.3 Line to Line Fault 131
4.3.4 Double Line to Ground Fault 132
4.4 Fault Currents for Transformers with 2 Terminals
per Phase 133
4.4.1 3-Phase Line to Ground Fault 135
4.4.2 Single Phase Line to Ground Fault 136
4.4.3 Line to Line Fault 137
4.4.4 Double Line to Ground Fault 137
4.4.5 Zero Sequence Impedences 138
4.5 Fault Currents for Transformers with 3 Terminals
per Phase 140
4.5.1 3-Phase Line to Ground Fault 143
4.5.2 Single Phase Line to Ground Fault 143
4.5.3 Line to Line Fault 144
4.5.4 Double Line to Ground Fault 145
4.5.5 Zero Sequence Impedances 146
4.6 Asymmetry Factor 147
5 Rabins’ Method for Calculating Leakage Fields, Forces and
Inductances in Transformers 149
5.1 Introduction 149
5.2 Theory 150
5.3 Determining the B-Field 166
5.4 Determing the Winding Forces 167
5.5 General Method for Determing Inductances and Mutual
Inductances 169
5.6 Rabins’ Formula for Leakage Reactance 175
5.7 Rabins’ Method Applied to Calculate Self and Mutual
Inductances of Coil Sections 182
© 2002 by CRC Press
CONTENTS vii
6 Mechanical Design 185
6.1 Introduction 185
6.2 Force Calculations 188
6.3 Stress Analysis 190
6.3.1 Compressive Stress in the Key Spacers 193
6.3.2 Axial Bending Stress per Strand 193
6.3.3 Tilting Strength 197
6.3.4 Stress in Tie Bars 201
6.3.5 Stress in the Pressure Rings 208
6.3.6 Hoop Stress 209
6.3.7 Radial Bending Stress 211
6.4 Radial Buckling Strength 219
6.5 Stress Distribution in a Composite Wire-Paper
Winding Section 229
6.6 Additional Mechanical Considerations 235
7 Capacitance Calculations 237
7.1 Introduction 237
7.2 Theory 238
7.3 Stein’s Capacitance Formula 245
7.4 General Disk Capacitance Formula 252
7.5 Coil Grounded at One End with Grounded
Cylinders on Either Side 253
7.6 Static Ring on One Side of Disk 256
7.7 Terminal Disk without a Static Ring 257
7.8 Capacitance Matrix 258
7.9 Two End Static Rings 261
7.10 Static Ring between the First Two Disks 265
7.11 Winding Disk Capacitances with Wound-in Shields 266
7.11.1 Analytic Formula 266
7.11.2 Circuit Model 270
7.11.3 Experimental Methods 276
7.11.4 Results 277
7.12 Multi-Start Winding Capacitance 281
8 High Voltage Insulation Design 285
8.1 Introduction 285
8.2 Principles of Voltage Breakdown 286
8.3 Insulation Coordination 298
8.4 Continuum Model of Winding Used to Obtain
the Impulse Voltage Distribution 303
© 2002 by CRC Press
viii CONTENTS
8.5 Lumped Parameter Model for Transient
Voltage Distribution 313
8.5.1 Circuit Description 313
8.5.2 Mutual and Self Inductance Calculations 317
8.5.3 Capacitance Calculations 319
8.5.4 Impulse Voltage Calculations and
Experimental Comparisons 320
8.5.5 Sensitivity Studies 326
9 Electric Field Calculations 329
9.1 Simple Geometries 329
9.2 Electric Field Calculations Using Conformal Mapping 337
9.2.1 Physical Basis 337
9.2.2 Conformal Mapping 338
9.2.3 Schwarz-Christoffel Transformation 342
9.2.4 Conformal Map for the Electrostatic
Field Problem 344
9.2.4.1 Electric Potential and Field Values 349
9.2.4.2 Calculations and Comparison with
a Finite Element Solution 356
9.2.4.3 Estimating Enhancement Factors 360
9.3 ‘Finite Element Electric Field Calculations 363
10 Losses 369
10.1 Introduction 369
10.2 No-Load or Core Losses 370
10.2.1 Building Factor 375
10.2.2 Interlaminar Losses 375
10.3 Load Losses 379
2
10.3.1 I R Losses 379
10.3.2 Stray Losses 380
10.3.2.1 Eddy Current Losses in the Coils 383
10.3.2.2 Tieplate Losses 387
10.3.2.3 Tieplate and Core Losses Due to
Unbalanced Currents 397
10.3.2.4 Tank and Clamp Losses 404
10.3.2.5 Tank Losses Due to Nearby Busbars 407
10.3.2.6 Tank Losses Associated with
the Bushings 412
10.3.3 Winding Losses Due to Missing
or Unbalanced Crossovers 418
© 2002 by CRC Press
CONTENTS ix
11 Thermal Model of a Core Form Power Transformer
and Related Thermal Calculations 429
11.1 Introduction 429
11.2 Thermal Model of a Disk Coil with Directed
Oil Flow 431
11.2.1 Oil Pressures and Velocities 433
11.2.2 Oil Nodal Temperatures and Path
Temperature Rises 438
11.2.3 Disk Temperatures 440
11.3 Thermal Model for Coils without Directed Oil Flow 441
11.4 Radiator Thermal Model 444
11.5 Tank Cooling 448
11.6 Oil Mixing in the Tank 450
11.7 Time Dependence 453
11.8 Pumped Flow 454
11.9 Comparison with Test Results 455
11.10 Determining M and N Exponents 460
11.11 Loss of Life Calculation 462
11.12 Cable and Lead Temperature Calculation 466
11.13 Tank Wall Temperature Calculation 473
11.14 Tieplate Temperature Calculation 475
11.15 Core Steel Temperature Calculation 478
12 Load Tap Changers 481
12.1 Introduction 481
12.2 General Description of LTC 482
12.3 Types of Regulation 483
12.4 Principle of Operation 484
12.4.1 Resistive Switching 484
12.4.2 Reactive Switching with Preventative
Autotransformer 486
12.5 Connection Schemes 488
12.5.1 Full Transformers 488
12.5.2 Autotransformers 490
12.5.3 Use of Auxiliary Transformer 493
12.5.4 Phase Shifting Transformers 495
12.6 General Maintenance 495
13 Phase Shifting Transformers 499
13.1 Introduction 499
13.2 Basic Principles 503
© 2002 by CRC Press
x CONTENTS
13.3 Squashed Delta Phase Shifting Transformers 507
13.4 Standard Delta Phase Shifting Transformers 514
13.5 Two Core Phase Shifting Transformer 519
13.6 Regulation Effects 526
13.7 Fault Current Analysis 528
13.7.1 Squashed Delta Fault Currents 532
13.7.2 Standard Delta Fault Currents 535
14 Cost Minimization 543
14.1 Introduction 543
14.2 Geometric Programming 545
14.3 Non-Linear Constrained Optimization 552
14.3.1 Characterization of the Minimum 552
14.3.2 Solution Search Strategy 561
14.3.3 Practical Considerations 567
14.4 Application to Transformer Design 568
14.4.1 Design Variables 569
14.4.2 Cost Function 570
14.4.3 Equality Constraints 573
14.4.4 Inequality Constraints 577
14.4.5 Optimization Strategy 578
References 583
© 2002 by CRC Press
PREFACE
Many of the standard texts on power transformers are now over ten
years old and some much older. Much has changed in transformer design
since these books were written. Newer and better materials are now
available for core and winding construction. Powerful computers now
make it possible to produce more detailed models of the electrical,
mechanical and thermal behavior of transformers than previously
possible. Although many of these modern approaches to design and
construction are found scattered in the literature, there is a need to have
this information available in a single source as a reference for the designer
or power engineer and as a starting point for the student or novice.
It is hoped that the present work can serve both purposes. As a text for
beginners, we emphasize the physical basis of transformer operation. We
also discuss the physical effects which result from various fault conditions
and their implications for design. Physical principles and mathematical
techniques are presented in a reasonably self-contained manner, although
references are provided to additional material. For the specialist such as
a power or transformer design engineer, detailed models are presented
which focus on various aspects of a transformer under normal or
abnormal conditions. Cost minimization techniques, which form the
starting point for most designs, are also presented.
Although this book primarily deals with power transformers, many
of the physical principles discussed or mathematical modeling
techniques presented apply equally well to other types of transformers.
The presentation is kept as general as possible so that designers or users
of other transformer types will have little difficulty applying many of
the results to their own designs. The emphasis on fundamentals should
make this process easier and should also foster the development of new
and more powerful design tools in the future.
The International System of Units (SI) is used throughout the text.
However, an occasional figure, graph, or table may show quantities in
the British system of units. Sometimes a quantity is given in British units
in parentheses after its metric value.
References are referred to generally by the first three letters of the first
author’s name followed by the last two digits of the publication date, e.g.
[Abc98]. In cases where this format cannot be followed, an appropriate
substitute is made. They are listed alphabetically at the end of the book.
We wish to thank Harral Robin for guidance throughout the course of
this work. We would also like to acknowledge many helpful suggestions
from power industry representatives and consultants over the years.
© 2002 by CRC Press