Table Of ContentTransformer Design Principles
Third Edition
Transformer Design Principles
Third Edition
Robert M. Del Vecchio, Bertrand Poulin,
Pierre T. Feghali, Dilipkumar M. Shah,
and Rajendra Ahuja
CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2018 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
International Standard Book Number-13: 978-1-4987-8753-6 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been
made to publish reliable data and information, but the author and publisher cannot assume responsibility for the
validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copy-
right holders of all material reproduced in this publication and apologize to copyright holders if permission to publish
in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know
so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or
utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, i ncluding pho-
tocopying, microfilming, and recording, or in any information storage or retrieval system, w ithout written permission
from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://
www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA
01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users.
For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been
arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for
identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Names: Del Vecchio, Robert M., author.
Title: Transformer design principles / Robert M. Del Vecchio, Bertrand
Poulin, Pierre T. Feghali, Dilipkumar M. Shah, and Rajendra Ahuja.
Description: Third edition. | Boca Raton : Taylor & Francis, CRC Press, 2018.
| Revised edition of: Transformer design principles / [authors], Robert M.
Del Vecchio ... [et al.]. 2010. | Includes bibliographical references and
index.
Identifiers: LCCN 2017011211| ISBN 9781498787536 (hardback : alk. paper) |
ISBN 9781315155920 (ebook)
Subjects: LCSH: Electric transformers--Design and construction.
Classification: LCC TK2551 .T765 2018 | DDC 621.31/4--dc23
LC record available at https://lccn.loc.gov/2017011211
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface ...........................................................................................................................................xiii
Authors ...........................................................................................................................................xv
1. Introduction .............................................................................................................................1
1.1 Historical Background ..................................................................................................1
1.2 Uses in Power Systems .................................................................................................2
1.3 Core-Form and Shell-Form Transformers .................................................................7
1.4 Stacked and Wound Core Construction ....................................................................8
1.5 Transformer Cooling ...................................................................................................10
1.6 Winding Types .............................................................................................................11
1.7 Insulation Structures ...................................................................................................13
1.8 Structural Elements .....................................................................................................16
1.9 Modern Trends ............................................................................................................19
2. Magnetism and Related Core Issues .................................................................................21
2.1 Introduction .................................................................................................................21
2.2 Basic Magnetism..........................................................................................................22
2.3 Hysteresis .....................................................................................................................25
2.4 Magnetic Circuits ........................................................................................................27
2.5 Inrush Current .............................................................................................................32
2.6 Fault Current Waveform and Peak Amplitude ......................................................34
2.7 Optimal Core Stacking ...............................................................................................39
3. Circuit Model of a 2-Winding Transformer with Core ..................................................43
3.1 Introduction .................................................................................................................43
3.2 Circuit Model of the Core ..........................................................................................43
3.3 2-Winding Transformer Circuit Model with Core ..................................................46
3.4 Approximate 2-Winding Transformer Circuit Model without Core ...................50
3.5 Vector Diagram of a Loaded Transformer with Core ............................................53
3.6 Per-Unit System ...........................................................................................................54
3.7 Voltage Regulation ......................................................................................................56
4. Reactance and Leakage Reactance Calculations .............................................................59
4.1 Introduction .................................................................................................................59
4.2 General Method for Determining Inductances and Mutual Inductances ..........60
4.2.1 Energy by Magnetic Field Methods ............................................................61
4.2.2 Energy from Electric Circuit Methods ........................................................63
4.3 2-Winding Leakage Reactance Formula ..................................................................65
4.4 Ideal 2-, 3-, and Multi-Winding Transformers ........................................................69
4.4.1 Ideal Autotransformer ...................................................................................72
4.5 Leakage Reactance for 2-Winding Transformers Based
on Circuit Parameters .................................................................................................73
4.5.1 Leakage Reactance for a 2-Winding Autotransformer .............................76
4.6 Leakage Reactances for 3-Winding Transformers ..................................................77
4.6.1 Leakage Reactance for an Autotransformer
with a Tertiary Winding ...............................................................................81
v
vi Contents
4.6.2 Leakage Reactance between 2 Windings Connected in Series
and a Third Winding .....................................................................................85
4.6.3 Leakage Reactance of a 2-Winding Autotransformer with X-Line
Taps ..................................................................................................................86
5. Phasors, 3-Phase Connections, and Symmetrical Components ...................................89
5.1 Phasors ..........................................................................................................................89
5.2 Y and Delta 3-Phase Connections .............................................................................92
5.3 Zig-Zag Connection ....................................................................................................97
5.4 Scott Connection..........................................................................................................98
5.5 Symmetrical Components........................................................................................101
6. Fault Current Analysis .......................................................................................................107
6.1 Introduction ...............................................................................................................107
6.2 Fault Current Analysis on 3-Phase Systems ..........................................................108
6.2.1 3-Phase Line-to-Ground Fault ....................................................................110
6.2.2 Single-Phase Line-to-Ground Fault ...........................................................111
6.2.3 Line-to-Line Fault ........................................................................................112
6.2.4 Double Line-to-Ground Fault ....................................................................112
6.3 Fault Currents for Transformers with Two Terminals per Phase .......................113
6.3.1 3-Phase Line-to-Ground Fault ....................................................................116
6.3.2 Single-Phase Line-to-Ground Fault ...........................................................116
6.3.3 Line-to-Line Fault ........................................................................................117
6.3.4 Double Line-to-Ground Fault ....................................................................118
6.3.5 Zero-Sequence Circuits ...............................................................................119
6.3.6 Numerical Example for a Single Line-to-Ground Fault .........................120
6.4 Fault Currents for Transformers with Three Terminals per Phase ....................120
6.4.1 3-Phase Line-to-Ground Fault ....................................................................123
6.4.2 Single-Phase Line-to-Ground Fault ...........................................................124
6.4.3 Line-to-Line Fault ........................................................................................126
6.4.4 Double Line-to-Ground Fault ....................................................................128
6.4.5 Zero-Sequence Circuits ...............................................................................130
6.4.6 Numerical Example .....................................................................................131
6.5 Asymmetry Factor ....................................................................................................134
7. Phase-Shifting and Zigzag Transformers .......................................................................135
7.1 Introduction ...............................................................................................................135
7.2 Basic Principles ..........................................................................................................136
7.3 Squashed Delta-Phase-Shifting Transformer ........................................................139
7.3.1 Zero Sequence Circuit Model .....................................................................142
7.4 Standard Delta-Phase-Shifting Transformer .........................................................144
7.4.1 Zero Sequence Circuit Model .....................................................................147
7.5 2-Core Phase-Shifting Transformer ........................................................................148
7.5.1 Zero Sequence Circuit Model .....................................................................152
7.6 Regulation Effects .....................................................................................................153
7.7 Fault Current Analysis .............................................................................................154
7.7.1 Squashed Delta Fault Currents ..................................................................156
7.7.2 Standard Delta Fault Currents ...................................................................157
7.7.3 2-Core Phase-Shifting Transformer Fault Currents ................................159
Contents vii
7.8 Zigzag Transformer ..................................................................................................160
7.8.1 Calculation of Electrical Characteristics ...................................................161
7.8.2 Per-Unit Formulas .......................................................................................164
7.8.3 Zero Sequence Impedance ..........................................................................166
7.8.4 Fault Current Analysis ................................................................................167
8. Multiterminal 3-Phase Transformer Model ...................................................................169
8.1 Introduction ...............................................................................................................169
8.2 Theory .........................................................................................................................170
8.2.1 Two-Winding Leakage Inductance ............................................................170
8.2.2 Multi-Winding Transformer .......................................................................171
8.2.3 Transformer Loading ...................................................................................174
8.3 Transformers with Winding Connections within a Phase ..................................174
8.3.1 Two Secondary Windings in Series ...........................................................174
8.3.2 Primary Winding in Series with a Secondary Winding .........................175
8.3.3 Autotransformer ..........................................................................................176
8.4 Multiphase Transformers .........................................................................................178
8.4.1 Delta Connection..........................................................................................180
8.4.2 Zigzag Connection .......................................................................................181
8.5 Generalizing the Model ............................................................................................183
8.6 Regulation and Terminal Impedances ...................................................................185
8.7 Multiterminal Transformer Model for Balanced and Unbalanced
Load Conditions ........................................................................................................187
8.7.1 Theory ............................................................................................................188
8.7.2 Admittance Representation ........................................................................190
8.7.2.1 Delta Winding Connection .........................................................191
8.7.3 Impedance Representation .........................................................................193
8.7.3.1 Ungrounded Y Connection .........................................................194
8.7.3.2 Series-Connected Windings from the Same Phase ..................196
8.7.3.3 Zigzag Connection .......................................................................197
8.7.3.4 Autoconnection ............................................................................198
8.7.3.5 Three Windings Joined ................................................................199
8.7.4 Terminal Loading .........................................................................................199
8.7.5 Solution Process ...........................................................................................200
8.7.5.1 Terminal Currents and Voltages .................................................200
8.7.5.2 Winding Currents and Voltages .................................................201
8.7.6 Unbalanced Loading Examples .................................................................201
8.7.6.1 Autotransformer with Buried Delta Tertiary and
Fault on LV Terminal ...................................................................201
8.7.6.2 Power Transformer with Fault on Delta Tertiary ....................202
8.7.6.3 Power Transformer with Fault on Ungrounded Y
Secondary ......................................................................................203
8.7.7 Balanced Loading Example ........................................................................204
8.7.7.1 Standard Delta Phase Shifting Transformer .............................204
8.7.8 Discussion .....................................................................................................205
8.8 2-Core Analysis..........................................................................................................206
8.8.1 2-Core Parallel Connection .........................................................................207
8.8.2 2-Core Series Connection ............................................................................208
8.8.3 Terminal Loading .........................................................................................209
viii Contents
8.8.4 Example of a 2-Core Phase Shifting Transformer ...................................209
8.8.4.1 Normal Loading ...........................................................................210
8.8.4.2 Single Line-to-Ground Fault .......................................................211
8.8.5 Discussion .....................................................................................................212
9. Rabins’ Method for Calculating Leakage Fields, Inductances, and Forces
in Iron Core Transformers, Including Air Core Methods ...........................................213
9.1 Introduction ...............................................................................................................213
9.2 Theory .........................................................................................................................214
9.3 Rabins’ Formula for Leakage Reactance ................................................................226
9.3.1 Rabins’ Method Applied to Calculate the Leakage Reactance
between Two Windings Which Occupy Different Radial Positions .....226
9.3.2 Rabins’ Method Applied to Calculate the Leakage Reactance
between Two Axially Stacked Windings ..................................................229
9.3.3 Rabins’ Method Applied to Calculate the Leakage Reactance
for a Collection of Windings .......................................................................231
9.4 Rabins’ Method Applied to Calculate the Self-Inductance of and Mutual
Inductance between Coil Sections ..........................................................................232
9.5 Determining the B-field ............................................................................................234
9.6 Determining the Winding Forces ............................................................................236
9.7 Numerical Considerations .......................................................................................238
9.8 Air Core Inductance ..................................................................................................238
10. Mechanical Design .............................................................................................................243
10.1 Introduction ...............................................................................................................243
10.2 Force Calculations .....................................................................................................245
10.3 Stress Analysis ...........................................................................................................246
10.3.1 Compressive Stress in the Key Spacers ....................................................248
10.3.2 Axial Bending Stress per Strand ................................................................249
10.3.3 Tilting Strength .............................................................................................252
10.3.4 Stress in the Tie Bars ....................................................................................255
10.3.5 Stress in the Pressure Ring ..........................................................................259
10.3.6 Hoop Stress ...................................................................................................260
10.3.7 Radial Bending Stress ..................................................................................261
10.4 Radial Buckling Strength .........................................................................................267
10.4.1 Free Unsupported Buckling .......................................................................268
10.4.2 Constrained Buckling ..................................................................................270
10.4.3 Experiment to Determine Buckling Strength ...........................................272
10.5 Stress Distribution in a Composite Wire–Paper Winding Section .....................276
10.6 Additional Mechanical Considerations .................................................................279
11. Electric Field Calculations .................................................................................................283
11.1 Simple Geometries ....................................................................................................283
11.1.1 Planar Geometry ..........................................................................................283
11.1.2 Cylindrical Geometry ..................................................................................286
11.1.3 Spherical Geometry .....................................................................................288
11.1.4 Cylinder–Plane Geometry ..........................................................................289
11.2 Electric Field Calculations Using Conformal Mapping.......................................295
11.2.1 Mathematical Basis ......................................................................................295
Contents ix
11.2.2 Conformal Mapping ....................................................................................296
11.2.3 Schwarz–Christoffel Transformation ........................................................299
11.2.4 Conformal Map for the Electrostatic Field Problem ...............................300
11.2.4.1 Electric Potential and Field Values ............................................305
11.2.4.2 Calculations and Comparison with a Finite Element
Solution ..........................................................................................313
11.2.4.3 Estimating Enhancement Factors ...............................................314
11.3 Finite Element Electric Field Calculations .............................................................318
12. Capacitance Calculations ..................................................................................................325
12.1 Introduction ...............................................................................................................325
12.2 Distributive Capacitance along a Winding or Disk .............................................325
12.3 Stein’s Disk Capacitance Formula ..........................................................................331
12.3.1 Determining Practical Values for the Series and Shunt
Capacitances, C and C .............................................................................334
s dd
12.4 General Disk Capacitance Formula ........................................................................338
12.5 Coil Grounded at One End with Grounded Cylinders on Either Side .............339
12.6 Static Ring on One Side of a Disk ...........................................................................341
12.7 Terminal Disk without a Static Ring.......................................................................342
12.8 Capacitance Matrix ...................................................................................................343
12.9 Two End Static Rings ................................................................................................345
12.10 Static Ring between the First Two Disks ................................................................348
12.11 Winding Disk Capacitances with Wound-in-Shields ...........................................349
12.11.1 A nalytic Formula .........................................................................................349
12.11.2 C ircuit Model ................................................................................................352
12.11.3 E xperimental Methods ................................................................................357
12.11.4 R esults ...........................................................................................................358
12.12 Multi-Start Winding Capacitance ...........................................................................361
13. Voltage Breakdown Theory and Practice .......................................................................363
13.1 Introduction ...............................................................................................................363
13.2 Principles of Voltage Breakdown ............................................................................364
13.2.1 Breakdown in Solid Insulation ..................................................................368
13.2.2 Breakdown in Transformer Oil ..................................................................369
13.3 Geometric Dependence of Transformer Oil Breakdown .....................................372
13.3.1 Theory ............................................................................................................373
13.3.2 Planar Geometry ..........................................................................................374
13.3.3 Cylindrical Geometry ..................................................................................376
13.3.4 Spherical Geometry .....................................................................................378
13.3.5 Comparison with Experiment ....................................................................379
13.3.6 Generalization ..............................................................................................380
13.3.6.1 Breakdown for the Cylinder-Plane Geometry .........................381
13.3.6.2 Breakdown for the Disk–Disk-to-Ground Plane Geometry ...382
13.3.7 Discussion .....................................................................................................385
13.4 Insulation Coordination ...........................................................................................386
13.5 Continuum Model of Winding Used to Obtain the Impulse Voltage
Distribution ................................................................................................................389
13.5.1 Uniform Capacitance Model ......................................................................389
13.5.2 Traveling Wave Theory ...............................................................................392
