[13]-transformer design and principles PDF

Preview

Summary

With Applications toCore-Form Power TransformersSecond EditionTransformerDesignPrinciplesWith Applications toCore-Form Power TransformersSecond EditionTransformerDesignPrinciples3PCFSU
.
%FM
7FDDIJP
t
#FSUSBOE
1PVMJO1JFSSF
5
'FHIBMJ
t
%JMJQLVNBS
.
4IBI3BKFOESB
"IVKBMATLAB® is a trademark of The M...

Description

Transformer Design Principles With Applications to Core-Form Power Transformers

Second Edition

Transformer Design Principles With Applications to Core-Form Power Transformers

Second Edition

MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book ’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 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 Version Date: 20140514 International Standard Book Number-13: 978-1-4398-0583-1 (eBook - PDF) 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 responsibilit y for the validit y of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright 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, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without 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. 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 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

Contents

vi

4.4 4.5

4.6

Ideal Two-, Three-, and Multiwinding Transformers .................... 80 4.4.1 Ideal Autotransformers ......................................................... 83 Leakage Reactance for Two-Winding Transformers Based on Circuit Parameters ......................................................................... 84 4.5.1 Leakage Reactance for a Two-Winding Autotransformer .................................................................... 88 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

6.5

vii

6.4.5 Zero-Sequence Circuits ....................................................... 149 6.4.6 Numerical Examples ........................................................... 149 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

Contents

viii

8.7.3

8.7.4 8.7.5

8.7.6

8.7.7 8.7.8

8.7.2.1 Delta Winding Connection ................................. 220 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 Terminal Loading ................................................................ 229 Solution Process ................................................................... 230 8.7.5.1 Terminal Currents and Voltages ........................ 230 8.7.5.2 Winding Currents and Voltages......................... 230 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 Balanced Loading Example ................................................ 234 8.7.7.1 Standard Delta Phase-Shifting Transformer.... 234 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 Self-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 Spacers ............................275 10.3.2 Axial Bending Stress per Strand ....................................... 276 10.3.3 Tilting Strength .................................................................... 279 10.3.4 Stress in the Tie Bars ........................................................... 283 10.3.5 Stress in the Pressure Ring ................................................. 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 Strength ................... 303 10.5 Stress Distribution in a Composite Wire–Paper Winding Section ................................................................................ 307 10.6 Additional Mechanical Considerations ......................................... 312 11. Electric Field Calculations......................................................................... 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 Transformation ................................ 332 11.2.4 Conformal Map for the Electrostatic Field Problem ................................................................................. 334 11.2.4.1 Electric Potential and Field Values ....................338 11.2.4.2 Calculations and Comparison with a Finite Element Solution .................................................. 346 11.2.4.3 Estimating Enhancement Factors....................... 350 11.3 Finite Element Electric Field Calculations..................................... 352 12. Capacitance Calculations .......................................................................... 357 12.1 Introduction ....................................................................................... 357 12.2 Distributive Capacitance along a Winding or Disk ..................... 358 12.3 Stein’s Disk Capacitance Formula .................................................. 363 12.4 General Disk Capacitance Formula ................................................ 371 12.5 Coil Grounded at One End with Grounded Cylinders on Either Side .......................................................................................... 372 12.6 Static Ring on One Side of a Disk ...............................