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TRANSFORMER

DESIGN PRINCIPLES

2002 by CRC Press

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TRANSFORMERDESIGN PRINCIPLES

With Applications toCore-Form Power Transformers

Robert M.Del VecchioBertrand PoulinPierre T.Feghali

Dilipkumar M.ShahRajendra Ahuja

Boca Raton London New York Washington, D.C.

CRC PRESS

2002 by CRC Press

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This book contains information obtained from authentic and highly regarded sources.Reprinted material is quoted with permission, and sources are indicated. A widevariety of references are listed. Reasonable efforts have been made to publish reliabledata and information, but the authors and the publisher cannot assume responsibilityfor the validity of all materials or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or byany means, electronic or mechanical, including photocopying, microfilming, andrecording, or by any information storage or retrieval system, without prior permissionin writing from the publisher.

The consent of CRC Press does not extend to copying for general distribution, forpromotion, for creating new works, or for resale. Specific permission must be obtainedin writing from CRC Press for such copying.

Direct all inquiries to CRC Press, 2000 N.W. Corporate Blvd., Boca Raton, Florida33431.

Trademark Notice: Product or corporate names may be trademarks or registeredtrademarks, and are used only for identification and explanation, without intent toinfringe.

Visit the CRC Press Web site at www.crcpress.com

2002 by CRC Press

No claim to original U.S. Government worksInternational Standard Book Number 90-5699-703-3

Printed in the United States of America 1 2 3 4 5 6 7 8 9 0Printed on acid-free paper

2002 by CRC Press

http://www.crcpress.com/http://www.crcpress.com/

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CONTENTS

Preface xi

1 Introduction to Transformers 11.1 Historical Background 11.2 Uses in Power Systems 31.3 Core-Form and Shell-Form Transformers 91.4 Stacked and Wound Core Construction 101.5 Transformer Cooling 151.6 Winding Types 171.7 Insulation Structures 201.8 Structural Elements 241.9 Three-Phase Connections 29

1.10 Modern Trends 36

2 Transformer Circuit Models, Including Magnetic CoreCharacteristics and Applications 392.1 Introduction 392.2 Basic Magnetism 422.3 Hysteresis 462.4 Magnetic Circuits 49

2.5 Inrush Current 542.6 Transformer Circuit Model with Core 662.7 Per Unit System 802.8 Voltage Regulation 82

3 Reactance Calculations 873.1 Introduction 873.2 Ideal Transformers 88

3.2.1 Ideal Autotransformer 923.3 Leakage Impedance for 2-Winding Transformers 94

3.3.1 Leakage Impedance for a 2-WindingAutotransformer 98

3.4 Leakage Impedances for 3-Winding Transformers 993.4.1 Leakage Impedances for an

Autotransformer with Tertiary 104

2002 by CRC Press

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CONTENTSvi

3.4.2 Leakage Impedance between 2 WindingsConnected in Series and a Third Winding 109

3.4.3 Leakage Impedance of a 2-WindingAutotransformer with X-Line Taps 110

3.4.4 More General Leakage Impedance Calculations 1133.5 Two Winding Leakage Reactance Formula 114

4 Fault Current Calculations 1194.1 Introduction 1194.2 Symmetrical Components 121

4.3 Fault Analysis on 3-Phase Systems 1274.3.1 3-Phase Line to Ground Fault 1294.3.2 Single Phase Line to Ground Fault 1304.3.3 Line to Line Fault 1314.3.4 Double Line to Ground Fault 132

4.4 Fault Currents for Transformers with 2 Terminalsper Phase 1334.4.1 3-Phase Line to Ground Fault 135

4.4.2 Single Phase Line to Ground Fault 1364.4.3 Line to Line Fault 1374.4.4 Double Line to Ground Fault 1374.4.5 Zero Sequence Impedences 138

4.5 Fault Currents for Transformers with 3 Terminalsper Phase 1404.5.1 3-Phase Line to Ground Fault 1434.5.2 Single Phase Line to Ground Fault 143

4.5.3 Line to Line Fault 1444.5.4 Double Line to Ground Fault 1454.5.5 Zero Sequence Impedances 146

4.6 Asymmetry Factor 147

5 Rabins Method for Calculating Leakage Fields, Forces andInductances in Transformers 1495.1 Introduction 149

5.2 Theory 1505.3 Determining the B-Field 1665.4 Determing the Winding Forces 1675.5 General Method for Determing Inductances and Mutual

Inductances 1695.6 Rabins Formula for Leakage Reactance 1755.7 Rabins Method Applied to Calculate Self and Mutual

Inductances of Coil Sections 182

2002 by CRC Press

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CONTENTS vii

6 Mechanical Design 1856.1 Introduction 1856.2 Force Calculations 1886.3 Stress Analysis 190

6.3.1 Compressive Stress in the Key Spacers 1936.3.2 Axial Bending Stress per Strand 1936.3.3 Tilting Strength 1976.3.4 Stress in Tie Bars 2016.3.5 Stress in the Pressure Rings 2086.3.6 Hoop Stress 209

6.3.7 Radial Bending Stress 2116.4 Radial Buckling Strength 2196.5 Stress Distribution in a Composite Wire-Paper

Winding Section 2296.6 Additional Mechanical Considerations 235

7 Capacitance Calculations 2377.1 Introduction 237

7.2 Theory 2387.3 Steins Capacitance Formula 2457.4 General Disk Capacitance Formula 2527.5 Coil Grounded at One End with Grounded

Cylinders on Either Side 2537.6 Static Ring on One Side of Disk 2567.7 Terminal Disk without a Static Ring 257

7.8 Capacitance Matrix 2587.9 Two End Static Rings 2617.10 Static Ring between the First Two Disks 2657.11 Winding Disk Capacitances with Wound-in Shields 266

7.11.1 Analytic Formula 2667.11.2 Circuit Model 2707.11.3 Experimental Methods 2767.11.4 Results 277

7.12 Multi-Start Winding Capacitance 281

8 High Voltage Insulation Design 2858.1 Introduction 2858.2 Principles of Voltage Breakdown 2868.3 Insulation Coordination 2988.4 Continuum Model of Winding Used to Obtain

the Impulse Voltage Distribution 303

2002 by CRC Press

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CONTENTSviii

8.5 Lumped Parameter Model for TransientVoltage Distribution 3138.5.1 Circuit Description 3138.5.2 Mutual and Self Inductance Calculations 3178.5.3 Capacitance Calculations 3198.5.4 Impulse Voltage Calculations and

Experimental Comparisons 3208.5.5 Sensitivity Studies 326

9 Electric Field Calculations 329

9.1 Simple Geometries 3299.2 Electric Field Calculations Using Conformal Mapping 337

9.2.1 Physical Basis 3379.2.2 Conformal Mapping 3389.2.3 Schwarz-Christoffel Transformation 3429.2.4 Conformal Map for the Electrostatic

Field Problem 3449.2.4.1 Electric Potential and Field Values 349

9.2.4.2 Calculations and Comparison witha Finite Element Solution 356

9.2.4.3 Estimating Enhancement Factors 3609.3 Finite Element Electric Field Calculations 363

10 Losses 36910.1 Introduction 369

10.2 No-Load or Core Losses 37010.2.1 Building Factor 37510.2.2 Interlaminar Losses 375

10.3 Load Losses 37910.3.1 I2R Losses 37910.3.2 Stray Losses 380

10.3.2.1 Eddy Current Losses in the Coils 38310.3.2.2 Tieplate Losses 387

10.3.2.3 Tieplate and Core Losses Due toUnbalanced Currents 397

10.3.2.4 Tank and Clamp Losses 40410.3.2.5 Tank Losses Due to Nearby Busbars 40710.3.2.6 Tank Losses Associated with

the Bushings 41210.3.3 Winding Losses Due to Missing

or Unbalanced Crossovers 418

2002 by CRC Press

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CONTENTS ix

11 Thermal Model of a Core Form Power Transformerand Related Thermal Calculations 42911.1 Introduction 42911.2 Thermal Model of a Disk Coil with Directed

Oil Flow 43111.2.1 Oil Pressures and Velocities 43311.2.2 Oil Nodal Temperatures and Path

Temperature Rises 43811.2.3 Disk Temperatures 440

11.3 Thermal Model for Coils without Directed Oil Flow 441

11.4 Radiator Thermal Model 44411.5 Tank Cooling 44811.6 Oil Mixing in the Tank 45011.7 Time Dependence 45311.8 Pumped Flow 45411.9 Comparison with Test Results 45511.10 Determining M and N Exponents 46011.11 Loss of Life Calculation 462

11.12 Cable and Lead Temperature Calculation 46611.13 Tank Wall Temperature Calculation 47311.14 Tieplate Temperature Calculation 47511.15 Core Steel Temperature Calculation 478

12 Load Tap Changers 48112.1 Introduction 481

12.2 General Description of LTC 48212.3 Types of Regulation 48312.4 Principle of Operation 484

12.4.1 Resistive Switching 48412.4.2 Reactive Switching with Preventative

Autotransformer 48612.5 Connection Schemes 488

12.5.1 Full Transformers 488

12.5.2 Autotransformers 49012.5.3 Use of Auxiliary Transformer 49312.5.4 Phase Shifting Transformers 495

12.6 General Maintenance 495

13 Phase Shifting Transformers 49913.1 Introduction 49913.2 Basic Principles 503

2002 by CRC Press

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CONTENTSx

13.3 Squashed Delta Phase Shifting Transformers 50713.4 Standard Delta Phase Shifting Transformers 51413.5 Two Core Phase Shifting Transformer 51913.6 Regulation Effects 52613.7 Fault Current Analysis 528

13.7.1 Squashed Delta Fault Currents 53213.7.2 Standard Delta Fault Currents 535

14 Cost Minimization 54314.1 Introduction 543

14.2 Geometric Programming 54514.3 Non-Linear Constrained Optimization 552

14.3.1 Characterization of the Minimum 55214.3.2 Solution Search Strategy 56114.3.3 Practical Considerations 567

14.4 Application to Transformer Design 56814.4.1 Design Variables 56914.4.2 Cost Function 570

14.4.3 Equality Constraints 57314.4.4 Inequality Constraints 57714.4.5 Optimization Strategy 578

References 583

2002 by CRC Press

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PREFACEMany of the standard texts on power transformers are now over tenyears old and some much older. Much has changed in transformer designsince these books were written. Newer and better materials are nowavailable for core and winding construction. Powerful computers nowmake 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 andconstruction are found scattered in the literature, there is a need to havethis information available in a single source as a reference for the designeror 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 forbeginners, we emphasize the physical basis of transformer operation. Wealso discuss the physical effects which result from various fault conditionsand their implications for design. Physical principles and mathematical

techniques are presented in a reasonably self-contained manner, althoughreferences are provided to additional material. For the specialist such asa power or transformer design engineer, detailed models are presentedwhich focus on various aspects of a transformer under normal orabnormal conditions. Cost minimization techniques, which form thestarting point for most designs, are also presented.

Although this book primarily deals with power transformers, manyof 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 usersof other transformer types will have little difficulty applying many ofthe results to their own designs. The emphasis on fundamentals shouldmake this process easier and should also foster the development of newand 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 unitsin parentheses after its metric value.

References are referred to generally by the first three letters of the firstauthors name followed by the last two digits of the publication date, e.g.[Abc98]. In cases where this format cannot be followed, an appropriatesubstitute is made. They are listed alphabetically at the end of the book.

We wish to thank Harral Robin for guidance throughout the course ofthis work. We would also like to acknowledge many helpful suggestions

from power industry representatives and consultants over the years.