Transformer design principles : with applications to core-form power transformers / Robert M. Del Vecchio ... [and others]

Includes bibliographical references and index

Contents Preface 1 Introduction 1.1 Historical Background 1.2 Uses in Power Systems 1.3 Core-Form and Shell-Form Transformers 1.4 Stacked and Wound Core Construction 1.5 Transformer Cooling 1.6 Winding Types 1.7 Insulation Structures 1.8 Structural Elements 1.9 Modern Trends 2 Magnetism and Related Core Issues 2.1 Introduction 2.2 Basic Magnetism 2.3 Hysteresis 2.4 Magnetic Circuits 2.5 Inrush Current 2.6 Distinguishing Inrush from Fault Current 2.7 Optimal Core Stacking 3 Circuit Model of a Two-Winding Transformer with Core 3.1 Introduction 3.2 Circuit Model of the Core 3.3 Two-Winding Transformer Circuit Model with Core 3.4 Approximate Two-Winding Transformer Circuit Model without Core 3.5 Vector Diagram of a Loaded Transformer with Core 3.6 Per-Unit System 3.7 Voltage Regulation 4 Reactance and Leakage Reactance Calculations 4.1 Introduction 4.2 General Method for Determining Inductances and Mutual Inductances 4.2.1 Energy by Magnetic Field Methods 4.2.2 Energy from Electric Circuit Methods 4.3 Two-Winding Leakage Reactance Formula 4.4 Ideal Two-, Three-, and Multiwinding Transformers 4.4.1 Ideal Autotransformers 4.5 Leakage Reactance for Two-Winding Transformers Based on Circuit Parameters 4.5.1 Leakage Reactance for a Two-Winding Autotransformer 4.6 Leakage Reactances for Three-Winding Transformers 4.6.1 Leakage Reactance for an Autotransformer with a Tertiary Winding 4.6.2 Leakage Reactance between Two Windings Connected in Series and a Third Winding 4.6.3 Leakage Reactance of a Two-Winding Autotransformer with X-Line Taps 4.6.4 More General Leakage Reactance Calculations 5 Phasors, Three-Phase Connections, and Symmetrical Components 5.1 Phasors 5.2 Wye and Delta Three-Phase Connections 5.3 Zig-Zag Connection 5.4 Scott Connection 5.5 Symmetrical Components 6 Fault Current Analysis 6.1 Introduction 6.2 Fault Current Analysis on Three-Phase Systems 6.2.1 Three-Phase Line-to-Ground Fault 6.2.2 Single-Phase Line-to-Ground Fault 6.2.3 Line-to-Line Fault 6.2.4 Double Line-to-Ground Fault 6.3 Fault Currents for Transformers with Two Terminals per Phase 6.3.1 Three-Phase Line-to-Ground Fault 6.3.2 Single-Phase Line-to-Ground Fault 6.3.3 Line-to-Line Fault 6.3.4 Double Line-to-Ground Fault 6.3.5 Zero-Sequence Circuits 6.3.6 Numerical Example for a Single Line-to-Ground Fault 6.4 Fault Currents for Transformers with Three Terminals per Phase 6.4.1 Three-Phase Line-to-Ground Fault 6.4.2 Single-Phase Line-to-Ground Fault 6.4.3 Line-to-Line Fault 6.4.4 Double Line-to-Ground Fault 6.4.5 Zero-Sequence Circuits 6.4.6 Numerical Examples 6.5 Asymmetry Factor 7 Phase-Shifting and Zig-Zag Transformers 7.1 Introduction 7.2 Basic Principles 7.3 Squashed Delta Phase-Shifting Transformer 7.3.1 Zero-Sequence Circuit Model 7.4 Standard Delta Phase-Shifting Transformer 7.4.1 Zero-Sequence Circuit Model 7.5 Two-Core Phase-Shifting Transformer 7.5.1 Zero-Sequence Circuit Model 7.6 Regulation Effects 7.7 Fault Current Analysis 7.7.1 Squashed Delta Fault Currents 7.7.2 Standard Delta Fault Currents 7.7.3 Two-Core Phase-Shifting Transformer Fault Currents 7.8 Zig-Zag Transformer 7.8.1 Calculation of Electrical Characteristics 7.8.2 Per-Unit Formulas 7.8.3 Zero-Sequence Impedance 7.8.4 Fault Current Analysis 8 Multiterminal Three-Phase Transformer Model 8.1 Introduction 8.2 Theory 8.2.1 Two-Winding Leakage Inductance 8.2.2 Multiwinding Transformers 8.2.3 Transformer Loading 8.3 Transformers with Winding Connections within a Phase 8.3.1 Two Secondary Windings in Series 8.3.2 Primary Winding in Series with a Secondary Winding 8.3.3 Autotransformer 8.4 Multiphase Transformers 8.4.1 Delta Connection 8.4.2 Zig-Zag Connection 8.5 Generalizing the Model 8.6 Regulation and Terminal Impedances 8.7 Multiterminal Transformer Model for Balanced and Unbalanced Load Conditions 8.7.1 Theory 8.7.2 Admittance Representation 8.7.2.1 Delta Winding Connection 8.7.3 Impedance Representation 8.7.3.1 Ungrounded Y Connection 8.7.3.2 Series-Connected Windings from the Same Phase 8.7.3.3 Zig-Zag Winding Connection 8.7.3.4 Autoconnection 8.7.3.5 Three Windings Joined 8.7.4 Terminal Loading 8.7.5 Solution Process 8.7.5.1 Terminal Currents and Voltages 8.7.5.2 Winding Currents and Voltages 8.7.6 Unbalanced Loading Examples 8.7.6.1 Autotransformer with Buried Delta Tertiary and Fault on Low-Voltage Terminal 8.7.6.2 Power Transformer with Fault on Delta Tertiary 8.7.6.3 Power Transformer with Fault on Ungrounded Y Secondary 8.7.7 Balanced Loading Example 8.7.7.1 Standard Delta Phase-Shifting Transformer 8.7.8 Discussion 9 Rabins' Method for Calculating Leakage Fields, Leakage Inductances, and Forces in Transformers 9.1 Introduction 9.2 Theory 9.3 Rabins' Formula for Leakage Reactance 9.3.1 Rabins' Method Applied to Calculate the Leakage Reactance between Two Windings That Occupy Different Radial Positions 9.3.2 Rabins' Method Applied to Calculate the Leakage Reactance between Two Axially Stacked Windings 9.3.3 Rabins' Method Applied to Calculate the Leakage Reactance for a Collection of Windings 9.4 Application of Rabins' Method to Calculate the Self-Inductance of and Mutual Inductance between Coil Sections 9.5 Determining the B-Field 9.6 Determination of Winding Forces 9.7 Numerical Considerations 10 Mechanical Design 10.1 Introduction 10.2 Force Calculations 10.3 Stress Analysis 10.3.1 Compressive Stress in the Key Spacers 10.3.2 Axial Bending Stress per Strand 10.3.3 Tilting Strength 10.3.4 Stress in the Tie Bars 10.3.5 Stress in the Pressure Ring 10.3.6 Hoop Stress 10.3.7 Radial Bending Stress 10.4 Radial Buckling Strength 10.4.1 Free Unsupported Buckling 10.4.2 Constrained Buckling 10.4.3 Experiment to Determine Buckling Strength 10.5 Stress Distribution in a Composite Wire-Paper Winding Section 10.6 Additional Mechanical Considerations 11 Electric Field Calculations 11.1 Simple Geometries 11.1.1 Planar Geometry 11.1.2 Cylindrical Geometry 11.1.3 Spherical Geometry 11.1.4 Cylinder-Plane Geometry 11.2 Electric Field Calculations Using Conformal Mapping 11.2.1 Physical Basis 11.2.2 Conformal Mapping 11.2.3 Schwarz-Christoffel Transformation 11.2.4 Conformal Map for the Electrostatic Field Problem 11.2.4.1 Electric Potential and Field Values 11.2.4.2 Calculations and Comparison with a Finite Element Solution 11.2.4.3 Estimating Enhancement Factors 11.3 Finite Element Electric Field Calculations 12 Capacitance Calculations 12.1 Introduction 12.2 Distributive Capacitance along a Winding or Disk 12.3 Stein's Disk Capacitance Formula 12.4 General Disk Capacitance Formula 12.5 Coil Grounded at One End with Grounded Cylinders on Either Side 12.6 Static Ring on One Side of a Disk 12.7 Terminal Disk without a Static Ring 12.8 Capacitance Matrix 12.9 Two Static Rings 12.10 Static Ring between the First Two Disks 12.11 Winding Disk Capacitances with Wound-in Shields 12.11.1 Analytic Formula 12.11.2 Circuit Model 12.11.3 Experimental Methods 12.11.4 Results 12.12 Multistart Winding Capacitance 13 Voltage Breakdown and High-Voltage Design 13.1 Introduction 13.2 Principles of Voltage Breakdown 13.2.1 Breakdown in Solid Insulation 13.2.2 Breakdown in Transformer Oil 13.3 Geometric Dependence of Transformer-Oil Breakdown 13.3.1 Theory 13.3.2 Planar Geometry 13.3.3 Cylindrical Geometry 13.3.4 Spherical Geometry 13.3.5 Comparison with Experiment 13.3.6 Generalization 13.3.6.1 Breakdown for the Cylinder-Plane Geometry 13.3.6.2 Breakdown for the Disk-Disk-to-Ground Plane Geometry 13.3.7 Discussion 13.4 Insulation Coordination 13.5 Continuum Model of Winding Used to Obtain the Impulse-Voltage

Distribution 13.5.1 Uniform Capacitance Model 13.5.2 Traveling Wave Theory 13.6 Lumped-Parameter Model for Transient Voltage Distribution 13.6.1 Circuit Description 13.6.2 Mutual and Self-Inductance Calculations 13.6.3 Capacitance Calculations 13.6.4 Impulse-Voltage Calculations and Experimental Comparisons 13.6.5 Sensitivity Studies 14 Losses 14.1 Introduction 14.2 No-Load or Core Losses 14.2.1 Building Factor 14.2.2 Interlaminar Losses 14.3 Load Losses 14.3.1 I2R Losses 14.3.2 Stray Losses 14.3.2.1 Eddy Current Losses in the Coils 14.3.2.2 Tieplate Losses 14.3.2.3 Tieplate and Core Losses Due to Unbalanced Currents 14.3.2.4 Tank and Clamp Losses 14.3.3 Stray Losses Obtained from 3D Finite Element Analyses 14.3.3.1 Shunts on the Clamps 14.3.3.2 Shunts on the Tank Wall 14.3.3.3 Effects of Three-Phase Currents on Losses 14.3.3.4 Stray Losses from the 3D Analysis versus Analytical and Test Losses 14.4 Tank and Shield Losses Due to Nearby Busbars 14.4.1 Losses Obtained with 2D Finite Element Study 14.4.2 Losses Obtained Analytically 14.4.2.1 Current Sheet 14.4.2.2 Delta Function Current 14.4.2.3 Collection of Delta Function Currents 14.4.2.4 Model Studies 14.5 Tank Losses Associated with the Bushings 14.5.1 Comparison with a 3D Finite Element Calculation 15 Thermal Design 15.1 Introduction 15.2 Thermal Model of a Disk Coil with Directed Oil Flow 15.2.1 Oil Pressures and Velocities 15.2.2 Oil Nodal Temperatures and Path Temperature Rises 15.2.3 Disk Temperatures 15.3 Thermal Model for Coils without Directed Oil Flow 15.4 Radiator Thermal Model 15.5 Tank Cooling 15.6 Oil Mixing in the Tank 15.7 Time Dependence 15.8 Pumped Flow 15.9 Comparison with Test Results 15.10 Determining m and n Exponents 15.11 Loss of Life Calculation 15.12 Cable and Lead Temperature Calculation 15.13 Tank Wall Temperature Calculation 15.14 Tieplate Temperature 15.15 Core Steel Temperature Calculation 16 Load Tap Changers 16.1 Introduction 16.2 General Description of Load Tap Changers 16.3 Types of Regulation 16.4 Principles of Operation 16.4.1 Resistive Switching 16.4.2 Reactive Switching with Preventive Autotransformers 16.5 Connection Schemes 16.5.1 Power Transformers 16.5.2 Autotransformers 16.5.3 Use of Auxiliary Transformers 16.5.4 Phase-Shifting Transformers 16.5.5 Reduced versus Full-Rated Taps 16.6 General Maintenance 17 Miscellaneous Topics 17.1 Setting the Impulse Test Generator to Achieve Close to Ideal Waveshapes 17.1.1 Impulse Generator Circuit Model 17.1.2 Transformer Circuit Model 17.1.3 Determining the Generator Settings for Approximating the Ideal Waveform 17.1.4 Practical Implementation 17.2 Impulse or Lightning Strike on a Transformer through a Length of Cable 17.2.1 Lumped Parameter Model 17.2.1.1 Numerical Example 17.2.2 Traveling Wave Theory 17.3 Air Core Inductance 17.4 Electrical Contacts 17.4.1 Contact Resistance 17.4.2 Thermal Considerations 17.4.3 Practical Considerations References Index