Electromagnetic Computation Methods for Lightning Surge Protection Studies

Professor Yoshihiro Baba, Associate Professor, Department of Electrical Engineering, Doshisha University, Kyoto, Japan Professor Baba received his PhD from the University of Tokyo. He was Visiting Scholar at the Lightning Laboratory at the University of Florida, USA, and is currently Editor of the IEEE Transactions on Power Delivery. His areas of interest include computational electromagnetics, electromagnetic compatibility and lightning protection. Professor Vladimir A. Rakov, Department of Electrical and Computer Engineering, University of Florida, USA Professor Rakov studied for his PhD at Tomsk University in Russia and has written extensively on the subject of lightning in numerous international journals and conference proceedings. He is A Fellow of the IEEE and a Fellow of the American Meteorological Society, amongst others, and his areas of interest cover lightning, lightning protection and atmospheric electricity.

Preface xi

1 Introduction 1

1.1 Historical Overview of Lightning Electromagnetic-Field and Surge Computations 1

1.2 Overview of Existing Electromagnetic Computation Methods 2

1.2.1 Method of Moments 2

1.2.2 Partial-Element Equivalent-Circuit Method 4

1.2.3 Finite-Element Method 4

1.2.4 Transmission Line Modeling Method 4

1.2.5 Constrained Interpolation Profile Method 5

1.2.6 Finite-Difference Time Domain Method 6

1.3 Summary 7

References 7

2 Lightning 11

2.1 Introduction 11

2.2 Thundercloud 12

2.2.1 Formation of Thunderclouds 12

2.2.2 Mechanism of Cloud Electrification 14

2.3 Lightning Discharges 15

2.3.1 Categories of Lightning Discharges 15

2.3.2 Classification of Cloud-to-Ground Lightning Discharges 15

2.3.3 Downward Negative Lightning Discharges to Ground 16

2.3.4 Positive Lightning Discharges 23

2.3.5 Upward Lightning Discharges 23

2.3.6 Rocket-Triggered Lightning Discharges 25

2.4 Lightning Electromagnetic Fields 26

2.4.1 Measured Lightning Return-Stroke Electromagnetic Fields 26

2.4.2 Mathematical Expressions for Calculating Electric and Magnetic Fields 29

2.5 Lightning Surges 31

2.5.1 Surges Due to Direct Lightning Strike 31

2.5.2 Surges Induced by a Nearby Lightning Strike 32

2.5.3 Surges Coming from Grounding Due to Its Potential Rise 33

2.6 Lightning Surge Protection 34

2.6.1 Insulation Coordination 34

2.6.2 Protection against Direct Lightning Strikes 35

2.6.3 Back-Flashover Phenomena 37

2.6.4 Lightning Surge Protection Measures 38

2.7 Summary 40

References 41

3 The Finite-Difference Time Domain Method for Solving Maxwell's Equations 43

3.1 Introduction 43

3.2 Finite-Difference Expressions of Maxwell's Equations 44

3.2.1 3D Cartesian Coordinate System 44

3.2.2 2D Cylindrical Coordinate System 49

3.3 Subgridding Technique 51

3.4 Absorbing Boundary Conditions 55

3.5 Representation of Lumped Sources and Lumped Circuit Elements 57

3.5.1 Lumped Voltage Source 57

3.5.2 Lumped Current Source 57

3.5.3 Lumped Resistance 59

3.5.4 Lumped Inductance 59

3.5.5 Lumped Capacitance 60

3.6 Representation of Thin Wire 61

3.7 Representation of Lightning Return-Stroke Channel 63

3.7.1 Lightning Return-Stroke Channel 63

3.7.2 Excitations 66

3.8 Representation of Surge Arresters 67

3.9 Summary 69

References 70

4 Applications to Lightning Surge Protection Studies 73

4.1 Introduction 73

4.1.1 Overview 73

4.1.2 Lightning Electromagnetic Fields at Close and Far Distances 73

4.1.3 Lightning Surges on Overhead Power TL Conductors and Towers 75

4.1.4 Lightning Surges on Overhead Distribution and Telecommunication Lines 76

4.1.5 Lightning Electromagnetic Environment in Power Substations 77

4.1.6 Lightning Surges in Wind-Turbine-Generator Towers 77

4.1.7 Lightning Surges in Photovoltaic Arrays 78

4.1.8 Lightning Electromagnetic Environment in Electric Vehicles 78

4.1.9 Lightning Electromagnetic Environment in Airborne Vehicles 78

4.1.10 Lightning Surges and the Electromagnetic Environment in Buildings 79

4.1.11 Surges on Grounding Electrodes 79

4.2 Electromagnetic Fields at the Top of a Tall Building Associated with Nearby Lightning Return Strokes 80

4.2.1 Introduction 80

4.2.2 Methodology 81

4.2.3 Analysis and Results 85

4.2.4 Summary 96

4.2.5 Appendix: Comparison of Fields in the Absence of a Building Computed Using the FDTD Method and Thottappillil et al.'s (2001) Analytical Expressions 96

4.2.6 Appendix: Enhancement Factors Due to the Presence of Hemisphere or Rectangular Building in a Uniform Static Electric Field 97

4.3 Influence of Strike Object Grounding on Close Lightning Electric Fields 100

4.3.1 Introduction 100

4.3.2 Methodology 103

4.3.3 Analysis and Results 105

4.3.4 Discussion 122

4.3.5 Summary 128

4.3.6 Appendix: Comparison of Fields Due to a Lightning Strike to Flat Ground Calculated Using the FDTD Method in the 2D Cylindrical Coordinate System and Thottappillil et al.'s (2001) Analytical Expressions 128

4.4 Simulation of Corona at Lightning-Triggering Wire: Current, Charge Transfer, and Field Reduction Effect 129

4.4.1 Introduction 129

4.4.2 General Approach 135

4.4.3 Model 136

4.4.4 Analysis and Results 141

4.4.5 Discussion 145

4.4.6 Summary 149

4.4.7 Appendix: Geometry of a Wire Corona Sheath 149

4.5 On the Interpretation of Ground Reflections Observed in Small-Scale Experiments Simulating Lightning Strikes to Towers 151

4.5.1 Introduction 151

4.5.2 Current Pulses Propagating along a Conical Conductor Excited at Its Apex or Base 153

4.5.3 FDTD Simulation of Small-Scale Experiments 157

4.5.4 Interpretation of Ground Reflections Arriving at the Tower Top 162

4.5.5 TL Representation of a Tall Object on the Ground Plane 164

4.5.6 Summary 169

4.5.7 Appendix: FDTD Representation of Tower Models 170

4.6 On the Mechanism of Attenuation of Current Waves Propagating along a Vertical Perfectly Conducting Wire above Ground: Application to Lightning 171

4.6.1 Introduction 171

4.6.2 Incident Current (Iinc), Incident E-field (Einc): Analytical Solution 174

4.6.3 Total Current (Itot), Total E-field (Etot): Numerical Solution 176

4.6.4 Scattered Current (Iscat), Scattered E-field (Escat): Iscat = Itot − Iinc, Escat = −Einc 179

4.6.5 Dependences of Current Attenuation on the Source Length, Conductor Thickness, and Frequency 181

4.6.6 Nonuniform TL Approximation 184

4.6.7 Summary 186

4.6.8 Appendix: Incident E-field for Two Parallel Vertical Phased Current Source Arrays—Analytical Solution 187

4.6.9 Appendix: Total Current for Horizontal Configurations—Numerical Solution 188

4.6.10 Appendix: Comparison of FDTD Simulation with an Analytical Solution 190

4.6.11 Appendix: E-field Structure around a Vertical Nonzero-Thickness Perfect Conductor 191

4.6.12 Appendix: Vertical E-field Produced by an Electrically-Short Vertical Dipole 192

4.7 FDTD Simulation of Lightning Surges on Overhead Wires in the Presence of Corona Discharge 193

4.7.1 Introduction 193

4.7.2 Modeling 195

4.7.3 Results and Discussion 199

4.7.4 Summary 209

4.8 FDTD Simulation of Insulator Voltages at a Lightning-Struck Tower Considering the Ground-Wire Corona 212

4.8.1 Introduction 212

4.8.2 Methodology 212

4.8.3 Analysis and Results 215

4.8.4 Summary 224

4.9 Voltages Induced on an Overhead Wire by Lightning Strikes to a Nearby Tall Grounded Object 224

4.9.1 Introduction 224

4.9.2 Methodology 228

4.9.3 Analysis and Results 231

4.9.4 Discussion 238

4.9.5 Summary 240

4.9.6 Appendix: Testing the Validity of the FDTD Calculations against Experimental Data (Strikes to Flat Ground) 242

4.9.7 Appendix: Comparison with Rusck's Formula (Strikes to Flat Ground) 243

4.9.8 Appendix: Testing the Validity of the FDTD Calculations against Experimental Data (Strikes to a Tall Object) 245

4.10 3D-FDTD Computation of Lightning-Induced Voltages on an Overhead Two-Wire Distribution Line 247

4.10.1 Introduction 247

4.10.2 Methodology 249

4.10.3 Analysis and Results 252

4.10.4 Summary 260

4.11 FDTD Simulations of the Corona Effect on Lightning-Induced Voltages 260

4.11.1 Introduction 260

4.11.2 Methodology 261

4.11.3 Analysis and Results 263

4.11.4 Discussion 269

4.11.5 Summary 277

4.12 FDTD Simulation of Surges on Grounding Electrodes Considering Soil Ionization 277

4.12.1 Introduction 277

4.12.2 Representation of Soil Ionization and De-ionization 278

4.12.3 Analysis and Results 279

4.12.4 Conclusions 288

4.13 Summary 288

References 288

Appendix: 3D-FDTD Program in C++ 299

Index 311