Lightning return stroke current from a new distributed circuit model and electromagnetic fields generated by tortuous lightning channels

LIGHTNING RETURN STROKE CURRENT FROM A NEW DISTRIBUTED CIRCUIT MODEL AND ELECTROMAGNETIC FIELDS GENERATED BY TORTUOUS LIGHTNING CHANNELS CHIA KOK LIAN DARWIN B Eng (1st class Hons.), NUS A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2007 ACKNOWLEDGEMENTS I am deeply indebted to Professor Liew Ah Choy, my supervisor, who has guided me with his patience and knowledge His understanding, encouragement and personal guidance have been inspirational towards the completion of this thesis I would like to express my heartfelt gratitude to Professor Walid Tabbara, my co-supervisor, for his care and help Great appreciation goes to his teachings and constructive criticism, which have been of great value My sincere thanks to all the colleagues at the Power Systems Laboratory at NUS and SONDRA at Supélec for their kind friendship and support I would also like to express my appreciation to the Singapore Millennium Foundation for the scholarship funding received I cannot end without thanking my family and friends for their love and encouragement throughout the extended period of my scholarship i TABLE OF CONTENTS ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY v LIST OF PUBLICATIONS vii LIST OF TABLES viii LIST OF FIGURES ix LIST OF SYMBOLS xi CHAPTER INTRODUCTION 1.1 BACKGROUND AND OBJECTIVE 1.1.1 1.1.2 1.2 Overview on Lightning Objective and Contribution of Work Undertaken ORGANISATION OF THESIS CHAPTER THE LIGHTNING DISCHARGE 2.1 TYPES OF LIGHTNING DISCHARGES 2.2 LIGHTNING DISCHARGE MECHANISM 2.2.1 Preliminary Breakdown 2.2.2 Stepped Leader 10 2.2.3 Attachment Process 11 2.2.4 Return Stroke 12 2.2.5 Subsequent and Multiple Strokes 15 2.3 LIGHTNING CURRENT 15 2.4 LIGHTNING ELECTROMAGNETIC FIELDS 18 ii CHAPTER LIGHTNING RETURN STROKE MODELS 21 3.1 MODELLING 21 3.2 BRUCE-GOLDE (BG) MODEL 25 3.3 TRANSMISSION LINE (TL) MODEL 26 3.4 27 MASTER-UMAN-LIN-STANDLER (MULS) MODEL 3.5 TRAVELLING CURRENT SOURCE (TCS) MODEL 29 3.6 DIENDORFER-UMAN (DU) MODEL 30 3.7 33 PAN-LIEW (PL) MODEL 3.8 LUPO ET AL.’S MODEL CHAPTER DEVELOPMENT OF DISTRIBUTED CIRCUIT MODEL 4.1 ASSUMPTIONS 36 38 38 4.1.1 Discharge Current Components 38 4.1.2 Charge Distribution along the Leader Channel 39 4.1.3 Height of Lightning Channel 4.1.4 Return Stroke and Discharge Current Speeds 40 PROPOSED MODEL 40 4.2.1 Equivalent Circuit 4.2 40 41 4.2.2 Derivation of Equations Defining Return Stroke Current 44 4.2.3 Profile of Circuit Elements 4.3 EVALUATION OF PROPOSED MODEL 56 58 CHAPTER APPLICATION OF DISTRIBUTED CIRCUIT MODEL ON SEMICONDUCTOR LIGHTNING EXTENDER 5.1 SEMICONDUCTOR LIGHTNING EXTENDER (SLE) 66 67 5.1.1 Physical Structure 67 5.1.2 Characteristics 68 iii 5.1.3 Field Measurement Results on SLE 69 5.2 MODELLING OF THE SLE 71 5.2.1 Circuit Representation 71 5.2.2 73 Flashover Voltage 5.3 RESULTS AND EVALUATION 73 CHAPTER IMPROVED MODEL FOR ELECTROMAGNETIC FIELDS GENERATED BY TORTUOUS LIGHTNING CHANNELS 81 6.1 MATHEMATICAL FORMULATION 6.1.1 82 Electromagnetic Fields due to a Straight Vertical Segment 82 6.1.2 Geometrical Transformation for a Segment of Arbitrary Location and Orientation 6.1.3 6.2 87 Comparison with Lupò et al.’s Model 89 PROPOSED MODEL 90 6.2.1 Lightning Parameters 90 6.2.2 Random Tortuous Lightning Channel 90 6.2.3 Results and Observations 91 CHAPTER CONCLUSION 110 7.1 DISTRIBUTED CIRCUIT MODEL (CHIA-LIEW MODEL) 110 7.2 TORTUOUS LIGHTNING CHANNELS 111 7.3 SCOPE FOR FUTURE WORK 112 BIBLIOGRAPHY 113 APPENDIX A DERIVATION OF ELECTROMAGNETIC FIELD EQUATIONS 118 iv SUMMARY In contribution to the field of lightning research, two lightning return stroke models are developed The distributed circuit model contrived to produce the lightning return stroke current at ground and the mathematical formulation for the electromagnetic fields generated by tortuous lightning channels are presented The distributed circuit model is made up of resistive, capacitive and inductive elements which represent the lightning channel The inclusion of inductances addresses the limitation of the Pan-Liew model While simulating the discharge mechanism, the lightning return stroke current at ground was produced to match the 5th-percentile, median and 95th-percentile recorded values of the peak current, charge lowered and front duration reported by Berger et al At the same time, reference to the theoretical waveshape proposed by the Diendorfer-Uman model was kept A key function of the distributed circuit model is its applicability in the evaluation of resistive lightning protection terminals in mitigating the lightning return stroke current Such protection systems can be easily represented by resistive circuit elements and a study was conducted on the Semiconductor Lightning Extender (SLE) From the waveforms of the voltage and current through the SLE, the peak of the return stroke current was shown to be significantly reduced This demonstrates the efficacy of resistive lightning protection terminals and highlights a major function of the model in such studies, while enforcing the validity of the distributed circuit model v In the formulation for the electromagnetic fields due to tortuous lightning channels, a flaw identified in Lupò et al.’s model was improved upon with a more appropriate current description The formulation allows for the determination of lightning radiated electromagnetic fields at any distance and height The resulting waveforms from a randomly generated lightning stroke path demonstrated the sharp initial peak and zerocrossing for fields at far distance, which are key characteristics observed by Lin et al from measured waveforms Furthermore, while the electromagnetic fields calculated from models adopting the straight vertical lightning channel approximation fail to exhibit the fine structure representing more significant high frequency components in actual measurements, the tortuous channel model clearly displays this attribute It was also noted that for a lightning channel that does not deviate much from a straight path, which was less than 100 m in both the x- and y-directions for the randomly generated lightning channel, the straight channel approximation adopted by most lightning models is adequate Potential applications of this model include the reconstruction of the lightning stroke path from remote electromagnetic field measurements and also the study of electromagnetic coupling to systems vi LIST OF PUBLICATIONS K L Chia and A C Liew, “Modeling of Lightning Return Stroke Current with Inclusion of Distributed Channel Resistance and Inductance,” IEEE Trans Power Del., vol 19, no 3, pp 1342–1347, Jul 2004 D K L Chia and A C Liew, “Analysis of Effect of Resistive Lightning Protection Terminal on Lightning Return Stroke Current,” IEEE Trans Power Del., vol 20, no 3, pp 2307–2314, Jul 2005 D K L Chia, A C Liew and W Tabbara, “An Improved Model for the Electromagnetic Fields Generated by Tortuous Lightning Channel,” IEEE Trans Electromagn Compat (under review) vii LIST OF TABLES Table 2.1 Lightning Current Parameters 16 Table 3.1 Constants Used to Calculate Return Stroke Current in the DU Model 31 Table 3.2 Circuit Parameter Values Used in the PL Model 35 Table 4.1 Circuit Element Values for 14 kA Return Stroke Current 63 Table 4.2 Circuit Element Values for 30 kA Return Stroke Current 64 Table 4.3 Circuit Element Values for 80 kA Return Stroke Current 65 Table 5.1 Lightning Current Measured by Xie et al 70 Table 5.2 Cumulative Probability Distribution of Currents Larger Than I 71 viii LIST OF FIGURES Figure 2.1 Categorisation of lightning Figure 2.2 Single stroke lightning flash 14 Figure 2.3 Typical vertical electric field intensity and azimuthal magnetic flux density waveforms for the first and subsequent return strokes at distances of 1, 2, 5, 10, 15, 50 and 200 km 20 Figure 3.1 Lumped parameter transmission line representation of lightning return stroke 23 Figure 3.2 Geometrical parameters used in the models 25 Figure 3.3 Channel-base return stroke current in the DU model 31 Figure 3.4 Equivalent circuit of leader channel in the PL model 33 Figure 3.5 Channel-base current waveform for the PL model 35 Figure 4.1 Equivalent circuit of lightning channel 42 Figure 4.2 Lightning return stroke currents in proposed model 60 Figure 4.3 14 kA return stroke current 63 Figure 4.4 30 kA return stroke current 64 Figure 4.5 80 kA return stroke current 65 Figure 5.1 3-rod SLE 67 Figure 5.2 Single SLE rod with needles 68 Figure 5.3 Equivalent circuit of lightning channel with inclusion of SLE 72 Figure 5.4 Volt-time curve for 2-m rod gap 73 Figure 5.5 Voltage and current through SLE for 30 kA stroke 75 Figure 5.6 Voltage and current through SLE for 14 kA stroke 76 ix Appendix A Derivation of Electromagnetic Field Equations APPENDIX A DERIVATION OF ELECTROMAGNETIC FIELD EQUATIONS Using the vector magnetic potential A and scalar electric potential φ expressions, the electric field strength E and magnetic field strength H are given by ∂A ∂t E = −∇ϕ − H= (A.1) ∇×A (A.2) µ The vector magnetic potential dA and scalar electric potential dϕ associated with an infinitesimal element of length dz', at z = z', traversed by a current i(z',t) and carrying a charge λ(z',t)dz' are ( µ i z′, t − dA = 4π Rr Rr c ) dz′a ( λ z′, t − dϕ ( r , φ , z ) = 4πε Rr Rr c (A.3) z ) dz′ (A.4) where az is the unit vector along the z-axis, c is the speed of light and Rr = ( x′ − x ) + ( y ′ − y ) + ( z ′ − z ) 2 (A.5) 118 Appendix A Derivation of Electromagnetic Field Equations The current profile assumed is a unit step function z′   i + ( z′, t ) = u  t −  v  (A.6) where v is the return stroke front speed and u is the Heaviside step function Applying the continuity equation ∇⋅J = − ∂ρ ∂t (A.7) the corresponding charge distribution is ⌠ ∂i ( z′,τ ) dτ λ ( z′, t ) = λ + ( z′,0 ) −  ⌡0 ∂z′ t + + t z′   ⌠   = −  − δ τ −   dτ v  ⌡0  v  z′   = ut −  v  v (A.8) where δ is the Dirac delta function and the initial charge distribution λ+(z',0) is assumed to be zero Substituting equations (A.6) and (A.8) into equations (A.3) and(A.4), dA + = à0 z R I ì u  t − − r  dz′a z Rr  v c  4π dϕ + ( r , φ , z ) = − 4πε × (A.9) I0   z′ R   × − u  t − − r  Rr  v  v c  (A.10) To find H + , take the curl of dA and integrate along the segment Since x' = y' = 0, Rr = x + y + ( z′ − z ) = r + ( z′ − z ) 2 (A.11) 119 Appendix A Let F + = Derivation of Electromagnetic Field Equations  z′ R ut − − r Rr  v c  ,   ∂F + ∂F +   ∂F + ∂F +  ∂F +  1 ∂ ∇ × F + = a r  z − φ  + aφ  r − z  + a z  ( rFφ+ ) − r   r∂φ ∂z  ∂r  ∂φ  r  ∂r  ∂z   ∂F + = −aφ z ∂r ∂F + ∂R = −aφ z × r ∂Rr ∂r (A.12)   z′ R   z′ R   r δ  t − − r  × = −aφ  − u  t − − r  − v c  cRr  v c   Rr  Rr   r  z′ R  r z′ R    δ  t − − r  = aφ  u  t − − r  + v c  cRr  v c   Rr  Hence H+ = à0 I = ( ì dA ) h + ∫ h (A.13) ∇ × F ( t )  dz′   + Therefore H r+ = H z+ = (A.14) h H φ+ = ∫ g + ( t ) dz′ where g + (t ) = 4π  r  z′ Rr  r z′ R    δ  t − − r   ut − −  + v c  cRr  v c   Rr  (A.15) Similarly, to find E+ , substitute equations (A.9) and (A.10) into equation (A.1) and integrate along the segment 120 Appendix A Derivation of Electromagnetic Field Equations h  z′ R A = ì u t − − r  dz′a z v c  ⌡0 4π Rr  + (A.16) h ∂A + µ0 ⌠  z′ R  = δ  t − − r  dz′a z  4π ⌡0 Rr  v c  ∂t ⇒ ∇ϕ + = a r (A.17) ∂ϕ + ∂ϕ + ∂ϕ + + aφ + az ∂r r∂φ ∂z (A.18) For the r-component, h ⌠ ∂  1  z′ R      u  t − − r    dz′    v c   ⌡0  ∂r  4π vε Rr    h ⌠   =   ⌡0  4π vε  −1  z′ Rr  ut − − v c  Rr  h ⌠  −1  r  z′ Rr  =   ut − − v c  ⌡0  4π vε  Rr  z′ Rr    r   −1    +  δ  t − −  ×  v c    Rr  Rr  c   r z′ R   δ t − − r + v c  cRr      dz′         dz′    h = − ∫ m + ( t ) dz′ (A.19) where m+ ( t ) =  r  z′ Rr  ut − − 4π vε  Rr  v c  z′ R  δ t − − r + v c  cRr     (A.20) 121 Appendix A Derivation of Electromagnetic Field Equations For the z-component, h ⌠ ∂  1  z′ R      u  t − − r    dz′    v c     ⌡0  ∂z  4π vε Rr  h ⌠   −1  z′ Rr  =   ut − − v c  ⌡0  4π vε  Rr  z′ Rr    z − z′      −1     dz′  +  δ  t − −  ×  v c    Rr    Rr  c    h ⌠  −1  z − z ′  z′ Rr  =   ut − − v c   ⌡0  4π vε  Rr z′ R      z − z′  δ  t − − r    dz′ + v c    cRr   (A.21) Grouping with the n+ (t ) = ∂A + term, let ∂t  z − z′  z′ Rr  ut − − 4π vε  Rr v c  z′ R  z − z′  δ t − − r + v c  cRr  z′ R    µ0  δ t − − r   − v c    4π Rr  (A.22) Hence h Er+ = ∫ m + ( t ) dz′ h Ez+ = ∫ n + ( t ) dz′ (A.23) Eφ+ = where m + ( t ) and n + ( t ) are defined in equations (A.20) and(A.22) As with most lightning models, a perfectly conducting ground is assumed and thus, the method of images can be applied A similar analysis is followed to obtain the contributions of the image below ground The quantities for the image is superscripted with the negative sign ‘–’ for differentiation with the segment above ground which is superscripted with the positive sign ‘+’ 122 Appendix A Derivation of Electromagnetic Field Equations For the image, − h ≤ z′ ≤ , z′   i − ( z′, t ) = u  t +  v  (A.24) and t − ⌠ ∂i ( z′,τ ) dτ λ ( z′, t ) = λ ( z′,0 ) −  ⌡0 ∂z′ − − t z′   ⌠ 1  = −  δ τ +   dτ v  ⌡0  v  z′   1  = −  u  t +  v  v  (A.25) = −λ + ( − z′, t ) Similarly, the initial charge distribution λ– (z',0) was set to zero Substituting equations (A.24) and (A.25) into equations (A.3) and (A.4), dA − = µ0  z′ Rr × ut + − 4π Rr  v c dϕ − ( r , φ , z ) = 4πε ×   dz′a z    z′ R   × − u  t + − r  Rr  v  v c  (A.26) (A.27) H − is found as previously with the integral limits changed to from − h to 123 Appendix A Let F − = Derivation of Electromagnetic Field Equations  z′ R ut + − r Rr  v c  ,   ∂F − ∂F −   ∂F − ∂F −  1 ∂ ∂F −  ∇ × F − = a r  z − φ  + aφ  r − z  + a z  ( rFφ− ) − r   r ∂φ r  ∂r ∂z  ∂r  ∂φ   ∂z   ∂F − = −aφ z ∂r ∂F − ∂R = −aφ z × r ∂Rr ∂r (A.28)   z′ R   z′ R   r = −aφ  − u  t + − r  − δ  t + − r  × v c  cRr  v c   Rr  Rr   r  z′ R  r z′ R    = aφ  u  t + − r  + δ  t + − r  v c  cRr  v c   Rr  Hence, H− = µ0 = 4π ∫ ( ∇ × dA ) − −h ∫ (A.29) ∇ × F ( t )  dz′   − −h Therefore H r− = H z− = (A.30) H φ− = ∫ g − ( t ) dz′ −h where g − (t ) = 4π  r  z′ Rr  r z′ R    δ  t + − r   ut + −  + v c  cRr  v c   Rr  (A.31) For Ε− , substitute equations (A.26) and (A.27) into equation (6.1) and integrate along the channel from from − h to 124 Appendix A Derivation of Electromagnetic Field Equations  z′ R  ⌠ µ0 A = I × u  t + − r  dz′a z Rr  v c  ⌡− h 4π − (A.32) ∂A − µ0 I ⌠  z′ R  = δ  t + − r  dz′a z  ∂t 4π ⌡− h Rr  v c  ⇒ (A.33) ∂ϕ − ∂ϕ − ∂ϕ − + aφ + az ∂r r ∂φ ∂z (A.34) ∇ϕ − = a r For the r-component, ⌠ ∂  1   z′ R     u  t + − r   × u ( z′ + h ) − u ( z′ )   dz′      v c   ⌡− h  ∂r  4π vε Rr    ⌠   −1  z′ Rr  =   ut + − v c ⌡− h  4π vε  Rr   z′ Rr     −1    +  δ  t + −  v c   Rr  c     r   ×   × u ( z′ + h ) − u ( z′ )   dz′ (A.35)     Rr   ⌠  −1  =   ⌡− h  4π vε  r  r z′ R    z′ Rr    δ  t + − r    dz′  ut + −  + v c   v c  cRr   Rr   = − ∫ m − ( t ) dz′ −h where m− ( t ) = 4π vε  r  z′ Rr   z′ R   δ  t + − r   ut + −  + v c  cRr  v c   Rr  (A.36) 125 Appendix A Derivation of Electromagnetic Field Equations For the z-component, ⌠ ∂  1   z′ R     u  t + − r   × u ( z′ + h ) − u ( z′ )   dz′      v c     ⌡− h  ∂z  4π vε Rr  ⌠   −1  z′ Rr  =   ut + − v c  ⌡− h  4π vε  Rr  z′ Rr     −1    +  δ  t + −  v c   Rr  c     z − z′   ×  × u ( z′ + h ) − u ( z′ )   dz′     Rr   ⌠  −1  z − z′  z′ Rr  =   ut + − v c  ⌡− h  4π vε  Rr  (A.37) z′ R      z − z′  δ  t + − r    dz′ +  v c    cRr   ∂A − Grouping with the term, let ∂t n− ( t ) =  z − z′  z′ Rr  z − z′  z′ R   µ  z′ R  δ  t + − r  + δ  t + − r   ut + −  + 4π vε  Rr v c  cRr v c   4π Rr  v c    (A.38) Hence Er− = − ∫ m − ( t ) dz′ −h E = − ∫ n − ( t ) dz′ − z −h (A.39) Eφ− = where m − ( t ) and n − ( t ) are defined in equations (A.36) and(A.38) For the integrals ∫ g ( t ) dz′ , ∫ m ( t ) dz′ and ∫ n ( t ) dz′ , the integrand contains either ± ± ± the Heaviside step function or the Dirac delta function with the argument z′ v ψ ± (t ) = t m − Rr c (A.40) 126 Appendix A Derivation of Electromagnetic Field Equations The zeros of this function can by found as below: tm tm ζ 0± v − ζ 0± v − Rr =0 c r + (ζ 0± − z ) =0 c v r + (ζ 0± − z ) = cvt mcζ 0± 2 v  r + (ζ 0± − z )  = c 2v 2t m2c 2vtζ 0± + c (ζ 0± )     v r + v (ζ 0± ) − 2v zζ 0± + v z = c 2v 2t m2c 2vtζ 0± + c (ζ 0± ) (c 2 − v )(ζ 0± ) + ( 2v z m2c 2vt ) ζ 0± + c 2v 2t − v r − v z = 2 t z r2 z2   t z   1  ± − m  m  −  − t2 − −  v c c  c c  v c  v ζ 0± ( t ) = 1 − v2 c2 (A.41) Note that the other root of the quadratic equation is neglected as it was introduced when the square of both sides was taken A simple back-substitution can verify the choice of roots For Ε+ and H + , the Dirac delta term is equal to zero except when z′ = ζ 0+ ( t ) and the Heaviside step term is equal to zero for z′ > ζ 0+ ( t ) Therefore, when ζ 0+ ( t ) < , both Heaviside step and Dirac delta functions are zero and when ζ 0+ ( t ) > h , the Dirac delta function is zero while the Heaviside step function has a value of for ≤ z′ ≤ h For ≤ ζ 0+ ( t ) ≤ h , the upper limit for the integral can be changed to ζ 0+ ( t ) Similarly for Ε− and H − , the Dirac delta term is equal to zero except when z′ = ζ 0− ( t ) and the Heaviside step term is equal to zero for z′ < ζ 0− ( t ) Therefore, when ζ 0− ( t ) > , 127 Appendix A Derivation of Electromagnetic Field Equations the integrals for both Heaviside step and Dirac delta functions are zero and when ζ 0− ( t ) < −h , the integral for the Dirac delta term is zero while the integral for the Heaviside step function has a value of for − h ≤ z′ ≤ For − h ≤ ζ 0− ( t ) ≤ , the lower limit for the integral can be changed to ζ 0− ( t ) To further evaluate the integrals, let z′ − z = r tan θ dz′ = r sec θ dθ (A.42)  zi′ − z    r  θ ( zi′ ) = tan −1  ′ z2 ⌠ ⌠ r dz′ =    ⌡z1′ Rr ⌡z1′ ′ z2 (A.43) ( r r + ( z′ − z ) ′ θ ( z2 ) ⌠ =  ⌡θ ( z1′ ) ( ) r r + r tan θ dz′ ) r sec θ dθ ( 2) sec θ =⌠ dθ  ⌡θ ( z1′ ) r sec3 θ θ z′ (A.44) ′ θ ( z2 ) ⌠ = cosθ dθ ⌡θ ( z1′ ) r ′ 1   −1  z1 − z     z′ − z   = sin  tan −1    − sin  tan    r   r   r    128 Appendix A Derivation of Electromagnetic Field Equations ′ z2 ⌠ ⌠ z − z′ ′  dz =   ⌡z1′ Rr ⌡z1′ ′ z2 z − z′ ( r + ( z′ − z ) ′ θ ( z2 ) ⌠ =  ⌡θ ( z1′ ) ) −r tan θ ( r + r tan θ dz′ ) r sec θ dθ ( 2) −1 tan θ sec θ =⌠ dθ  sec3 θ ⌡θ ( z1′ ) r θ z′ (A.45) ′ θ ( z2 ) −1 =⌠ sin θ dθ  ⌡θ ( z1′ ) r ′ 1   −1  z1 − z     z′ − z   = cos  tan −1    − cos  tan    r   r   r    δ ( z − zi ) ′ ′ where zi are roots of p , for z1 ≤ ζ 0± ( t ) ≤ z2 , Since δ  p ( z )  = ∑   p′ ( zi ) i z′ r z′ R  ⌠  δ  t m − r  dz′ =  v c  ⌡z1′ cRr  r c(R ) z − ζ 0± ( t ) z′ z′ R  ⌠ z − z′  δ  t m − r  dz′ =  v c   ⌡z1′ cRr ± ζ (t )− z m − v cR ± r ,0 ± r ,0 c ( Rrm ) ,0 z′ z′ R  ⌠  δ  t m − r  dz′ =  v c  ⌡z1′ Rr  ± ζ (t )− z R (A.47) m − v cR ± r ,0 ± r ,0 (A.46) ± ζ (t )− z m − v cR ± r ,0 (A.48) where Rr±,0 = r + ζ 0± ( t ) − z    (A.49) Therefore the expressions for the integrals of g ± ( t ) , m ± ( t ) and n ± ( t ) can be written as shown in equations (A.50) to (A.55) 129 Appendix A  0,    h  + ∫0 g ( t ) dz′ =  4π      4π  Derivation of Electromagnetic Field Equations ζ 0+ ( t ) <  sin θ (ζ 0+ ( t ) )  − sin θ ( )  r      + ζ + t −z r  c ( Rr+,0 ) − − (+)  v cRr ,0   sin θ ( h )  − sin θ ( )       ,   r       , ≤ ζ 0+ ( t ) ≤ h    ζ 0+ ( t ) > h (A.50)  0,    h  + ∫0 m ( t ) dz′ =  4π vε      4π vε  ζ 0+ ( t ) <  sin θ (ζ 0+ ( t ) )  − sin θ ( )  r      + ζ + t −z r  c ( Rr+,0 ) − − (+)  v cRr ,0   sin θ ( h )  − sin θ ( )       ,   r       , ≤ ζ 0+ ( t ) ≤ h    ζ 0+ ( t ) > h (A.51)  0,     4π vε  h  n + ( t ) dz′ =  ∫0        4π vε  ζ 0+ ( t ) <  cos θ (ζ 0+ ( t ) )  − cos θ ( )  z − ζ 0± ( t )      + ζ + t −z  r c ( Rr+,0 ) − − (+)  v cRr ,0  µ − , + 4π R + − − ζ ( t ) − z r ,0  cos θ ( h )  − cos θ ( )       ,   r     v      ≤ ζ 0+ ( t ) ≤ h + cRr ,0 ζ 0+ ( t ) > h (A.52) 130 Appendix A Derivation of Electromagnetic Field Equations Note that the limits ζ 0+ ( t ) < , ≤ ζ 0+ ( t ) ≤ h and ζ 0+ ( t ) > h correspond respectively to t < ∫ −h r2 + z2 , c    4π    f − ( t ) dz′ =   4π   0,   r + ( z − h) r2 + ( z − h) r2 + z2 h h ≤t ≤ + and t > + c v c v c  sin θ ( )  − sin θ ( −h )       ,   r     −  sin θ ( )  − sin θ (ζ ( t ) )  r   +   ζ − t −z r  c ( Rr−,0 ) − (−)  v cRr ,0  ζ 0− ( t ) < − h   , − h ≤ ζ 0− ( t ) ≤    ζ 0− ( t ) > (A.53)    4π vε    g − ( t ) dz′ =  ∫− h  4π vε   0,    sin θ ( )  − sin θ ( − h )       ,   r     −  sin θ ( )  − sin θ (ζ ( t ) )  r   +   ζ − t −z r  c ( Rr−,0 ) − (−)  v cRr ,0  ζ 0− ( t ) < − h   , −h ≤ ζ 0− ( t ) ≤    ζ 0− ( t ) > (A.54) 131 Appendix A Derivation of Electromagnetic Field Equations  I   4π vε   I   4π vε 0  − ∫− h p ( t ) dz′ =       0,    cos θ ( )  − cos θ ( − h )       , ζ 0− ( t ) < − h   r     −  cos θ ( )  − cos θ (ζ ( t ) )   z − ζ 0− ( t )    +   ζ − t −z  r c ( Rr−,0 ) − (−)   v cRr ,0    −h ≤ ζ 0− ( t ) ≤ µI + , − 4π R − − ζ ( t ) − z r ,0 v − cRr ,0 ζ 0− ( t ) > (A.55) Similarly, the limits ζ 0− ( t ) < −h , − h ≤ ζ 0− ( t ) ≤ and ζ 0− ( t ) > correspond respectively t< to r + ( z + h) h t> + v c , r2 + ( z + h) r2 + z2 h ≤t ≤ + c v c and r2 + z2 c 132 ... Negative Upward Figure 2.1 Categorisation of lightning Categories and are relatively rare and generally occur from mountain tops and tall man-made buildings And because the leaders move upward from. .. formulation allows for the determination of lightning radiated electromagnetic fields at any distance and height The resulting waveforms from a randomly generated lightning stroke path demonstrated... the lightning return stroke current at ground and the mathematical formulation for the electromagnetic fields generated by tortuous lightning channels are presented The distributed circuit model