COMPUTATIONS OF ELECTROMAGNETIC FIELDS RADIATED FROM COMPLEX LIGHTNING CHANNELS

Pogess In Electomagnetics Reseach, PIER 73, 93 105, 2007 COMPUTATIONS OF ELECTROMAGNETIC FIELDS RADIATED FROM COMPLEX LIGHTNING CHANNELS T.-X. Song, Y.-H. Liu, and J.-M. Xiong School of Mechanical Engineeing Hubei Univesity of Technology Wuhan 430068, China Abstact In this pape, thee methods fo calculating electomagnetic fields adiated fom complex lightning channels ae discussed, which includes channel obliqueness, banches and totuosity. By otating and moving the coodinate system of the vetical channel, diffeential expessions of electomagnetic fields fom the iegula channel can be deived fom the conclusions of the staight and vetical channel. Though analyzing calculation esults of two examples eveals that channel totuosity and banch is to intoduce the highe fequency content above 100 khz into lightning electomagnetic fields. 1. INTRODUCTION In most computations of lightning electomagnetic fields, the etunstoke channel is assumed to be staight and vetical 1, while it is known to be totuous on scales anging fom less than 1 m to ove 1 km 2. In the case of natual lightning, only subsequent stokes wee consideed. The effects of channel totuosity on etun-stoke adiation fields may be studied theoetically using a piecewise linea epesentation of the lightning channel. In geneal, the effect of totuosity is to intoduce fine stuctue into the time-domain adiation field wavefom and consequently to incease the highe fequency content above 100 khz. At each kink, that is, point at which the linea segments joint, thee is a change in the diection of the popagation of the cuent wave and such changes will intoduce apid vaiations in the adiation field. If consideation fo banches, adiation fields should be ovelaid with electomagnetic fields fom main channel and banches. In Refeence 3, we assume the lightning channel is a staight and vetical antenna, its electomagnetic fields ae calculated with dipole method. Based on Refeence 3, we pesent the methods to calculate

94 Song, Liu, and Xiong electic fields and magnetic fields adiated fom a totuous channel with banches in this pape. 2. OBLIQUE LIGHTNING CHANNEL In lightning engineeing model, we fist assume a lightning channel is oblique, and oblique angle is α see Fig. 1. Afte otating cylindical coodinate system see Fig. 2, channel s length is H and height of etun-stoke cuent is h, the infinitesimal cuent element dz can be looked as an electic dipole at a height z, which moves upwad along the channel at a speed v. The obsevation point P has a Cloud H* h* z '* dz' v iz'*,t R R 0 R H z* P,φ, z z α Gound * Figue 1. Model of oblique lightning channel. x* z H dz' z' Φ z* H* R P z' α E z * * Φ E * E y x * y* Figue 2. Rotating cylindical coodinate system.

Pogess In Electomagnetics Reseach, PIER 73, 2007 95 hoizontal distance fom the lightning channel and a height z, and R is the distance fom the electic dipole. The gound has a pefect conductivity and the ai has a pemittivity ε 0 and pemeability u 0. The cuent iz,t in channel is a function vaying with channel height and time 4. If we find out coesponding elations between two coodinate systems, and substitute them into the fomulations fo the vetical channel in Refeence 3, we will be able to obtain equations to calculate electomagnetic fields of oblique lightning channel. Accoding to geometic elations in Fig. 2, we can obtain = 2 + z 2 sinα tg 1 z 1 2 z = tgα tg 1 z = + z 2 sinα tg 1 z tgα tg 1 z 2 h = h sin α 3 H = H sin α 4 z = z sin α 5 whee z,, h, and z denote the coelative distance in oiginal coodinate system x, y, z espectively. So the distance fom Point P to the bottom of etun-stoke cuent is R 0 = 2 + z 2 6 The distance fom Point P to the top of etun-stoke cuent is R H = 2 +h z 2 { = 2 + z 2 sin 2 α tg 1 z + h 2 sin α + z 2 sin α tg 1 z 2 tg α tg 1 z The distance fom Point P to the dipole dz is R = z z 2 + 2 { = 2 + z 2 sin 2 α tg 1 z + 2 + z 2 sin α tg 1 z 2 tg α tg 1 z z sin α 1/2 1/2 7 8

96Song, Liu, and Xiong We still use dipole method to solve Maxwell s equations. Substituting, h into the fomulations fo the vetical channel in Refeence 3, analytical expessions of electic fields and magnetic fields at Point P, φ, 0 in otating coodinate system x,y,z can be deived as following H φ, 0,t= I 0 2π h 2 + 2 1/2 + E z, 0,t= I 0 th + 2h 2 v + 2 v 2πε 0 h 2 + 2 3/2 1 v h c v h 2 + 2 +h h 2 + 2 1/2 9 2 c 2 h 2 + 2 3/2 1 v + 10 h c h 2 + 2 whee h denotes the height of lightning etun-stoken cuent in otating coodinate system, it can be calculated with Equation 3; denotes hoizontal distance between Point P and the point that lightning stik falls to the gound, it can be calculated with Equation 1. Consequently, in ode to evaluate electomagnetic fields of any point in lightning space, substituting z, z, R into the fomulations in Refeence 3, we can get diffeential expessions of electic fields and magnetic fields in the otating cylindical coodinate system as following de = dz 3 z z t 4πε 0 R 5 it R/cdt + 3 z z 0 cr 4 it R/c + z z it R/c c 2 R 3 11 de z = dz 2z z 2 2 t 4πε 0 R 5 it R/cdt 0 + 2z z 2 2 cr 4 it R/c 2 it R/c c 2 R 3 12 db φ = µ 0dz it R/c it R/c+ 4π R3 cr 2 13 whee R can be computed with Equation 8, z can be computed with Equation 2, z can be computed with Equation 5, can be computed with Equation 1.

Pogess In Electomagnetics Reseach, PIER 73, 2007 97 3. LIGHTNING CHANNEL BRANCHES 3.1. Calculation Method As we know, a lightning channel is iegula, so its geneal field intensity is the sum of electomagnetic fields fom main channel and banches 5. But the diection of eithe coodinate system is diffeent, thus electic fields and magnetic fields should be added in vecto. Assuming the main channel is staight and vetical, the banch is oblique see Fig. 3, electomagnetic fields fom the main channel can still be calculated with the fomulations in Refeence 3, while field intensity expessions of the banch is deived as following. z H P z R* L z'* α O' * z' O R z* z Figue 3. Cylindical coodinate system fo banches. In Fig. 3, Point O is oiginal point, its location is diffeent fom the oblique channel, we need to calculate the displacement at diection once again. The hoizontal distance between Point P and main channel is, the distance between Point P and Point O is +L cos α, theefoe = + L cos α 2 + z 2 sin α tg 1 z 14 + L cos α z = + L cos α 2 + z 2 cos α tg 1 z 15 + L cos α R = z z 2 + 2 { = + L cos α 2 + z 2 sin 2 α tg 1 z + +L cos α 2 +z 2 cos + L cos α α tg 1 z 2 } 1/2 z 16 +L cos α sin α

98 Song, Liu, and Xiong Replacing, z and R in Equations 11, 12, 13 with above thee expessions, field intensity fom the banch at Point P can be witten as de = dz 3 z z t 4πε 0 R 5 it R /cdt + 3 z z 0 cr 4 it R /c + z z it R /c c 2 R 3 17 de z = dz 2z z 2 2 t 4πε 0 R 5 it R /cdt 0 + 2z z 2 2 cr 4 it R /c 2 it R /c c 2 18 R 3 db φ = µ 0dz 4π R 3 it R /c+ it R c cr 2 19 In above thee expessions, R can be calculated with Equation 16, z can be calculated with Equation 15, can be calculated with Equation 14. Assuming E z, E and B φ epesent electomagnetic fields of main channel, Ez, E and Bφ epesent electomagnetic fields of banches see Fig. 4. Fom this figue we may know that electonic fields of both channels have diffeent diections, while the diections of magnetic fields ae same. Theefoe, geneal field intensity at Point P can be evaluated as following. E z B Φ* B Φ E z * P α α E * E Figue 4. Ovelay of electomagnetic fields. Electonic field at z diection: Ez = E z + E z sin α E cos α 20

Pogess In Electomagnetics Reseach, PIER 73, 2007 99 magnetic field at diection: E = E + E z cos α E sin α 21 geneal electonic field: E = Ez 2 + E 2 22 geneal magnetic field: Bφ = B φ + B φ 23 3.2. An Evaluation Example fo Banches We choose a vetical main channel with a banch as an example to evaluate field intensity and compae calculation esults with a vetical channel without banches see Fig. 5. The banch s oblique angle is 60 and its height is 100 m, whole main channel s height is 200 m. At fist we solve electomagnetic fields of both channels espectively, then add thei field intensity in vecto, thus we will be able to get geneal field intensity. The banch s electomagnetic fields can be evaluated accoding to above oblique channel s fomulations, while main channel s electomagnetic fields ae still calculated with the method fo vetical channel. The calculation esults ae shown in Fig. 6 and Fig. 7 fom which we can know that values of main channel with a banch is moe than those who have no banches, thee ae some high fequency contents at stating point of the wavefom, and moe ae banches, moe ae high fequency contents. Main channel H 2 =100m Banch L H 1 =100m 60 o Figue 5. A vetical channel with a banch.

100 Song, Liu, and Xiong Vetical Channel with Banches Vetical Channel Figue 6. Vetical electonic field fom banches = 10 km, z = 0. Vetical Channel with Banches Vetical Channel Figue 7. Azimuthal magnetic field fom banches = 10 km, z = 0. 4. LIGHTNING CHANNEL TORTUOSITY 4.1. Calculation Method Actually natue lightning is totuous and has many banches. In ode to calculate its electomagnetic fields we may divide a totuous channel into many linea segments which may be looked as oblique channels to poceed, so geneal field intensity can be obtained though adding electonic fields and magnetic fields of all linea segments. In Fig. 8, we assume that thee is a totuous lightning channel including two oblique linea segments which length ae L and L 1, oblique angle ae

Pogess In Electomagnetics Reseach, PIER 73, 2007 101 α and α 1 espectively in cylindical coodinate system. Thus Point P s electomagnetic fields fom OO 1 can be evaluated diectly with Equations 11, 12, 13, while field intensity fom O 1 O 2 can not be calculated diectly with these thee equations. Only afte the coodinate system is moved to Point O 1, these thee equations can be used. z z 1 O 2 E z B Φ L 1 R 1 P, z E R Lsinα L O 1 α O Lcosα α 1 1 Figue 8. Coodinate system fo totuous channel. In coodinate system z-o-, Point O 1 s coodinate is L cos α, L sin α, Point P s coodinate is, z. Afte this coodinate system is moved to Point O 1, Point O 1 s coodinate is 0, 0, Point P s coodinate is -L cos α, z-l sin α. Theefoe, eplacing in Equation 1 though 5 with -L cos α, eplacing z with z-l sin α, we can obtain the values of z, z, R 1, and h egading channel O 1 O 2 as following = L cos α 2 +z L sin α 2 1 z L sin α sin α 1 tg 24 L cos α L cos α 2 +z L sin α 2 sin z = tg α 1 tg 1 z L sin α L cos α z L sin α a 1 tg L cos α 25 z = z z L sin α 26 sin α 1 R 1 = 2 +z z 2 { = L cos α 2 +z L sin α 2 sin 2 1 z L sin α a 1 tg L sin α L cos α 2 +z L sin α 2 sin α 1 tg 1 z L sin α L cos α tg α 1 tg 1 z L sin α L cos α

102 Song, Liu, and Xiong z z L sin α 2 } 1/2 27 sin α 1 h = = h sin α 1 ξ 1 ξ 2 ct ξz L sin α sin α 1 ξct z L sin α 2 + L cos α 2 1 ξ 2 28 whee ξ = v/c, c is the tavel speed of electomagnetic wave which equals light speed 3 10 8 m/s in ai medium. Substituting above expessions into Equations 11 12 13, Point P s field intensity fom channel O 1 O 2 can be witten as db φ = µ 0dz 4π R1 3 it R 1 /c+ it R 1 /c cr1 2 29 de = dz 3 z z t 4πε 0 R1 5 it R 1 /cdt 0 + 3 z z cr1 4 it R 1 /c+ z z it R 1 /c c 2 R1 3 30 de z = dz 2z z 2 2 t it R 1 /cdt 4πε 0 0 R 5 1 + 2z z 2 2 cr1 4 it R 1 /c 2 c 2 R1 3 it R 1 /c 31 whee R 1 can be got fom Equation 27, z can be got fom Equation 25, z can be got fom Equation 26, can be got fom Equation 24. Consequently, Point P electonic fields and magnetic fields may be calculated espectively with diffeent equations. Equations 11 12 13 should be used in the case of etun-stoke cuent going though L, o time t L/v v is the speed of etun-stoke cuent, Equations 29 30 31 should be used in the case of etun-stoke cuent going though L 1, o time t > L/v. 4.2. An Evaluation Example fo Totuous Channel In Fig. 9 we assume that a totuous channel is composed of two linea oblique channels. Fom above discussion we know the fist linea

Pogess In Electomagnetics Reseach, PIER 73, 2007 103 segment may be calculated with the equations of oblique channel diectly, while the second segment can be calculated only afte the coodinate system is moved. Figue 9. A totuous channel. L 1 NO.2 H 2 =100m o 60 L NO.1 H 1 =100m 30 o In ode to compae evaluation esults with a vetical channel, we assume the oblique angle of NO. 1 segment α is 30, the angle of NO. 2 segment α 1 is 60, the height of each segment is 100 m and the height of the vetical channel is 200 m, thei lightning cuent paametes ae the same as Refeence 6, etun-stoke speed is 1.5 10 8 m/s 7. Evaluation esults ae shown as Fig. 10 and Fig. 11, the eal line denotes electomagnetic fields fom the totuous channel, the dashed line denotes electomagnetic fields fom the vetical channel. Fom these two figues we can see that initial values of electonic fields and magnetic fields ae diffeent and the values at stating pat of wavefom of totuous channel ae much geate, which illuminate that thee ae Totuous Channel Vetical Channel Figue 10. Vetical electonic field fom totuous channel = 10 km, z = 0.

104 Song, Liu, and Xiong Totuous Channel Vetical Channel Figue 11. Azimuthal magnetic field fom totuous channel = 10 km, z = 0. high fequency contents above 100 khz emeged in electomagnetic fields. While initial values fom the vetical channel ae almost zeo, which illuminate that thee ae no high fequency contents coming fom a vetical channel. Othewise, the values of electomagnetic fields fom the vetical channel ae geate than totuous channel on the same conditions. Theefoe channel totuosity will intoduce highe fequency contents into electomagnetic fields, and moe ae totuosity, moe ae high fequency contents 8, 9. 5. CONCLUSION In this pape, thee methods fo calculating electomagnetic fields adiated fom complex lightning channels have been discussed, which includes channel obliqueness, banches and totuosity. By otating and moving the coodinate system of the vetical channel, diffeential expessions of electomagnetic fields fom the oblique channel can be deived fom the conclusions of the vetical channel. Electomagnetic fields fom the channel banch may be evaluated though ovelaying electonic fields and magnetic fields fom a vetical main channel and oblique channels. The totuous channel can be divided into many linea oblique segments fom which field intensity can be added to obtain geneal electomagnetic fields fom whole totuous channel. Though analyzing calculation esults of two examples eveals that channel totuosity and banches ae to intoduce the highe

Pogess In Electomagnetics Reseach, PIER 73, 2007 105 fequency contents above 100 khz into electomagnetic fields, and moe ae totuosity and banches, moe ae highe fequency contents. Meanwhile, field intensity fom a channel with banches is geate than single vetical channel. REFERENCES 1. Rubinstein, M. and M. A. Uman, Methods fo calculating the electomagnetic fields fom a known souce distibution: application to lightning, IEEE Tansactions on Electomagnetic Compatibility, Vol. 31, No. 2, 183 189, 1989. 2. Uman, M. A., Natual lightning, IEEE Tansactions on Industy Applications, Vol. 30, No. 3, 785 790, 1994. 3. Song, T.-X. and C. Wang, Two numeical methods fo calculating electomagnetic fields adiated fom natue lightning, Jounal of Electomagnetic Waves and Applications, Vol. 19, No. 4, 513 528, 2005. 4. Nucci, C. A., et al., Lightning etun stoke cuent models with specified channel-base cuent: a eview and compaison, J. Geophys. Res., Vol. 95, 20395 20408, 1990. 5. Rakov, V. A. and M. A. Uman, Review and evaluation of lightning etun stoke models including some aspects of thei application, IEEE Tansactions on Electomagnetic Compatibility, Vol. 40, No. 4, 403 426, 1998. 6. Zhao, X. and K. Huang, Calculation of pobability distibution of Maximal eceived powe of electonic eceive in lightning electomagnetic envionment, Jounal of Electomagnetic Waves and Applications, Vol. 19, No. 2, 221 230, 2005. 7. Mu, M. K., J. T. Huangfu, L. X. Ran, and K. Zang, Design of lightning potecto compatible fo both 2G and 3G cellula systems, Jounal of Electomagnetic Waves and Applications, Vol. 20, No. 15, 2167 2175, 2006. 8. Li, J. Y. and Y.-B. Gan, Multi-band chaacteistic of open sleeve antenna, Pogess In Electomagnetics Reseach, PIER 58, 135 148, 2006. 9. Sijhe, T. S. and A. A. Kishk, Antenna modeling by infinitesimal dipoles using genetic algoithms, Pogess In Electomagnetics Reseach, PIER 52, 225 254, 2005.