On the Influence of Channel Tortuosity on Electric Fields Generated by Lightning Return Strokes at Close Distance

Transcription

1 Pogess In Electomagnetics Reseach B, Vol. 74, 6 75, 27 On the Influence of Channel Totuosity on Electic Fields Geneated by Lightning Retun Stokes at Close Distance Calo Petaca *, Simone Minucci, and Amedeo Andeotti Abstact In this pape the esults of the estimated electic field associated with totuous lightning paths at close distance (5 m to 5 m) ae shown. Such esults ae compaed with expeimental data available in the liteatue and ae illustated along with a quantitative analysis of the field wavefoms and thei fequency specta. The limits of the usual staight-vetical channel assumption and the influence of totuosity at diffeent azimuth and distances fom the lightning channel base ae also highlighted.. INTRODUCTION A model fo the estimation of electic and magnetic fields geneated by lightning channels is extemely impotant [ 3]. Indeed, the computed fields can be used as input to models devoted to the calculation of induced voltages and cuents, and thei popagation, e.g., [4 8], can suppot the design of electical potection system o help to size outdoo electical components [9 ] and can be used to estimate the lightning potection level of existing stuctues [2]. Moeove, suitable compute models can help eseaches in undestanding the physics of lightning by deiving some typical physical paametes fom measuements [3], such as the shape and ise of the etun stoke cuent, its attenuation, distotion, and speed, and they can also impove the identification of the elationship of these paametes with electic field chaacteistics. Many models have been poposed to study the cloud-to gound lightning and to calculate the fields. In paticula, lightning etun stoke models have been intensely studied, since the etun stoke flashes ae the most poweful known lightning pocesses [4, 5]. A significant limitation of most of these models is that the lightning channel is usually assumed to be staight and vetical although, actually, the path followed by a etun stoke is totuous [6]. Moeove, it is woth noticing that the lightning channel usually stikes the gound with a cetain inclination with espect to the vetical, and its effects should not be neglected [7], since they can significantly affect both EM fields [8 2] and induced voltages [2 23]. Consequently, these simplified models ae of geat elevance but can miss some peculia aspects, such as the fine stuctue and fequency content of the fields, o they can even undeestimate thei amplitude, both in natual and tiggeed lightning. It is theefoe impotant to study the effects of channel totuosity on the fields, by adopting a suitable geometical stuctue of the path and a pope lightning etun-stoke model. A pioneeing pape on modeling lightning electomagnetic fields poduced by totuous channels was pesented by Le Vine and Meneghini [24], who used a piecewise linea model fo a channel geneated by inteconnecting a seies of segments automatically geneated basing on assigned statistics. In ode to pedict the fields, the authos employed the Faunhofe appoximation, and consequently thei esults wee limited to fa-field computation (i.e., km fom the base of the channel). A significant contibution to ovecome such a limitation was published in [25], whee a closed-fom solution fo the Received 27 Januay 27, Accepted 27 Mach 27, Scheduled Apil 27 * Coesponding autho: Calo Petaca (petaca@unina.it). The authos ae with the Depatment of Electical Engineeing and Infomation Technology, Univesity of Naples Fedeico II, Naples 825, Italy.

2 62 Petaca, Minucci, and Andeotti fields geneated by a lightning channel with abitay location and slope was given, without intoducing any appoximation in the analytical field calculation. In [26], the esults wee also extended to a compute-geneated totuous channel, made of a seies of abitaily oiented staight segments, teated individually. A ecent pape [27] detailed the influence of totuous channels and, in paticula, the contibution to distant electic fields of the etun-stoke cuent speed of the lightning channel geomety and of its distance fom the obseve. Chia and Liew [28] obtained simila esults on andomly geneated totuous lightning channels even though they limited thei study only to one type of channel. Song et al. [29] also computed electomagnetic fields adiated fom complex lightning channels. They showed that the totuosity intoduces fequency contents above khz and that the field intensity geneated by a channel with banches is geate than the one geneated by a single vetical channel. Thei analysis was limited to vey schematic shapes fo the channel to epoduce totuosity and also to a small numbe of obsevation points. Moe ecently, Meedith et al. [3] studied the effect of totuosity by adopting an atificially geneated path. Thei finding was that totuosity is only significant at elatively close distances and that the electomagnetic field poduced by a totuous channel becomes indistinguishable fom that poduced by a staight and vetical channel fo distances geate than m. Thei esults wee patial since they wee extemely influenced by the shape of the geneated channel, having a zig-zag symmetical stuctue. The effects poduced by a eal lightning channel (actually, only the x-z plane of a eal channel was digitized fom published photogaphs, while the y-z segments wee atificially geneated by compute) wee then studied in [3]. Convesely, it was shown that, at close distances, tilt of the lowe segments of the channel stongly affects the electic field amplitude (but not the waveshape) while only at lage distances, totuosity poduces jagged field wavefoms. The pesent pape has the aim to povide a futhe contibution to the undestanding of the elationship between the lightning channel shape and the geneated electic fields, by extending the analysis to close electic fields, which ae fields geneated at distances between 5 m and 5 m fom the channel base. As in [3], the lightning channels have been digitized fom eal pictues [2]. The field wavefoms and thei fequency specta have been computed and analyzed, and the esults have been discussed and compaed both to data obtained fom numeical simulations on staight vetical lightning paths and to expeimental data available in the liteatue, which ae indeed vey few. It is woth noticing that the model is geneal and able to calculate the fields at any point of the fee space, but compaisons ae limited to obsevation points at gound level whee most of the expeimental data ae collected. Although the numbe and type of channels chosen fo the simulations cannot be a epesentative set fo a statistical desciption of the electic field wavefoms, the esults clealy show the contibution that a moe detailed epesentation of the lightning channel geomety can give the chaacteization of the lightning electomagnetic envionment. In the following, Section 2 shows the lightning channels used fo the simulations and povides the analytical solution fo the electic field adiated by an abitaily oiented channel; Section 3 pesents numeical esults and discussions, and the conclusions ae dawn in Section ELECTRIC FIELDS PRODUCED BY LIGHTNING CHANNELS 2.. Totuous Lightning Channels The channels adopted fo the simulations have been obtained by digitizing channel geometies of natual flashes ecoded by Idone and Oville [32]; they ae epoduced in Fig.. Since in [32] only a 2D epesentation in the xz plane is given, the yz plane pojection of the path is geneated hee by imposing two main constaints, on the mean absolute value [ ΔΩ of the deviation angle fom the aveage diection, and on the length L b of each segment of the channel. In paticula, fo each channel, we have assumed ΔΩ =7 and 5 m L b m, accoding to data epoted in [32]. Even though the numbe of constaints may appea highe than the degees of feedom of each segment of the channel, it is woth stessing that the deviation angle is computed by using a Gaussian distibution whose median is ΔΩ. Fig. 2 shows the obtained 3D paths of all the channels shown in Fig..

3 Pogess In Electomagnetics Reseach B, Vol. 74, z [m] Ch Ch2 Ch3 Ch4 Ch x [m] φ P y [m] 7 Figue. Lightning channels used fo simulations (see Idone et al. [32]). Figue 2. 3-D view of lightning channels. P is the obsevation point, φ is the azimuth angle Calculation of Electic Field Fo EM fields calculation, the analytical appoach adopted in [25] has been implemented, needing a simple analytical epesentation of the single dischage cuent, tavelling an abitaily oiented channel C as a function of time and space. In paticula, it is possible to calculate the step esponse analytically by assuming that a unit step-function cuent tavels along the channel: { ( ) ( ) [ ( i z,t = u t z u z ) ( u z h)] } () v whee v is the etun stoke font speed, and u is the Heaviside step function. The oveall effect of the totuous channel can then be found by summing up the effects of the cuent i(z,t) tavelling though all its segments. Once the electic field due to a unit step cuent has been calculated, the field y(t) associated with an abitay cuent wavefom i(t) can be obtained by convolution integation y (t) = t di (τ) s (t τ) dτ (2) dτ whee s(t) is the field component geneated by a unit step cuent u(t z v ) tavelling along C, andy(t) is the field component geneated at the same point by the geneic cuent i(t). By adopting a pope cylindical efeence system R (Fig. 3), with the z-axis coincident with the axis of the channel segment and the oigin coincident with the stating point O, the mathematical expession of the fields can be simplified since only and z components of the electic field ae pesent. The geneated electic field in the cylindical efeence system R at the obsevation point P (, φ, z) assumes the fomulation epoted in Eq. (3), whee c is the speed of light. The detailed calculations of the field can be found in [25] and [26]. [ ( )] [ )] E = E ϕ = E z = tu t 2 +z 2 c 4πɛ ( 2 +z 2 ) 3 + [ ( z tu t 2 +z 2 c )] 4πɛ ( 2 +z 2 ) 3 + ( t v l [ ( (z l) t v l ) u (t lv 2 +(z l) 2 c 4πɛ ( 2 +(z l) 2) 3 + ) u (t lv 2 +(z l) 2 c )] 4πɛ ( 2 +(z l) 2) 3 + l g (t) dz l p (t) dz (3)

4 64 Petaca, Minucci, and Andeotti z'' x l z' O ϕ dz' C z R z y'' P(, ϕ,z) y O'' x'' Figue 3. Geneic dischage channel C and main geometical paametes. whee the integals l l l g (t) dz = p (t) dz = l g(t)dz and p(t)dz have the following values: if z (t) < 4πɛ v 4πɛ v sin(ϕ 2) sin(ϕ ) + cr 2 v (z (t) z) cr [ ( ( )) ] sin actg l z sin(actg( z )) if z (t) l if z (t) >l if z (t) < 4πɛ v 4πɛ v cos(ϕ 2) cos(ϕ ) (z z + (t)) cr 2 v (z (t) z) cr [ ( ( )) ] cos actg l z cos(actg( z )) if z (t) l if z (t) >l (4) (5) in which and z (t) = t ( v z c t 2 v z ) 2 ( c 2 v ) ( 2 c t ( ) c z2 2 c 2 ) (6) v 2 c 2 ϕ =actg ( z ) ( ) ϕ 2 =actg z (t) z (7) R = 2 +(z z (t)) 2 Since the electic field has to be calculated in the pesence of a soil-ai inteface, the image pinciple has been adopted [33], tuning the analysis in two diffeent media into an equivalent analysis caied out in one homogeneous medium, by popely placing a set of souces into the whole fee space. Thei magnitude is calculated in such a way to keep the same inteface conditions on the sepaation suface between the two media. Thanks to the uniqueness of the solution of Maxwell s equation with pescibed bounday conditions, the supeposition of the two image solutions gives the total electic field as in the eal geomety, in which the soil-ai inteface is taken into account. The gound is modelled as a pefectly conducting plane (i.e., its conductivity is infinite) and the contibution of the image souces has been obtained in the same way, by adopting a cylindical coodinate system with the z-axis coincident with the axis of the image channel.

5 Pogess In Electomagnetics Reseach B, Vol. 74, i (t) [ka] Time [ μs] Figue 4. Channel base etun stoke cuent adopted in the simulations. Concening the analytical expession of the channel base cuent, the sum of a Heidle function [34] and a double exponential function has been used in Eq. (8) because it is well suited fo successive etun stokes. Theefoe, it can be used as souce at the channel base fo the popagation along the photogaphed channels of Fig.. i (,t)= I η ( t τ ) n +( t τ ) n e t ( τ 2 + I 2 e t τ 3 e t τ 4 ) If the values poposed in [35] ae adopted (τ =.72 µs, τ 2 = 5. µs, τ 3 = µs, τ 4 = 6. µs, I =9.9kA, I 2 =7.5kA, η =.845 and n = 2), we get the wavefom shown in Fig. 4. The cuent is assumed to popagate at constant speed v = 2 m/µs (β = v c =.4) and to decay with the height accoding to the Modified Tansmission Line Linea (MTLL) model [36], which is appopiate fo electic field wavefoms at close distances. A linea cuent decay along the channel is consideed, fom a maximum value at the channel base to zeo at the channel top. The paamete H is 7.5 km. 3. RESULTS AND DISCUSSION The electic field has been calculated at the gound suface (z = ) at diffeent distances d (5 m, 5 m, 25 m and 5 m) fom the channel base and at diffeent azimuth values φ. In paticula, fo each distance d, 24 obsevation points P wee selected, stating fom φ = up to φ = 36,insteps of Δφ =5 (see Fig. 2). In this way, it is possible to show the dependence of the fields on the elative position between the channel and the obsevation point and to focus on the diffeences with the fields calculated by adopting a simple vetical channel model, which ae φ independent. Befoe caying out the analysis of the fields, it should be consideed that expeimental data fo electic fields geneated at close distance by downwad natual lightning dischages to gound ae few and limited. In fact, fo evident difficulties, most of the measuements ae taken duing tiggeed lightning, fomed by an upwad-extending leade, diffeent fom natual lightning. Fo this eason, we emak that, in what follows, expeimental data should only be consideed as a valuable help in intepeting the esults fom simulation but cannot be used as a benchmak fo testing the validity of the model. 3.. Electic Field E(t) Electic field waveshapes fom etun stokes measued at to 2 km ae pesented in [37] and classified in [38]. Electic and magnetic field waveshapes geneated by leade and etun stoke, measued at close distance, can be found in [39 4]. A typical electic field wavefom shows: (i) an initial slow font lasting fom five up to ten micoseconds; (8)

6 66 Petaca, Minucci, and Andeotti (ii) a successive shap incease in few micoseconds; (iii) the pesence of a fist peak (not always visible at vey close distances); (iv) a chaacteistic flattening, due to the main contibution given by the electostatic component. Sometimes the wavefoms exhibit small humps, whose oigin is uncetain, but that can be pobably due to chage motion in channels with majo bends (Fig. 5). Figue 5. Measued E-field wavefoms. Adapted fom Jeauld et al. [39]. The electic field wavefoms calculated fom the simulations on totuous channels, as shown in Fig. 6, ae able to epoduce these typical chaacteistics. As an example, in Fig. 6 the time evolution of the vetical electic field calculated at gound, at fou diffeent distances (5 m, 5 m, 25 m and 5 m) and fou diffeent obsevation points (φ =9, φ = 35, φ = 65, φ = 27 ) ae illustated, in the case of lightning channel n.. The field evolution pesents both the shap incease and humps, but not the slow font chaacteistic; this is pobably due to the pesence of an upwad connecting leade [38] which, as discussed above, is not accounted in the pesent model, but is typical fo tiggeed channels. In Fig. 6, the electic field wavefom geneated by a vetical lightning path is also plotted as a benchmak, in ode to show effects of the channel geomety in clea detail. At vey close distances (up to 5 m), the effect of channel totuosity is negligible; indeed, the electic field wavefoms show a egula shape, with no pesence of a fist peak and no jaggedness. Such a behavio can be explained by consideing the contibutions to the E-field of each segment composing the channel: the lowe sections of the lightning path (i.e., those nea the gound) dwaf the effects of the uppe sections. A slight effect of totuosity is given by small humps visible in Fig. 6(b) at position φ = (35, 27 ). Moeove, the effect of the channel inclination (mainly of the lowe segments of the channel) is consideable, because the electic field is now stongly dependent on the obsevation point and can be significantly diffeent fom the field calculated in the same point and poduced by a vetical channel. As an example, if in Fig. 6(a) we calculate the electic field amplitude E p at t =2µs afte the stoke initiation and compae it to the value obtained fo a vetical channel (E pv =25.6kVm ), we find that such a value is much highe (E p =6.4kVm )atφ =9 and sensibly lowe (E p =4.6kVm )at the opposite obsevation point φ = 27. Such an aspect puts in evidence a geat limitation of the staight vetical channel model; it will be discussed in detail in subsection 3.2 of the pesent section. When looking at distances highe than 25 m, the fine stuctue of the fields due to channel totuosity stats to appea, and the small humps ae much moe ponounced (see Figs. 6(c) and 6(d)). In fact, the uppe segments of the lightning channel ae no longe masked by the lowe sections and give thei significant contibution to the electic field. At these distances, the totuous model is also able to epoduce anothe inteesting featue of close electic fields, namely a vey fast tansition in a time inteval of a few hundeds of nanoseconds towads a fist peak, followed by a slowe incease of the field [38]. The fist peak is due to the field adiation component, it is almost independent of the obsevation point and can be now moe easily identified, if compaed to close wavefoms. This aspect cannot be consideed in the staight vetical channel model. The calculation of the electic fields geneated by all the channels epesented in Fig. shows that all the coesponding wavefoms exhibit simila behavios, although the channels can be vey diffeent fom each othe. Fo example, it is possible to compae the fields geneated by channel n. (shown in Fig. 6) and those geneated by channel n.2 (shown in Fig. 7).

7 Pogess In Electomagnetics Reseach B, Vol. 74, =9 =35 =65 =27 vet =9 =35 =65 =27 vet (a) (b) =9 =35 =65 =27 vet. 3 =9 =35 =65 =27 vet (c) Figue 6. Electical field E(t) at(a)d =5m,(b)d = 5 m, (c) d = 25 m and (d) d = 5 m as a function of the azimuth angle φ (channel n.). The ed line coesponds to E(t) geneated by a vetical channel. Also in this case it is possible to obseve that thee is always an obsevation point P M (φ = 3 in Fig. 7(a)) whee the highest value of the electic field is eached and a point P m, on the opposite side (φ = 2 in Fig. 7(a)), with the lowest value at close distances. Moeove, the effects of totuosity and humps ae evident at distances above 25 m, whee the pesence of the fist peak in the electic field can also be noticed Effects of Channel Inclination The effects of the inclination of the lowe segments of the channel can be descibed by intoducing the paamete DE φ %, which descibes the distance between the time evolution of the electic field E pφ (t) (poduced by a totuous lightning path and calculated at azimuth φ) and the electic field E pv (t) (geneated by a vetical channel and calculated in the same obsevation point). In ode to evaluate the paamete DE φ %, we have to conside the two time-seies E pφ (k) and E pv (k), calculated in a fixed time inteval T s,wheek =, 2,...,N s, N s being the numbe of samples. The paamete is defined by the following Equation (9): DE φ %= N s ( ) E pφ (k) E pv (k) N s E pv (k) (9) k= (d)

8 68 Petaca, Minucci, and Andeotti 5 =2 =2 =27 =3 vet. 5 =2 =2 =27 =3 vet (a) (b) 7 =2 =2 =27 =3 vet. 3 =2 =2 =27 =3 vet (c) (d) Figue 7. Electical field E(t) at(a)d =5m,(b)d = 5 m, (c) d = 25 m and (d) d = 5 m as a function of the azimuth angle φ (channel n.2). The ed line coesponds to E(t) geneated by a vetical channel. If the fields ae equal, then DE φ % = ; othewise, DE φ % inceases as the diffeence in the calculated fieldsisgeate. Figue 8 shows DE φ % calculated in the time inteval T s =µs as a function of the azimuth angle φ at vaious distances, fo channels to 5. Fom the figue we can infe that the electic field can be sensibly diffeent fom the field poduced by a staight vetical channel, depending on the obsevation point and on the shape of the lightning path. Fo instance, at d = 5 m (Fig. 8(a)), fo channel we find a maximum (DE φ max % = 3%) at φ =9 and a minimum (DE φ min %=7%)atφ = and φ = 65 ; fo channel 5, the maximum (DE φ max % = 6%) is calculated at φ =75 while the minimum DE φ min %=2.7% can be found, again, at φ = and φ = 65. Less ponounced ae the maxima calculated fo channel 2 and channel 3 (DE φ max % = 38% and DE φ max = 4%, espectively) while only fo channel 4, DE φ % is vey small and almost independent of φ. In the latte case the field amplitude is almost equal to that geneated by a lightning cuent flowing along a vetical path. Such behavio can be explained by caefully obseving Fig. 9. In this figue we have shown: i) on the left side the elative position of the obseve with espect to the lightning channels (channel, pictued in black, and channel 4, depicted in magenta); ii) on the ight side the time evolution of the electic field calculated in the position of the obseve (5 m) and poduced by the same cuent flowing in the afoementioned channels and in the vetical one. At viewpoint φ = (Fig. 9(a)) both channels and 4 appea nealy vetical to the obseve: in

9 Pogess In Electomagnetics Reseach B, Vol. 74, DE% Azimuth angle Azimuth angle DE% DE% (a) (b) Azimuth angle (c) (d) Figue 8. Paamete DE φ % as a function of azimuth angle at vaious distances, (a) d =5m,(b) d = 5 m, (c) d = 25 m and (d) d = 5 m. such position DE φ % is vey low, and the electical field amplitude is vey simila to that calculated fo a vetical lightning path, as confimed by the wavefoms shown in Fig. 9(e). The electic field wavefoms show analogous behavio (Fig. 9(g)) if being calculated at viewpoint φ = 65 (Fig. 9(c)); in fact in that position the shape of the channels still esembles that of a vetical path. At position φ =9 (Fig. 9(b)), channel 4 is still seen as vetical by the obseve, while channel is extemely inclined towads it. The espective electic fields, as depicted in Fig. 9(f), ae significantly diffeent: i) the field poduced by channel 4 is almost equal to the field poduced by the vetical channel, and DE φ % in that position is vey low (DE φ % = 7%); ii) the field poduced by channel is sensibly highe and DE φ % 3%. In paticula at time instant t =2µs, the electic field amplitude due to a vetical channel is E =25.5 kv/m, while the field poduced by channel is E = 6 kv/m, i.e., 235% highe. The limits of the vetical channel model ae in this case clealy pointed out. In Fig. 9(d), coesponding to the viewpoint φ = 27, the lightning channel moves away fom the obseve, and the esulting electic field is again diffeent fom the field poduced by a vetical path (DE φ % = 4%), but in this case, its amplitude is much lowe (E =2.3kV/matt =2µs) than the field computed with the vetical channel (E= 25.5 kv/m). Again, the poposed model shows the limitations of using a simplified cylindically symmetic electic field distibution. Anothe inteesting featue can be obtained by analyzing the electic field poduced by channel and channel 5 at a distance d = 5 m and d = 5 m and calculated at the viewpoint φ =9. Fom Figs. 8(b) and 8(d), we deive that the index DE φ % deceases with distance fo channel (fom 45% at d = 5 m to 49% at d = 5 m) while it inceases fo channel 5 (fom 8% at d = 5 m to 95%

10 7 Petaca, Minucci, and Andeotti z-axis [m] z-axis [m] z-axis [m] z-axis [m] x φ-axis [m] (a) -5 5 x φ-axis [m] (b) -5 5 x φ-axis [m] (c) -5 5 x φ-axis [m] (d) Electic Field [kv/m] Electic Field [kv/m] Electic Field [kv/m] Electic Field [kv/m] ch ch4 vet ch ch4 vet (f) 8 7 ch ch4 vet (g) 8 7 (e) ch ch4 vet (h) Figue 9. Channel (blue) and channel 4 (magenta) seen fom diffeent obsevation points at d =5m. (a) φ =,(b)φ =9,(c)φ = 65 and (d) φ = 27 and (e), (f), (g), (h) coesponding time evolution of the electic field.

11 Pogess In Electomagnetics Reseach B, Vol. 74, ch ch5 vet x -axis [m] (a) (c) 5 ch ch5 vet x -axis [m] (b) (d) Figue. Channel (blue) and channel 5 (geen) seen obsevation point φ =9 at (a) d = 5 m, (b) d = 5 m and (c), (d) coesponding time evolution of the electic field. at d = 5 m). The explanation of such a behavio can be found again in the elative position of the obseve with espect to the lightning path. At close distance (d = 5 m) channel is moe inclined than channel 5 with espect to the obseve, and the poduced electic field diffes fom that of a vetical one much moe than the latte (Fig. (c)). At d = 5 m, the inclination of lowe channel segments has a mino influence, and the wavefom of the electic field depends on the shape of the entie lightning path, as confimed by Figs. (b) and (d) Electic Field Specta The knowledge of the electic field specta can be of geat impotance to chaacteize the lightning phenomena, since it can be a vey useful tool both fo scientific investigations on lightning dischage physics and fo engineeing assessment theat on electonic devices. Lightning specta have been geneally obtained by two main techniques: using naow-band eceives, which measue the enegy adiated at the tuning fequency o ecoding the electic field wavefom with wide bandwidth instuments and then applying the Fouie tansfomation [42, 43]. Expeimental data geneally efe to distant fields (up to some km) and show that etun stokes ae the stongest souce of adio-fequency adiation in the inteval fom [.2 2] MHz. Within this inteval, the specta ae quite simila fo vaious lightning events, and thei amplitude mainly deceases as the invese of fequency (f ) [, 42 44]. Figue shows the specta obtained as the aveage amplitude fo 5 cloud-to-gound dischages obseved in 22 in Tibet aea [42].

12 72 Petaca, Minucci, and Andeotti The esults of the simulations fo channel ae shown in Fig. 2. The electic field specta at close distance (fom d =5mtod = 25 m) in the ange [.7 7] MHz ae simila to that of a vetical channel; moeove, they do not depend on azimuth angles, and thei amplitude tends to decease oughly as f. Figue. Measued E-field specta at km. Adapted fom Chen et al. [42]. (a) (b) (c) (d) Figue 2. Electical field specta at (a) d =5m,(b)d = 5 m, (c) d = 25 m, and (d) d = 5 m as a function of azimuth angle (Channel n.). The ed line coesponds to the specta of a vetical channel.

13 Pogess In Electomagnetics Reseach B, Vol. 74, (a) (b) Figue 3. Electical field specta by channels to 5 at azimuth angle ϕ =9 at (a) d =5m,(b) d = 5 m. The ed line coesponds to the electical field spectum of a vetical channel. At geate distance, the effect of totuosity intoduces an incease in the fequency content above 2 khz. In fact, each kink geneates a change in the diection of popagation of the cuent which ceates a apid change in the electic field. With espect to the specta poduced by a vetical channel, the main changes in the amplitude geneally occu at fequencies in the ange aound 3 khz 3MHz, iespectively of the azimuth angle. The esults ae confimed by Fig. 3 whee the fequency specta fo all the 5 lightning channels, calculated at azimuth position φ =9 and at distance d =5mandd = 5 m ae plotted. At close distance the specta ae simila to that of a vetical channel, although scaled, whilst fathe fom the channel base changes in the fequency content ae obseved in the ange 3 khz 3 MHz. 4. CONCLUSIONS Channel totuosity can significantly affect the electic field geneated by cloud-to-gound lightning. At close distance, the field amplitude mainly depends on the inclination of the bottom segments of the channel and on the elative position of the obseve. Fields can be even doubled if being compaed to the field poduced by a vetical channel. At longe distances the effect of the oveall totuosity can be bette seen, because of the effect of a longe potion of the lightning channel which is affected by the etun stoke cuent. Fequency spectum is dependent on totuosity, and diffeences with a vetical channel can be found in the ange 3 khz 3 MHz. These consideations point out the limits of the evaluation of electic fields based on a simplified model in which the lightning path is consideed vetical. Such aspects should be consideed in compute models used fo the estimation of the fields geneated by lightning. REFERENCES. Uman, M. A. and V. A. Rakov, Lightning Physics and Effects, Cambidge Univesity Pess, Uman, M. A., The At and Science of Lightning Potection, Cambidge Univesity Pess, New Yok, Cooay, V., (edito), Lightning Electomagnetics, IET Powe and Enegy Seies 62, Rachidi, F., C. A. Nucci, and M. Ianoz, Tansient analysis of multiconducto lines above a lossy gound, IEEE Tans. Powe Delivey, Vol. 4, No., , Jan Diendofe, G., Induced voltage on an ovehead line due to neaby lightning, IEEE Tans. on Electomagnetic Compatibility, Vol. 32, No. 4, , Nov Høidalen, H. K., J. Slebtak, and T. Heniksen, Gound effects on induced voltages fom neaby lightning, IEEE Tans. on Electomagnetic Compatibility, Vol. 32, No. 4, , Nov. 99.

14 74 Petaca, Minucci, and Andeotti 7. Andeotti, A. P. and V. A. Rakov, An analytical appoach to calculation of lightning induced voltages on ovehead lines in case of lossy gound Pat I: Model development, IEEE Tans. Powe Delivey, Vol. 28, No. 2, , Nov Andeotti, A. P. and V. A. Rakov, an analytical appoach to calculation of lightning induced voltages on ovehead lines in case of lossy gound Pat II: Compaison with othe models, IEEE. Tans. Powe Delivey, Vol. 28, No. 2, , Nov Boghetti, A., S. Moched, F. Napolitano, C. A. Nucci, and M. Paolone, Lightning-induced ovevoltages tansfeed though distibution powe tansfomes, IEEE Tans. Powe Delivey, Vol. 24, No., , Jan Tesche, F. M., A. W. Kälin, B. Bändli, B. Reusse, M. Ianoz, D. Taba, and P. Zweiacke, Estimates of lightning-induced voltage stesses with buied shielded conduits, IEEE Tans. on Electomagnetic Compatibility, Vol. 4, , Petache, E., F. Rachidi, M. Paolone, C. A. Nucci, V. A. Rakov, and M. A. Uman, Lightning induced distubances in buied cables Pat I: Theoy, IEEE Tans. on Electomagnetic Compatibility, Vol. 47, No. 3, , IEEE Guide fo Impoving the Lightning Pefomance of Electic Powe Ovehead Distibution Lines, IEEE Standad 4, Cummins, K. L., Lightning infomation fo use in powe systems analysis: How much moe do we need to know?, Tansmission and Distibution Confeence and Exhibition 22: Asia Pacific, IEEE/PES, Vol., , Yokohama, Japan, Oct Rakov, V. A. and M. A. Uman, Review and evaluation lightning etun stoke models including some aspects of thei application, IEEE Tans. on Electomagnetic Compatibility, Vol. 4, No. 4, , Rakov, V. A. and F. Rachidi, Oveview of ecent pogess in lightning eseach and lightning potection, IEEE Tans. on Electomagnetic Compatibility, Vol. 5, No. 3, , Uman, M. A., The Lightning Dischage, Academic Pess, San Diego, CA, Uman, M., J. Schoene, V. Rakov, K. J. Rambo, and G. H. Schnetze, Coelated time deivatives of cuent, electic field intensity and magnetic flux density fo tiggeed lightning at 5 m, Jounal of Geophysical Reseach, Vol. 7, , Izadi, M., M. Z. A. A. Kadi, and C. Gomes, Evaluation of electomagnetic fields associated with inclined lightning channel using second ode FDTD-hybid methods, Pogess In Electomagnetics Reseach, Vol. 7, , Gomes, C., V. Cooay, and M. Z. A. Ab Kadi, Vetical electic fields and field change paametes due to patly inclined lightning leade channels, Pogess In Electomagnetics Reseach, Vol. 35, 55 8, Amaasinghe, D., U. Sonnadaa, M. Beg, and V. Cooay, Channel totuosity of long laboatoy spaks, Jounal of Electostatics, Vol. 65, No. 8, , Andeotti, A., U. De Matinis, C. Petaca, V. A. Rakov, and L. Veolino, Lightning electomagnetic fields and induced voltages: Influence of channel totuosity, 3th URSI Geneal Assembly and Scientific Symposium, URSIGASS, pape 6572, Tukey, Andeotti, A., C. Petaca, V. A. Rakov, and L. Veolino, Calculation of voltages induced on ovehead conductos by nonvetical lightning channels, IEEE Tans. on Electomagnetic Compatibility, Vol. 54, No. 4, 86 87, Jan Andeotti, A., C. Petaca, and A. Pieno, On the effects of channel totuosity in lightning-induced voltages assessment, IEEE Tans. on Electomagnetic Compatibility, Vol. 57, No. 5, 96 2, Oct Le Vine, M. and R. Meneghini, Simulation of adiation fom lightning etun stokes: The effects of totuosity, Radio Sci., Vol. 3, No. 5, 8 89, Sep./Oct Lupò, G., C. Petaca, V. Tucci, and M. Vitelli, EM fields geneated by lightning channels with abitay location and slope, IEEE Tans. on Electomagnetic Compatibility, Vol. 42, No., 39 53, Feb. 2.

15 Pogess In Electomagnetics Reseach B, Vol. 74, Lupò, G., C. Petaca, V. Tucci, and M. Vitelli, EM fields associated with lightning channels: On the effect of totuosity and banching, IEEE Tans. on Electomagnetic Compatibility, Vol. 42, No. 4, , Nov Petaca, C., Geometical and physical paametes affecting distant electic fields adiated by lightning etun stokes, Pogess In Electomagnetics Reseach B, Vol. 58, 67 8, Chia, K. L. and A. C. Liew, Effect of totuosity of lightning stoke path on lightning electomagnetic fields, Asia-Pacific Symposium on EMC, , Singapoe, Song, T. X., Y. H. Liu, and J. M. Xiong, Computations of electomagnetic fields adiated fom complex lightning channels, Pogess In Electomagnetics Reseach, Vol. 73, 93 5, Meedith, S. L., S. K. Eales, I. N. Kostanic, N. E. Tune, and C. E. Oteo, How lightning totuosity affects the electomagnetic fields by augmenting thei effective distance, Pogess In Electomagnetics Reseach B, Vol. 25, 55 69, Andeotti, A., G. Lupo, and C. Petaca, Evaluation of EM fields fom etun stoke fo indiect Lightning potection of wind tubines, 23 Intenational Confeence on Clean Electical Powe (ICCEP), , Algheo, Italy, Jun Idone, V. P. and R. E. Oville, Channel totuosity vaiation in Floida tiggeed lightning, Geophysical Reseach Lettes, Vol. 5, No. 7, , Jul Jackson, J. D., Classical Electodynamics, John Wiley & Sons, Heidle, F., Taveling cuent souce model fo LEMP calculation, 6th Int. Zuich Symposium on Electomagnetic Compatibility, 57 62, Zuich, Switzeland, Nucci, C. A., G. Diendofe, M. A. Uman, F. Rachidi, M. Ianoz, and C. Mazzetti, Lightning etun stoke cuent models with specified channel-base cuent: A eview and compaison, Jounal of Geophysical Reseach, Vol. 95, No. D2, , Nov Rakov, V. A. and A. A. Dulzon, A modified tansmission line model fo lightning etun stoke field calculations, 9th Int. Symposium on Electomagn. Compat., , Zuich, Switzeland, Ma Baba, Y. and V. Rakov, Electic and magnetic fields pedicted by diffeent electomagnetic models of the lightning etun stokes vesus measued fields, IEEE Tans. on Electomagnetic Compatibility, Vol. 5, No. 3, , Nov Lin, Y. T., M. A. Uman, J. A. Tille, R. D. Bantley, W. H. Beasley, E. P. Kide, and C. D. Weidman, Chaacteization of lightning etun stoke electic and magnetic fields fom simultaneous two-station measuements, Jounal of Geophysical Reseach, Vol. 84, , Jeauld, J., M. A. Uman, V. A. Rakov, K. J. Rambo, D. M. Jodan, and G. H. Schnetze, Electic and magnetic fields and field deivatives fom lightning stepped leades and fist etun stokes measued at distances fom to m, Jounal of Geophysical Reseach, Vol. 3, No. D7, 5, Sep Uman, M., V. Rakov, G. H. Schnetze, K. J. Rambo, D. E. Cawfod, and R. J. Fishe, Time deivative of the electic field, 4 and 3 m fom tiggeed lightning stokes, Jounal of Geophysical Reseach, Vol. 5, No. D2, , Rubinstein, M., F. Rachidi, M. A. Uman, R. Thottappillil, V. A. Rakov, and C. A. Nucci, Chaacteization of vetical electic fields 5 m and 3 m fom tiggeed lightning, Jounal of Geophysical Reseach, Vol., No. D5, , Chen, M., Y. Du, and W. Dong, Some new obsevations of lightning specta in the bands above 25 MHz, Radio Science Confeence, , Aug Le Vine, M., Review of measuements of the RF spectum of adiation fom lightning, Meteoology and Atmospheic Physics, No. 37, 95 24, Willett, J. C., J. C. Bailey, J. C. Leteintuie, and E. P. Kide, Lightning electomagnetic adiation field specta in the inteval fom.2 to 2 MHz, Jounal of Geophysical Reseach, Vol. 95, No. 2, , 99.