Relativistic Doppler effect in an extending transmission line: Application to lightning

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi:1.129/21jd15279, 211 Relatiisti Doppler effet in an extending transmission line: Appliation to lightning Abdolhamid Shoory, 1 Farhad Rahidi, 1 and Maros Rubinstein 2 Reeied 2 Noember 21; reised 24 Marh 211; aepted 7 April 211; published 6 July 211. [1] We present in this paper a thorough analysis of urrent wae propagation with arbitrary speed along an extending transmission line. We derie rigorous analytial equations in the time and frequeny domains expressing the refletions of the urrent wae ourring at the extending end of the line. The deried equations reeal that it is not possible to represent urrent refletions ourring at the extending end of a transmission line using a onstant, frequeny independent refletion oeffiient, as preiously done in the literature. The refleted wae from the extending end of the line is shown to be affeted by the Doppler frequeny shift. In other words, the refleted wae from an extending transmission line suffers distortion, the amount of whih depends on the inident wae form, its frequeny ontent, and the speed of the extending end of the line. The deried expression is in agreement with the relatiisti Doppler effet and is onsistent with the Lorent transformation. Finally, engineering models for return strokes are generalied and losed form analytial expressions are deried for the spatial temporal distribution of the urrent along the hannel aounting for refletions at ground and at the return stroke wae front taking into aount the Doppler effet. Citation: Shoory, A., F. Rahidi, and M. Rubinstein (211), Relatiisti Doppler effet in an extending transmission line: Appliation to lightning, J. Geophys. Res., 116,, doi:1.129/21jd Introdution and Bakground [2] Lightning return strokes models an be lassified, depending on the type of goerning equation, into four lasses of models [Rako and Uman, 1998], namely, (1) gas dynami models, (2) eletromagneti models, (3) distributediruit models, and (4) engineering models. [3] Among these lasses of models, the engineering models hae been extensiely used sine the 194s to study eletromagneti radiation from lightning return strokes (see Nui et al. [199] for a reiew). In these models the spatial and temporal distribution of the hannel urrent (or the hannel harge density) is speified as a funtion of the urrent at the hannel base, the return stroke speed, and a number of adjustable parameters [Thottappillil et al., 1997; Rako and Uman, 1998]. These models inlude the Brue Golde model [Brue and Golde, 1941], the traeling urrent soure (TCS) model [Heidler, 1985], the transmission line (TL) model [Uman and MLain, 1969], the modified transmission line with exponential urrent deay with height (MTLE) model [Nui et al., 1988; Rahidi and Nui, 199], the modified transmission line with linear urrent deay with height (MTLL) model [Rako and Dulon, 1987], and the 1 Eletromagneti Compatibility Laboratory, Swiss Federal Institute of Tehnology, Lausanne, Switerland. 2 Department of Eletrial Engineering and Computer Siene, Uniersity of Applied Sienes of Western Switerland, Yerdon les Bains, Switerland. Copyright 211 by the Amerian Geophysial Union /11/21JD15279 Diendorfer and Uman (DU) model [Diendorfer and Uman, 199] and its modified ersion (MDU) [Thottappilil et al., 1991]. Reent deelopments on lightning return stroke models an be found in the work of Rako and Rahidi [29]. [4] The engineering models an be grouped in two ategories, the lumped soure (also referred to as transmissionline type or urrent propagation) models and the distributedsoure (also referred to as traeling urrent soure type or urrent generation) models [Rako and Rahidi, 29]. Cooray [23] showed that any lumped soure (LS) model an be formulated in terms of soures distributed along the hannel and progressiely atiated by the upward moing return stroke front, as preiously demonstrated for the MTLE model by Rahidi and Nui [199]. None of the engineering models in their original formulations onsidered the possibility of urrent refletions from the extending return stroke wae front. (This assertion applies to all lasses of return stroke models.) [5] For lightning strikes to tall towers, possible urrent refletions from the return stroke wae front hae been onsidered in a few studies [e.g., Janishewskyj et al., 1998; Shostak et al., 1999; Shostak et al., 2; Napolitano and Nui, 29; Mosaddeghi et al., 21]. Speifially, Shostak et al. [2] extended the MTLE model for the modeling of the lightning attahment to the 553 tall CN tower assuming a onstant urrent refletion oeffiient of.9 at the moing front of the lightning return stroke. These studies hae shown that taking into aount possible refletions at the return stroke wae front results in a better reprodution of the fine struture of the lightning urrent and radiated eletromagneti fields 1of8

2 Figure 1. An extending transmission line along the axis. (Z M is the mathing impedane). L a (, t) is the apparent length of the line seen by an obserer (shown with an eye symbol) at spatial position and at time t and L d (t) isthe dynami spatial position of the upper end of the line at any time t. assoiated with strikes to the CN Tower. Note that an inomplete refletion at the return stroke wae front indiates impliitly that part of the urrent is transmitted on to the leader region aboe the front. [6] Further, the lassial refletion mehanism employed in these studies is only alid when the return stroke wae front is stationary or at least moing with speeds signifiantly smaller than the speed of upward waes transmitted into the hannel due to transient phenomena inside the tower. In an attempt to gie a more realisti aount of the boundary onditions at the moing return stroke wae front, Heidler and Hopf [1994a, 1994b] deried an expression for the urrent refletion oeffiient using the TCS model and onsidering ground initiated lightning return strokes. The deried expression for the urrent refletion oeffiient is solely a funtion of the return stroke speed and the speed of light (( )/( + )). The same expression for the urrent refletion oeffiient was later used by Shul and Diendorfer [1995] who proposed an extended ersion of the DU return stroke model [Diendorfer and Uman, 199] to alulate radiated fields at different distanes from the lightning hannel. Interestingly, their simulation results show that the urrent was disontinuous at the return stroke wae front. [7] More reently, Mosaddeghi et al. [21] presented an extension of the engineering return stroke models for lightning strikes to tall strutures that takes into aount the presene of refletions at the return stroke wae front and the presene of an upward onneting leader. They used a similar approah as in the work of Shostak et al. [2] but using the expression of Heidler and Hopf [1994a, 1994b] for the urrent refletion oeffiient at the return stroke wae front. Simulation results for the magneti fields were ompared with experimental wae forms assoiated with lightning strikes to the CN Tower (553 m) and the preditions taking into aount refletions at the wae front and the presene of upward onneting leaders were found to be in better agreement with experimental obserations. Raysaha et al. [21] presented an analysis taking into aount nonlinear hannel dynamis and orona effets along the hannel. Their results suggest that the transmitted waes from the tall tower to the hannel undergo signifiant attenuation in the region near the return stroke wae front resulting in negligible refletion. [8] In this paper, we will present a rigorous analysis of wae propagation along an extending transmission line with an arbitrary speed. The analysis will then be applied to lightning return stroke modeling. The presene of eleated strike objets and/or upward onneting leaders is disregarded in the present analysis. [9] The paper is organied as follows. In setion 2 we will present a theoretial analysis of the urrent wae refletion from an extending transmission line both in the time domain (setion 2.1) and in the frequeny domain (setion 2.2). The proposed formulation will be examined from the point of iew of the relatiisti Doppler effet in setion 2.3. Setion 3 will present the extension of the engineering return stroke models taking into aount refletions at the extending return stroke wae front. The extension is based on the distributed soure representation of the engineering models whih allows a straightforward inlusion of refletions at both ends of the return stroke hannel (ground leel and the return stroke wae front). A disussion will be proided in setion 4 and, finally, a summary and onlusions will be gien in setion Pulse Propagation in an Extending Transmission Line [1] Consider a lossless transmission line along the axis as shown in Figure 1. The lower termination of the line is fixed at =. It has an initial length of L at time t = and lengthens upwards along the positie axis with a onstant speed. The basi problem of the radiation from suh a line, but with a stati upper end and a square pulse waeform was studied by Rubinstein and Uman [1991]. In this paper, an arbitrary urrent soure with a waeform f(t) exites the line at its bottom end. The wae form propagates up along the line with a speed (note that although we hae used to represent the speed of the upward moing wae front in the presented deriation, it does not hae to be the speed of light; it only has to be greater than ; howeer, in this ontext, it is usually assumed to be the speed of light) greater than and will eentually ath up with the moing upper end of the line (assumed to be an open iruit). The wae will then be refleted and it will begin to propagate bak down the line with the same speed. For the sake of simpliity we will ignore in this setion any refletions from the lower end of the line. In other words, we assume a perfetly mathed termination at that end obtained by using the line s harateristi impedane. Suh refletions will be onsidered in setion 3 where the proposed formulation will be applied to lightning return stroke modeling. Further, we assume that the wae form suffers no distortion as it propagates up and down along the line. Two different deria- 2of8

3 tions will be gien in the following subsetions, the first in the time domain and the seond in the frequeny domain Time Domain Deriation [11] We will follow an instantaneous alue f (t ) in the wae form f (t) as it traels up the line, passes a spatial point, and omes bak to this point after being refleted at the extending termination of the line. The spatial position of this instantaneous alue f(t ) at any time t until it athes up with the upper end of the line is gien by L p ðþ¼t t ð t Þut ð t Þ ð1þ where u(t) is the unit Heaiside step funtion and the subsript p indiates the length along the line traersed by the spatial position of the instantaneous alue f(t ). On the other hand, the dynami spatial position of the upper end of the line at any time t an be written as L d ðþ¼l t þ tuðþ t The height at whih the two spatial positions gien by (1) and (2) are idential is in fat the enounter point of the upward moing wae form instant and the extending end of the line, whih an be obtained by soling the following equation for the enounter time t t ð t Þut ð t Þ ¼ L þ tuðþ t Sine the enounter will happen neessarily at a time t > t, we an rewrite (3) dropping out the step funtions as follows: ð2þ ð3þ t ð t Þ ¼ L þ t ð4þ from whih we an sole for the time t at whih the onsidered instantaneous alue f(t ) of the wae form f(t) reahes the top of the line, t ¼ L þ t Although t represents the ath up time and thus is a partiular alue for this ariable, we purposely did not inlude a subsript to aoid the partiulariation, sine suh a time exists for any instant in the exiting wae form. [12] On the other hand, soling (4) or (5) for t gies t ¼ t ð ÞL The instantaneous inident wae form seen at the upward extending end of the line, f i (L d (t), t) is therefore gien by t ð ÞL f i ðl d ðþ; t tþ ¼ fðt Þ ¼ f ð7þ This funtion will produe the right alue of the exiting wae form reahing the moing front of the line whih is generally appliable to the whole wae form. It is this funtion that needs to be refleted bak from the top of the extending line. Assuming a omplete refletion (openiruit ondition), we get the refleted wae form at the line top L d (t) as follows t ð ÞL f r ðl d ðþ; t tþ ¼ f ð8þ ð5þ ð6þ To find the refleted wae form at an arbitrary spatial point along the line, we an proeed as we did to alulate the time dependene of the wae form at the ath up point. In doing so, we take (5), whih expresses the time at whih an instantaneous alue f (t ) of the wae form f (t) launhed at instant t from the bottom meets with the moing front. We will now add to that time the interal needed for the refleted wae to reah the obseration point. To obtain this interal, we first use (2) and (5) to alulate the height of the enounter H e ¼ L d ðþ¼l t þ L þ t The time interal required for the refleted wae to trael from this height to point at the speed is then gien by Dt ¼ H e L þ L þ t ¼ ð9þ ð1þ The total time that the instantaneous alue at t = t o takes to trael from the base of the line up to the upward moing end and bak down to position (using as a referene the time t = at whih the exitation at the bottom of the line is initiated) is gien by the sum of (5), whih is the elapsed time until the upward exitation reahes the rising end of the line and (1) whih is the time from that enounter point bak down to position : L þ t t ¼ L þ t L þ þ Soling (11) for t, we get t ¼ þ t þ 2L þ ð11þ ð12þ whih represents the instant of the exiting wae form f(t) seen at position at any time t due to the first refletion off the moing top end of the line. Note that by using instant we mean the time orresponding to a gien point in the exiting wae form f (t). [13] The refleted wae form at position is therefore gien by the exiting wae form f(t) ealuated at the time t gien by (12), f r ð; tþ ¼ fðt Þ ¼ f þ t þ 2L þ ð13þ On the other hand, the inident wae form at point and time t is simply the retarded alue of the exiting wae form gien by f i ð; tþ ¼ f t ð14þ The total wae form at spatial point an then be obtained by superposition, adding the inident and refleted wae forms gien by (14) and (13), respetiely, as follows: f t ð; tþ ¼ f t f þ t þ 2L þ ð15þ It an be seen from (15), howeer, that the refleted wae form does undergo distortion as a result of the refletion at 3of8

4 the extending end of the transmission line. Equation (15) anbeomparedinfatwithbergeronequations[teshe et al., 1997], with the only differene that Bergeron equations apply to stationary lines. The presene of an extending end, howeer, results in the dispersie fator ( )/( + ). [14] In setion 2.2 we present an equialent frequeny domain analysis, and we show that this distortion is atually the Doppler effet. Note that in any stage of the deelopment of the aboe formulation, the ausality of the wae forms should be maintained. In other words, any resulting wae form annot hae nonero alues before the onset of the exiting wae form at time t = Frequeny Domain Deriation [15] Let us assume that the exiting wae form at the bottom end of the line is a sinusoid of onstant amplitude A i and frequeny w gien by A i e jwt. Then, with referene to Figure 1, the inident wae form at spatial position due to this exitation an be written as F i ð; t;! Þ ¼ A i e j!t e j! ð16þ Assuming that the apparent length of the line seen by an obserer at spatial position and at time t is L a (, t), we an write t ¼ L að; tþl þ L að; tþ ; L > ð17þ The first term in (17) is the time delay taken by the line to extend from its initial length L to its apparent length L a (, t) and the seond term is the retardation time from the extending end of the line to the obseration point at. Soling (17) for L a (, t) yields L a ð; tþ ¼ t þ L þ þ ð18þ Note that the apparent length gien by (18) is learly different from the dynami length gien by (2) and also the spatial position of the instantaneous alue f (t ) in the inident wae form gien by (1). The inident wae gien by (16) traels up along the line, passes point, and reflets bak from the extending end of the line to the position. The refleted wae seen by the obserer at suh a position an then be written as F r ð; t;! Þ ¼ A r e j!t e ;t j!la ð Þ La ;t e j! ð Þ ð19þ Assuming a omplete refletion at the moing end of the line (open iruit ondition), the total wae, whih is the sum of the inident and refleted waes, should anish at this end, i.e., F i ðl a ð; tþ; t;! ÞþF r ðl a ð; tþ; t;! Þ ¼ ð2þ Inserting (16) and (19) into (2) and after straightforward mathematial manipulations, we obtain A r ¼A i e j! La ;t ð Þ ð Þ ð21þ Replaing (21) into (19), replaing L a (, t) from (18), and again after straightforward mathematial manipulations, we obtain F r ð; t;! Þ ¼ A i e j! þð tþ Þ e j! 2L þ ð22þ The total wae at spatial position an then be written as the sum of the inident (16) and refleted (22) waes as follows: F t ð; t;! Þ ¼ A i e j! ð t Þ Ai e j! þð tþ Þ 2L þ ð Þ ð23þ Equation (23) was obtained assuming a single frequeny harmoni exitation at the bottom end of the line. It is lear that sine any gien wae form in the time domain an be represented using its Fourier transform, equation (23) an be easily transformed into the time domain to gie equation (15) for a general time domain exitation, namely f (t) Relation to Relatiisti Doppler Effet [16] It an be readily seen from (23) that the refleted wae from the extending end of the line has a frequeny that is shifted in spetrum from the soure frequeny. This is the so alled Doppler effet usually understood in its lassial form [Cheng, 1983]. To explain suh a frequeny shift from a relatiisti Doppler effet point of iew and to show that it is onsistent with Lorent transformation, we first fix the soure emitting a signal with frequeny w at = and let the obserer moe with the extending end of the line at speed. Aording to the relatiisti Doppler effet (see, for instane, hapter 11 of Jakson [1999]), the obserer reeies the soure signal at a different frequeny gien by rffiffiffiffiffiffiffiffiffiffi! o1 ¼! þ ð24þ Now let us assume another obserer loated at =. The extending end of the line, after reeiing the inident wae and transmitting it bak through refletion ats as another soure emitting a signal with frequeny w o1 toward this obserer. Sine this soure is again moing away from the obserer with speed, the reeied frequeny by the obserer at = an then be written as rffiffiffiffiffiffiffiffiffiffi! o2 ¼! o1 þ ¼! þ whih is the frequeny deried in (23). 3. Reision of Return Stroke Models ð25þ [17] In this setion, the formulation we deeloped for the Doppler effet in an extending transmission line will be used to reise engineering return stroke models taking into aount the refletions from the return stroke wae front. We will first onsider the MTLE model for ground initiated lightning return strokes and then we will generalie the formulation to other models. [18] The spatial temporal distribution of the return stroke urrent along a ertial hannel (see Figure 2) aording to the MTLE model [Nuietal., 1988; Rahidi and Nui, 199] is gien by i; ð t Þ ¼ e i ; t u t ð26þ 4of8

5 Figure 2. Distributed soure representation of the lightning hannel in engineering return stroke models for the ase of no strike objet but onsidering refletions at ground (adopted from Rahidi et al. [22]). (a) The urrent soure loated at is aboe the obseration point at and (b) the urrent soure loated at is below the obseration point at. where is the height aboe the ground, l is the attenuation height onstant, i(,t) is the urrent at the hannel base, and is the return stroke speed assumed to be onstant. The spatial temporal distribution of the urrent (26) an be iewed as being due to the ontribution of distributed soures along the hannel [Rahidi and Nui, 199]. Eah soure is swithed on when the return stroke wae front reahes its altitude and deliers a urrent whih flows down the hannel at the speed of light. The general expression for suh urrent soure loated at height is gien by Rahidi and Nui [199] as 8 >< t < di s ð; tþ ¼ >: g t e d t ð27þ where g(t) an be an arbitrary funtion. Assuming a omplete math at the hannel base similar to the situation shown in Figure 1 in whih the line is onneted to its harateristi impedane, the expression for the urrent distribution at a gien obseration point along the hannel was obtained by integrating the ontributions of all urrent soures aboe it as follows: i; ð tþ ¼ ZH ð;tþ di s ; t d ¼ ZH ð;tþ g t e d ð28þ where H(, t) is the apparent height of the return stroke wae front as seen by an obserer at height, whih is gien by Hð; tþ ¼ þ t þ ð29þ In partiular, the urrent at the hannel base an be obtained from (28) letting = ið; tþ ¼ ZH ð;tþ g t e d ð3þ 5of8

6 Table 1. P() and * for Fie of the Engineering Return Stroke Models a Model P() * BG 1 TCS 1 TL 1 MTLL 1 /H tot MTLE Exp( /l) a Rako and Uman [1998]. H tot is the total return stroke hannel height and l is the attenuation height onstant in the MTLE model. For the ase where > (Figure 2b), the elemental urrent seen by an obserer at due to a urrent soure at an be written as di 2 ð; ; tþ¼ e d g g t g g k t þ 2 gg k t þ 2 g g k2 t þ þ 3 g g k2 t 3 g g k3 t þ þ... ð35þ Combining (26) and (28) we an write e i ; t ¼ ZH ð;tþ g t e d ð31þ Regrouping similar terms, we an write ( X di 2 ð; ; tþ¼ e d ð1þ nþ1 n g g kn1 t 2 ð1þ n n g g kn t ) ð36þ Now, let us inlude refletions both at ground leel and at the return stroke wae front in the analysis. In doing so, the return stroke hannel is assumed to be a transmission line whose bottom end is fixed at ground leel and features a onstant, frequeny independent refletion oeffiient, r g, for downward urrent waes and its upper end is extending with speed, featuring refletions harateried by the Doppler effet for upward propagating urrent waes formulated in setion 2. Any downward wae, when refleted upward from the hannel base, ats a soure loated at the lower end of the transmission line model of the lightning return stroke hannel in a similar way as shown in Figure 1. For the ase where > (Figure 2a), the elemental urrent seen by an obserer at due to a urrent soure at an be written as di 1 ð; ; tþ¼ e d g t þ þ g g t g g k t þ 2 gg k t þ 2 g g k2 t þ þ 3 g g k2 t 3 g g k3 t þ þ... where k is gien by k ¼ þ ð32þ ð33þ Regrouping similar terms, we an write di 1 ð; ; tþ ¼ e d g t ð1þ nþ1 n g g kn1 t 2 ð1þ n n g g kn t ð34þ The total urrent at a height due to suh a distributed urrent soure representation an then be obtained by integrating (34) and (36) as follows: Z i; ð tþ ¼ di 2 ð; ; tþd þ ZH ð;tþ Replaing (34) and (36) into (37), we obtain i; ð tþ¼ ZH ð;tþ þ þ dg t e ZH ð;tþ ZH ð;tþ e d X e d X ð1þ nþ1 n g g kn1 t ð1þ n n g g kn t di 1 ð; ; tþd ð37þ 2 Finally, using (31), we an simplify (38) to yield i; ð tþ ¼ e i ; t ð1þ nþ1 n g i ; kn1 t 2 ð38þ ð1þ n n g i ð ; kn tþ ð39þ Equation (39) is a generaliation of the MTLE model in whih refletions at ground leel and at the return stroke wae front are both taken into aount. In generaliing the aboe deriation to other engineering models, we follow the approah used by Rahidi et al. [22]. In this regard, we note that the following expression an be used to express the spatial temporal distribution of the urrent for most of the engineering return stroke models [Rako and Uman, 1998] i; ð tþ ¼ P ðþi ; t u t ð4þ * where P() is the attenuation funtion, is the return stroke wae front speed defined earlier, and * is the urrent wae 6of8

7 speed. P() and * for fie of the engineering return stroke models to be disussed in what follows are shown in Table 1. [19] Applying the same proedure that led to (39) for the other four models, we arrie at the following general expression for the urrent wae form along the hannel aounting for refletions at ground and at the return stroke wae front taking into aount the Doppler effet: i; ð tþ ¼ P ðþi ; t ð1þ nþ1 n g * i ; kn1 t 2 4. Disussion ð1þ n n g i ð ; kn tþ ð41þ [2] The deried equations (15) and (23) imply that it is not possible to represent refletions ourring at the extending end of a transmission line using a onstant, frequenyindependent refletion oeffiient, as preiously done in the lightning literature. The refleted wae from an extending transmission line suffers distortion, the amount of whih depends on the inident wae form and its frequeny ontent. [21] When the speed of the extending transmission line is muh smaller than that of the propagating pulses,,itis easy to see that equations (15) and (23) redue to the expressions for a lassial refletion from a stati openiruited transmission line for whih the refletion oeffiient is equal to 1. On the other hand, if the speed of the extending transmission line is assumed equal to the speed of the propagating pulses ( = ), areful examination of (15) and (23) shows that no refletion would our at the extending end of the transmission line. [22] The reised expression for engineering return stroke models (41) aounts rigorously for the boundary ondition at the extending return stroke wae front and guarantees therefore the urrent ontinuity. Also, the formulation is shown to be onsistent with the relatiisti Doppler effet, whih was not aounted for in preious studies. Indeed, in suh a speed range (speeds near the speed of light), any formulation should satisfy the speial theory of relatiity and the Lorent transformation. 5. Conlusions [23] The possibility of urrent refletions ourring at the extending end of a return stroke hannel has been onsidered in seeral reent studies and inluded in the return stroke models assuming a onstant refletion oeffiient at the return stroke wae front. [24] In this paper, we presented a thorough analysis of urrent wae propagation with arbitrary speed along an extending transmission line. We deried rigorous analytial equations in the time and the frequeny domains expressing the refletions ourring at the extending end of the line. The deried equations reealed that it is not possible to represent refletions ourring at the extending end of a transmission line using a onstant, frequeny independent refletion oeffiient, as preiously done in the lightning literature. The refleted wae from the extending end of the line was shown to be affeted by the Doppler frequeny shift. In other words, the refleted wae from an extending transmission line suffers distortion, the amount of whih depends on the inident wae form, its frequeny ontent, and the speed of the extending end of the line. The deried expression is found to be in agreement with the relatiisti Doppler effet and is onsistent with the Lorent transformation. [25] Finally, engineering models for return strokes were generalied aounting for refletions at the ground and at the return stroke wae front taking into aount the Doppler effet. Closed form analytial expressions were deried for the spatial temporal distribution of the urrent along the hannel. [26] The extension of the presented analysis to inlude the presene of a tall strike objet and an upward onneting leader is straightforward. Work is in progress to analye the effet of refletions on radiated eletromagneti fields. [27] Aknowledgments. This work is supported by the European COST Ation P18 (The Physis of Lightning Flash and Its Effets). Thanks are owed to M. A. Uman, V. A. Rako, G. Diendorfer, and the anonymous reiewers for their preious suggestions. Referenes Brue, C. E. R., and R. H. Golde (1941), The lightning disharge, J. Inst. Eletr. Eng., 88(6), Cheng, D. K. (1983), Field and Wae Eletromagnetis, Addison Wesley, Boston, Mass. Cooray, V. (23), On the onepts used in return stroke models applied in engineering pratie, IEEE Trans. Eletromagn. Compat., 45(1), 11 18, doi:1.119/temc Diendorfer, G., and M. A. Uman (199), An improed return stroke model with speified hannel base urrent, J. Geophys. Res., 95(D9), 13,621 13,644, doi:1.129/jd95id9p Heidler, F. (1985), Traeling urrent soure model for LEMP alulation, paper presented at 6th Symposium and Tehnial Exhibition on Eletromagneti Compatibility, Swiss Fed. Inst. of Tehnol., Zurih, Switerland. Heidler, F., and C. Hopf (1994a), Lightning urrent and Lightning eletromagneti impulse onsidering urrent refletion at the Earth s surfae, paper presented at 22nd International Conferene on Lightning Protetion, Teh. Uni. of Budapest, Budapest, Hungary. Heidler, F., and C. Hopf (1994b), Influene of the lightning hannel termination on the lightning urrent and lightning eletromagneti impulse, paper presented at International Aerospae and Ground Conferene on Lightning and Stati Eletriity, Bund. Fuer Wehrerwaltung und Wehrtehnik, Mannheim, Germany. Jakson, J. (1999), Classial Eletrodynamis, 3rd ed., Wiley, New York. Janishewskyj, W., V. Shostak, and A. M. Hussein (1998), Comparison of lightning eletromagneti field harateristis of first and subsequent return strokes to a tall tower: 1. Magneti field, paper presented at 24th International Conferene on Lightning Protetion, Staffordshire Uni., Birmingham, U.K., Sept. Mosaddeghi, A., F. Rahidi, M. Rubinstein, F. Napolitano, D. Paanello, V. Shostak, W. Janishewskyj, and M. Nyffeler (21), Radiated fields from lightning strikes to tall strutures: Effet of refletions at the return stroke waefront and an upward onneting leader, IEEE Trans. Eletromagn. Compat., 53(2), Napolitano, F., and C. A. Nui (29), An engineering model aounting multiple refletions at the waefront of return strokes to tall towers, in 4th International Workshop on Eletromagneti Radiation from Lightning to Tall Strutures, Uni. of Toronto, Montreal, Quebe, Canada. Nui, C. A., C. Maetti, F. Rahidi, and M. Iano (1988), On lightning return stroke models for LEMP alulations, paper presented at 19th International Conferene on Lightning Protetion, Aust. Eletr. Asso., Gra, Austria, May. Nui, C. A., G. Diendorfer, M. Uman, F. Rahidi, M. Iano, and C. Maetti (199), Lightning return stroke urrent models with speified hannelbase urrent: A reiew and omparison, J. Geophys. Res., 95(D12), 2,395 2,48, doi:1.129/jd95id12p2395. Rahidi, F., and C. A. Nui (199), On the Master, Uman, Lin, Standler and the modified transmission line lightning return stroke urrent models, J. Geophys. Res., 95(D12), 2,389 2,393, doi:1.129/jd95id12p of8

8 Rahidi, F., V. A. Rako, C. A. Nui, and J. L. Bermude (22), Effet of ertially extended strike objet on the distribution of urrent along the lightning hannel, J. Geophys. Res., 17(D23), 4699, doi:1.129/ 22JD2119. Rako, V. A., and A. A. Dulon (1987), Calulated eletromagneti fields of lightning return strokes, Tekh. Elektrodin., 1, Rako, V. A., and F. Rahidi (29), Oeriew of reent progress in lightning researh and lightning protetion, IEEE Trans. Eletromagn. Compat., 51(3), , doi:1.119/temc Rako, V. A., and M. A. Uman (1998), Reiew and ealuation of lightning return stroke models inluding some aspets of their appliation, IEEE Trans. Eletromagn. Compat., 4(4), , doi:1.119/ Raysaha, R. B., U. Kumar, V. Cooray, and R. Thottappillil (21), Speial ase of lightning strike to tall objets on ground, paper presented at 3th International Conferene on Lightning Protetion, Uni. of Cagliari, Cagliari, Italy, Sept. Rubinstein, M., and M. A. Uman (1991), Transient eletri and magneti fields assoiated with establishing a finite eletrostati dipole, reisited, IEEE Trans. Eletromagn. Compat., 33(4), , doi:1.119/ Shul, W., and G. Diendorfer (1995), Properties of an extended Diendorfer Uman return stroke model [lightning], paper presented at Ninth International Symposium on High Voltage Engineering, Inst. of High Voltage Eng., Gra, Austria. Shostak, V., W. Janishewskyj, A. M. Hussein, J. S. Chang, and B. Kordi (1999), Return stroke urrent modeling of lightning striking a tall tower aounting for refletions within the growing hannel and for upwardonneting disharges, paper presented at 11th International Conferene on Atmospheri Eletriity, Global Hydrol. Clim. Cent., Guntersille, Ala. Shostak, V., W. Janishewskyj, A. Hussein, and B. Kordi (2), Eletromagneti fields of lightning strikes to a tall tower: A model that aounts for upward onneting disharges, paper presented at 25th International Conferene on Lightning Protetion, Uni. of Patras, Rhodes, Greee. Teshe, F. M., M. Iano, and T. Karlsson (1997), EMC Ananlysis Methods and Computational Models, Wiley Intersi., New York. Thottappillil, R., D. K. MLain, M. A. Uman, and G. Diendorfer (1991), Extension of the Diendorfer Uman lightning return stroke model to the ase of a ariable upward return stroke speed and a ariable downward disharge urrent speed, J. Geophys. Res., 96(D9), 17,143 17,15, doi:1.129/91jd1764. Thottappillil, R., V. A. Rako, and M. A. Uman (1997), Distribution of harge along the lightning hannel: Relation to remote eletri and magneti fields and to return stroke models, J. Geophys. Res., 12(D6), , doi:1.129/96jd3344. Uman, M. A., and D. K. MLain (1969), Magneti field of the lightning return stroke, J. Geophys. Res., 74(28), , doi:1.129/ JC74i28p6899. F. Rahidi and A. Shoory, Eletromagneti Compatibility Laboratory, Swiss Federal Institute of Tehnology, CH 115 Lausanne, Switerland. (farhad.rahidi@epfl.h; abdolhamid.shoory@epfl.h) M. Rubinstein, Department of Eletrial Engineering and Computer Siene, Uniersity of Applied Sienes of Western Switerland, CH 14 Yerdon les Bains, Switerland. (maros.rubinstein@heig d.h) 8of8