On return stroke currents and remote electromagnetic fields associated with lightning strikes to tall structures: 1. Computational models

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, D1311, doi:1.129/26jd7958, 27 On return stroke currents and remote electromagnetic fields associated with lightning strikes to tall structures: 1. Computational models D. Paanello, 1 F. Rachidi, 1 M. Rubinstein, 2 J. L. Bermudez, 3 W. Janischewskyj, 4 V. Shostak, 5 C. A. Nucci, 6 A. M. Hussein, 4,7 and J. S. Chang 8 Receied 24 August 26; reised 8 February 27; accepted 6 April 27; published 3 July 27. [1] In this paper, analytical expressions relating far fields and currents associated with lightning strikes to tall towers are deried. The deried equations are general and can be used with any engineering model. It is shown that the far field can be decomposed into three terms, namely (1) contribution of the main return stroke pulse along the lightning channel, (2) contribution of the multiple-reflection process along the eleated strike object, including the contribution of upward propagating pulses transmitted into the channel, and (3) contribution of the so-called turn-on terms, associated with the current discontinuity at the return stroke waefront. This discontinuity is associated with neglecting reflections at the upward moing front. It is also shown that only the first term is model-dependent and that the far field current relationship does not significantly depend on the adopted engineering model. It is therefore possible to use the analytical equation relating the current peak and the associated distant electric or magnetic fields, deried by Bermudez et al. (25) for the transmission line (TL) model, for any engineering model extended to include a tall strike object. It is also shown that the peak amplitude of the electromagnetic field radiated by a lightning strike to a tall structure is relatiely insensitie both to the alue of the return stroke speed and the top reflection coefficient. These results and findings of this paper emphasize the key differences between return strokes initiated at ground leel (or from short strike objects) and those striking tall towers: (1) The electromagnetic field from lightning strikes to tall towers is largely determined by the tower and only to a lesser extent by the channel, (2) electromagnetic fields associated with tall strike objects are less model-sensitie than those corresponding to a strike to ground; in particular, the early time response of the field is nearly model-independent, and (3) unlike ground-initiated strikes, for which the far field peak is strongly dependent on the return stroke speed (proportional according to the TL model), far field peaks associated with strikes to tall strike objects are little sensitie to the return stroke speed. This is a particularly interesting result when lightning currents are measured directly on instrumented towers to calibrate the performance of lightning location systems, since in most practical cases the alue of the return stroke speed is unknown. Note that points (2) and (3) can be considered as corollaries of point (1). Citation: Paanello, D., F. Rachidi, M. Rubinstein, J. L. Bermudez, W. Janischewskyj, V. Shostak, C. A. Nucci, A. M. Hussein, and J. S. Chang (27), On return stroke currents and remote electromagnetic fields associated with lightning strikes to tall structures: 1. Computational models, J. Geophys. Res., 112, D1311, doi:1.129/26jd Introduction 1 EMC Group, Swiss Federal Institute of Technology (EPFL), Lausanne, Switzerland. 2 IICT Department, Uniersity of Applied Sciences of Western Switzerland (HEIG-VD), Yerdon-les-Bains, Switzerland. 3 Diagnostic and Monitoring Group Serice, ABB Sécheron SA, Genèe, Switzerland. Copyright 27 by the American Geophysical Union /7/26JD7958 [2] The determination of the peak return stroke current from remotely measured electric or magnetic fields considerably facilitates the collection of data on the lightning return stroke current without haing to instrument towers or 4 Department of Electrical and Computer Engineering, Uniersity of Toronto, Toronto, Ontario, Canada. 5 Department of Electrical Engineering, Kyi Polytechnic Institute, Kyi, Ukraine. 6 Department of Electrical Engineering, Uniersity of Bologna, Bologna, Italy. 7 Electrical and Computer Engineering Department, Ryerson Uniersity, Toronto, Ontario, Canada. 8 Department of Engineering Physics, McMaster Uniersity, Hamilton, Ontario, Canada. D1311 1of1

2 D1311 D1311 between lightning return stroke currents and far electromagnetic fields, taking into account the presence of an eleated strike object. [5] The paper is organized as follows: In section 2, the generalized return stroke models taking into account the presence of an eleated strike object are summarized. In section 3, equations relating lightning return stroke currents and far electromagnetic fields are deried. Simulation results obtained using engineering models are presented and compared in section 4. Section 5 is deoted to the relation between far field peak and current peak. In particular, the sensitiity of the inferred current peak (from far field peak) to the parameters inoled (e.g., the return stroke speed) will be discussed. General conclusions are presented in section Generalized Return Stroke Models Taking Into Account an Eleated Strike Object [6] Considering the eleated strike object as a ertically extended transmission line characterized by constant reflection coefficients at its top r t and at its bottom r g (see Figure 1), the general expressions for the spatial-temporal current distribution along the lightning channel and along the tower, respectiely, are gien by Rachidi et al. [22]: iz ð ; tþ ¼ Pz ð hþi o h; t z h * r t i o h; t z h c þ ð1 r t Þð1 þ r t Þ X1 r nþ1 r n i g t o h; t h þ z 2nh # c c n¼ u t z h for h < z < H ð1þ Figure 1. Strike object and channel geometry (H is the actual height of the return stroke waefront). trigger the lightning artificially, and without the inherent relatie inefficiency associated with those methods. This is especially true nowadays because of the widespread use of lightning location systems. Indeed, such systems are already used to proide also estimates of lightning current parameters [e.g., Cummins et al., 1998; Herodotou et al., 1993]. [3] The theoretical estimation of return stroke currents from remote electromagnetic fields depends on the adopted return stroke model. Expressions relating radiated fields and return stroke channel-base currents hae been deried for arious engineering return stroke models for a ground initiated lightning [e.g., Rachidi and Thottappillil, 1993]. [4] Recently, Bermudez et al. [25] deried expressions relating lightning return stroke currents and far radiated electromagnetic fields, taking into account the effect of an eleated strike object, whose presence is included as an extension to the transmission line (TL) model. The aim of this paper is to generalize the work of Bermudez et al. and proide, for arious engineering models, namely Bruce- Golde (BG), transmission line (TL), traeling current source (TCS), and modified transmission line models MTLL and MTLE [Rako and Uman, 1998], the analytical relationship iz ð ; tþ ¼ ð1 r t Þ X1 r n t rn g i h z o h; t 2nh c c n¼ þ r n t rnþ1 g i o h; t h þ z 2nh c c u t h þ z 2nh for z h c c where h is the height of the tower, H is the height of the extending return stroke channel, i o (h, t) is the so-called undisturbed current, defined as the idealized current that would be measured at the tower top if the current reflection coefficients at its both extremities were equal to zero, u(t) is the Heaiside function, P(z ) is the height-dependent current attenuation factor within the lightning channel, is the return stroke speed, * is the current-wae propagation speed and c is the speed of light. Table 1 summarizes P(z ) and * for the considered fie engineering models (BG, TCS, TL, MTLL, and MTLE) [Rako and Uman, 1998], in which, H tot is the total channel height and l is the current decay constant. [7] Note that equations (1) and (2), which are obtained by using a distributed source representation of the lightning return stroke channel [Rachidi et al., 22], can be equialently expressed in terms of the short-circuit current i sc (t) =2i o (h, t) [Baba and Rako, 25b; Rako, 21], or the reference current [Shigihara and Piantini, 26]. ð2þ 2of1

3 D1311 D1311 Table 1. P(z ) and * For Different Return Stroke Models a Model b P(z ) * BG 1 1 TCS 1 c TL 1 MTLL 1 z /H tot MTLE exp( z /l) a Adapted from Rako and Uman [1998]. b BG is Bruce-Golde; TCS is traeling current source; TL is transmission line; MTLL and MTLE are modified transmission line models. Note that the reference current and the short-circuit current coincide when the reflection coefficient at ground is equal to 1. Note also that equation (2) does not include any dependence on P(z ) and * because the current distribution along the tower does not depend upon the adopted engineering model [Paanello et al., 24c]. 3. Far Field Current Relationship [8] The general expressions for the ertical electric field and the azimuthal magnetic field from a ertical antenna aboe a perfectly conducting ground, for an obseration point at ground leel (see Figure 2), are gien by Uman [1985] 2 E z ðr; tþ ¼ pe o þ 2z 2 r 2 R 5 Z t R=c 2 z 2 r 2 cr 4 iz ð ; t R=cÞdtdz iz ð ; t R=cÞdz r ð ; t R=c c 2 R Þ dz 3 5 ð3þ H 8 ðr; tþ ¼ 1 2p 2 4 r ð R 3iz ; t R=cÞdz þ ð ; t R=cÞ cr dz 3 5 ð4þ where r is the horizontal distance between the channel and the obseration point, R is the distance between a single dipole located at a height pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z aboe ground and the obseration point (R = r 2 þ z 2 ). H is the height of the return stroke waefront as seen by the obserer (at a gien time t, H is the solution of the equation H/ + R(H)/c = t, where R is the distance between the return stroke waefront and the obseration point). Contributions to remote electric and magnetic fields from currents in the lightning channel and in the CN tower hae been preiously studied by Kordi et al. [23] who used an Antenna Theory (AT) model. [9] Let us consider here only the radiated electromagnetic field. For distant obseration points, neglecting the static and induction components of the electric field, and considering R ffi r and r H, the general expression for the electric and magnetic field for an obseration point located at ground leel (see Figure 2) reduces to E z ðr; tþ ffi 1 2pe o c 2 r H 8 ðr; tþ ffi 1 ; t ; t r=cþ [1] Introducing the general expressions for the spatialtemporal distribution of the current (1) and (2) into (5) ð5þ ð6þ Figure 2. Geometry for the far field calculation (H is the height of the return stroke waefront, as seen by the obserer). 3of1

4 D1311 D1311 and (6), and after appropriate mathematical manipulations, it can be obtained [Paanello et al., 24a] E z ðr; t H 8 ðr; t Þ ffi E rs ðr; t z Þ ffi H8 rs ðr; t ÞþEz eso ÞþH eso ðr; tþþe to ðr; tþ ð7þ z 8 ðr; tþþh8 to ðr; tþ ð8þ in which [11] 1. E z rs (r, t) and H 8 rs (r, t) are electric and magnetic fields due to the main return stroke pulse within the lightning channel, that is the first term on the right hand side of equation (1), gien by Ez rs ðr; tþ ¼ 1 2pe o c 2 i H8 rs ðr; tþ ¼ h i o h Pz ð hþ h; t z h * Pz ð hþ h; t z h * u t z h u t z h dz dz ð9þ ð1þ [12] 2. E z eso (r, t) and H 8 eso (r, t) are electric and magnetic fields resulting from the contribution of multiple-reflection process along the eleated strike object, including the contribution of pulses transmitted into the channel. They are gien by Bermudez et al. [25] E eso z ðr; t þ r=cþ ¼ 1 2pe o c 2 ð r 1 2r tþi o ðh; tþ 8 ð 1 r r tþx 1 g 1 r n g rn t i o h; t ð2n þ 1Þh 9 >< c >= 2pe o cr n¼ þ2r nþ1 g r nþ1 t i o h; t 2ðn þ 1Þh >: >; c H8 eso ðr; t þ r=cþ ¼ 1 ð 2pr 1 2r tþi o ðh; tþ 8 ð þ 1 r r tþx 1 g 1 r n g rn t i o h; t ð2n þ 1Þh 9 >< c >= 2pe o cr n¼ þ2r nþ1 g r nþ1 t i o h; t 2ðn þ 1Þh >: >; c ð11þ ð12þ [13] 3. Finally, E z to (r, t) and H 8 to (r, t) are the so-called turn-on electric and magnetic field terms [Paanello et al., 24b], associated with the current discontinuity at the return stroke waefront. Indeed, the current distribution associated with the extended models (models extended to take into account the presence of the eleated strike object) exhibits a discontinuity at the return stroke waefront. This discontinuity arises from the fact that the current injected into the tower from its top is reflected back and forth at its ends, and portions of it are transmitted into the channel; these transmitted pulses, which are assumed to trael at the speed of light, catch up with the return stroke waefront traeling at a lower speed and no current is allowed to flow in the leader region aboe the front [Paanello et al., 24b]. The return stroke waefront is assumed in this study not to produce any reflection when reached by fastermoing current waes produced by transient processes along the tower and injected into the lightning channel traeling upward at the speed of light. The effect of including reflections at the return stroke waefront has been described in preious works [e.g., Heidler and Hopf, 1994; Shostak et al., 1999]. [14] The discontinuity which affects the current distribution at the waefront, although not physically conceiable, is predicted by the extended models which take into account the presence of an eleated strike object and needs to be carefully treated when calculating the radiated electromagnetic field through an additional turn-on term in the electromagnetic field equations, gien by E to z H to ¼ I frontðhþ 2pe c 2 r 1 1 þ H ð13þ cr 8 ¼ I frontðhþ 1 2pcr 1 þ H ð14þ cr where I front is the amplitude of the current discontinuity at the waefront [Paanello et al., 24b]. [15] It is interesting to note that the second and the third terms of the electromagnetic fields, namely, E z eso (r, t), H 8 eso (r, t) and E z to (r, t), H 8 to (r, t) are independent of the adopted model. The only model-dependent terms are E z rs (r, t) for the electric field, and H 8 rs (r, t) for the magnetic field. Tables 2 and 3 summarize the specific expressions for E z rs (r, t) and H 8 rs (r, t) deeloped for arious models [e.g., Rachidi and Thottappillil, 1993]. [16] Additionally, it is important to note that the BG and the TCS models exhibit an inherent discontinuity at the return stroke waefront. This discontinuity gies rise to a turn-on term which is already included in the main pulse (E z rs (r, t) and H 8 rs (r, t)) contributions. 4. Simulation Results [17] Figures 3 and 4 present simulation results for the electric field at a distance of 1 km, for the fie considered models and considering two different eleated strike objects. At this distance, the fields are essentially radiation fields, and electric and magnetic fields hae the same waeshape. The computations are performed using the same undisturbed current i o (t), gien by Nucci et al. [199]: i o ðh; tþ ¼ I o1 h ðt=t 1 Þ 2 1 þ ðt=t 1 Þ 2 e t=t2 ð Þ þ I o2 e t=t3 e t=t4 ð15þ 4of1

5 D1311 D1311 Table 2. Expressions to Calculate E z rs (r, t) for Different Return Stroke Models, Far Field Conditions a Model E rs z (r, t) BG 2pe o c 2 r i oðh; tþþt di oðh; tþ dt TL 2pe o c 2 r i oðh; tþ TCS 2pe o c 2 ½ r ki oðh; ktþ i o ðh; tþš MTLL 2pe o c 2 ½ r i oðh; t r=cþ 3 Z t i o ðh; t r=cþdt5 H tot MTLE Solution of a Note: k =1+/c. Ez rs ðr; t þ r=cþ þ 1 l 1 2pe o c 2 r ðr; t þ r=cþ ¼ dt di o ðh; tþ dt de rs z where the alues of the parameters chosen are I o1 = 9.9 ka, h =.845, t 1 =.72 ms, t 2 =5.ms, I o2 = 7.5 ka, t 3 = 1. ms, t 4 = 6. ms. These alues correspond to the channel-base current adopted by Nucci et al. [199] (and in seeral other papers) to compare ground-initiated lightning return stroke models. Other considered parameters are H tot = 8 km (for the MTLL model), l = 2 km (for the MTLE model) and the return stroke front speed = 15 m/ms. [18] The two considered eleated strike objects correspond respectiely to simplified models of the Peissenberg Tower in Germany and the CN Tower in Toronto. The Peissenberg Tower is characterized by a height h = 168 m and reflection coefficients of r t =.53 and r g =.7 [Heidler et al., 21]. The adopted parameters of the CN Tower are gien by h = 553 m, r t =.366 and r g =.8 [Janischewskyj et al., 1996]. [19] In the same figures (Figures 3 and 4), the contributions of the three terms in equation (7) are also reproduced. The following conclusions can be drawn from the results presented in Figures 3 and 4: [2] 1. No significant differences are found among the arious models, especially in the early time region where the peak field occurs (see also Table 4). This conclusion confirms the findings of Paanello et al. [24c]. Note that the field peaks associated with the 168-m tall tower are slightly larger than those associated with the 553-m tall tower. Since the risetime of the adopted current (about.5 ms) is comparable to the traeltime along the 168-m tall tower, the contributions of the current and its reflection at ground add constructiely to the field peak. Whereas for the 553-m tall tower, the current peak occurs before the current has reached the ground. [21] 2. As predicted by the theory, the contributions E z eso (r, t) and E z to (r, t) are independent of the considered models. [22] 3. The main contribution (about 7%) to the field peak is gien by the eleated strike object E z eso (r, t). Then comes the contribution of the main return stroke pulse E z rs (r, t) (about 2 25%), and finally, the contribution due to the turn-on term is only about 15% or less. [23] 4. The fast decay of the field right after the first peak occurs, respectiely, at 1.1 ms for the 168-m tall tower and 3.7 ms for the 553-m tall one (namely, twice the traeltime along the respectie towers) and such a decay is produced by the reflection of the current wae at ground leel. 5. Far Field Peak Current Peak Relationship [24] As can be seen from the expressions in Tables 2 and 3, among the considered models, the TL is the only model for which it is possible to derie a simple, closed form expression relating the distant electric and magnetic field peaks and the associated return stroke current peaks. This has recently been done by Bermudez et al. [25] who deried specific expressions for tall and short strike objects. [25] For the case when the round-trip propagation time of the tall structure is greater than the zero-to-peak risetime t f, the relation between far electric and magnetic field peaks and the associated undisturbed current peak at the top of the eleated object is gien by Bermudez et al. [25]. (Note that Bermudez et al. [25] disregarded in their deriation the effect of the discontinuity at the return stroke waefront.) E far z peak ¼ h 2pe o c 2 r 1 þ c i ð 1 2r tþ H far 8 peak ¼ h 2pcr 1 þ c i ð 1 2r tþ I o peak I o peak ð16þ ð17þ Table 3. Expressions to Calculate H 8 rs (r, t) for Different Return Stroke Models, Far Field Conditions a Model H rs 8 (r, t) BG TL TCS MTLL MTLE 2pcr i oðh; tþþt di oðh; tþ dt 2pcr i oðh; tþ ½ 2pcr ki oðh; ktþ i o ðh; tþš ½ 2pcr i oðh; t r=cþ 3 Z t i o ðh; t r=cþdt5 H tot H8 rsð r; t þ r=c Þ þ 1 dh8 rsð r; t þ r=c Þ ¼ l dt Solution of 1 di o ðh; tþ 2pcr dt a Note: k =1+/c. 5of1

6 D1311 D1311 Figure 3. Electric field calculated at a distance of 1 km from a lightning return stroke to a 168-m tower for fie different engineering models. 6of1

7 D1311 D1311 Figure 4. Electric field calculated at a distance of 1 km from a lightning return stroke to a 553-m tower for fie different engineering models. 7of1

8 D1311 D1311 Table 4. Electric Field Peaks at 1 km as Predicted by Engineering Models 168-m Tall Tower 553-m Tall Tower Model E z peak (V/m) E z peak (V/m) TL MTLL MTLE BG TCS where I o peak is the first peak of the undisturbed current i o (h, t). Note that, because of the condition t f < 2h/c imposed on the current, the aboe are independent of the structure s height h and of the ground reflection coefficient r g [Bermudez et al., 25]. [26] For the case of a strike to ground when the reflections at ground leel are taken into account, the relationships between far field peaks and the associated undisturbed current peak become [Bermudez et al., 25] E far z peak ¼ h 2pe o c 2 r 1 þ c r ch g i I o peak H far 8 peak ¼ h 2pcr 1 þ c i r ch g I o peak ð18þ ð19þ in which r ch g = (Z ch Z g )/(Z ch + Z g ) is the current reflection coefficient at ground leel. [27] Comparing (16) (17) with (18) (19), we can define the far field enhancement factor due to the presence of a tall strike object as follows k tall ¼ 1 þ c ð 1 2r tþ 1 þ c r ch g ð2þ [28] Baba and Rako [25a] hae deried a similar expression for the far field enhancement factor using their model, and they hae presented a thorough analysis of the distance dependences of electric and magnetic fields due to a lightning strike to a tall object and due to the same lightning strike to flat ground. (The expression for the far field enhancement factor in the model of Baba and Rako is slightly different from (2). As shown by Baba and Rako [25a], the difference is due to the fact that the speed of current waes propagating along the lightning channel is assumed to be the speed of light in the present study, which is based on the work of Rachidi et al. [22], whereas in the model by Baba and Rako, the current waes along the channel are assumed to trael at the return stroke speed.) [29] In what follows, let us express the relationship between the far electromagnetic field peak and the peak current that would be measured at the top of a tall strike object [Bermudez et al., 25] where k tall is defined as ktall ¼ 1 þ ð 1 2r tþc= ð23þ 1 r t and I peak is the peak amplitude of the current that would be measured at the top of the eleated strike object. [3] A short discussion on the terminology is in order. Bermudez et al. [25] used the notation k tall instead of k tall and called this factor the tower enhancement factor. In the present paper, we followed the notation and definition of Baba and Rako [25a] in which the term tower enhancement factor is attributed to the ratio of the far field associated with a strike to a tall tower to the far field associated with a similar strike to ground, expressed by k tall, and gien by (2) within the model adopted in this study. Results obtained using (22) will be compared in part 2 [Paanello et al., 27] with experimental data consisting of simultaneous measurements of magnetic fields and of the return stroke current associated with lightning strikes to the Toronto CN Tower (553 m) obtained during the summer of 25. [31] Figure 5 presents the ariation of the factor k tall as a function of the reflection coefficient at the tower top. The adopted alue for the return stroke speed is = 15 m/ms. It can be seen that a ariation of r t from to 1 results in a ariation of k tall in the range 3 to 3.5. [32] Figure 6 presents the ariation of the factor k tall as a function of the return stroke speed. This time, the alue of r t has been chosen to be equal to.4. Note that, in Figure 6, the factor k tall is somewhat more sensitie to the return stroke speed. Howeer, it is important to realize that, in the expression relating the far electric field peak and the current peak (see (22)), the return stroke speed appears not only within k tall but also as a separate proportionality factor. The oerall effect of the return stroke speed on the far field current relation is presented in Figure 7. In this figure, two typical alues for the return stroke speed hae been considered, namely, 1 m/ms and 2 m/ms. It can be seen that the peak E field and H field radiated by a lightning return E far z peak ¼ 2pe o c 2 r k tall I peak H far 8 peak ¼ 2pcr k tall I peak ð21þ ð22þ Figure 5. Variation of k tall for a tall tower as a function of the top reflection coefficient r t. The return stroke speed is assumed to be = m/s. 8of1

9 D1311 D1311 the top of an eleated strike object is relatiely insensitie both to the alue of the return stroke speed and the top reflection coefficient. The factor k tall could find a useful application in correcting current peak estimates obtained from remote field measurements for strikes to tall structures, particularly when these are used to calibrate the performance of lightning location systems. [38] Results and findings of this paper emphasize the differences between return strokes initiated at ground leel (or from short strike objects) and those striking tall towers: (1) Radiated electromagnetic field peaks can be significantly enhanced by the presence of a tall strike object. (2) Electromagnetic fields from lightning strikes to tall towers are largely determined by the tower and only to a lesser extent by the channel. Consequently, electromagnetic fields asso- Figure 6. Variation of k tall for a tall tower as a function of the return stroke speed. The top reflection coefficient r t is assumed to be equal to.4. stroke to a tall structure is little sensitie to the return stroke speed. 6. Conclusions [33] In this paper, analytical expressions relating far fields and currents associated with lightning strikes to tall towers hae been deried. The deried equations are general and can be used with any engineering model expressed in terms of P(z ) (height-dependent current attenuation factor), and * (current-wae propagation speed). [34] It is shown that the far field can be decomposed into three terms, namely (1) contribution of the main return stroke pulse along the lightning channel, (2) contribution of the multiple-reflection process along the eleated strike object, including the contribution of upward propagating pulses transmitted into the channel, and (3) contribution of the so-called turn-on terms, associated with the current discontinuity at the return stroke waefront. This discontinuity is associated with neglecting reflections at the upward moing front. Only the first term is model-dependent. [35] It is found that the computed electromagnetic fields associated with a strike to a tall tower are not ery sensitie to the adopted model for the return stroke. This is especially true for the early time response of the field and, in particular, the peak field alue. It is also found that the electromagnetic field from lightning strikes to tall towers is largely determined by the tower and only to a lesser extent by the channel. [36] It is demonstrated, in addition, that the TL is the only model, among the considered ones, for which it is possible to derie simple analytical formulas relating current peak and far field peak alues. Such deriation has been recently presented by Bermudez et al. [25]. The electromagnetic field peak alue being nearly independent of the adopted model, the TL expression becomes a general expression that can be applied for any of the considered engineering return stroke models extended to include a tall strike object. [37] It is shown that the relationship between far electromagnetic fields and the current that would be measured at Figure 7. (a) Electric field peak as a function of return stroke current peak and (b) magnetic field peak as a function of return stroke current peak. The obseration point is located at a distance of 1 km from the stricken tall structure. The top reflection coefficient r t is assumed to be equal to.4. The adopted return stroke speeds are =2 1 8 m/s (solid line) and =11 8 m/s (dashed line). 9of1

10 D1311 D1311 ciated with strikes to tall objects are less model-sensitie than those corresponding to a strike to ground. In particular, the early time response of the field is nearly model-independent. In addition, and unlike ground-initiated strikes, for which the far field peak is strongly dependent on the return stroke speed (proportional, according to the TL model), far field peaks associated with strikes to tall objects, where a major contribution comes from the current propagating practically at the speed of light within the tall object, are not ery sensitie to the return stroke speed within the channel. This is an interesting result when lightning currents are measured directly on instrumented towers to calibrate the performance of lightning location systems, since in most practical cases the alue of the return stroke speed is unknown. [39] Acknowledgments. This work has been financially supported by the Swiss National Science Foundation (grant ) and the Natural Sciences and Engineering Research Council of Canada (NSERC). The authors would like to thank V. Rako and two anonymous reiewers for their aluable comments and suggestions. References Baba, Y., and V. A. Rako (25a), Lightning electromagnetic enironment in the presence of a tall grounded strike object, J. Geophys. Res., 11, D918, doi:1.129/24jd555. Baba, Y., and V. A. Rako (25b), On the use of lumped sources in lightning return stroke models, J. Geophys. Res., 11, D311, doi:1.129/24jd522. Bermudez, J. L., F. Rachidi, W. Janischewskyj, V. Shostak, M. Rubinstein, D. Paanello, A. M. Hussein, J. S. Chang, C. A. Nucci, and M. Paolone (25), Far-field Current relationship based on the TL model for lightning return strokes to eleated strike objects, IEEE Trans. Electromagn. Compat., 47, Cummins, K. L., E. P. Krider, and M. D. 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